Introducing the EDHEC-Risk Solvency Benchmarks Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints

Transcription

1 An EDHEC-Risk Institute Publication Introducing the EDHEC-Risk Solvency Benchmarks Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints January 2012 with the support of Institute

2 Table of Contents Executive Summary 7 Part I. Challenges related to Equity Investments within the Solvency II Regulatory Framework Solvency II: Boosting the on-going cultural revolution of insurance companies Solvency II: A risk-based approach through a modular structure Market risk module in Solvency II Pillar I Implementation of a partial internal model: Solvency II and financial risk management Implementation of partial internal models: Auditing and validating under Solvency Conclusion 74 Part II. Introducing the Solvency II Investment Benchmarks The LCI/RCI paradigms A formal dynamic allocation model in the presence of Solvency II constraints From stylised strategies to real world solutions Calibration of the model Numerical analysis Conclusion 111 Appendix 115 References 125 About Russell Investments 131 About EDHEC-Risk Institute 133 EDHEC-Risk Institute Publications and Position Papers ( ) Printed in France, January Copyright EDHEC This research has benefited from the support of the Russell Investments Solvency II research chair at EDHEC-Risk Institute. The opinions expressed in this study are those of the author and do not necessarily reflect those of EDHEC Business School. The author can be contacted at research@edhec-risk.com.

3 Foreword The publication that we are pleased to present here, Introducing the EDHEC- Risk Solvency Benchmarks Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints, represents the first-year research work from the EDHEC-Risk/ Russell Investments research chair on Solvency II Benchmarks. The aim of the research chair is to enable European insurance companies that do not have a full internal risk mitigation model to be able to avail of an objective academic reference in order to manage the risk of their equity investments. As the EDHEC-Risk Solvency II Benchmarks in cooperation with Russell Investments are based on dynamic allocation of sources of risk (equity and cash) and a rules-based approach, they constitute an easilyreplicable independent external reference, which facilitates the implementation of a partial internal model to assess equity risk, and its internal and external control. Moreover, the capital freed up from the optimisation of equity risk can be reallocated to cover other risks, such as sovereign debt. Putting in place these dynamic risk management strategies implies implementing partial internal models which capture insurance companies specific characteristics, particularly the benefits of these allocation strategies. These benefits include the following: responsiveness to changes in market environment, better exposure to the returns of the equity markets, and respect of Solvency II constraints. As you will be able to observe in the present publication, the benchmarks framework is public, totally transparent, well documented and grounded in a rules-based approach with solid, objective academic foundations. We would like to extend particular thanks to our research chair partner, Russell Investments, for enabling this important research to be developed. Russell Investments has a history of strong relationships with academic institutions and we are delighted with the decision to partner with EDHEC-Risk Institute in order to provide solutions for the insurance industry in the face of the challenges represented by the Solvency II Directive. We are confident that the approach described in this document can make a genuine contribution to the welfare of European insurance companies and the quality of their investment approach. Wishing you an interesting and informative read, Noël Amenc Professor of Finance Director of EDHEC-Risk Institute An EDHEC-Risk Institute Publication 3

4 About the Authors Noël Amenc is professor of finance and director of EDHEC-Risk Institute. He has a masters in economics and a PhD in finance and has conducted active research in the fields of quantitative equity management, portfolio performance analysis, and active asset allocation, resulting in numerous academic and practitioner articles and books. He is a member of the editorial board of the Journal of Portfolio Management, associate editor of the Journal of Alternative Investments and a member of the scientific advisory council of the AMF (French financial regulatory authority). François Cocquemas is a PhD in Finance candidate and research assistant at EDHEC-Risk Institute. His research specialises in market microstructure, asset pricing and the evaluation of regulatory impacts. François holds a Master in Finance from Sciences Po Paris and a Master in Economics and Public Policy from École Polytechnique, ENSAE and Sciences Po Paris. Before joining EDHEC, he worked as a lecturer in economics at Sciences Po Paris and as an equity research analyst at Natixis Securities. Romain Deguest is a Senior Research Engineer at EDHEC-Risk Institute. His research on portfolio selection problems and continuous-time assetpricing models has been published in leading academic journals and presented at numerous seminars and conferences in Europe and North America. He holds masters degrees in Engineering (ENSTA) and Financial Mathematics (Paris VI University), as well as a PhD in Operations Research from Columbia University and Ecole Polytechnique. Philippe Foulquier PhD, EFFAS, is Professor of Finance and Accounting, Director of EDHEC Financial Analysis and Accounting Research Centre. He is also actively involved in consulting in both IFRS-Solvency II and corporate valuation issues. He began his career in 1990 in the Scientific Department of the insurer UAP, notably in Asset Liability Management. He left UAP in 1996 and spent 10 years as a sell-side financial analyst in brokerage firms. He was head of the Pan-European insurance sector at Credit Lyonnais Securities Europe, Enskilda in 2000 and Exane BNP in He carried out several IPO and international M&A operations. Lionel Martellini is professor of finance at EDHEC Business School and scientific director of EDHEC-Risk Institute. He has graduate degrees in economics, statistics, and mathematics, as well as a PhD in finance from the University of California at Berkeley. Lionel is a member of the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. An expert in quantitative asset management and derivatives valuation, his work has been widely published in academic and practitioner journals and has co-authored textbooks on alternative investment strategies and fixed-income securities. 4 An EDHEC-Risk Institute Publication

5 About the Authors Samuel Sender has participated in the activities of EDHEC-Risk Institute since 2006, firstly as a research associate at the same time he was a consultant to financial institutions on ALM, capital and solvency management, hedging strategies, and the design of associated tools and methods. He is now applied research manager at EDHEC-Risk Institute. He has a degree in statistics and economics from ENSAE (Ecole Nationale de la Statistique et de l'administration Economique) in Paris. An EDHEC-Risk Institute Publication 5

6 About the Authors 6 An EDHEC-Risk Institute Publication

7 Executive Summary An EDHEC-Risk Institute Publication 7

8 Executive Summary The Solvency II Directive introduces a prudential framework for the computation of regulatory capital requirements for insurers. It specifically defines a standard formula that must be applied by default and serves as a reference point for more advanced approaches, notably partial or full-blown internal models. Solvency II regulators have detailed the design, construction, calibration and implementation of this standard formula. For the insurance sector, the capital requirement associated with equity investments remains prohibitive. Some large insurance companies have already reacted by starting to reduce their exposure to equity risk. This is not just the result of the recent market downturns, but also in preparation for the new regulatory constraints. Others firms are likely to follow suit and make a clear move away from equity in the near future. This forced shift from equity is not good news for the industry. The basis for sound investment should be proper diversification and risk management, without shying away from capturing the equity risk premium altogether. The stringency of the standard formula will be particularly prejudicial to firms that do not have the resources to develop an internal risk model that would make equity investment less costly. In order to alleviate this issue, EDHEC- Risk Institute is introducing a framework for designing dedicated dynamic asset allocation solutions, as part of a research chair supported by Russell Investments. These Solvency II dynamic allocation benchmarks, or Solvency II benchmarks in short, should be regarded as substitutes for static equity investments by insurance companies. They can be used by insurance companies to achieve a substantial exposure to equity risk and the associated premium, while maintaining strict and explicit control over the implied Solvency II charge. The conceptual foundation for this approach relies upon two main paradigms known as life-cycle investing (LCI) and risk-controlled investing (RCI) respectively. Thanks to these two components, it meets the challenge of reconciling long-term performance objectives and short-term solvency constraints. After recapping the relevant items from Solvency II, this executive summary introduces the theoretical foundation and the implementation of this proposed solution. I. Challenges related to equity investments within the Solvency II regulatory framework The European Commission published the Solvency II Directive proposal in July The European Parliament adopted it in April 2009 as did the Council of Finance Ministers of the European Union (ECOFIN) in May The first and main objective of the Solvency II regulation concerns the protection of policyholders and beneficiaries of insurance contracts (Art. 27, Directive 2009). The second is preserving financial stability and fair and stable markets. This means that control authorities have to take into account the impact of their decisions on the stability of financial markets and the pro-cyclical effects of their actions (Art. 28, Directive 2009). 8 An EDHEC-Risk Institute Publication

9 Executive Summary The Solvency II framework draws lessons from the Basel II banking regulation, but it includes a number of distinct features. Firstly, Solvency II allows diversification not only within risk types, but also across risk types. Secondly, Solvency II has arguably given more focus to risk management by allowing partial or full internal models that fully reflect the risks and management actions of insurance companies. Solvency II is a principle-based way of catalysing the adoption of modern risk management practices by insurance companies To attain its objectives, Solvency II is encouraging a move from an off-therack system built on minimalist rules to a custom-made system by broadening the notion of risk, and transferring the analysis of their risks to the companies themselves. This new solvency system is better suited to the specific features of each firm. The aim is not to lay down new rules for capital or provisions but to encourage companies to put in place more sophisticated partial or total internal models for the analysis, management and control of risks. It is in this context that we propose the development of equity benchmarks, which could be a starting point for building a partial internal model for defining risk management strategies. The Solvency Capital Requirement (SCR) corresponds to a Value at Risk (VaR) of basic own funds with a 99.5% confidence level over a one-year horizon calculated under the assumption of continuity of business. It covers unexpected losses arising from the insurer s current and future business that will be written over the coming 12 months. The SCR has been calibrated to take into account all quantifiable risks to which an insurance company can be exposed. Among those risks, it has to cover at least the following: life, nonlife, health underwriting, market, credit and operational risks. It can also include the risk mitigation techniques under the condition that the risks arising from the use of those techniques (credit risk for example) are properly taken into account in the calculation of the SCR. Insurance companies are required to calculate their SCR at least once a year and to report it to the supervisory authorities. However, the monitoring of the level of the SCR, as well as the amount of own funds covering it should be performed on a continuous basis. After reporting to the supervisory authorities, the insurer is required to recalculate the SCR if the risk profile of the company deviates significantly from its last SCR reporting. The solvency capital requirement is calculated using a standard formula. It is the sum of the Basic Solvency Capital Requirement (BSCR), the solvency capital requirement for operational risk and the adjustments for the loss absorbency capacity of technical provisions and deferred taxes. The Basic SCR is made up of six risk modules, each made up of submodules that lead to the calculation of required capital, usually following a type of shock for each risk. Aggregation of these shocks, in keeping with correlation matrices that reveal the dependence and the diversification of risks, leads to the An EDHEC-Risk Institute Publication 9

10 Executive Summary 1 - Stress testing involves analysing the changes in the valuation (often net asset value) resulting from changes in risk factors that reflect extreme events (crisis scenarios). final regulatory capital requirement. The calibration for each of these risk modules and sub-modules is defined for each type of risk and in accordance with its own framework. One of the drawbacks of VaR, highlighted by the banking and insurance regulators, is that in general these measures rely on the assumption of normal events. Yet extreme events are of greater magnitude than the normal distribution would have it (fat tails). The advantage of stress testing 1 is that by offering the opportunity to choose the magnitude of the event, whatever the odds of its occurring, it makes it possible to get around resorting to fat tail distributions. Insurance companies (like in the Solvency II framework) thus combine stress testing and VaR. Solvency II sets another capital requirement: the Minimum Capital Requirement (MCR). The calculation of the MCR is based on an 85% confidence level over a one-year horizon and is calculated using a linear function of the following variables: technical provisions, written premiums, capital at risk, deferred taxes and administrative expenses. Moreover, the MCR is subject to two thresholds: it should not be lower than 25% or exceed 45% of the SCR. If the MCR equals one of these two limits, the insurance company has to justify this result to the supervisory authority. The frequency of the MCR calculation is once every three months and the result should be reported to the supervisory authority. The Solvency II regime, unlike Solvency I, does not require insurance companies to invest in defined asset categories but defines general principles for investment. The aim of the regulation is to let insurers manage their investments according to the prudent person principle. This means it requires insurers to carefully choose their investments without limiting their alternatives. With Solvency II, the prudential regulator transfers the analysis of risk to insurers. To achieve this objective, the regulator has defined the quantitative requirements for assessing risks in Pillar I, and qualitative requirements in Pillar II. More precisely, under Pillar II, insurance companies are required to demonstrate that they have an organisation that ensures good knowledge of their risk exposure, its coherent assessment, an operational mechanism to manage risks and a reporting stream for decision-making. Under the ORSA (Own Risk and Solvency Assessment) framework, insurers are required to assess the risk arising from holding these securities and should make sure that they have the funds necessary to cover it. Implementing a partial internal model should better reflect the real risk exposure This study proposes a methodological framework, based on objective and thoroughly tested academic references, to design dynamic risk management strategies in the form of benchmarks that allow for exposure to equity markets, while maintaining a target solvency capital requirement. Because dynamic hedging is not recognised as a risk mitigation technique in the standard formula, it will be necessary to implement a partial internal model in order to better reflect the real risk exposure. 10 An EDHEC-Risk Institute Publication

11 Executive Summary The EDHEC-Risk benchmarks serve as a good point of reference for developing a partial internal model that supports a dynamic approach for equity investments. The Solvency II benchmarks framework is public, totally transparent, thoroughly tested, well documented and grounded in a rules-based approach and on solid academic foundations. Benchmarks constitute an independent external reference; they are easily replicable and ensure the rules-based approach is actually respected by the insurance companies applying them. This transparency also allows for Solvency II compliance and facilitates internal and external (auditors and regulators) control of these partial internal models. Consequently, they could both serve to reduce the cost of capital for equity investment (pillar I), as well as optimise the consumption of economic capital stemming from the real risks of the company (ORSA approach in pillar II). Regulators are, in principle, suspicious of the very notion that risk can be totally managed. This is probably due to the historical track record of complex risk management approaches, notably with the failure of sophisticated VaR estimation models, including those from prestigious institutions with considerable capital. However, regulators should also recognise that for some risks where management cannot expect to have the required expertise, a controlled externalisation in the form of rule-based strategies can be helpful in ensuring that risks are managed. In this sense, the existence of an external reference such as the Solvency II benchmarks, which are thoroughly tested, grounded in a rulesbased approach and based on sound academic foundations, could help the regulator in the validation and control processes. As such, these benchmarks can be easily implemented by insurance companies and used operationally by third party asset managers within a strict mandate which situates compliance with risk budget management rules as a firstorder fiduciary responsibility. Moreover, they could be helpful in the validation of partial internal models where investment strategies are dynamic while satisfying both the Solvency II Directive (article 126, Directive 2009) and CEIOPS level II advice on calibration paper 56 (10.11 and following). Auditing and validation of partial internal models under Solvency II so far rely on broad principles rather than on detailed procedures The Solvency II benchmarks are located upstream in the process of creating and validating partial internal models. Insofar as the Solvency II benchmarks are used for building partial internal models, a vital aspect of our approach is that it should allow for easy auditing and validation, both internally and externally. The regulator has acknowledged that total or partial internal models are developed through a combination of internal and external models and data and has allowed their use under Solvency II. However, models need to be documented and auditable on a continuous basis by the insurance company itself, while being regularly controlled by statutory auditors and supervisors. Pillar II principles for total or partial internal models require insurers to have an internal cycle for validation of the models to demonstrate that they An EDHEC-Risk Institute Publication 11

12 Executive Summary are appropriate for use within the risk management and decision-making process, as well as to demonstrate to the supervisory authority that the capital requirements calculated with (partial) internal models are adequate. In the case where the supervisor concludes that an insurer s profile significantly deviates from the underlying assumptions used in the internal model (quantifiable risks insufficiently captured, model adaptation to better reflect risk fails within an appropriate time framework), a capital add-on is required. The Solvency II benchmarks used within the context of Pillar II may contribute to the validation of partial internal models thanks to their transparency and their academic soundness. Validation includes a range of methods, techniques and verifications, which involve all players in the chain, both internally (notably internal control, risk management, compliance and the actuarial function) and externally (statutory auditors and regulators). For now, however, some aspects of auditing and validation processes for insurers under Solvency II are not completely fleshed out by the legislators. Ultimately, we believe some implementing measures will define the requirements more precisely. It is not unlikely that we will see some convergence towards Basel II processes and the international accounting standards rules. The Solvency II validation process is part of Pillar II and corresponds to a set of tools and processes used by the undertaking to gain confidence over the results, design, workings and other processes within the internal model (Tests and Standards for Internal Models Approval, EIOPA). Since models are complex, their validation provides a certain degree of confidence that methods and assumptions are appropriate and that outputs are reliable. As internal models are used in the own risk management assessment as well as in the calculation of the regulatory capital requirements, a misestimation might lead to policyholders protection being reduced and it might also affect the company s risk management and decision processes. If banking supervision is any indication, however, a single unified process cannot emerge. On the contrary, insurers should use a number of different layers and tools to assess the performance of their internal model and its surrounding processes. No unequivocal test can determine the adequacy of a model with certainty. Furthermore, not all tests have the same power for all areas of decision. This state of facts makes the qualitative processes almost as crucial as the quantitative toolbox in the general assessment of an internal model. Indeed, while quantitative validation can be at least partly based on an external toolbox especially for a strategy built around an external benchmark the qualitative validation process cannot be as standardised. The qualitative review process is a fundamental step in assessing the validity of an internal model and it involves every link in the chain. Solvency II has not set out explicit requirements at this stage. The qualitative review will be more difficult to outsource; however, it will be greatly simplified if the strategy follows an 12 An EDHEC-Risk Institute Publication

13 Executive Summary external benchmark. The use of external models and data should be accompanied by articulated strategies for validating and regularly reviewing the performance of these models and data. This strategy should include expert judgement, a use test, documentation that is sufficiently detailed and comprehensive enough to allow knowledgeable third parties to understand the internal model and finally some data policies to ensure the accuracy, completeness and appropriateness of the data used by the internal model. The quantitative processes for model validation under Solvency II are not yet set in stone, and it is not clear how comprehensive the testing should be for a firm to gain approval from the supervisor. However, some general principles and methods are already being discussed and should become clearer as insurance companies achieve conformity. They include input validation, model replication, benchmarking, backtesting, stress testing, sensitivity testing and profit and loss attribution. In any case, it is very likely that some discretion will be left to national supervisory bodies. II. Introducing the Solvency II investment benchmarks The conceptual foundations behind the Solvency II benchmarks are the LCI and RCI paradigms Although the incentives for building an internal model are great, it is likely that certain companies will not be able to do it without taking advantage of some external toolbox with a systematic approach. EDHEC-Risk Institute introduces a set of dynamic benchmarks that can fulfil this need. Our approach is based on a longterm optimisation programme for an insurance company that faces Solvency II risk-budget constraints and stochastic investment opportunities. In a nutshell, it is about designing a set of asset allocation strategies that should not only maximise the risk/return trade-off in terminal asset levels, but also satisfy certain Solvency II budgets. The idea here is to have separate control for risk-aversion (i.e. aversion with respect to uncertainty in long-term asset levels) and loss-aversion (i.e. aversion with respect to short-term losses). The EDHEC Solvency II benchmarks are based on an industrialisation of the two key aforementioned paradigms: risk-control investing (RCI) to take the presence of Solvency II risk budgets into account, and life-cycle investing (LCI) to take insurance company s typically long horizons into account and immunise the long-term portfolio against changes in key risk factors. A key insight from the LCI paradigm is that long-term equity investors should invest more in equities than their short-horizon counterparts, so as to benefit from the presence of a mean-reverting equity risk premium, but they should also consider periodic revisions of this allocation as market conditions change. In this context, Solvency II constraints should be incorporated ex-ante as additional key ingredients in the design of the optimal investment solutions. These Solvency II risk budgets are managed through the RCI paradigm, which is designed to maximise the probability of reaching the long-term An EDHEC-Risk Institute Publication 13

14 Executive Summary investment objectives while respecting the short-term risk constraints. In terms of investment solutions, it leads to the design of dynamic risk-controlled allocation strategies. In turn, these strategies lead to giving up the smallest possible part of the upside potential of the performance-seeking portfolio (and more specifically here, to the equity portfolio) in exchange for protection on the downside. The practical implication of the introduction of short-term constraints is that optimal investment in the risky equity index is a function not only of risk aversion and market conditions, but also of risk budgets, as well as of the likelihood of the risk budget being spent before the horizon. From the technical standpoint, our approach builds upon an abundant stream of research on long-term investment decisions in the presence of a stochastic opportunity set. In particular, we propose a comprehensive long-horizon dynamic allocation model in presence of stochastic inflation and interest rates, mean-reverting equity risk premium and stochastic volatility. The LCI component satisfies the longterm objectives of the investor We consider an investor with finite horizon T, which can be thought of as the duration of the insurance company s liabilities, and facing several sources of risk (interest risk, stock price, volatility and Sharpe ratio risk). The pure optimal LCI portfolio is designed for long-term investors in order to generate the optimal risk-return trade-off in terminal wealth. In this context, the optimal allocation to equities at any point in time is defined by the following expression (with the rest of the portfolio being invested in cash): where ƒ (,,T t)->0, when t > T. and are the equity risk premium (a.k.a. equity Sharpe ratio) and the volatility of the equity index respectively, and γ is the investor s risk aversion which is calibrated to reach a target level of average exposure to equity markets. The first term of this expression is speculative demand the Markowitz classical term of mean-variance portfolios. The speculative demand increases when the risk-return ratio of the stock is more attractive, and when it has reduced volatility. The second term is the hedging demand ( Merton term), denoting that investors also hedge against changes in the equity risk premium, which implies that allocation to stocks increases with the time-horizon. The mix between speculative and hedging demand depends on risk aversion γ. With the LCI dimension, the investor takes profit of the induced term-structure of equity risk implied by the presence of mean-reversion in equity returns (lower volatility in the long-term). The RCI component ensures that short-term regulatory constraints are met Our intention is to offer benchmarks, based on dynamic equity/cash allocation, that are compliant with the principles of the Directive for the calculation of the SCR. Therefore, we add a risk control component to our strategy that An EDHEC-Risk Institute Publication 14

15 Executive Summary 2 - Regarding the multiplier, increasing the m value will lead to increasing the upside potential (more aggressive spending of the risk budget) while also increasing the probability of hitting the floor or even underperforming the floor because of the presence of residual gap risk when the strategy is implemented in discrete time (here with a monthly frequency). 3 - Note that if the Solvency constraint disappears (that is if δ goes to 1), then we recover the pure LCI strategy. is specifically designed to control the drawdown. This RCI component is simple and rule-based: the weight of equity depends on the difference between the targeted Solvency II floor wealth and the current wealth. The Solvency II requirement is that the wealth generated by the strategy at any time within a year should never fall below a fraction 1 δ of the capital invested at the beginning of the year, where can be interpreted as a given Solvency II equity charge risk budget. Typical values for δ are 5%, 10%, 15%, and 20%. Accounting for the RCI term, the weight allocated to stocks is finally of the following form: where is the optimal weight coming from the LCI paradigm described above, m is a constant multiplier 2 and is the current value of the wealth. This formula says that the euro amount allocated to the unconstrained strategy is an increasing function of the risk budget A c F. If the risk budget shrinks to zero, the portfolio is entirely invested in cash so as to avoid over-spending of the pre-defined risk budget. 3 Since the cash investment grows faster than the floor, which is constant, a positive risk budget is recovered just after wealth has landed on the floor. Consequently, this strategy guarantees that the wealth is at least always equal to the floor, and if the floor is attained, the portfolio does not remain invested in cash at later dates. It is interesting to note that we have two opposite forces that regulate the level of equity in Solvency II benchmarks (the risk budget constraint is pro-cyclical and the LCI underlying strategy is anti-cyclical). The domination of one versus the other depends on the prevailing risk budget, but also on horizon T, and market conditions (estimated equity risk premium and estimated equity volatility ). From stylised strategies to real world solutions: Description of Solvency II benchmarks In order to satisfy the Solvency II requirements (short-term constraints at each time step) and since profits and losses are computed at the end of each calendar year, it seems natural to design the Solvency II benchmarks as roll-overs of strategies with a given horizon (e.g. 3, 5, 10 and 15 years in our study), stopped after one year. In a nutshell, we design 16 Solvency II benchmarks (combination of T=3, 5, 10, 15 years and solvency capital requirement δ of 5%, 10%, 15%, and 20%) of which the stock allocation is a function of market conditions, time-horizon and of the solvency risk budget. The benchmarks are rebalanced on a monthly basis, based on parsimonious dynamic estimates for and. The risk budget for these benchmarks is reset at the end of each year in order to meet the target capital requirement level. The benchmarks involve a time- and time horizon-dependent allocation between a stock index (proxied here by the Russell Global Equity Index, or the Russell Developed Equity Index in their euro-hedge version) and cash proxied by the EURIBOR 1M. The practical An EDHEC-Risk Institute Publication 15

16 Executive Summary implementation of these benchmarks is done in discrete time, because continuous trading would incur prohibitively high transaction costs. Moreover, Solvency II benchmarks are parameterised by the relative risk aversion γ and the constant multiplier m. These two parameters are calibrated in such a way that the average allocation to the equity index is sufficiently high to obtain significant exposure to the equity risk premium, but also in such a way that the allocation never gets so high that at any point in time it might lead to a violation of the Solvency II floor in the empirical analysis. Finally, short-selling is not a desirable feature in investment solutions for insurance companies. We impose a condition that the weight allocated to the stock index has to be between 0 and 1. The Solvency II benchmarks perform consistently across a number of methodologies The results of an analysis based on 10,000 Monte Carlo simulations show that the average returns achieved by the Solvency II benchmarks have an increased capital charge, which was expected since the average stock allocation also increases in the Solvency II risk budget. The wealth allocated to equities, and therefore the average performance, is also an increasing function of the timeto-horizon, which can be explained by the decreasing term-structure of equity risk implied by the presence of meanreversion in equity returns. Finally, even though the dynamic portfolio strategy is implemented in a discrete time (monthly), there is no violation of the target Solvency II risk budgets at the 99.5% confidence level, and in fact none at the 100% level given our scenarios. In fact, the risk budget is not entirely spent in most cases, and it is only for extreme parameter values that the risk budgets are close to being spent. In that sense, the Solvency II benchmarks achieve the initial objective allowing for a substantial allocation to equities while respecting given Solvency II risk budgets. A comparison to static benchmarks shows a significant improvement We compute the constant equity allocation such that the average returns of the static allocation match those of the Solvency II benchmarks. We then analyse what the maximum losses are at the 99.5% and 100% confidence levels, and also what Solvency II capital charges would correspond to these equity allocations. 16 An EDHEC-Risk Institute Publication 1Y-return statistics, and risk measures of the 10Y-Solvency II Benchmark δ=5% δ=10% δ=15% δ=20% Average Return 4.92 % 6.32 % 7.53 % 8.38 % Standard Deviation of returns 4.53 % 7.96 % % % Max Loss at 99.5% % 8.48 % % % Max Loss % 9.31 % % % Proba of Violating Floor 0 % 0 % 0 % 0 % Av. Stock allocation % % % % This table displays the performances of four Solvency II Benchmarks together with measures of risk, and average allocation in the Russell index over the 1Y period. The initial asset value is equal to 100. (1) The max losses have been computed in percentage of the initial asset value A 0.

17 Executive Summary In order to have a better understanding of the opportunity costs involved in following standard static asset allocation strategies, as opposed to using dedicated dynamic asset allocation benchmarks that have been specifically engineered to allow for the optimal spending of the regulatory risk budgets, we turn to the dual analysis. We consider the static benchmark that has an equity allocation leading to δ % of the Solvency II capital requirement (using the standard formula of 39% for equity), and we then look at the corresponding average returns. The results obtained for such strategies show that the static allocations have a performance level substantially lower than that of the comparable Solvency II benchmarks. Moreover, we see that the 99.5% max losses computed from our 10,000 Monte Carlo simulations are always higher than the capital requirement obtained from the Solvency II standard formula. This suggests that lower stock allocations should be used, leading to even lower performances for the static benchmarks. We will observe the same results with the historical datasets, which illustrates that these results are not mere artefacts of our simulated scenarios. Backtesting based on historical data more than meets the Solvency II requirements We also perform backtesting based on historical data using the longest available daily time-series of the Russell Equity Index in US$. We find an average performance that increases with the maturity T, and also with the risk budget δ, as was observed in the Monte Carlo simulations. Moreover, we see that with the same choice of parameter values, calibrated from the four Monte Carlo stress-tests, the budget constraints are always satisfied (with a 99.5% probability on a daily basis). When we deal with the second data set over each 1Y-period starting in January 2000 up to the end of 2010 (the Russell Global Equity and Developed Equity indexes in their euro-hedged version), one important difference we observe is that the performances no longer always increase with T or δ because the sample period was dominated by the impact of bear equity markets. Obviously, the Solvency II benchmarks performance strongly depends on the relative performance of the equity market. Since the sample period contains two substantial declines in equity markets ( and 2008), in tandem with 1Y-return statistics and risk measures of Static Allocations that respect the same Solvency II Capital Requirements δ=5% δ=10% δ=15% δ=20% Average Return 4.22 % 5.34 % 6.45 % 7.56 % Standard Deviation of returns 2.60 % 5.15 % 7.74 % % Max Loss at 99.5% % % % % Max Loss % % % % Proba of Violating Floor 0.68 % 1.40 % 1.77 % 1.84 % Stock allocation % % % % Capital Requirement (39% for equity) 5 % 10 % 15% 20% This table displays the performances of different Static Allocations together with measures of risk over 1Y periods. The initial asset value is equal to 100, and the allocation to the stock index is calibrated so that the standard formula (using 39% charge for equity) gives a capital requirement of δ%. (1) The max losses have been computed as a percentage of the initial asset value A 0. An EDHEC-Risk Institute Publication 17

18 Executive Summary Backtest results of the 10Y-Solvency II Benchmark with monthly rebalancing δ=5% δ=10% δ=15% δ=20% Average Performance 4.44 % 5.80 % 6.36 % 6.19 % Max Loss at 99.5% % 9.21 % % % Max Loss % % % % Proba of Violating Floor 0.03 % 0.03 % 0.07 % 0.07 % This table displays the average annual performance over each 1Y-period from January 2000 up to the end of The dataset includes the Russell Global equity index EURO-hedged and the 1-month EURIBOR rate. (1) The max losses have been computed as a percentage of the initial asset value A 0, and on a daily basis. a substantial drop in short-term interest rates, it is not obvious that having access to a higher risk budget will generate higher performance. Nonetheless, the presence of the two aforementioned extremely severe bear markets again does not lead to any constraint violation (with 99.5% confidence), confirming the robustness of our dynamic equity strategies. The maximum losses exceed the risk budgets less than 0.1% of the time, which is below the 0.5% threshold required by the Solvency II regulations on portfolio loss computations. Impact of tracking error constraints for active portfolio management In practice, the allocation to equities by insurance companies can take the form of investment in active managers. This is a possible source of concern as the tracking error of the active manager return will imply a deviation of the portfolio return from the passive index returns, which are used to generate the Solvency II benchmark allocations. The methodology we use involves applying the allocation recommendations emanating from the Solvency II benchmarks while modelling the added-value by the manager as a Gaussian process, independent from the return on the underlying equity index, with a mean (alpha) of 100bps (resp. 200bps) and a volatility (tracking error) of 2% (resp. 4%). Over the long US$ history, we find no violation of the risk budgets for the 2% and 4% tracking errors, even if we set the alpha down to zero. Violations start to occur when the tracking error reaches 6% and alpha is 0%. These results suggest that the ability to identify active managers who are likely to generate a stable level of tracking error is a key source of added value when attempting to implement the Solvency II benchmarks with non-passive vehicles. Conclusion The objective of the new Solvency II regulatory framework, which should come into force at the beginning of 2013, is to introduce unified economic risk-based solvency requirements for insurance companies across all EU Member States. The Solvency II framework imposes severe conditions for investment in equities. The consumption of capital makes the investment in equities very costly. The results presented in our study, obtained from empirical analysis based on both stochastic simulations and historical track records, confirm that the proposed dynamic EDHEC Solvency II benchmarks, produced as part of a research chair in partnership with Russell Investments, can be designed so 18 An EDHEC-Risk Institute Publication

19 Executive Summary as to allow for a more efficient use of the Solvency II risk budget compared to standard static strategies. Evidently, this is because the pre-commitment to reduce the equity allocation, under difficult market conditions that require restrained Solvency II risk budgets, allows insurance companies to increasingly invest in equities. This is in contrast to a simple static strategy that is calibrated so as to respect the same Solvency II constraints. The welfare gains involved are found to be substantial for reasonable parameter values, and are robust with respect to implementation constraints such as the presence of turnover constraints and tracking error risk. An EDHEC-Risk Institute Publication 19

20 Executive Summary 20 An EDHEC-Risk Institute Publication

21 Part 2. I: xxxxxxxxxxxxxxxxxx Challenges related to Equity Investments within the Solvency II Regulatory Framework An EDHEC-Risk Institute Publication 21

22 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework The Solvency II framework is derived from the economic capital model, which aims at aggregating all sources of risk by using a single metric. Empirical studies, notably by Borio et al. (2001), have shown that there was little consistency across firms on the implementation and calibration of economic capital models. Hence, there certainly was a need to specify the standard formula that must (by default) be applied by all parties and that serves as a reference point for more advanced approaches. Solvency II regulators have detailed the design, construction, calibration and implementation of this standard formula. In this implementation, lessons have been drawn from Basel II, but the Solvency II framework involves a number of distinct features. Firstly, Solvency II allows diversification not only within, but also between risk types. Secondly, Solvency II has arguably given more focus to risk management by allowing internal models that fully reflect the risks and management actions of insurance companies. By contrast, Basel II s advanced internal ratings-based (IRB) approach, can be compared to the socalled undertaking-specific parameters in Solvency II, not to the internal models approach. Since 2005, one of the most controversial points with respect to Solvency II is the capital requirement set by the prudential regulator for equities. There was intense debate, and many calibrations and adjustments (especially the Dampener formula) were proposed each year, but in the end, the insurance sector reached a consensus and concluded that the capital requirement for equity risk remained prohibitive. As shown in the table below, while some large insurance companies have already started to reduce their exposure to equity risk not only as a result of the recent market downturns but also in the context of preparing for Figure 1: Ratio of shares in European insurers total investment portfolio compared with the stock markets Some large insurance companies have reduced their equity holdings, but the European insurance sector on aggregate has not yet. The risk is that Solvency requirements for equity risk trigger aggregate sales in equities. Source: European Insurance in Figures, CEA An EDHEC-Risk Institute Publication

23 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Ratio of shares in leading European insurers total investment portfolio Company ALLIANZ 15% 16% 19% 15% 9% 8% 7% CNP 13% 14% 13% 13% 12% 11% 12% AXA (listed shares) 10% 11% 11% 9% 4% 4% 4% MUNICH RE 14% 14% 15% 14% 4% 3% 4% Source: Companies the new regulatory environment, this is not yet visible for the insurance sector as a whole. In this context, further dramatic decreases in allocation to equities by insurance companies are to be expected in the years ahead. According to the Solvency II framework, an equity investment based on a 99.5% 1-year Value at Risk, requires 39% of capital. In a nutshell, this equity capital requirement is defined so that insurance companies own funds are able to absorb an instantaneous 39% drop in equity markets. This percentage is increased to 49% for equities listed in emerging countries, private equity and alternative assets. Moreover, these average rates of 39% and 49% are adjusted (a range of +/- 10%) each year, depending on the most recent three-year performance in financial markets in order to limit counter-cyclical effects. Thus, when the market cycle is at its highest point, the capital requirement for an equity investment is 49% or even 59% for an alternative asset. This shortterm solvency constraint makes equity investment prohibitively expensive, resulting in the necessity for insurance companies to essentially shy away from investing in equity, with high associated opportunity costs related to giving up almost entirely on the equity risk premium. Thus, the industry is witnessing a derisking trend where portfolios are increasingly concentrated in government bonds. In such a context, insurers have been asking themselves a vital question: As Solvency II comes into force how do we benefit from the equity market risk premium to offer attractive products for policyholders and guarantee adequate asset-liability management without generating prohibitive capital requirements for shareholders? The aim of this study is to propose a methodological framework based on objective academic references (life-cycle and risk-controlled investing paradigms) to design risk management strategies, in the form of benchmarks (called Solvency II dynamic asset allocation benchmarks) which allow exposure to equity markets to be gained while maintaining a reasonable solvency capital requirement (SCR). These benchmarks can be used as a reference point for equity risk that will no longer be managed statically, but dynamically. Therefore, to implement these benchmarks, we have to integrate the fact that the Solvency II framework recognises financial risk mitigation techniques, but that dynamic hedging is not recognised as a risk mitigation technique in the standard formula. However, in order to improve modelling of risk management practices, the Solvency II framework encourages the development of partial internal models. The use of partial internal models is allowed for the calculation of capital charges for one or several risk An EDHEC-Risk Institute Publication 23

24 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework modules and sub-modules, notably in order to better reflect the risk profile of the company. Moreover, the Solvency II Directive (art. 126, 2009) allows the use of data and models from a third party in partial internal models. In that case, insurance companies have to document and explain the extent to which these are used within their internal model processes. If based on the standard formula, the assessment of the Solvency II capital requirements for the Solvency II dynamic asset allocation benchmarks would lead to a completely erroneous market risk exposure. Thus, a partial internal model is necessary. Our intention is to offer benchmarks, based on dynamic equity/cash allocation, that are compliant with the principles of the Directive for the calculation of the SCR. These benchmarks constitute a reference for small and medium-sized European insurance companies to develop partial internal models by themselves. Therefore, our Solvency II benchmarks proposal does not involve creating a new regulatory category through a partial internal model, but this proposal constitutes a starting point for insurance companies to develop their own partial internal models. It is thus located upstream in the process of creating and validating internal models. Our proposed Solvency II benchmarks offer numerous advantages. First of all, they are based on dynamic allocation of the sources of risks in accordance with academic recommendations. This dynamic asset allocation is responsive to market risk and allows for better management of losses in comparison with a static approach. Moreover, the Solvency II benchmarks framework is public, totally transparent, thoroughly tested, well documented and grounded on a rules-based approach and on solid academic foundations. With Solvency II, the prudential regulator transfers the analysis of risk to insurers. To achieve this objective, the regulator has defined the quantitative requirements for assessing risks in Pillar I, and qualitative requirements in Pillar 2. More precisely, under Pillar II, insurance companies are required to demonstrate that they have (i) an organisation that ensures good knowledge of their risk exposure, and its coherent assessment, (ii) an operational mechanism to manage risks, and (iii) a reporting stream for decision-making. This assessment clearly goes beyond the risk modules identified in the standard formula in Pillar I. Indeed, insurers are required under Pillar II to assess all the risks they face or might face, including those that are either not considered or not adequately covered by the standard formula (Level 2 Advice System of Governance, 2009). They are also required to determine their own fund s needs accordingly. For instance, risks arising from investments in sovereign debt are not adequately taken into account in the standard formula as debt securities issued by member countries of the European Economic Area (EEA), such as Greece, do not require any capital charge for spread and concentration risks. Under the ORSA (Own Risk and Solvency Assessment) framework, insurers are required to assess the risk arising from holding these securities and should make sure that they have the funds necessary to cover it. 24 An EDHEC-Risk Institute Publication

25 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 4 - FDBs are an additional yield distributed to policyholders. They are a function of company profitability (rules for distribution rates differ among companies and European countries), and their amount and distribution timing are freely decided by each insurer. This discretionary feature allows the FDB to be reduced or suppressed in a stressful environment, which as a consequence decreases the solvency capital requirement. In short, the Solvency framework recognises the loss-absorbing capacity of best estimate in an adverse context thanks to the ability to reduce the yield distributed to policyholders in contracts with FDB. Consequently, regarding Pillar I and Pillar II, the EDHEC-Risk Solvency II Benchmarks, being based on dynamic allocation of sources of risk (equity and cash) and a rule-based approach, constitute an easilyreplicable independent external reference, which facilitates the implementation of a partial internal model to assess equity risk, and its internal and external control (auditors and regulators). Moreover, the capital freed up from the optimisation of equity risk can be reallocated to cover other risks, such as sovereign debt. Since the asset and liability structure is intrinsic to each firm, the capital requirement of the Solvency benchmarks based on a dynamic equity/cash allocation is calculated independently of liabilities and other assets. Consequently, the calculation of concentration risk which depends on the total assets held by the company is not considered in this calculation. Secondly, for insurance contracts with future discretionary benefits (FDB), the Solvency II framework requires the calculation of a best estimate to integrate the impact of assets shocks on the FDB policy. 4 Obviously, this discretionary dimension is not part of the calculation scope for the capital requirement of the Solvency II benchmarks proposed in this study. However, total transparency and documentation on the benchmarks components required by regulators will allow each insurer to measure risks in their entirety. Our benchmarks therefore constitute a reference point for developing a partial internal model that supports a dynamic approach for equity investments and which incorporates risk controlled investment (RCI) and life-cycle investment (LCI) approaches. The rest of this document is organised as follows. To put Solvency II dynamic asset allocation benchmarks back in their context, the first part of the document introduces the Solvency II process and framework (section 1 and 2) and also analyses the prudential constraints for equity management (section 3) according to the standard formula. The latter requires shocks in equities, exchange rates, and portfolio concentration and their impacts on insurance liabilities to be considered. Section 4 explains the advantages of rule-based financial risk management, the pillar II requirements on system of governance and the internal and external auditing and validation requirements for such rule-based strategies. This section concludes on a rule-based riskmanagement strategy for equity risk. The second part of the document describes a novel approach to risk management based on an attempt to optimise the spending of Solvency II risk budgets by insurance companies so as to maximise the benefits of equity investments. We first describe the conceptual foundations of this approach, which rely on three main paradigms respectively known as liabilitydriven investing, life-cycle investing and risk-controlled investing (section 1). We then introduce the formal dynamic asset allocation model (section 2) and discuss the implementation challenges related to practical implementation constraints (section 3), as well as the questions related to the calibration of the model (section 4). A fifth section presents the numerical and empirical results we obtain from simulations based on stochastic and historical scenarios. An EDHEC-Risk Institute Publication 25

26 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 5 - Companies excluded from the scope of the Solvency II Directive are insurance companies which are part of the legal social security system and companies which fulfil all of the following conditions: - Annual gross written premiums inferior to EUR 5M; - Total gross technical provisions (without deducting receivables from reinsurance contracts or special purpose vehicles (SPV)) lower than EUR 25M; - Companies belonging to a group with total group provisions lower than EUR 25M; - If the company s business does not include insurance or reinsurance activities covering liability, credit and suretyship risk unless they constitute ancillary risks (as defined in Art. 16 paragraph 1, Directive 2009). - Companies whose reinsurance business does not exceed EUR 0.5M of gross written premiums or EUR 2.5M of gross technical provisions or 10% of its gross written premiums or of its gross technical provisions. If one of the conditions previously mentioned is not fulfilled during three consecutive years, the Directive applies from the fourth year. 1. Solvency II: boosting the on-going cultural revolution of insurance companies In recent years, change in the complexity of risks has led to a real determination to adapt accounting and prudential rules in order to offer a better vantage point on any company, particularly on the risks it is running. Although the aims of each are different, the implementation of IFRS, Solvency II, Basel III, the new rules for financial conglomerates and Market Consistent Embedded Value (MCEV) are converging toward this objective. With Solvency II, the European Union is seeking to draw up solvency requirements that are more relevant to the risks actually taken by insurance companies, thereby encouraging them to evaluate, manage and control their risks. 1.1 Objectives of the Directive The European Commission published the Solvency II Directive proposal in July The European Parliament adopted it in April 2009, as did the Council of Finance Ministers of the European Union (ECOFIN) in May The first and main objective of the Solvency II regulation concerns the protection of policyholders and beneficiaries of insurance contracts (Art. 27, Directive 2009). The second is preserving financial stability and fair and stable markets. This means that control authorities have to take into account the impact of their decisions on the stability of financial markets and the pro-cyclical effects of their actions (Art. 28, Directive 2009). To attain these objectives, Solvency II is encouraging a move from an off-therack system built on minimalist rules to a custom-made system by broadening the notion of risk and transferring the analysis of their risks to the companies themselves. This new solvency system is better suited to the specific features of each firm. The aim is not to lay down new rules for capital or provisions, but to encourage companies to put in place more sophisticated partial or total internal models for the analysis, management and control of risks. It is not necessary to revolutionise existing models but to adapt models for asset allocation, asset liability management, provisions and mitigation policies to the reality of risks. It is in this context that we propose the development of equity benchmarks, which could be a starting point for building a partial internal model for defining risk management strategies. 1.2 Scope of the Directive We propose to review in this section the scope of application of the Directive, so as to keep in mind which companies will be subject to compliance with the Solvency II requirements, and therefore which companies would be concerned by the Solvency II benchmarks approach proposed by EDHEC-Risk. The Directive applies to direct insurance or reinsurance companies, in life or non-life business, established in a Member State of the European Union (Art 2, Directive 2009). 5 While applying the Solvency framework, the Directive allows for the application of a proportionality principle for small- and medium-sized companies. This principle states that the Directive requirements and the execution measures should be applied in a proportionate manner with 26 An EDHEC-Risk Institute Publication

27 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework respect to the nature, the scope and the complexity of the risks undertaken by insurers. In addition to the proportionality principle, the Directive also allows for the use of simplified calculations for risk modules or sub-modules where the application of the standard formula is judged to be disproportionate and when the nature, scope and complexity of the risks undertaken by insurers justify the use of these methods Branches of companies based in the European Union with the parent located outside the European Union Insurance or reinsurance companies with branches in the 27 EU Member States, but headquartered outside the EU, are required to comply with the requirements stated in the Directive for technical provisions, own funds, valuation of assets and liabilities, Solvency Capital Requirement (SCR) and Minimum Capital Requirement (MCR) (Art. 165 and Art. 166, Directive 2009). To determine the SCR and MCR, all the business activities of the branch are taken into account. The Directive prescribes some additional specific requirements for these branches, namely: The amount of eligible basic own funds should not be lower than half the absolute floor defined for the MCR (Article 129, paragraph 1, point d, Directive 2009); Assets representing the MCR should be located in the Member State where the subsidiary operates. The amount of assets in excess of the MCR covering the solvency capital requirement should be located in the European Community Group Control Principles Group control applies to the following companies: European (re)insurance companies holding stakes in at least one (re) insurance company located in or outside the European Union; (Re)insurance companies, located in or outside the EU, whose parent is a holding insurance company with its head office in the European Community; European (re)insurance companies whose parent is a holding insurance company with its head office outside the Community. Group control does not imply that the supervisors exercise their control over (re)insurance companies located outside the European Union, holding companies, and mixed activity holding companies at an individual level. It is possible for the group supervisor to decide not to include a company under group control based on an analysis done on a case-by-case basis. This decision may be taken when a company is located in a country outside the European Union where there are legal obstacles to transferring the information needed to assess the group s solvency; or when a company does not represent a significant interest for the objectives of group control; or when the inclusion of a company is not appropriate or constitutes a source of confusion with regard to the objectives of group control. The solvency of a group should be calculated at least once a year. This calculation may be done through two approaches: the accounting consolidation method and the deduction and aggregation method. An EDHEC-Risk Institute Publication 27

28 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework With the first approach, the group s solvency is equal to the difference between eligible own funds determined on a consolidated basis and the SCR calculated at group level based on consolidated accounts. Regardless of the method used to determine the SCR, if the related company is a subsidiary that does not have enough eligible own funds to cover the solvency capital requirement, when the supervisory authority considers that the parent responsibility is strictly limited to the share of capital it holds, then the group supervisor may impose that the deficit of the subsidiary be taken into account proportionally to the share of capital held. For the purpose of assessing the solvency of an insurance company that holds a stake in a subsidiary operating in a country outside the European Union, the subsidiary is treated like any other related company when applying the deduction and aggregation method. However, when the country where the company is headquartered imposes a solvency regime at least equal to the Solvency II Directive regarding Supervisory authorities and general rules, the group may provide the calculation of solvency as defined by the regulation of the country outside the European Union. The group supervisor is in charge of verifying the balance of the regime. In order to better understand the definition of the solvency capital requirement and its control, the next section presents the general principles of Solvency II. 1.3 General principles of the economic approach and process of control This section introduces the general principles and the control process of Solvency II. Compared to Solvency I, Solvency II is a more risk-sensitive framework based on a three-pillar structure which constitutes the basis for the control process under the new regime General principles As with that of IFRS, the philosophy of Solvency II is to foster principles rather than to issue precise rules. The Solvency II Directive establishes principles regarding the valuation of assets and liabilities and, in particular, on technical provisions. The new regulatory framework proposes the adoption of an economic approach based on risk, which aims at encouraging insurance companies to measure, manage and control their risks. This risk-based approach is a significant change compared to Solvency I whose foundation dates to the 1970 s (Directives 73/239/EEC and 78/473/EEC) and whose vision of prudential rules is very administrative and accounting-centred. Solvency II is meant to be i) homogeneous, although some countries have already adopted complementary systems of calculating the solvency margin; ii) more exhaustive, by broadening the definition of risks (for instance, investment risk and the mitigation linked to hedging are not currently taken into account in most European countries) and by making allowances for reinsurance and volatility of claims. To put in place Solvency II, the European Commission proceeded in three phases. 28 An EDHEC-Risk Institute Publication

29 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 6 - The Lamfalussy approach is a regulatory approach to adopt, implement and control the application of legislation and execution measures in the area of financial services. The legislative measures are divided into three levels. The level one measures correspond to principles reflecting basic political choices that can be translated into a framework. The level two measures are more detailed and technical measures needed for the implementation of the objectives defined in the level one text. The level three measures are guidelines, interpretative recommendations or best practices issued which are not legally binding and implementation remains voluntary. The fourth level concerns the reinforcement of control by the European Commission with regard to potential infractions. 7 - The process of adoption of a Directive in the European Union involves interaction between three institutions: the European Commission, the Council of the European Union and the European Parliament. The role of the European Commission is to propose a new directive to the Parliament and the Council. After analysing the proposal, the Parliament and the Council take a decision to adopt or reject the proposal. If it is adopted, it becomes an official Directive of the European Union and the next step is its implementation. For this purpose, the Directive must be transposed to the national law of Member States. The European Commission and the Member States are in charge of the application of the new rules. An initial phase focused on determining the foundations of Solvency II. This phase came to a close on 3 March 2003 after two years of work, with the implementation of a framework for the prudential oversight system built on three pillars, a framework similar to that used for Basel II. This phase concluded that the requirement for a regulatory solvency margin is the basis of a prudential regime suited to risks. All the same, a capital requirement may be created to define both the level of capital that leads to an acceptable likelihood of default and the critical threshold beneath which the company is at high risk of bankruptcy. As with the American or Canadian RBC (Risk-Based Capital) systems, a degree of intervention is planned that is a function of the required solvency margin (RSM) multiple depending on a lower limit and a target. The second phase focused on the definition of the Solvency II Directive, also known as the level one measures under the Lamfalussy 6 procedure. In July 2007, the European Commission proposed the elaboration of a directive on insurance company and group solvency with the aim of replacing the Solvency I legal framework. The Solvency II Directive will come into force on 1 January The third phase is more technical and corresponds to the level two measures under the Lamfalussy procedure. In 2004, the Committee of European Insurance and Occupational Pensions Supervisors, CEIOPS (replaced in January 2011 by the EIOPA, the European Insurance and Occupational Pensions Authority) was created to define these measures. This phase implies carrying out quantitative impact studies (QIS) and consultation papers (CP) to define the implementation measures for the level one text. These measures have been subject to consultation and should be presented for approval to the European Parliament and ECOFIN at the end of In parallel to the level two measures, EIOPA has been working on the level three measures General process of control Control under Solvency II is grounded in a prospective and risk-based approach to ensure the continuous good functioning of insurance companies as well as compliance with the principles regarding control. For this purpose, control authorities analyse and assess whether insurance companies satisfy the requirements set for the system of governance (internal risk and solvency assessment), technical provisions, capital requirements, investments, quantity and quality of own funds and the requirements for full or partial internal models. Regarding the control of capital requirements, there are specific actions that have to be taken by the insurer and by the supervisory authority to ensure that the company maintains a sound solvency position. If an insurance company does not comply with the level of SCR, or reckons that it will not be able to comply with the SCR level in the next three months, it should inform the supervisory authority. Within the two months after the authority has been informed, the insurer has to present a realistic recovery plan to the supervisory An EDHEC-Risk Institute Publication 29

30 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework authority and obtain its approval. The authority will then set a timeframe of six months for the insurer to increase the level of own funds covering the SCR or to reduce its risk exposure to comply with the SCR coverage. This timeframe may be extended by three additional months. In the case of an exceptional market decline, the supervisory authority may extend the six-month period to a more appropriate period taking into account all the relevant factors. During this time, the insurer has to submit a report every three months on the measures taken to improve the solvency position. If the supervisory authority, based on these reports, observes that there is no significant improvement in the solvency position, it may decide to withdraw the extension granted. Furthermore, in exceptional circumstances, if the supervisory authority thinks that the insurer s financial position will further deteriorate, it can restrain or prohibit the free disposal of certain assets (Art , Directive 2009) As for the SCR, if an insurance company does not comply with the MCR, it has to inform the supervisory authority immediately about the non-compliance or about the fact that it will not comply with the MCR within the following three months. The insurer then has a one-month period to present a realistic recovery plan to the supervisory authority in order to comply with the MCR within the following three months. The insurer can increase its level of own funds or decrease its risk profile to ensure coverage of the MCR. Additionally, the supervisory authority can restrain or prohibit the free disposal of certain assets Control under Solvency II is based on a three-pillar structure Control under Solvency II is based on a three-pillar structure, similar to that of Basel II. We recall that in 1998, the objective of the Basel Committee, made up of representatives of the central banks and banking supervisory authorities of twelve countries (the body of rules has since been adopted by more than one hundred countries) was to increase the solidity and stability of the international banking system and to reduce competition inequalities in the industry. The contribution of Basel II lies in its adaptation of the rules for bank capital to changes in the risks prevalent in banking. Work on Basel II began in 1998, the reform was published in 2004, and it came into force in The prudential objective of Solvency II is very different from that of Basel II. Solvency II, after all, focuses not on individual risks but on the entire set of risks facing each company. In addition, the primary motivation for this body of rules is the protection of policyholders from the risk of bankruptcy of any insurance company. For insurance, the structure and definition of the measures are represented as follows: 30 An EDHEC-Risk Institute Publication

31 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Figure 2: Solvency II pillar structure Solvency II Corporate and risk governance Quantitative requirements Measurement of assets, liabilities and capital Own Risk and Solvency Assessment (ORSA) Supervisory review process Disclosure requirements Pillar I Pillar II Pillar III Source: EIOPA According to the framework defined by the European Commission, the first pillar contains the quantitative requirements and should define the prudential rules for technical provisions, assets and capital. The calculation of technical provisions as a best estimate plus a risk margin will be one of the major changes in Solvency II. It should integrate a forwardlooking approach, make allowances for the risk of a drift in the factors used as assumptions in the calculation, be founded on a discount rate that depends on the kind of insurance contract and the method of pricing the assets and liabilities, and determine the provisions for supplementary guarantees (explicit valuation of options). Asset risks are now taken into account quantitatively in the evaluation of the capital requirement (as in the United States). The second pillar deals with qualitative requirements. It is a large extension to the statement of good management practices in Solvency I that the supervisory services would like all insurers to put in place internally. This pillar is grounded in the definition of rules for internal analysis and management of risks (assets and liabilities) with ALM tools and reinsurance. Although the Commission acknowledges that asset-liability management should be strengthened, for the moment there is no plan to change the capital requirement explicitly as a result of the quality of ALM and of the management of mismatches (except possibly some capital add-ons). For this pillar, discussion revolves around the requirements made of the means of tracking exposure to investment risks, requirements founded on an explicit definition of an investment strategy (degree of risk accepted, target composition of portfolio, use of derivatives, liquidity of assets, correlation with the risk profile of liabilities). The insurance supervisory authorities would like to increase their powers of inspection and intervention (through the capital An EDHEC-Risk Institute Publication 31

32 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework add-ons requirement, for instance), after the fashion of the increase in this power provided by Solvency I, at each link in the chain of the management of risks: typology, analysis, valuation, acceptance, transfer, or reduction and management of risks. The principles stated in Pillar II may be classified into three types of requirements: (i) the principles related to the system of governance; (ii) those related to the supervisory process; and (iii) those related to internal models. As the Solvency II benchmarks proposed in this study will be properly documented in order to be internally and externally auditable, they should facilitate compliance with these requirements. Section 4 explains the pillar II principles in more detail. The third pillar has to do with market discipline, with using rules reflecting market demands that companies be more transparent about their exposures and their risk management. The Solvency II Directive (Art. 51, Directive 2009) requires insurers to publish an annual report on their solvency and financial positions. This report must contain a description of the company s activities and results, a description of the system of governance and an assessment of its adequacy with respect to the insurer s risk profile, a separate description of each risk category, of the risk exposure, concentrations, mitigation and sensitivity. Furthermore, the report must contain a description of assets, technical provisions and other liabilities, as well as the methods and bases used for their valuation. The difference between these methods and bases (used for solvency purposes) and the methods used to evaluate those items in the financial statements must be explained in the report. The capital management policy should also be detailed in the report and it should at least contain: the structure, amount and quality of the capital, the amounts of SCR and MCR, the use of the option of the duration-based approach in the equity risk sub-module (the duration-based approach allows a shock of 22% to be used to assess the capital requirement in life insurance under certain conditions cf. section 4.2), and all information allowing a better understanding of the difference between the underlying assumptions used in the standard formula and the underlying assumptions of any of the company s internal or partially internal models used for the SCR computation. Additionally, if the insurer has not complied with the MCR or the SCR during the period of examination, it should report the amount of non-compliance in the report (even if the problem has been solved) and explain the reasons for noncompliance, the consequences and the corrective measures taken. 1.4 A profound break from Solvency I Solvency II favours a profound change in the risk culture of insurance companies and therefore constitutes a significant departure from the current solvency system. The current European regulation requires that an insurance company be solvent, which means it is sufficiently sound financially to meet its obligations towards its policyholders and its other creditors. Regulation also requires the following: Sufficient reserves, calculated prudently, that is, providing a margin large enough 32 An EDHEC-Risk Institute Publication

33 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework to absorb any unfavourable changes in the variables that constitute them. The interest rate is set in accordance with the rules of the supervisory organ of the Member State. Safe, liquid, diversified and profitable assets. Each Member State has drawn up a list of admissible asset classes. A minimum amount of own funds in excess of the RSM (required solvency margin). In France, for example, the RSM for euro-denominated life insurance contracts is equal to 4% of mathematical reserves and of reserves for the management of policies involving an investment risk, plus a percentage of the capital at risk, a percentage that depends on the term to maturity of the obligations (0.3% for more than five years; 0.15% for between three and five years; 0.1% for less than three years). Reinsurance contracts can reduce the excess by at most 15% in life insurance and 50% in property and casualty activities. Before going on to the drawbacks of the current system to demonstrate the need for European reform, let us emphasise that the existing rules have been particularly effective in Europe. In spite of the significant weakening of insurance company balance sheets as a result of the turmoil in the financial markets, in spite of a significant slowdown in economic growth and an often devastatingly high claims rate, the number of insurance companies going broke has remained very low. In France, for example, the frequency of bankruptcy is less than 0.25% (approximately one company a year). All the same, Solvency I has several drawbacks: The minimum capital requirement involves several paradoxes. As minimum capital is calculated as a percentage of technical provisions in life insurance, the less well provisioned a company is, the less the capital required of it will be. Another paradox, and not the least significant, is the asymmetry of the treatment of bond gains and losses. Unrealised capital gains are added to available capital, whereas unrealised capital losses are not deducted from the calculation of the solvency margin. So, when rates fall, the solvency margins are over-estimated and include unrealised wealth that is very sensitive to a new rise in rates. In addition, this creation of wealth is not offset, as it should be, by a revaluation of liabilities. The views of prudential rules having to do with provisions remain highly administrative and accounting-centred. The calculation of provisions does not always make allowances for the general risks of doing business (inadequate choice of markets or products, ineffective prevention of fraud or human error, legal, tax, or reputation risks, internal risks linked to information systems) and/or risks inherent to the insurance business (misguided choice or modelling of the underwritten business, changes in the competitive environment). So it is surprising that hidden options such as lower-limit benefits or guaranteed rates, a cause of earlier bankruptcies, are not always explicitly taken into account in the calculation of the solvency margin. As for the prudential rules for the allocation of assets, it is surprising that after the turmoil in the stock and credit markets, the notion of risk remains so An EDHEC-Risk Institute Publication 33

34 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework simplistic. For example, the investment risks - volatility of the stock markets, exchange rates, interest rates, risk of derivatives, liquidity, matching, or credit risks - are not always included in the calculation of the solvency margin. The calculation of the Solvency I margin does not really make it possible to take the specific features of a reinsurance programme into account. The standard 15% reduction in life insurance and 50% in property and casualty insurance in the event of reinsurance seems devoid of any economic foundation. According to the European Commission, the primary cause of bankruptcy is poor operational costs and/or financial asset management. So, it seems paradoxical that the current solvency rules do not have the flexibility to integrate this parameter. It may seem paradoxical that the monitoring and the rules for calculation are done on past financial statements (total decorrelation of the solvency margin and prospects). As it happens, most European countries are putting in place complementary forward-looking prudential rules to mitigate the effects of this paradox. The correlation and dispersion of risks, which are also the subject of lively debate, are still not directly taken into account in phase I of IFRS or in Solvency I, whereas most leading insurers have already made them part of their internal models, especially those that have economic capital allocation and provision models. Finally, every Member State has enacted its own rules. The lack of harmonised rules for the solvency of insurance companies can sometimes lead to biases that distort competition. It seems incongruous that the solvency margin of a company in one country should depend on domestic accounting standards rather than on an economic reality shared by all of Europe. It is evident that the current solvency system is a set of rigid rules, corresponding to an acceptable minimum, so much so that, in general, when a company is found to be in trouble it is often too late for it to recover. So the current system is an offthe-rack set of rules that destroys any incentives for a company to monitor its own risks. 2. Solvency II: a risk-based approach through a modular structure 2.1 Solvency II requires implementation of an economic balance sheet based on a market consistent approach Contrary to the Solvency I framework, which is very focused on an accounting approach, one of the requirements of the new regulatory regime is the definition of an economic balance sheet as a first step to determining solvency capital requirements. The valuation of assets and liabilities should be done through a market consistent approach. More precisely, assets should be valued at the amount for which they could be exchanged between knowledgeable willing partners in an arm s length transaction; and liabilities should be valued at the amount for which they could be transferred or settled between knowledgeable willing parties in an arm's length transaction. The valuation of liabilities should not allow for any adjustment regarding the credit quality of the insurer itself (Art. 75, Directive 2009). 34 An EDHEC-Risk Institute Publication

35 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework The approaches that may be used to evaluate assets and liabilities other than technical provisions are: IFRS values for assets and liabilities (other than technical provisions) are considered a suitable proxy to the extent that they reflect the economic valuation principles of Solvency II (QIS5). However, in the cases when the valuation approach proposed by the IFRS does not reflect the economic values according to the Solvency II Directive, an economic approach should be used; A default approach which is the markto-market approach based on quoted market prices in active markets; If the mark-to-market method cannot be applied, the mark-to-model technique may be used. While applying this method, insurers should maximise the use of relevant observable inputs. Technical provisions are defined under Solvency II as the current amount an insurer has to pay to immediately transfer its insurance obligations to another insurer (Art. 76, Directive 2009). The amount of technical provisions may be determined in two ways. First, if the future cash flows of insurance obligations can be replicated by financial instruments that have observable and reliable market prices, the value of the technical provisions is determined based on the market value of those instruments. Second, if the future cash flows cannot be replicated, then the amount of technical provisions is determined as the sum of a best estimate and a risk margin. The best estimate is the probabilityweighted average of future cash flows taking into account the time value of money. The latter is estimated on the basis of the term structure of the relevant risk-free rate. The computation of the best estimate requires the use of adequate and relevant actuarial and statistical methods and should be based on updated and credible information as well as on realistic assumptions. The future cash flows used to determine the best estimate include all cash in and cash out flows necessary to satisfy the obligations towards policyholders for the lifetime of the obligations. These cash flows should be calculated gross of receivables from reinsurance contracts and securitisation arrangements. The value of those items is calculated and taken into account separately on the asset side of the economic balance sheet. In addition, insurers have to take into account the value of options and guarantees embedded in insurance contracts to determine the amount of technical provisions. The assumptions underlying the valuation of those items should be based on current and credible information regarding the probability of exercising the options (lapses and surrenders included) and they should include the impact of financial and nonfinancial conditions on the probability. The risk margin corresponds to the cost of providing an amount of eligible own funds equal to the solvency capital requirement necessary to face insurance obligations over their lifetime (cost of capital method). The cost of capital rate used to determine the risk margin is the same for all companies and is equal to the additional rate above the risk-free rate incurred by the insurer in order to have an amount of eligible own funds equal to the An EDHEC-Risk Institute Publication 35

36 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework SCR. For the purposes of QIS 5, the cost of capital rate was fixed at 6%. Within the economic balance sheet, own funds are also subject to special treatment under Solvency II. According the Directive, own funds are defined as the sum of basic own funds and ancillary own funds (Art. 87, Directive 2009). Basic own funds correspond to the excess of assets over liabilities (not including own shares held by the insurer) and subordinated liabilities. Ancillary own funds contain elements other than basic own funds that may be called up to absorb losses. These funds are subject to prior approval from the supervisory authority and once they have been called up, they are no longer considered as ancillary own funds but as assets. Ancillary own funds are defined (as long as they are not already considered basic own funds) as: Non paid-up capital or initial funds that have not been called up; Letters of credit and guarantees; Any other legally binding commitments received by the insurer; In the case of a mutual insurer, ancillary own funds may also contain any type of claims it may have against its members through the call of additional contributions within the following 12 months. Solvency II classifies own funds into three tiers. This classification is based on whether own funds are basic or ancillary and on the degree to which they possess the following two characteristics: Permanent availability, which refers to the fact that the items must be available and may be called-up to cover losses under the continuity of business scenario as well as under the company liquidation scenario; Subordination, which means that in the case of liquidation, the items must be available to absorb losses and cannot be reimbursed until all other insurance obligations towards policyholders and beneficiaries are settled. To assess the degree to which own fund items comply with the abovementioned characteristics, the insurer should also consider several factors such as: The duration of the item: whether it is dated or undated. If the item is dated, its duration is compared to the duration of insurance obligations (sufficient duration); The absence of incentives to reimburse the nominal amount; The absence of mandatory financial charges; The absence of encumbrances. Based on these characteristics, the Directive determines the following classification of own funds: Items classified under tier 1 are basic own funds which substantially possess the characteristics of permanent availability and subordination; Items classified under tier 2 are basic own funds which substantially possess the characteristic of subordination plus ancillary own funds which substantially possess the characteristics of permanent availability and subordination; Items classified as tier 3 are all other basic or ancillary own funds which do not possess the characteristics of permanent availability and subordination. 36 An EDHEC-Risk Institute Publication

37 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework It is important to clarify that for each item, the factors of duration, absence of incentives to reimburse the nominal amount, absence of mandatory financial charges and absence of encumbrances have to be considered. In the case of mutual insurers, the claims they hold against its members through the call of additional contributions over the following 12 months may be classified in tier 2 when they substantially possess the characteristics of permanent availability and subordination. Expected profits included in future premiums (EPIFP) resulting from the inclusion in technical provisions of premiums of existing business that will be received in the future are considered as Tier 1 own funds in QIS5 (naturally if EPIFP respects the requirement of tier 1 basic own funds). Moreover, eligible own funds are subject to a set of thresholds for the coverage of the SCR and MCR. Those thresholds as stated in the Solvency II Directive are summarised below: For QIS 5, the thresholds proposed were the following: SCR MCR Tier 1 At least 50% At least 80% Tier 2 The difference The difference with tier 1 between tier 1 and 3 Tier 3 Less than 15% Source: Quantitative Impact Study 5 The SCR is covered by eligible own funds (basic or ancillary) resulting from the sum of the amount of tier 1 items and the amount of tier 2 and tier 3 eligible items. The MCR is covered by basic own funds corresponding to the sum of tier 1 items and tier 2 eligible basic own funds. The definition of an economic balance sheet was the first step to determine the solvency capital requirements. Solvency II has defined two levels of required capital. The first, the minimum capital requirement (MCR), is the minimum beneath which supervisory intervention is systematic. The second is the solvency capital requirement (SCR), which is a capital target sufficiently high to absorb any unusual shock. In this section, we will review the definition and principles for the calculation of the SCR and MCR. Tier 1 Tier 2 Tier 3 SCR More than one third of eligible own funds The difference between tier 1 and 3 Less than one third of eligible own funds Source: Directive 2009 MCR More than half of eligible basic own funds The difference with tier Solvency capital requirements The solvency capital requirement corresponds to a Value at Risk of basic own funds with a confidence level at 99.5% over a one-year horizon calculated under the assumption of continuity of business. It covers unexpected losses arising from the insurer s current and future business that will be written over the following 12 months. An EDHEC-Risk Institute Publication 37

38 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework The SCR has been calibrated to take into account all quantifiable risks to which an insurance company can be exposed. Among those risks, it has to cover at least the following risks: life, non-life, health underwriting risks, market, credit and operational risks. Operational risk covers legal risk but does not cover risks arising from strategic decisions and reputational risks. It can also include the risk mitigation techniques under the condition that the risks arising from the use of those techniques (credit risk for example) are properly taken into account in the calculation of the solvency capital requirement. Insurance companies are required to calculate their solvency capital requirement at least once a year and to report it to the supervisory authorities. However, the monitoring of the level of the SCR, as well as the amount of own funds covering it should be performed on a continuous basis. After the reporting of the SCR to the supervisory authorities, the insurer is required to recalculate the SCR under two circumstances: firstly, if the risk profile of the company deviates from the underlying assumptions of the last reported SCR and secondly, if the supervisory authority finds elements that demonstrate that the risk profile of the company has changed significantly compared to the last SCR reported. The solvency capital requirement is calculated using a standard formula. It is the sum of the Basic Solvency Capital Requirement (BSCR), the solvency capital Figure 3: Modular Structure of the Solvency Capital Requirement Source: Quantitative Impact Study 5 38 An EDHEC-Risk Institute Publication

39 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework requirement for operational risk and the adjustments for the loss absorbency capacity of technical provisions and deferred taxes (see graph below). As far as technical provisions are concerned, the adjustments come from the capacity of insurers to reduce future discretionary benefits of insurance contracts. This risk mitigation effect is obtained by comparing the value of stressed future discretionary benefits to their value under the assumptions used to calculate the best estimate. The risk mitigation effect of future discretionary benefits should not exceed the sum of technical provisions and deferred taxes linked to those benefits. The Basic SCR is made of six risk modules, each made up of sub-modules that lead to the calculation of required capital, usually following a type of shock for each risk. Aggregation of these shocks, in keeping with correlation matrices that reveal the dependence and the diversification of risks, leads to the final regulatory capital requirement. The calibration for each of these risk modules and sub-modules is defined for each type of risk and in accordance with its own framework. As mentioned before, Solvency II sets another capital requirement which is the minimum capital requirement. The calculation of the MCR is based on a confidence level of 85% over a oneyear horizon and is calculated using a linear function of the following variables: technical provisions, written premiums, capital at risk, deferred taxes and administrative expenses. Moreover, the MCR is subject to two thresholds: it should not be lower than 25% or exceed 45% of the SCR. If the MCR equals one of these two limits, the insurance company has to justify this result to the supervisory authority. The frequency for the calculation of the MCR is once every three months and the result should be reported to the supervisory authority. The minimum capital requirement is derived as the maximum amount between the combined MCR and the absolute floor MCR (AMCR). The combined MCR correspond to the MCRs computed using the linear function for life and non-life business subject to a floor of 25% and a cap of 45% of the SCR. The AMCR is defined per type of insurance company, for example: the AMCR for life insurance companies is set at EUR 3.2M (Art. 129, Directive 2009): Where and 2.3 Value at risk As we have seen in the previous section, the Solvency II framework uses the Value at Risk as a risk measure to determine the capital requirements (SCR and MCR). This section presents a review of the concept of Value at Risk and its limitations. VaR is a probability measure of the risk of the loss that will be borne by a company as a result of future changes in risk factors. It is equal to the maximum potential loss suffered by a company given a An EDHEC-Risk Institute Publication 39

40 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 8 - For a company with VaR of 1bn at a one-year horizon and at a 99.5% confidence interval, the chances of suffering a loss greater than 1bn over a year are, assuming trading is normal, 0.5%; that is, such a fall in value will occur once every two hundred years. 9 - TailVaRα (X) = TCEα (X) = E[X/X VaRα (X)] 10 - Stress testing involves analysing the changes in the valuation (often net asset value) resulting from changes in risk factors that reflect extreme events (crisis scenarios). particular horizon (often one year) and a confidence interval (99.97%, for example, is associated with an AA rating). 8 It is expressed VaRα (X) = inf{x: FX (x) α}. The distribution of losses can be estimated with an historical method (observation of past behaviour), a parametric method (probability distribution of the risk factors), or the Monte Carlo method (several thousand random draws in an attempt to ascertain the likelihood of the occurrence of each of the states of nature). VaR may be a simple concept, easy to compute and interpret, but it is often criticised, as it is not sub-additive and it does not take into account the severity of the loss. As it is not sub-additive, the overall VaR of a company is not necessarily less than the sum of the figures for VaR or each of its components (VaR(X1,, Xn) > VaR(X1) + + VaR(Xn)). The economic capital defined internally by the insurance companies is built precisely on a sum of figures for VaR. As a result, highly debatable correlation matrices have to be put in place; otherwise, no allowances would be made for diversification. In addition, the correlation of the risk factors is often variable over time and, as we are seeing now, it generally increases in periods of turmoil. Yet most economic capital models assume constant correlation over time. Finally, VaR reduces the view of the risk profile of the company to a single point on the loss distribution and fails to indicate the severity of the fall in value (fatness of the tail distribution). So, a measure of risk derived from VaR, but sub-additive, is attracting growing attention. This measure is tail VaR (also known as expected shortfall, conditional tail expectation, or conditional VaR). The tail VaR 9 of a confidence interval is the conditional expectation of the random variable of an amount less than VaR of confidence interval α. Its advantage is that it is more sensitive to the distribution tails and that it is a coherent measure of risk coherence as defined by Artzner et al. (1999). One of the drawbacks of VaR and of tail VaR, highlighted by the banking and insurance regulators, is that in general these measures rely on the assumption of normal events. Yet extreme events are of greater magnitude than the normal distribution would have it (fat tails). The advantage of stress testing 10 is that by offering the opportunity to choose the magnitude of the event, whatever the odds of it occurring, it makes it possible to get around resorting to fat tail distributions. Insurance companies (like in the Solvency II framework) thus combine stress testing and VaR. Of course, in a more general manner, it is worth recalling the limitations intrinsic to any method of risk evaluation, limitations that are the result of the quality of data, of the occurrence of rare events that are not present in the simulations, of tested asset and liability valuation problems, and of model risks. 2.4 Investments The Solvency II regime, unlike Solvency I, does not require insurance companies to invest in defined asset categories but defines general principles for investments. The aim of the regulation is to let insurers 40 An EDHEC-Risk Institute Publication

41 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework manage their investments according to the prudent person principle. This means it requires insurers to carefully choose their investments without limiting their alternatives. We consider it important to highlight some of the principles stated in the Directive: Under Solvency II, the use of financial derivative instruments is allowed for risk reduction or improvement of portfolio management efficiency; Asset or investments not traded in a regulated market should be kept at a prudent level; Assets should be subject to an appropriate diversification so as to limit excessive exposure to an asset, an issuer, a group of companies or a geographical area and to avoid excessive risk concentration in the overall portfolio; Investments in assets issued by the same entity or by entities that belong to the same group should not expose the company to an excessive risk concentration. Thanks to the principles described in these three subsections, we are now able to deal with the market risk module. 3. Market risk module in Solvency II Pillar I As mentioned in the introduction, the aim of this study is to propose a methodological framework based on objective and thoroughly tested academic references to design dynamic risk management strategies in the form of benchmarks that allow exposure to be gained to equity markets while maintaining a reasonable solvency capital requirement. As dynamic hedging is not recognised as a risk mitigation technique in the standard formula, it will be necessary to implement a partial internal model in order to better reflect the real risk exposure and the EDHEC-Risk benchmarks would constitute a reference for developing them. Before introducing the Solvency II benchmarks approach (part 2), in section 3 we propose dealing with the standard formula, and particularly with how the Solvency II framework calculates the market risk of equity portfolios (components of our benchmarks). The standard formula requires shocks on equities, exchange rates, concentration and their impacts on insurance liabilities to be considered. Market risk arises from the level and volatility of market prices of financial instruments. It is measured by the impact of movements of key financial variables such as interest rates, stock prices, real estate prices and exchange rates on the balance sheet of insurance companies. For the calculation of the market capital requirement, it is necessary to include hedging and transfer mechanisms. For instruments to be considered as financial risk mitigation techniques, they should meet certain conditions dealt with in section 3.7. The market risk module is divided into seven sub-modules (see SCR modular structure above), which measure seven different types of risks: interest rate risk, equity risk, property risk, spread risk, currency risk, concentration risk and illiquidity risk. Regarding our target to define relevant solutions in asset management, in this section, we will An EDHEC-Risk Institute Publication 41

42 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework focus only on the relevant prudential constraints for equity management. This means, we have to deal with the equity, concentration and possibly currency risks. This section presents, for each risk submodule, the approach for computing the solvency capital requirement. In addition to the calculation of the SCR, each risk sub-module requires the calculation of the net solvency capital requirement, which corresponds to the SCR calculated under the assumption that the insurer is able to change the value of future discretionary benefits and the assumptions in future bonus rates in order to offset part of the shock. 3.1 Equity risk module Equity risk arises from changes in the level or volatility of equity market prices. In this risk module, all assets and liabilities whose value is sensitive to changes in equity prices are analysed. The capital charge for equity risk (Mkt eq ) is the change in the net value of assets minus liabilities (NAV) under the scenario of a fall in equity prices. The magnitude of the price fall depends on the type of equity. Basically, equity instruments are classified in two categories: Global equities : equities listed in regulated markets in the countries which are members of the EEA or OECD; Other equities : equities listed in emerging markets, non-listed equity, hedge funds and any other investments not included elsewhere in the market risk module. The standard shocks considered for Global equities and Other equities are a fall of 39% and 49% of their values, respectively. These values are subject to a symmetric adjustment (called the Dampener effect) according to the requirements set in the Solvency II Directive (Article 106, Directive 2009). The adjustment has several purposes. According to the European Insurance and Occupational Pensions Authority (EIOPA), it should allow sufficient time for insurers to rebalance their profile under a stress scenario, avoid unintended pro-cyclical effects, ensure that equity shocks remain sufficiently risk sensitive, prevent fire sales of assets, avoid insurers having to adjust their risk profile frequently solely as a result of movements in equity capital charges, avoid any incentive to invest in one or other asset class and allow the adjustment to be set independently of the standard equity stress. The Dampener effect, calibrated according to a VaR 99.5% over a one-year horizon, is a function of the value of a relevant equity index and its weighted average level, over an appropriate period of time. It should not imply a capital charge 10 percentage points higher or lower than than the charge implied by the standard shocks. This means that the shocks are applied in a range of [29%, 49%] for Global and [39%, 59%] for Other equities. These capital charges are considered prohibitive by the insurance sector and entail a change in the investment behaviour of insurers, leading them to invest almost exclusively in fixed-income securities. The adjusted capital stress is defined as the sum of the standard capital stress and (adjustment * beta); where the adjustment term is given by: 42 An EDHEC-Risk Institute Publication

43 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework with I t being the value of the equity index at time t and the beta calculated from a regression of the index level onto the weighted average index level. Moreover, Solvency II considers the special case of participations and ring fenced funds. Once the capital requirements for each category of equities have been determined, it is possible to calculate the total capital requirement for equity risk (Mkt eq ) by aggregating the individual capital charges for each category through a correlation matrix (CorrIndex): Participations are subject to different shocks depending on their nature as shown in the table below: Equity type Equity shock 1. Participations in financial and credit institutions 0% 2. Strategic participations (whether they are Global or Other equities) 22% 3. Strategic participations in insurance or reinsurance companies subject to Solvency II 22% 4. Non-strategic participations in insurance or reinsurance companies subject to Solvency II Standard equity charge 5. Other participations Standard equity charge Source: Quantitative Impact Study 5 An equity shock of 22% may apply for life insurers providing occupationalretirement provisions business in accordance with Article 4 of Directive 2003/41/EC, or retirement benefits paid by reference to reaching, or the expectation of reaching, retirement where the premiums paid for those benefits have a tax deduction which is authorised to policyholders in accordance with the national legislation of the Member State that has authorised the undertaking. To do so, the two following conditions should be met: 1. All assets and liabilities corresponding to the business are ring fenced, managed and organised separately from other activities of the insurer with no possibility of transfer. 2. The average duration of the liabilities exceeds an average of 12 years. where Mkt r and Mkt c correspond to the capital requirements for the equity categories Global and Other. CorrIndex is defined by: CorrIndex Global Other Blobal 1 Other Source: Quantitative Impact Study 5 As mentioned below, the capital requirement for equity risk should be calculated under two assumptions: under the assumption that the insurer is not able to vary the value of future discretionary benefits in technical provisions Mkt eq, under the scenario in which the insurer is able to vary future discretionary benefits and the assumptions in future bonus rates. The resulting capital charge is known as the net solvency capital requirement (n Mkt eq ). An EDHEC-Risk Institute Publication 43

44 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 11 - For investments in listed equity, it is assumed that the equities are sensitive to the currency of the main listing and for non-listed equity and property it is assumed they are sensitive to the currency of their location For currencies pledged to the Euro shocks (upward and downward) differ: % Danish krone against Euros, or against Lithuanian litas or against Estonian kroon - 0% Estonian kroon against Euros or Lithuanian litas - 1% Latvian lats against any Euros, Lithuanian litas or Estonian kroon - 0% Lithuanian litas against Euros or Estonian kroon - 3.5% Latvian lats against Danish krone 13 - Local currency is defined as the currency in which the insurer prepares its financial statements. Where, nmkt r and nmkt c correspond to the capital charges of each equity category after the adjustment for changes in value of future discretionary benefits and in assumptions of future bonus rates. 3.2 Currency risk module Currency risk arises from changes in the level and volatility of currency exchange rates. Insurers are exposed to currency risk through their investment portfolios 11 as well as through liabilities. The capital requirement for currency risk (Mkt fx ) is calculated for all foreign currency exposures which are not hedged. Currency risk is assessed through two scenarios: An upward shock 12 (fx upward shock) in which the value of the foreign currency (C) increases by 25% against the local currency 13 ; A downward shock (fx downward shock) in which the value of the foreign currency (C) decreases by 25% against the local currency. First of all, to determine the capital charge for currency risk, it is necessary to calculate individual capital charges per currency C by applying the two previously mentioned shocks. Thus we have: These capital charges are also calculated under the assumption that the insurer is able to adjust future discretionary benefits and to adjust the future bonus rates to absorb part of the shocks: nmkt fx, Cup and nmkt fx,c down. For each foreign currency, the capital requirement nmkt fx c is determined as the maximum of the two values nmkt fx,c up and nmkt fx,c down. If the net capital charge for currency C (nmkt fx,c ) equals the capital charge derived from the upward (downward) scenario, the gross capital charge for currency C (Mkt fx, C ) would equal the capital charge under the upward (downward) scenario. The total capital charges for currency risk result from the summing of the individual capital charges per currency. 3.3 Market risk concentration module One of the Solvency II risk modules requires insurers to measure the concentration risk of their asset portfolios. Naturally, this risk is not part of the scope of our benchmark as it depends on the total assets held by insurance companies. However, thanks to the transparency of our benchmarks, the breakdown of their composition will be available to determine the capital charge for concentration risk. This section covers the steps for the calculation of concentration risk according to the Solvency II standard formula. The risk module of market risk concentration measures concentration regarding accumulation of financial exposures with the same counterparty. It applies to indirect and direct exposures resulting from all assets considered in the equity, spread and property risk submodule. However, the concentration risk module does not apply to borrowings issued or demonstrably guaranteed by the national government of a State of the European Economic Area, issued in 44 An EDHEC-Risk Institute Publication

45 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 14 - As defined in Article 212 of the Solvency II framework 15 - As defined in Article 2(14) of the financial conglomerate directive(2002/87/ec) 16 - If an insurer has several exposures to the same counterparty, the rating used is a weighted credit rating corresponding to the weighted average credit quality step (rounded average of the credit quality steps of individual exposures weighted by the net exposure at default in respect of that exposure to the counterparty) The total amount of assets considered in the concentration risk sub-module does not include: - assets where the investment risk is borne by policyholders (unit linked), - exposures to counterparties that belong to the same group if the following conditions are met: the counterparty is an insurance or reinsurance company, a financial holding company, an asset management company or ancillary services subject to prudential requirements; the counterparty is part of the consolidation on a full basis and there is no impediment (current or foreseen) to transfer own funds or to repay liabilities from the counterparty to the undertaking; - Assets covered in the counterparty risk module. the currency of the government, issued by a multilateral development bank, by an international organisation or by the European Central Bank. The first step to determine the capital requirement for market risk concentration is to group exposures at default (Ei) per counterparty. Grouping the exposures per counterparty allows the solvency capital requirement for each counterparty to be calculated. These are then aggregated to obtain the total capital charge for concentration risk. Exposures to companies which belong to the same group 14 or the same financial conglomerate 15 are grouped together and treated as if they constituted only one counterparty. In addition, exposures through investment funds (or entities managing insurers investments) should be treated using the look-through approach. The capital requirement for concentration risk per counterparty (Conc i ) is calculated as the change in the net value of assets minus liabilities resulting from the application of a concentration shock. It is a function of the excess exposure to counterparty i (XS i ), its credit rating 16 and a risk factor (g i ) reflecting the concentration shock. The shock corresponds to an instantaneous loss of value of XS i * g i. The excess exposure to counterparty is determined based on defined concentration thresholds: Where E i is exposure at default to counterparty i, Assets xl 17 the total amount of assets considered in the concentration risk sub-module and CT the concentration threshold given by the table below. Rating i Concentration threshold (CT) AA-AAA 3% A 3% BBB 1.5% BB or lower 1.5% Source: Quantitative Impact Study 5 For mortgage covered bonds and public sector covered bonds, the concentration threshold is set at 15%, if their credit quality is AA or better and if they meet the requirement of Article 22(4) of the UCITS directive (85/611/EEC). The risk factor g i depends on the credit rating of the counterparty. For the purpose of QIS 5, g i is defined as follows: Rating i Credit Quality Step g i AAA AA A BBB BB lower Source: Quantitative Impact Study 5 In general for unrated counterparties, the g i factor is set to However, for unrated counterparties which are insurance companies complying with the MCR, gi is a function of their solvency ratio: Solvency ratio g i >175% 0.12 >150% 0.21 >125% 0.27 <125% 0.73 Source: Quantitative Impact Study 5 The total capital requirement for concentration risk is the aggregation of An EDHEC-Risk Institute Publication 45

46 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework the capital requirements per counterparty assuming a correlation of zero: The concentration risk modules allow for special treatments for concentration in property risk and concentration in exposures to governments or central banks of States outside the European Economic Area. For concentration in property risk, this module concerns direct and indirect exposure to properties as well as ownership and other types of exposures such as mortgages. For the purpose of determining concentration risk, properties located in the same building are considered to be one property. The threshold is set at 10% of the total assets considered in this sub-module, including government bonds, and the risk factor is set at For exposures to governments and central banks denominated and funded in the domestic currency of countries which are not part of the European Economic Area, the risk factor g i depends on the credit rating of the counterparty and is given in the table below: Rating i Credit Quality Step g* i AAA 1A 0 AA 1B 0 A 2 12 BBB BB B or lower, unrated 5-6, Source: Quantitative Impact Study Look-through approach principles In order to assess the risk of collective investment funds and indirect exposures, Solvency II requires, whenever possible, the application of a special treatment called the look-though approach. This approach aims to examine the economic substance of the investments. It allows the risk of the underlying assets to be assessed. Each underlying asset is analysed under the relevant risk submodule. When an investment fund is invested in other investment funds, several iterations of the look-though approach are applied until the number of iterations is sufficient to ensure that all material risks are captured. If the fund is not sufficiently transparent to reasonably allocate the investments, then the insurer should look at the investment mandate of the fund. It should be assumed that the schemes invest according to the mandate in such a manner as to produce the maximum overall capital requirement. If the lookthrough approach and the mandatebased method cannot be applied, the collective investment scheme should be treated in the equity risk sub-module in the category Global equities if the fund assets are only listed in the EEA or OECD, or it should otherwise be considered in the Other Equities category. 3.5 Impact of changes in value of assets on the value of insurance liabilities Under Solvency II, the losses arising from the shocks on the asset side have an 46 An EDHEC-Risk Institute Publication

47 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework impact on the liability side through two items: future discretionary benefits (FDB) and surrender options. Therefore, this section does not concern all insurance contracts, but only those with FDB and/ or those that benefit from a surrender option. In practice, insurance companies have to integrate the impact of the changes in value of assets on the FDB and surrender rate policies in order to define a best estimate (insurance liabilities), which would be used to determine the capital requirement. FDB has several main drivers such as national regulations, global results of the company or of the book, the nature of the company, the balance sheet of the company, the policyholder contract, objectives and policy of the company and companies discretion. Regarding surrender options, Solvency II requires that insurers take into account the value of all contractual options in the calculation of the best estimate of technical provisions. In order to determine the rate that will be paid to the policyholder, insurers project their financial assets. The interest rate paid will depend on the minimum rate guaranteed by the insurer, market conditions, the rate paid in the past (low volatility of the rates served), reserves for future benefits and unrealised gains and losses. By nature, the impact of changes in value of assets on the value of insurance liabilities depends on the intrinsic features of each insurer s balance sheet, and thus cannot be considered for determining the capital requirements for the Solvency II benchmarks suggested in this study. However, total transparency and documentation on the benchmarks components, required by the Solvency II framework, will allow each insurer to include them in its balance sheet and to measure risks in their entirety. More precisely, our benchmarks are a reference for building partial internal models which supports a dynamic approach for equity investments and which incorporates RCI and LCI approaches. 3.6 Capital requirement for market risk Once the capital charges for each risk sub-module of the market risk module are calculated, it is possible to aggregate them to obtain the overall market risk capital requirement. Additionally, for the calculation of the market capital requirement, it is necessary to include hedging and transfer mechanisms. According to QIS5, financial risk mitigation techniques imply the purchase or issuance of financial instruments (such as financial derivatives) that transfer financial risk to the markets. Examples of financial risk mitigation techniques are: Put options bought to cover the risk of a fall in assets; Protection through credit derivatives or collateral bought to cover the risk of default or deterioration in credit standing; Currency swaps and forwards to cover risks related to assets and liabilities; Swaptions to cover variable/fixed risks. For instruments to be considered as financial risk mitigation techniques, they should meet certain conditions such as: To be legally effective and enforceable in all relevant jurisdictions and to be an effective transfer risk to a third party; The company must have the right to claim on the protection provider. There An EDHEC-Risk Institute Publication 47

48 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 18 - We recall that under the interest rate risk sub-module, two capital charges were obtained: one reflecting the change in the net value of assets minus liabilities under the scenario of an increase in interest rates and another under the scenario of a decrease in interest rates. must be explicit references to specific exposures (or pool of exposures) so that the scope of the protection is clearly defined and incontrovertible; The calculation of the SCR must allow for effects of financial risk mitigation techniques and for the risks embedded in the use of those techniques. For the purpose of calculating the solvency capital requirement, hedging instruments should be permitted with an average protection over the next year unless they are part of a rolling hedging programme. For example, if an equity option, not included in the rolling hedging programme, offers protection for the next six months it should be considered as only covering half of the current exposure. Moreover, dynamic hedging should not be treated as a risk-mitigating technique. With respect to the counterparties providing financial protection, QIS 5 requires that only counterparties rated at least BBB or higher should be considered. For unrated counterparties, it should be demonstrated that they comply with the standards of a BBB-rated company. In addition, QIS5 specifies some particular requirements for credit derivatives. These instruments should be capable of liquidation in a timely manner in case of default, insolvency or bankruptcy of the counterparty or other credit events set out in the transaction document. A credit derivative contract is recognised if the credit events specified by the two contracting parties cover at least: Failure to pay the amounts due that are in effect at the time of such failure; Bankruptcy, insolvency and incapacity of the debtor to pay its debts or bankruptcy or written admission of its general incapacity to pay its debt as they fall due, and analogous events; Restructuring of the underlying obligation, involving forgiveness or postponement of principal, interest or fees that results in a credit loss event. Where credit events specified in the credit derivative do not include restructuring of the underlying obligation, the protection is partially recognised: When the amount that the protection provider has undertaken to pay is not higher than the amount of the exposure, the value of the protection should be reduced by 40%; or When the amount that the protection provider has undertaken to pay is higher than the amount of the exposure, the value of the protection should not be higher than 60% of the exposure value. In the end, the market risk capital requirement is determined as the highest amount between the market risk capital requirement under the upward interest rate risk scenario and the downward interest rate risk scenario. 18 Thus, the capital requirement for market risk is given by: where, Mkt up,r and Mkt up,c are the capital requirements for the market risk submodules under the upward interest rate scenario and Mkt down,r and Mkt down,c are the capital requirements for the market 48 An EDHEC-Risk Institute Publication

49 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework risk module under the downward interest rate scenario. The correlation matrices used in the aggregation are: authority has defined the quantitative requirements for assessing risks. In Pillar II, the qualitative requirements are CorrMkt Down Interest Equity Property Spread Currency Concentration Illiquidity premium Interest 1 Equity Property Spread Currency Concentration Illiquidity premium Source: Quantitative Impact Study 5 CorrMkt Up Interest Equity Property Spread Currency Concentration Illiquidity premium Interest 1 Equity 0 1 Property Spread Currency Concentration Illiquidity premium Source: Quantitative Impact Study 5 The net capital charge for the market risk module is determined in the same way as the gross capital charge, but using the net capital charges for each risk sub-module. 3.7 General Solvency II principles for Pillar II With Solvency II, the regulator seeks to set in place a risk-sensitive regime by broadening the notion of risk and transferring the analysis of risk to insurers. To achieve this objective, we have seen that the regulator has defined a threepillar structure. In Pillar I, the prudential defined. More precisely, Pillar II covers the principles and actions expected from insurance companies with respect to risk management, risk measurement and own funds. Under Pillar II, insurance companies are required to demonstrate that they have an organisation that ensures good knowledge of their risk exposure, its coherent assessment, an operational mechanism to manage risks and a reporting stream for decision-making. This organisation is subject to a supervisor review process and could be subject to a capital add-on under exceptional circumstances: if there is a significant deviation in the governance standards with respect to the Level 1 and Level 2 principles and if these deviations prevent An EDHEC-Risk Institute Publication 49

50 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 19 - According to Article 47 of the Solvency II Directive (2009), a capital add-on can be required in the following cases: - There is a significant deviation of the risk profile with respect to the assumptions used in the calculation of the SCR with the standard formula and the use of an internal model is inappropriate/ineffective or an internal model is being developed; - There is a significant deviation of the risk profile with respect to the assumptions used in the calculation of the SCR with the full or partial internal model because certain quantifiable risks are captured insufficiently and the adaptation of the model to better reflect the risk profile has failed within an appropriate timeframe; - The system of governance of an insurance or reinsurance undertaking deviates significantly from the standards, that those deviations prevent it from being able to properly identify, measure, monitor, manage and report the risks that it is or could be exposed to, and that the application of other measures is in itself unlikely to improve the deficiencies sufficiently within an appropriate timeframe the identification, measuring, monitoring, managing and reporting of risk and other remedial measures are unlikely to produce sufficient improvement within an appropriate timeframe (Level 2 implementing measures for Solvency II: Capital Add-on, 2009). 19 Naturally, as we will demonstrate below, this organisation is key to implementing a partial internal model and the EDHEC- Risk Solvency II Benchmarks can strongly facilitate this process from both an internal and an external (assessment and validation by regulator) point of view. The principles of Pillar II may be classified into three types of requirements: The principles related to the system of governance; The principles related to the supervisory process; The principles related to internal models. Within the system of governance, the regulator requires implementation of an effective risk management system which includes the Own Risk and Solvency Assessment (ORSA). The ORSA process involves analysing all the risks the company is exposed to or might be exposed to. Its objective is to ensure that insurers continuously meet the regulatory requirements as well as the internal target they set themselves (Issues Paper ORSA, 2008). This assessment clearly goes beyond the risk modules identified in the standard formula used by the regulator. Indeed, the Solvency II standard formula is designed to take into account the risks to which an average undertaking is exposed. It is a standard calculation that is not tailored to the individual risk situation of a specific insurer (Issues Paper ORSA, 2008). However, insurers are required under Pillar II, and more specifically through ORSA, to assess all the risks they face or might face and determine their own funds needs accordingly, including those that are either not considered or not adequately covered by the standard formula. For instance, risks arising from investments in sovereign debt are not adequately taken into account in the standard formula as debt securities issued by member countries of the European Economic Area (EEA) such as Greece do not require any capital charge for spread and concentration risks. Under ORSA, insurers are required to assess the risk arising from holding these securities and should make sure they have the own funds necessary to cover it. In order to accurately evaluate risks in the ORSA process, insurers may need to develop and set in place full or partial internal models adapted to reflect companies specificities. In this context, the EDHEC-Risk Solvency II Benchmarks constitute a reference for developing them and for facilitating their assessment and validation by the regulator. Their framework is public, totally transparent, thoroughly tested, well documented and grounded in solid academic foundations. This allows them to be Solvency II compliant and facilitates internal and external (auditors and regulators) control of these partial internal models. Consequently, the use of Solvency II dynamic allocation benchmarks would be very useful for insurance companies who 50 An EDHEC-Risk Institute Publication

51 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework genuinely wish to manage the reality of their risk exposures properly, as requested by Pillar II. They would also allow insurers to optimise capital investment related to equity risk under Pillar I. The capital freed up from this optimisation would be reallocated to cover other risks, such as sovereign debt for instance. In short, the Solvency II benchmarks would allow insurers to have a reference to develop partial internal models and to better appreciate the reality of their capital requirements as part of their internal risk management process. 3.8 System of governance Solvency II introduces the notion of governance with the aim of having an efficient system that allows for sound and prudent management of the business (Art. 41, Directive 2009). This system basically consists of principles, functions and models that allow for an adequate decision-making process. The system of governance is made up of an organisational structure that clearly defines the roles and responsibilities of each participant and the rules for decision-making, internal reporting, communication, cooperation, remuneration and supervision. The system must contain written policies for at least the following components: risk management, internal control, internal audit and outsourcing. These policies should be in line with the overall strategy of the business and should define for each component of the system the objectives, the tasks to perform, the people responsible for those tasks and the procedures and reporting set in place. It is the role of management to approve those policies and ensure their implementation. Policies should be reviewed at least once a year and are modified if there are important changes in the system itself or in the company. The system of governance should be proportional to the nature, scale, and complexity of the insurer s operations. Additionally, insurers should develop and document contingency plans to ensure the continuity of business and the regular performance of their companies activities. Management is ultimately responsible for the system of governance, but the whole company is concerned. It is important that the company sets in place an organisational culture that allows and supports effective operation of the system of governance. Since the EDHEC-Risk Solvency II benchmarks are based on a rule-based approach, within a framework that is publicly available, totally transparent, thoroughly tested, well documented, grounded in solid academic foundations and easily replicable, they can constitute a very useful reference to implement a partial internal model to assess equity exposure, which is easily compliant in particular with the governance requirement and more generally with Pillar Risk management Under Solvency II, the risk management system and function are mandatory, efficient and integrated in the organisation. The risk management function is the core of risk management. Insurance companies have to set in place An EDHEC-Risk Institute Publication 51

52 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework an effective risk management system that contains strategies, information processes and procedures necessary to identify, measure, control, manage and report their risks on a continuous basis (and their interdependencies) at an aggregated and individual level (Art. 44, Directive 2009). This risk management system should be perfectly integrated in the organisation and especially in the company s decisionmaking process. An effective risk management system under Solvency II is required to contain the following: A clear and well documented risk strategy that defines risk management objectives, principles, risk appetite, responsibilities across activities and that is consistent with the overall business strategy; Written policies that define and categorise material risks faced by the company by type of risk and the levels of risk acceptance for each risk. Written policies should allow the implementation of the risk strategy and facilitate control mechanisms; Appropriate processes and procedures to identify, assess, manage, monitor, and report risks; Appropriate reporting procedures and feedback to ensure that the information on the risk management system is monitored and managed by all staff, management and supervisors; A suitable Own Risk and Solvency Assessment (ORSA). The scope of the risk management system covers all risks insurers are or might be exposed to, which means those included in the standard formula as well as other risks that are considered to be material for the business. Article 44 in the Directive specifies that the risks covered should at least be the following: underwriting and reserving, asset-liability management, investments, liquidity and concentration risk management, operational risk management, reinsurance and other riskmitigation techniques. For insurance companies using partial internal models, the risk management system should cover additional areas: design and implementation of the partial internal model, testing and validation of the model, documentation of the internal model and of subsequent changes, analysis of the performance, production of summary reports, reporting to management of internal model performance, areas of improvement, weaknesses and efforts to improve them. As mentioned in the previous section, the EDHEC-Risk Solvency II Benchmarks constitute a very useful point of reference to implement a partial internal model to assess equity exposure, which is easily compliant, particularly with the risk management requirement, and more generally with Pillar Own Risk and Solvency Assessment (ORSA) In the Solvency II Directive it is stated that insurers as part of their risk management system should perform their own risk and solvency assessment (ORSA) (Art. 45, Directive 2009). ORSA is set in place to demonstrate the suitability of the risk management process developed within the organisation. It is a part of the commercial strategy and of the business strategy. ORSA presents the vision of the company s own risks, not only the ones included in the standard formula but all the risks that may 52 An EDHEC-Risk Institute Publication

53 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework have an impact on the insurer s ability to meet its obligations. The standard formula takes into account risks faced by an average insurance company and measured by them accordingly; it is therefore not meant to capture some risks that are specific to insurers. However, insurers should, in their internal assessment process, adequately assess all the risks to obtain a real understanding of their risk profile and ensure that they continuously meet the regulatory capital requirements and their internal objectives. The processes should enable insurers to identify and measure risks they face in the short-term and the long-term and also to identify possible events or future changes in external factors such as changes in economic conditions. Thus, ORSA is forward-looking; it should take into account insurers plans and projections. Each insurer is free to decide how to design its ORSA process; but according to the EIOPA s Issues Paper on ORSA, it should at least include the following: Solvency needs taking into account the risk profile, the approved tolerance limits, and the business strategy; Compliance on a continuous basis with the solvency capital requirement and the technical provisions requirements; Significance of the deviation of the risk profile of the company with respect to the underlying assumptions of the standard formula or the internal model (partial or full). With respect to the risk profile, the assessment should focus on the difference between the amount of own funds the insurer needs to run its business and the regulatory capital requirement. These differences may be explained by: (i) the use of different confidence levels and time horizons, (ii) a different assessment of capital needed to cover certain risks, (iii) consideration of new risks which are not included in the SCR, and (iv) consideration of different management actions. The ORSA process has to be documented and integrated in the daily operations of the company. ORSA needs to be performed regularly and communicated to the regulator at least once a year and after a significant change in the insurer s risk profile. In ORSA, the use of partial or total internal models is allowed and insurers should use the parameters they feel best reflect the position of the company. The role of the partial or total internal model in the management of risks and capital needs should be described in ORSA. Finally, ORSA needs to comply with the requirements set under the three pillars of the Solvency II structure as it involves the whole process of risk assessment (from identification and measurement, as defined in Pillar 1, to reporting as defined in Pillar 3). More guidance regarding the ORSA process will be published by EIOPA in the Level 3 measures which are to be published by the end of Implementation of a partial internal model: Solvency II and financial risk management Solvency II requires insurance companies to implement a risk management framework. This section will give some insight about the regulatory approach to risk management. As previously mentioned, our aim is to design risk management strategies in the form An EDHEC-Risk Institute Publication 53

54 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework of benchmarks which allow exposure to be gained to equity markets while maintaining a reasonable solvency capital requirement. These benchmarks are based on dynamic allocation of the sources of risks in accordance with academic recommendations. This dynamic asset allocation enables better management of losses in comparison to a static approach, but it is not recognised by the standard Solvency formula. Thus, our benchmarks would better reflect the real exposure to equity risks. The benchmarks constitute a reference for developing a partial internal model that supports a dynamic approach for equity investments and which incorporates risk-controlled investment (RCI) and life-cycle investment (LCI) approaches. Moreover, the Solvency II benchmarks framework is public, totally transparent, thoroughly tested, well documented and grounded in a rules-based approach and on solid academic foundations. As a consequence, benchmarks constitute an independent external reference; they are easily replicable and ensure the rulesbased approach is actually respected by insurance companies applying them. This transparency also allows Solvency II compliance and facilitates internal and external (auditors and regulators) control of these partial internal models. Our study is limited to financial risk management, a focus which is possible both in practice and within the Solvency II framework. In many risk management processes, risks are firstly managed at the risk-type level, and only at a second stage are the aggregated risks managed. The modular approach taken by Solvency II also reflects this practice, and means that Figure 4: Diversified SCR, life undertakings (solo) Market risk is the most significant risk for life insurance companies. Life risk encompasses biometric risks and subscription risks (expenses and lapse risk). Lapse risk is (in)directly linked to financial risks because policyholders may exercise their put options when the asset value of the insurance company falls. So, the traditional life insurance risks linked to biometric risks longevity for the annuity business and mortality for the loans and life guarantees are smaller than could be read from the figure. Source: QIS 5 results (p67) 54 An EDHEC-Risk Institute Publication

55 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Figure 5: Market risk is the second most significant risk for non-life insurance undertakings. Diversified SCR, non-life undertakings (solo) Source: QIS 5 results (p67) Figure 6: Market risk composition (solo) Equity risk alone represents 42% of market risk. it is possible to have each risk managed independently in the first place, before adjustments are made at the group level. These adjustments can be viewed as the strategic policy and can involve aggregate decisions such as raising capital. It is also justified by the considerable magnitude of financial risks at insurance companies, irrespective of whether financial risk management is their core business or not. Financial risk is unsurprisingly the largest risk in the savings business (and thus, by extension, in the life business). After all, in a savings policy the insurance company does financial intermediation, so financial risk, as opposed to insurance risk, is its main risk. After non-life risks, financial risk is the second largest risk for non-life companies. After all, the insurance business model is dubbed "inversed" as companies receive An EDHEC-Risk Institute Publication 55

56 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 20 - Embedded Value (EV) has long been the preferred metric of management and shareholders, in particular in life insurance. The notion of fair value in Solvency II is more universal. Management, however, can still have EV as its target under Solvency II More formally, as an insurance company can manage simultaneously the risk of its assets and that of its liabilities in addition to its expected profits, there is never a uniquely defined way of modifying the risk profile of a company. premiums before benefits are paid out. So, companies hoard cash, and the investment of this cash generates financial (market) risks. 4.1 Solvency II risk-management: the regulatory point of view Solvency II requires insurance companies to implement a risk management framework. This section will give some insight into the regulatory approach to risk management, which could be summarised through the following requirements: Have comprehensive knowledge of the various risks faced; Define own strategy or risk appetite; Define a risk management process and manage the risks. The default or preferred regulatory option to manage risks is a discrete-time approach known as economic capital Importance of comprehensive knowledge of the various risks faced The primary regulatory requirement is that management understand the risks and sources of value in the company it manages: Solvency II should not simply be a reporting tool designed by an IT department while the company remains managed as under Solvency I. Solvency II and the risk management concepts should lie at the heart of a company s decisions. For this, the primary requirement made to insurance companies is that they know their risks, which in practical terms could mean that senior management should understand the sensitivity of the company s assets, liabilities, net asset value, 20 and solvency capital requirements with regard to changes in risk factors and to management actions. Management should also understand the impact on the aggregated capital requirements, as well as the impact for each risk-type (which we call unaggregated SCR). Only in full knowledge of its risks and value drivers can an insurance company develop an optimal risk management policy, so regulators and supervisors attach great importance to management being knowledgeable and in control Importance of the definition of a risk appetite or strategy An insurance company, like other institutional investors, faces not only institutional constraints (regulatory, accounting and tax), but also reports to different stakeholders (shareholders, debt holders, current and future clients). As there are many levers around which a strategy can be articulated, 21 it is expected that insurance companies be able to build a strategy that responds to both the constraints it faces and to the expectations of its main stakeholders. In the implementation, management is supposed to be able to define risk limits, and allocate the available risk budget across business units and risk factors. In this paper, we are mainly interested in financial risk management and examine how it should be implemented and accounted for once a risk budget or available capital has been defined for equity risk Importance of following academic and regulatory prescriptions Both academic prescriptions and Solvency 56 An EDHEC-Risk Institute Publication

57 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 22 - Even with the stochastic volatility linked to the evolution of the stock market, when the surplus vanishes, the allocation to stocks remains primarily driven by portfolio insurance and eventually shrinks to zero For financial risks, there is always a choice between deleveraging by selling risky assets or by buying protection on these assets; the decision may depend on both the price of derivatives and the necessary liquidity of the market to sell part of the portfolio. In some cases, equilibrium will be restored by actions of other risk factors such as insurance risks if market conditions mean that action is made easier and cheaper for these risks. II requirements focus on adequate risk management practices for each individual insurance company. The first necessary ingredient of any adequate ALM strategy, liability-driven investing, requires that liabilities are formally taken into account. This requirement is explicit both in academic prescriptions and in Solvency II. The second ingredient is risk management. The Solvency II standard formula can be labelled as economic capital, and has similarities with some specific optimal risk-controlled programme (see Van Binsbergen and Brandt, 2007). Insurance companies, however, should define their own way of managing risks, depending on their objectives and risk tolerance. A third ingredient available to long-term investors is the term structure of risk. Authors such as Campbell and Shiller (1988), Campbell and Viceira (2005), Hoevenaars et al. (2008) have argued that equities were less risky over the long run than they are over the short run because of mean-reversion in stock returns. This idea has also been extensively discussed in the Solvency II process, and the Dampener effect incorporates such mean reversion effects. 4.2 Solvency II risk management: towards efficient implementation The Dampener effect in QIS5 makes the equity risk a function of past equity returns, with risk falling when past 3-year returns are negative and rising when returns are positive. For an insurance company that works with the standard formula, the Dampener effect attenuates the amount by which it needs to sell equities during downturns. The same would happen in a continuous-time version of the standard formula or in internal models that build on similar features: the change in volatility modifies the local leverage or the speed at which the investment strategy reverts to the liability-hedging portfolio, with higher exposure to stocks when these are low relative to historical averages. 22 The default or preferred regulatory option to manage risks is a discrete-time approach known as economic capital, a flexible tool that requires a form of portfolio insurance to be implemented ex-post, but does not require a proper ex-ante definition. Practitioners can thus either define ex-ante a portfolio insurance strategy that can be implemented with a dynamic strategy and/or options, or modify ex-post their asset allocation to conform to the prescribed rules The benefits of rule-based strategies Preparation of possible strategies is necessary to avoid delays. Adequate preparation involves identifying the impact of possible actions. 23 Management must have a clear understanding in advance of the available alternatives so as be able to decide quickly. The preparation and documentation of derivativesbased hedges (the recognition of hedges is a subject in itself and is not always straightforward) is important for changes to be implemented quickly. So, a technical framework needs to be put in place to allow real-time management. An efficient risk management system will rely on a proxy of the evolution of An EDHEC-Risk Institute Publication 57

58 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework the company s balance sheet, where the premium flows, claims pay-outs and evolution of asset prices allow for a close to real-time snapshot. There is an increased risk of delays when management acts on a complex set of decisions. This risk is also increased by inadequate risk management systems, the time needed to assemble information and to perform ALM studies, infrequent management committee meetings, a large number of validation steps required and unprepared or insufficiently knowledgeable management. The main benefit of rules is that they allow for close to real-time responses to market developments. After all, rules are usually part of risk management, whether these involve continuous monitoring of a risk budget or risk limits. Rules form a general guideline which management should follow by default. Within the Solvency II framework, or more generally in an economic capital framework, rules should ensure compliance with the assumptions of the model they should ensure, on a dynamic basis, that capital requirements are respected at all times. Rules thus imply a form of risk management that is close to traditional portfolio insurance (in fact, this is discrete-time portfolio insurance). The establishment of rules does not mean that management loses control of the implementation of the strategy. On the contrary, the establishment of a framework can be conceived either as general guidelines for actions, or simply as a default option from which management can opt out at any time in favour of any other action that seems preferable at the time. Financial derivatives vs. dynamic strategies in risk management In frictionless complete markets, a dynamic strategy is equivalent to a structured product or a fit-for-purpose derivative. In incomplete markets, there are pros and cons for each option; the pros depending on the source of incompleteness considered and the size of the investor. The main drawback of (equity) derivatives is that the more liquid and least expensive ones are simple put and call options on the main cap-weighted equity indices, so their payoff cannot be considered a proxy for many classes of dynamic strategies that investors would want to carry out. The main advantage of derivatives is that they protect from the risks and costs associated with transactions, market jumps and model risk, because all of these risks are transferred to the investment bank that produces them. Derivatives in particular protect against market liquidity risk (i.e., the inability to implement dynamic strategies at the time they are most needed) The needs of smaller insurance companies The need for simplification depends on the scale and granularity of the organisation. Smaller insurance companies should expect to focus primarily on managing their core business insurance risks. The large teams necessary for a fully-fledged management of financial risks are only expected at larger insurance companies. The smaller companies should instead adopt rigorous but simplified approaches 58 An EDHEC-Risk Institute Publication

59 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 24 - Profit-sharing is partly discretionary in with-profit funds; it is usually decided at the end of the year. to financial risk management, relying on proxies and externalising its implementation. Smaller insurance companies should give great importance to rules, because they make it easy to have appropriate, responsive and easy-to-implement management of market risk without the need for a specific financial risk management team: the process can be followed by a more global risk management team. Fully rule-based strategies defined by service providers or investment firms are thus appropriate for risks where management devotes less time and resources on a daily basis. In this sense, the Solvency II dynamic allocation benchmarks will offer insurance companies an objective reference for developing partial internal models. 4.3 The use and recognition of rule-based strategies in Solvency II The treatment of management actions in the Solvency II standard formula From a valuation perspective, in the Solvency II fair value framework, all options and guarantees need to be valued and all future management actions need to be taken into account. In these, future changes in asset allocation decisions are treated just like discretionary benefits: 24 Where the future discretionary benefits depend on the assets held by the undertaking, the calculation of the best estimate should be based on the current assets held by the undertaking. Future changes of the asset allocation should be taken into account according to the requirements on future management actions. (TP.2.92, QIS 5 Technical Specifications, 2010) The standard formula recommends that for the calculation of capital requirements, future changes in investment decisions simply play no role: To the extent that the scenario stress under consideration is considered to be an instantaneous stress, no management actions may be assumed to occur during the stress. (SCR.1.8, QIS5 Technical Specifications, 2010) The incentives to build internal models and better recognition of companies risk-management Solvency II aims to provide incentives for risk management by having their effects recognised in the Pillar I quantitative requirements: In accordance with the risk-oriented approach to the Solvency Capital Requirement, it should be possible, in specific circumstances, to use partial or full internal models for the calculation of that requirement rather than the standard formula. In order to provide policyholders and beneficiaries with an equivalent level of protection, such internal models should be subject to prior supervisory approval on the basis of harmonised processes and standards. (Solvency II Directive, par. 68, p.7). In (partial) internal models, insurance companies should model the way they manage, and manage the way they model. Internal consistency within a formal risk management framework is rewarded with lower capital requirements. This is because regulators favour internal consistency. After all, the lack of consistency is a risk, and in particular leads to the usual banking habit of regulatory arbitrage. As far as financial risk is concerned, risk management is globally synonymous with An EDHEC-Risk Institute Publication 59

60 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework risk-controlled investing. In a partial internal model, there is no other solution than forms of structured investment strategies, be they based on a derivatives overlay or on dynamic investment strategies Regulatory reticence towards rule-based strategies Regulators think of economic capital as a measure that encompasses all risks and which allows for an infinite number of actions to restore equilibrium. They think of it as a central tool for consistent measurement and management of all risks. The requirement that management remain in control explains the reluctance towards automation of procedures. This can be seen in the requirement to have a risk-management function that is formally in charge of testing, validating, updating the model and informing management about changes to the model. The idea of partial internal models is that they fully reflect the characteristics of the insurance companies, their management objectives, and the way they implement their models and actually manage risks. But the very notion that risk can be totally managed is suspicious in principle to regulators, probably because prestigious regulated institutions with considerable capital have failed without regulators being able to protect investors and clients, except for the latter through government support and public insurance. However, regulators should also recognise that for some risks where management cannot expect to have the required expertise, a controlled externalisation in the form of rule-based strategies can be helpful in ensuring that risks are managed. In this sense, the existence of an external reference such as the Solvency II benchmarks which are thoroughly tested, grounded in a rules-based approach and in sound academic foundations, could help the regulator in the validation and control processes. As such, these benchmarks can be easily implemented by insurance companies and used operationally by asset managers. They can also be helpful in the validation of partial internal models where investment strategies are dynamic while satisfying both the Solvency II Directive (article 126, Directive 2009) and CEIOPS level II advice on calibration paper 56 (10.11 and following). 4.4 Risk management practices: Where does the industry stand and what are the likely next steps? Lack of preparation of insurance companies The lack of attention to risk management and the static approach taken by Solvency II have for the moment given little incentive to adopt more rule-based, quantitativelydriven risk-management techniques. After all, QIS exercises have measured risk through a somewhat backward-looking view (i.e., according to the static approach that conforms to the traditional way that insurance companies have of modelling). The natural consequence of the lack of attention to risk management is a lack of preparation of insurance companies. So, insurance companies may not be ready to implement dynamic strategies in their ALM practices. The requirement of being knowledgeable about risks may explain the considerable amount of time devoted since QIS 1 in 2005 to the quantitative Pillar I calculations. After all, know your risks in a quantitative 60 An EDHEC-Risk Institute Publication

61 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 25 - Formal processes for the design and implementation, testing, validation and documentation of the internal model; Management or supervisory body must put in place systems which ensure that the internal model operates properly on a continuous basis. See section 6 for details. framework supposes an ability to evaluate them properly. Paradoxically, however, this focus on quantitative calculations has resulted in many companies experiencing considerable delays in their preparation for risk management. After all, risk measurement can be seen as a preliminary to risk management, but it is also a distinct process. In some companies, Solvency II has so far been considered a pure IT and reporting challenge, but it has led to no transformation in risk management practices: in these companies, risk measurement takes the place of risk management. Improvement in risk management practices is, however, necessary. We recall that the lack of a quantitative approach to risk management under Solvency I has already led to considerable capital and regulatory problems in the insurance sector after Some bank insurance companies had to request capital injections from their shareholders because of mismanagement of equity risks and subsequent latent or realised losses; regulators have subsequently adapted the way equity losses can be accounted for, providing more generous treatment to avoid significant problems at some large institutions. Under Solvency II, such arrangements will be less likely for outliers, as risk management is expected from everyone. Once adequate investment practices are put in place, insurance companies should ideally try to have their true risk profile recognised (i.e., to have the benefits of risk management recognised in their capital requirements). For this, (partial) internal models allow for much more flexibility compared to the standard formula, and, in particular, dynamic strategies are allowed, subject to prior supervisory approval. 25 In the second part of this study, we will present the framework based on objective academic references for the design of the Solvency II dynamic allocation benchmarks and will proceed to perform different tests to demonstrate how these benchmarks may be used as a starting point to build partial internal models A product approach We argue that most unprepared insurance companies should start with a product approach and then switch to internal models. Over the short run, unprepared insurance companies need to star adopting rule-based financial risk management for all of their assets and liabilities, even if they are not technically ready for fullydeveloped financial risk management. From the liability viewpoint, we recall that savings liabilities are complex to model because of their profit-sharing component which links the liability value to the performance of the assets, and that the definition of proxies to be used in risk management systems requires thoughtful preparation, whereas the liabilities of non-life companies are easier to model since they are usually independent of the investment strategy. A very straightforward option is to have only part of the assets managed with rule-based risk-controlled strategies. This also avoids having to commit to a particular way of managing the entire asset base of a company that is unfamiliar with the proposed techniques. An EDHEC-Risk Institute Publication 61

62 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework When such investment practices are implemented, the question of their risk quantification under Solvency II, presumably the standard formula, must be raised and resolved. An investment in such a strategy must be recognised as less risky than an investment in equities, with an equity charge that corresponds to its maximum loss. Frictionless strategies in complete markets would guarantee perfect control of risk. If the model could guarantee that losses are controlled perfectly, then the capital requirement for each risk factor would never exceed the model s theoretical maximum loss. In reality, however, (see Cont and Tankov, 2009), there is always a possibility to hit the designed floor of the strategy, leading to gap risk, because market incompleteness and discrete-time as opposed to continuous-time trading may provoke additional un-modelled losses. So, the loss to any strategy structured as a product must be modelled in a realistic manner for its incorporation in the Solvency II framework. Risk-controlled types of products simplify some of the risk management and organisational requirements. In particular, the required commitment to follow a strategy is fully externalised to the investment manager, who follows prescribed rules. For prescribed rules to be known impartially, they can be defined by an external entity, research centre or provider of indices and benchmarks. And qualified asset managers could be in charge of implementing such benchmarks and ensuring adequate risk-management processes The recognition of strategies as internal models Management actions can be recognised, especially if they take place over a sufficiently short time period, so that they can be credibly implemented by management. Naturally, in practice, the implementation of rule-based strategies is delegated to the asset management arm of the investment company. Again, it can be expected that such models will be built more easily in non-life companies where the liability is not related to the investments, or in smaller life insurance companies and mutual companies where the regulator explicitly recommends the use of proxies for the valuation of liabilities. In these situations, the financial liability can be perfectly replicated and the optimal strategy must indeed result in dynamic allocation between the liability-hedging portfolio and the performance-seeking portfolio. Risk-controlled strategies are therefore not only appropriate as a proxy for the short-term decisions related to risk control, but also as a suitable description of the strategy that should actually be performed over the long run. In short, the design and implementation of EDHEC-Risk s Solvency II Benchmarks have to be an integrated part of the risk management framework, which comprises strategies, processes and reporting procedures. In this perspective, the next section presents the Solvency II requirements for validation and auditing of partial internal models. 62 An EDHEC-Risk Institute Publication

63 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 5. Implementation of partial internal models: Auditing and validating under Solvency II As previously mentioned, the Solvency II benchmarks are located upstream in the process of creating and validating partial internal models. Insofar as the Solvency II benchmarks are used for building partial internal models, a vital aspect of our approach is that it should allow for easy auditing and validation, both internally and externally. In our view, this is perfectly in line with the Solvency II Directive. The regulator has acknowledged that total or partial internal models are developed through a combination of internal and external models and data, and has thus allowed their use under Solvency II. However, models need to be documented and auditable on a continuous basis by the insurance company itself, and regularly controlled by statutory auditors and supervisors. Pillar II principles for total or partial internal models require insurers to have an internal cycle for validation of the models to demonstrate that they are appropriate for use within the risk management and decision-making processes, as well as to demonstrate to the supervisory authority that the capital requirements calculated with (partial) internal models are adequate. In the case where the supervisor concludes that an insurer s profile significantly deviates from the underlying assumptions used in the internal model (quantifiable risks insufficiently captured, model adaptation to better reflect risk fails within an appropriate time framework) (Directive, 2009), a capital add-on is required. The Solvency II benchmarks used within the context of Pillar II may contribute to the validation of internal models thanks to their transparency and their academic soundness. Some specific rules apply to outsourced functions and may be especially important for small insurers who might not be equipped to deal with every requirement in-house, or might choose to implement a strategy based on the Solvency II benchmark that we propose in part II. EIOPA published the Level 2 Advice on Tests and Standards for Internal Model Approval in October This document provides more details on the requirements for the supervisory approval of internal models and focuses mainly on the requirements for the use test, internal model governance, statistical quality standards, calibration standards, profit and loss attribution, validation, documentation and external models and data (Articles from 120 to 126 of the Solvency II Directive, 2009). Additionally, in January 2010 EIOPA published the Level 2 Implementing Measures on Solvency II Partial Internal Models. This document treats the requirements specific to partial internal model approval such as the scope, the way they are integrated in the standard formula, the concept of major business unit, the integration of risks not covered in the standard formula and the adaptations of Articles of the Directive (Articles 120 to 125) for partial internal models. It is important to clarify that the validation process as presented in the Solvency II framework is the responsibility of the insurer and that it is different from the approval process, where the supervisor An EDHEC-Risk Institute Publication 63

64 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework decides whether to authorise the use of the internal model or not. The internal validation process is part of the elements that the supervisor assesses during its approval process. What are the advantages of the Solvency II dynamic asset allocation benchmarks and where do they rank with respect to partial internal models? Context: Solvency II encourages insurance companies to measure and control their risks better. Solvency II allows the development of partial internal models for the calculation of capital charges for one or several risk modules and sub-modules in order to better reflect the risk profile of the company. The Solvency II Directive (art. 126, 2009) allows the use of data and models from a third party in partial internal models. Insurance companies have to document and explain the extent to which they are used within their internal model processes. Proposal: A methodological framework based on objective academic references (life-cycle and risk-controlled investing paradigms) to design risk management strategies in the form of benchmarks (called Solvency II dynamic asset allocation benchmarks) which allow exposure to be gained to equity markets while maintaining a reasonable solvency capital requirement. The benchmarks constitute a reference for developing a partial internal model that supports a dynamic approach for equity investments. Scope: The Solvency II benchmarks are a starting point for insurance companies to develop their own partial internal models themselves. The benchmarks are located upstream in the process of creating and validating internal models. The capital requirement of the Solvency II benchmarks is calculated independently of liabilities and other assets as the asset and liability structure is intrinsic to each firm. Therefore, the scope excludes: - The calculation of concentration risk, which depends on the total assets held by the company; - The loss-bsorbing capacity effect arising from the ability to reduce the yield distributed o policyholders in contracts with FDB, as the distribution policy is intrinsic to each company; - The impact of shocks in assets on surrender rate assumptions in the calculation of the best estimate, as scenario modelling is intrinsic to each company; 64 An EDHEC-Risk Institute Publication

65 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Main Advantages: The Solvency II benchmarks are based on dynamic allocation of the sources of risks in accordance with academic recommendations. This dynamic asset allocation improves loss volatility management in comparison with a static approach. The framework of the Solvency II benchmarks is public, totally transparent, thoroughly tested, well documented and grounded in a rules-based approach and in solid academic foundations. The transparency of the Solvency II benchmarks allows Solvency II compliance and facilitates internal and external (auditors and regulators) control of these partial internal models This definition is by the Basel Committee on Banking Supervision, but we believe it can also be relevant to insurers. In this section, we consider the broad perimeter for the validation of internal models, namely all the processes that provide evidence-based assessment about a model s fitness for purpose. 26 Validation therefore includes ranges of methods, techniques and verifications, which involve all players in the chain, both internally (notably internal control, risk management, compliance, and the actuarial function) and externally (statutory auditors and regulators). For now, however, some aspects of auditing and validation processes for insurers under Solvency II have not yet been completely fleshed out by the legislators. Ultimately, we believe some implementing measures will define the requirements more precisely; it is not unlikely that we will see some convergence with Basel II processes and the international accounting standards rules. While always keeping in mind the specific characteristics of insurers with respect to banks, as well as the differences between what stems from international standards and what stems from European law, we think that the validating practices of banks can shed light on what is on the cards for insurance companies. 5.1 Players in the validation chain Internal control Internal control is defined in Solvency II as at least comprising administrative and accounting procedures, an internal control framework, appropriate reporting arrangements at all levels of the undertaking and a compliance function (Solvency II Directive, article 46, Internal control). According to EIOPA, an effective internal control system needs to take into account an effective and robust control of all activities in the company. In this sense, internal control ensures compliance with the laws, regulations and administrative provisions as well as the reliability and availability of financial information. The first step towards the validation and documentation of internal models is of course to have a strong system of governance within the insurance company. The regulator explicitly defines a wide perimeter: The system of governance includes the risk-management function, the compliance function, the internal audit function and the actuarial function (Solvency II Directive, 30, 2009). The compliance and actuarial function do An EDHEC-Risk Institute Publication 65

66 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework not play a specific role with respect to internal models, however, both should bring their technical expertise, overseeing the methodology and data used for all calculations and processes. The Solvency II Directive sets rules for each of these functions to work hand-in-hand with the control of internal models. Some detailed implementing measures are still missing, in which case we can turn to banking regulations (Basel II) for comparison. Finally, regulation of outsourced functions is particularly important for small- to medium-sized insurers who might want to delegate some parts of their business. Risk management Being the first stage in designing, auditing and validating models, risk management entails strategies, processes and reporting procedures necessary to identify, measure, monitor, manage and report, on a continuous basis the risks, at an individual and at an aggregated level, to which they are or could be exposed, and their interdependencies (Solvency II Directive, article 44, Risk management, 1, 2009). Risk management should generally supervise all types of risks, from those applying to SCR, ALM and operational risks. As specifically regards internal models, major additional tasks are explicitly assigned to the risk-management function: notably everything regarding the design and implementation of the internal model, then its testing and validation and regular performance analysis. Risk management also includes communication surrounding the model, including documentation, periodic reporting, internal information, as well as dialogue with supervisors. It is required to be proactive in suggesting updates and improvements to the models. In short, Solvency II gives a central role to the risk-management function for all aspects of internal models. Implementing measures are to be adopted by the Commission regarding both elements of the systems and the risk-management function itself (Solvency II Directive, article 50, Implementing measures, 2009). Internal audit Solvency II calls for an internal audit function that is objective and independent from the operational functions, and which should in particular provide an evaluation of the adequacy and effectiveness of the internal control system and other elements of the system of governance (Solvency II Directive, article 47, Internal audit, 2009). This responsibility applies, in particular, to internal model control. Internal auditors should also make sure appropriate actions are taken to remedy any issue they might have encountered. As before, pending the adoption of implementing measures (Solvency II Directive, article 50, Implementing measures, 2009), we believe that internal audit requirements for a partial model based on an external benchmark should be much less stringent than for a full-scale internal model. Outsourced functions Outsourcing of some functions is likely to be a widespread practice, especially for smaller firms, which may not want to develop the in-house resources to deal with Solvency II requirements. However, this is severely restricted by the Solvency II Directive so as to limit potential abuses. The insurer 66 An EDHEC-Risk Institute Publication

67 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework will indeed retain the full responsibility for satisfying the obligations described in the directive. It will be allowed to outsource only if it can ensure that neither governance, nor operations, nor supervision, nor continuous and satisfactory service to policy holder will suffer from it (Solvency II Directive, article 49, Outsourcing, 2009). Finally, the supervisor needs to be notified of any decision to outsource an important function or activity, and in case of material developments. In our view, these requirements do not necessarily preclude the outsourcing of parts of the design and implementation of the full or partial internal model, as long as strict governance and reporting processes are followed. As long as the appropriate due diligence is conducted, following a rule-based strategy around an external benchmark should generally make the validation process simpler External audit and validation Statutory audit The lack of transparent financial information (within banks and the financial system) is considered one of the causes of the financial crisis. Reliance on external audits has become an important element for market confidence, especially if they are performed in accordance with high quality auditing and ethical standards. A statutory audit is important for encouraging consistent and meaningful disclosures about valuation processes and enhancing transparency. Aside from their traditional role, under Solvency II statutory auditors have an obligation to report promptly to the supervisory authorities any material breach or non-compliance they might encounter, or any reservation they might have on the accounts (Solvency II Directive, article 72, Duties of auditors). While nothing is specifically set out regarding internal models, in our view they ought to be inspected as part of statutory audits of annual accounts and consolidated accounts since they have a serious effect on the financial situation or the administrative organisation. This is different from the Basel II framework for banking supervision where a more comprehensive role is set out for external auditors. For instance, the internal model approach for market risk under Basel II specifies that the validation of models accuracy by external auditors and/or supervisors should cover at least the verification of models internal validation process, of the independent (from trading area) validation of the formulas used in the calculation process and pricing of complex products, of the adequacy of the structure of internal models with respect to the bank s activities and geographical coverage, of the back-testing results for the bank s internal measurement system, and of the transparency and accessibility of data flows and processes. As supervisors rely upon financial information to evaluate the condition of banks and to determine regulatory capital, an accurate valuation of loans, securities and other assets is necessary. Supervisors expect management to have sound processes for valuation and may potentially rely on valuations contained in the financial statements that have been subject to external audit. The same applies to the capital components reported in the financial statements which are subject to external auditing and are used as a basis for determining regulatory capital. An EDHEC-Risk Institute Publication 67

68 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Supervision The approach to supervision is to be prospective and risk-based, and comprises an appropriate combination of off-site activities and on-site inspections (Solvency II Directive, article 29, General principles of supervision, 2009). Supervision itself is devolved to the State level. Member States shall ensure that the supervisory authorities review and evaluate the strategies, processes and reporting procedures which are established by the insurance and reinsurance undertakings to comply with the laws, regulations and administrative provisions (Solvency II Directive, article 36, Supervisory review process, 2009). The fifth Quantitative Impact Study (QIS5) surveys the industry regarding the implementation of the Solvency II Directive, notably with regard to internal models and related issues. Regulated firms are asked to make clear when and where they have been working with consultants on developing their internal model, which external model or data they used and what adjustments and limitations they might entail. Firms will need to highlight how independent reviews, internal or external, played a part in the validation process. External models and data used for the development of internal models are subject to the validation process and insurers need to demonstrate their complete understanding. They are required to document and explain their preference for them, their role and the extent to which they are used within their internal model processes. Insurers should also make sure that they are consistent with the framework set out for the calculation of the capital requirements through internal models. Supervisors will intervene at every stage of the process and departures from the standard formula, namely internal models, will be especially scrutinised. Insurers will have to justify the scope of their internal model, and special approval will have to be obtained from the supervisor for partial internal models. According to the level 3 measures Pre-application Process for Internal Models published by EIOPA on March 2010, the supervisory authority will look for a complete understanding of the technical characteristics of the internal model, its documentation and its usage in the business. This means that quantitative and qualitative elements of the internal model will be considered. For this purpose, the supervisor will review several aspects of the model such as the structure, parameters and they way they are determined, the methods and data used as well as the validation process of the output. Additionally, the ORSA results might also be part of the revision process in order to examine the risk profile and the solvency assessment, analyse the deviation of the company s risk with respect to internal model assumptions, risk tolerance limits and compliance with the Directive s requirements. With respect to external models and data the supervisor would verify the method used, the appropriateness of incorporating them into the internal model, how they are used in the calculation of the SCR, and how the data integrity has been verified. Given these requirements, relying on transparent external references based on solid academic foundations such as the Solvency II Benchmarks developed by EDHEC-Risk Institute would facilitate the supervision of partial internal models and the internal validation process. 68 An EDHEC-Risk Institute Publication

69 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 5.2 The validation processes The Solvency II validation process is part of Pillar II and corresponds to a set of tools and processes used by the undertaking to gain confidence over the results, design, workings and other processes within the internal model (Tests and Standards for Internal Models Approval, EIOPA). Since models are complex, their validation allows a degree of confidence that methods and assumptions are appropriate and that outputs are reliable. As internal models are used in the own risk management assessment as well as in the calculation of the regulatory capital requirements, a misestimation might lead to policyholders protection being reduced and also affect the risk management and decision process of the company. This internal validation process would serve to demonstrate to the regulator that the model and its output are understood, appropriate and trusted. In order to achieve this, the Solvency II benchmarks may constitute a sound reference for validation of internal models. They will be useful to prove that the model and its output are understood and trusted. In Solvency II, validation of the internal model should not only apply to the calculation kernel, but should encompass the qualitative and quantitative processes of the internal model. This means that the validation includes an assessment of how the model is used within the company and how it is integrated in the overall risk management and system of governance. The validation process may be carried out by the internal audit function of the company or may be outsourced. The areas that need to be validated are several but should at least comprise data, methods, assumptions, expert judgement, documentation, systems IT, model governance and the use test (Technical specifications for QIS5, IM.C.9 Validation). This comprehensive task can and should be accomplished thanks to a variety of methods, which are still being defined. In 2009, CEIOPS published the level 2 advice of Tests and Standards for Internal Model Approval which sheds some light on the Directive principles on model validation. If banking supervision is any indication, however, a single unified process cannot emerge. On the contrary, insurers should use a number of different layers and tools to assess the performance of their internal model and its surrounding processes. No unequivocal test can determine the adequacy of a model with certainty; at various stages assumptions have to be made which are sometimes untestable, or for which the conceptual foundations may seem sound, but they remain hypotheses. Furthermore, not all tests have the same power for all areas of decision. Hence, they should be used complementarily, and the final decision should involve a judgmental evaluation of possibly ambivalent results. This state of facts makes the qualitative processes almost as crucial as the quantitative toolbox in the general assessment of an internal model. Indeed, while quantitative validation can be at least partly based on an external toolbox especially for a strategy built around an external benchmark, the qualitative validation process cannot be as standardised. The Basel II regulation standards with respect to market risk model validation An EDHEC-Risk Institute Publication 69

70 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework state that the banks need to set in place processes to ensure that their internal model have been adequately validated by suitable qualified parties independent of the development process to ensure that they are conceptually sound and adequately capture all material risks (Basel Committee on Banking Supervision, 2006). The validation process not only includes back-testing but also tests to verify the appropriateness of assumptions, hypothetical changes in portfolio value, tests carried out for longer periods or different confidence intervals, and the use of hypothetical portfolios to ensure that the model is able to account for particular features Qualitative processes Qualitative review and oversight The qualitative review process is a fundamental step in assessing the validity of an internal model. While it should be clear from the previous section that every link in the chain is involved in this process, Solvency II has not set out very explicit requirements at this stage. The qualitative review will be more difficult to outsource; however, it will be greatly simplified if the strategy follows an external benchmark. The use of external models and data should be accompanied by articulated strategies for validating and regularly reviewing the performance of these models and data. As a comparison, Basel II is generally more specific on this type of qualitative review procedure. It is also very comprehensive. This process could entail review of documentation, review of development work, dialogue with model developers, review and derivation of any formulae, comparison with what other firms are known to do, comparison with publicly available information (Basel Committee on Banking Supervision, Range of practices and issues in economic capital modelling, Consultative Document, August 2008). Human control of model validation is also an important feature in Basel II. Expert judgment Expert judgment is recognised to be an indispensable element within Solvency II internal model validation. It is especially important for evaluating extreme risks, for which a reliable estimate might be difficult to obtain if little data is available. It could also be the case, though, that even when a large number of observations is available, expert judgment may also be required, for example, if the observations of past events are not suitable for predicting the future occurrence of events (Technical specifications for QIS5, IM.C.6 Statistical quality Expert judgment). Considering the risks inherent in expert judgment, any resorting to it should be thoroughly documented and any assumption or parameter estimates based on expert judgment should be justified within the qualitative validation process. This is similar to the Basel II procedure (see Basel II Revised Framework, Comprehensive Version, 428). Use test Insurers should pass the use test, which means that they need to be able to demonstrate to the supervisor that their internal model is widely used and plays an important role in the system of governance, risk management and decision making (Technical specifications for QIS5, IM.C.5 Use test). It should also be used in 70 An EDHEC-Risk Institute Publication

71 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework the process of measuring and allocating economic and solvency capital, as well as be included in the own risk and solvency assessment. It is not enough that a model is carefully designed and tested, if it is not used internally. This rule is supposed to limit any kind of window dressing whereby an insurer would conceive a model to meet the Solvency II specifications, but not trust the model enough to use it internally. It is also the likely case that in such a configuration, the results of the Solvency II internal model and the one actually used for decisionmaking would diverge in certain types of risky scenarios. Under the Level 2 measures on Tests and Standards for internal model approval (2009), EIOPA states a list of principles for the supervisory authority to assess the use test. The founding principle states that the use of the internal model should be sufficiently material to encourage the improvement of its quality. Including the use test prominently in the model validation process would also provide incentives for people who run the undertaking to actually consider, rely on and improve the internal model in line with their day-to-day job. The articulation with Solvency II benchmarks and the accompanying toolbox is therefore natural, as it would become an element within the validation dashboard. However, the permanent adequacy of the design and functioning of the internal model and the correct reflection of the risk profile of the company will remain the responsibility of the administrative, management and control bodies of the insurance company. Documentation In terms of documentation, the criteria set out by the regulators are very extensive. It must be sufficiently detailed and comprehensive enough to allow knowledgeable third parties to understand the internal model (Technical specifications for QIS5, IM.C.10 Documentation). Documentation should demonstrate that the internal model (including external models and data) complies with all the requirements set in the Directive and is sufficient to fully reproduce the outputs of the internal model, based on the inputs. It should therefore include a complete description of the governance process, including model change policy, technical specifications of the model, data, statistical quality standards, expert judgement, calibration, profit and loss attribution, validation, scope and finally, known shortcomings and mitigation techniques used. Where Solvency II states general principles, Basel II is again very specific on what an appropriate documentation for internal models should contain, and every detail of the elements to document is fleshed out in the framework. Data policy The model validation process [ ] shall also include an assessment of the accuracy, completeness and appropriateness of the data used by the internal model. (Solvency II Directive, article 124, Validation standards, 2009). Accuracy is defined as the absence of material mistakes, while completeness means there are available data for all relevant model variables and no relevant data are excluded (Technical specifications for QIS5, IM.C.6 Statistical quality). Both An EDHEC-Risk Institute Publication 71

72 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework of these criteria should limit estimation errors of the parameters. Appropriateness entails consistency with the underlying assumptions of the actuarial and statistical techniques so as to reflect the risks to which the undertaking is exposed. All of these criteria are of a qualitative nature; even though some quantitative statistical tests might be involved, they cannot completely be a substitute for human judgment. However, an acceptable degree of standardisation (within the specific requirements of the company) should make it easier to pass validation. Thus, the insurer will have to demonstrate to the regulator that the data used is accurate, complete and appropriate. To do so, each insurer should specify a policy on data quality and data updates. This policy is agreed with the regulator as part of the initial internal model approval and will serve as a basis for supervision. The data policy covers several aspects. Firstly, insurers should define their own data quality concept based on the regulator s requirements. Secondly, insurers need to define quality checks which consist of processes designed to provide assurance of the quality of data used to operate, validate and develop internal models. These quality checks should also specify the actions needed to be taken when the data no longer complies with the Directive requirements. Thirdly, the policy should contain guidance on the frequency of regular data updates and circumstances that trigger unexpected data updates or recalculation of the SCR. Finally, a detailed description of the methods or methodologies used, the definition of responsibilities, and the frequency of application are also required Quantitative processes The quantitative processes for model validation under Solvency II are not yet set in stone, and it is not clear how comprehensive the testing should be for a firm to gain approval from the supervisor. However, some general principles and methods are already being discussed and should become clearer as insurance companies achieve conformity. In any case, it is very likely that some discretion will be left to national supervisory bodies. Input validation Validating input parameters themselves is of course a primordial task for internal model assessment. This part, like much of the design of the internal model, is still largely unspecified. In principle, it should be in line with the risk management requirements. It should be noted that no validation of parameter estimation is enough to compensate for model uncertainty and potentially incorrect assumptions on the underlying data processes. In our view, it would make sense for an insurer to rely on externally-provided parameters so far as they are appropriate to the actual risk exposures undertaken by the company. Model replication As we have mentioned, under Solvency II outputs of the internal model should in principle be reproducible on the basis of the documentation if all the inputs of the internal model were available (Technical specifications for QIS5, IM.C.10 Documentation). However, in principle suggests that this might not always be practically possible, as internal models can generally be very complex to run at least for large organisations. 72 An EDHEC-Risk Institute Publication

73 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Replicating portfolios can be a means to work around the complexity of the internal model. Finding proxies for the actual model can simplify computations and provide a useful approximation quickly. It is not, however, a valid method to substitute for full internal modelling, be it complex: [f] or the calculation of technical provisions, however, undertakings should rely on their more accurate valuation techniques (Technical specifications for QIS5, IM.C.6 Statistical quality). On the matching side of banking supervision, Basel II tackles the more general issue of model replication as a good practice for validation. However, replication simply by re-running a set of algorithms to produce an identical set of results would not be sufficient model validation due diligence (Basel Committee on Banking Supervision, Range of practices and issues in economic capital modelling, Consultative Document, August 2008). Benchmarking, backtesting and stress testing Benchmarking, backtesting and stress testing are three useful tools for validating a model. Benchmarking involves comparing the internal model to a reference model or testing it on some reference portfolios. Backtesting is testing the model on historical data to check how well it performs. Finally, stress testing implies simulating extreme conditions for both the model input and its assumptions. Those three tools are commonplace in the risk management function. The only relevant mention in Solvency II is that [s] upervisory authorities may require insurance and reinsurance undertakings to run their internal model on relevant benchmark portfolios and using assumptions based on external rather than internal data in order to verify the calibration of the internal model and to check that its specification is in line with generally accepted market practice (Solvency II Directive, article 122, Calibration standards, 2009). This is very much in line with the external benchmark approach that we propose in the next section. If we turn to the practices of the Basel II framework for comparison, benchmarking is also a crucial tool for quantitative validation (see Basel Committee on Banking Supervision, Range of practices and issues in economic capital modelling, Consultative Document, August 2008). Backtesting is also used; but the BCBS notes that it is not yet a key component of banks' validation practices for economic capital purposes. Stress testing is of course common for banks, and gained even more prominence with the 2010 EU-wide testing conducted by the Committee of European Banking Supervisors. Sensitivity testing and profit and loss attribution Sensitivity testing is one of the only methods explicitly mentioned in the Solvency II Directive. The model validation process shall include an analysis of the stability of the internal model and in particular the testing of the sensitivity of the results of the internal model to changes in key underlying assumptions (Solvency II Directive, article 124, Validation standards, 2009). This should be part of any toolbox used to test a partial model. An EDHEC-Risk Institute Publication 73

74 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework Finally, profit and loss attribution receives a mention in the QIS5 technical specifications, namely how the categorisation of risk chosen in the internal model explains the causes and sources of profits and losses (Technical specifications for QIS5, IM.C.8 Profit and loss attribution). The risk categorisation, combined with profit and losses, allows the risk profile of the company to be understood. Contrary to sensitivity testing, it is not, however, an explicit requirement for validation. The BCBS notes that on the banking side, [a]ttribution is not widely used except for market risk pricing models (Basel Committee on Banking Supervision, Range of practices and issues in economic capital modelling, Consultative Document, August 2008). 6. Conclusion Solvency II requires capital buffers for all risks, in contrast to Solvency I where capital requirements where mainly a function of technical provisions and premiums. For instance, under Solvency I, market risks were not taken into account while under Solvency II, the market risk capital charge considers shocks on key financial variables such as interest rates, spreads, equities value, exchange rates and concentration. With respect to equity risk, the constraint set by the regulator makes equity investment prohibitively expensive (a SCR of 39% of the value of equities listed in OCED countries and 49% for equities listed in emerging countries, non-listed equity and alternative investments; plus an adjustment ranging from -10% to 10%), resulting in the necessity for insurance companies to essentially shy away from investing in equity, with a prohibitive associated opportunity costs related to giving up almost entirely on the equity risk premium. This study focuses on analysing risk reduction management strategies that allow insurance companies to keep a reasonable capital requirement while providing attractive products to policyholders. Setting in place these dynamic risk management strategies implies implementing partial internal models which capture insurance companies specific characteristics and, in particular, the benefits of these allocation strategies. Those benefits are numerous, including: responsiveness to changes in market environment, increased returns while respecting Solvency II constraints and asset diversification. There are two main requirements for such dynamic asset allocation solutions. Firstly, insurance companies facing long-term horizons should also try to account for the presence of a non-trivial term structure of equity risk (i.e., the fact that equities are less risky over the long run than they are over the short run because of mean-reversion in the equity risk premium). Secondly, of course, dedicated Solvency II solutions should account for the presence of the short-term constraints imposed by the regulation, which for financial risks, means the control of downside risk, not in absolute terms, but with respect to the liabilities. Simple products such as constant proportion portfolio insurance (CPPI) strategies, sometimes popular in the asset management industry, fall short of meeting the two aforementioned requirements for an adequate Solvency II solution. Integrating dynamic risk allocation is more in line with the logic of an internal model, 74 An EDHEC-Risk Institute Publication

75 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework rather than a product logic, and is more consistent with the spirit of Solvency II. Partial internal models are more flexible than the standard formula and they can target specific risk components such as equity risk. Their implementation requires fewer resources compared to a full internal model and therefore allows smaller insurance companies to regain competitiveness. However, their implementation is not an easy task and one of the major challenges is their compliance with Solvency II. Under Pillar II, insurance companies are required to demonstrate that they have an organisation (governance, risk management systems) that ensures good knowledge of their risk exposure, its coherent assessment, an operational mechanism for managing risks and a reporting stream for decisionmaking. This organisation is a key issue in implementing a partial internal model. Additionally, insurers need to implement an internal cycle for internal model validation to demonstrate that they are appropriate for the use within risk management and decision-making processes, as well as to demonstrate to the supervisory authority that the capital requirements calculated with internal models are adequate. Compliance with these Pillar II requirements determines the approval of internal models by the Supervisory Authority. Part II of the present document presents a methodological framework based on objective academic references to design risk management strategies in the form of dynamic allocation benchmarks which allow exposure to be gained to equity markets while maintaining a reasonable solvency capital requirement. The benchmarks framework is public, totally transparent, well documented and grounded in a rules-based approach and in solid academic foundations. These features turn the benchmarks into an independent external reference that can be used as the fundamentals for the development of partial internal models by insurance companies. Since Solvency II provides incentives for risk management by recognising the benefits of full or partial internal risk management models, it is in the company s interest to have such internal models recognised. Our view is that there is value in relying on third parties for the definition and documentation of internal models for market risks, and in relying on asset managers for their implementation. In this context, the EDHEC-Risk Solvency II Benchmarks can serve as a very useful point of reference to implement a partial internal model to assess equity exposure which is easily compliant, particularly with the governance requirement and more generally with Pillar 2. An EDHEC-Risk Institute Publication 75

76 Part I: Challenges related to Equity Investments within the Solvency II Regulatory Framework 76 An EDHEC-Risk Institute Publication

77 Part II: Introducing the Solvency II Investment Benchmarks An EDHEC-Risk Institute Publication 77

78 Part II: Introducing the Solvency II Investment Benchmarks 27 - The terminology life cycle investing is also sometimes used in a retail money management context, where it relates to retirement products (also known as target date funds) recommending deterministic decreases in equity allocation when moving towards retirement date. Having the same allocation for investors with a given time-horizon regardless of market conditions is however entirely inconsistent with common sense and also with academic prescriptions (see more details and references in the body of the text). Our implementation of the life-cycle investing concept in the context of Solvency II benchmarks will be very different from these sub-optimal heuristic strategies. In the second part of this document, we present a framework for designing dedicated dynamic asset allocation solutions, which we refer to as Solvency II dynamic allocation benchmarks, or Solvency II benchmarks in short, and which can be used by insurance companies to achieve a substantial exposure to equity risk, and the associated premium, with a strict and explicit control over the implied Solvency II charge. We first describe (in section 1) the conceptual foundations for this approach, which rely upon two main paradigms respectively known as life-cycle investing (LCI) and risk-controlled investing (RCI). We then introduce the formal dynamic asset allocation model (section 2) and discuss the implementation challenges related to practical implementation constraints (section 3), as well as the questions related to the calibration of the model (section 4). A fifth section presents the numerical and empirical results we obtain from simulations based on stochastic and historic scenarios. 1. The LCI/RCI paradigms In what follows, we review the conceptual foundations behind the Solvency II dynamic allocation benchmarks, and also discuss how to implement these principles. The Solvency II allocation benchmarks are meant to be regarded as substitutes for static equity investments by insurance companies. Overall, the key question faced by insurance companies with respect to equity investment can be summarised as follows. On the one hand, short-term solvency constraints make equity investment prohibitively expensive. On the other hand, shying away from investing in equity involves a substantial opportunity cost, especially for long-term investors. This is because the equity risk premium is: (i) positive, hence attractive for all investors and (ii) mean-reverting, hence even better for long-term investors. So the challenge is how to reconcile long-term performance objectives and short-term solvency constraints. In what follows, we argue that meeting this challenge requires relying on two different approaches to risk management, namely risk hedging to generate access to optimal long-term risk-adjusted performance, and risk insurance to ensure the respect of short-term risk budgets. As we will see, implementing efficient risk hedging and risk insurance requires the use of dynamic, as opposed to static, asset allocation techniques. In a nutshell, risk-hedging involves reaching the highest possible asset levels at horizon with minimum uncertainty, which can be achieved through a time- and statedependent mixture of safe and risky assets that are revised as a function of market conditions and time-horizon. This approach is known as life cycle investing in the academic literature. 27 One remaining limit of the approach is that hedging risk away symmetrically decreases both downside and upside potential. In other words, extremely risk-averse investors will focus on hedging away the impact of risk factors impacting the portfolio value, without any access to the upside potential. In this context, regulatory constraints, which can be formalised as Solvency II risk budgets, should be incorporated ex-ante as key ingredients in the design of the optimal investment solutions. These constraints are not managed through hedging strategies, which are dedicated to immunising the 78 An EDHEC-Risk Institute Publication

79 Part II: Introducing the Solvency II Investment Benchmarks long-term portfolio value against changes in key risk factors, but through insurance strategies, which are designed to maximise the probability of reaching the long-term investment objectives while respecting the short-term risk constraints. In other words, meeting this challenge is based on managing risks by insuring them away. In terms of investment solutions, it leads to the design of dynamic risk-controlled allocation strategies that lead to giving up part of the upside potential of the performance-seeking portfolio, and more specifically, to the equity portfolio, in exchange for protection on the downside. The practical implication of the introduction of short-term constraints is that optimal investment in the risky equity index is a function not only of risk aversion but also of risk budgets (margin for error), as well as of the likelihood of the risk budget being spent before the horizon. In a nutshell, a pre-commitment to risk management allows one to adjust risk exposure in an optimal state-dependent manner and therefore to generate the highest exposure to the equity risk premium and respect risk constraints. We argue in this document that dedicated Solvency II allocation benchmarks can be designed to be used as references by insurance companies. These dynamic asset allocation benchmarks are based on an industrialisation of the two key aforementioned paradigms: life-cycle investing (LCI), to take insurance companies typically long horizons into account, and risk-control investing (RCI), to take the presence of Solvency II risk budgets into account. Implementing such optimal strategies in a real world context is a serious challenge, which requires a parsimonious partition of the insurance company/market conditions that will allow for different allocation strategies. Broadly speaking, there are two sets of attributes that should be used to define the various categories of asset allocation decisions, namely the subjective attributes and the objectives attributes. The subjective attributes are related to each particular insurance company, and include most importantly the duration of the liabilities faced by the insurance company. The objective attributes, on the other hand, apply to all insurance companies, and relate instead to market conditions, with a proposed asset allocation decision that will be a function of the following three state variables: the current (estimated) level of risk premium (typically proxied by a function of dividend yield or priceearning ratios), the current level of interest rates and the current volatility level. Here, a discrete partition of the states of the world can be used for these three variables, with suitably defined high, median and low values for risk premium, interest rate and volatility levels. From the technical standpoint, our approach builds upon an abundant stream of research on long-term investment decisions in the presence of a stochastic opportunity set. This question was first formally analysed in the seminal paper by Merton (1971), who shows that the presence of risk factors that impact the productivity of wealth justifies the introduction of intertemporal hedging demands in an investor s optimal allocation. Subsequent papers have shown that when maximising utility from nominal wealth, an investor only hedges those state variables that impact the nominal short-term rate An EDHEC-Risk Institute Publication 79

80 Part II: Introducing the Solvency II Investment Benchmarks 28 - In an individual money management context, another possible rationale for an allocation to equities increasing in the horizon is the need for young investors to compensate for the bond-like nature of human capital. One notable exception is Benzoni et al. (2007), who question the bond-like nature of human capital, and show that the presence of co-integration between labour income and stock dividends results in a high long-term correlation between stocks and human capital. This substantially reduces the optimal demand for stocks by young investors, and can even results in short positions in the equity market. and the market prices of risk (see Lioui and Poncet (2001) and Detemple et al. (2003) for recent references). Other papers have explicitly solved the portfolio choice problem when only one state variable is stochastic (e.g., the nominal interest rate in Sørensen (1999), Lioui and Poncet (2001), Bajeux-Besnainou et al. (2003) and Munk and Sørensen (2004), or the stock index Sharpe ratio in Kim and Omberg (1996), Campbell and Viceira (1999) or Wachter (2002). More realistic models have also been developed to account for the presence of both state variables. This challenge has been addressed both in the context of discrete-time VAR models (see for example Campbell et al. (2003), and in the context of continuous-time models, either by numerically solving the Hamilton-Jacobi- Bellman (HJB) equation obtained through dynamic programming (see Brennan et al., 1997), or, more recently, by exploiting the affine structure of the model (in the sense of Dai and Singleton (2000) and Liu (2007)) and solving the HJB equation explicitly (Munk et al., 2004) or quasi-explicitly up to the solution of a series of ordinary differential equations (Sangvinatsos and Wachter, 2005). Fewer papers have considered the impact of stochastic volatility. The reason is that volatility is not an issue, unless it affects the interest rate or Sharpe ratio. Kim and Omberg (1996) allow volatility to be stochastic in their model, but do not assume a specific form of dynamic evolution. Therefore, the Sharpe ratio is modelled independently and the optimal allocation does not contain any hedging demand against the volatility risk. In a different framework, Chacko and Viceira (2005) solve for the optimal policy, when the Sharpe ratio is inversely proportional to the volatility and the squared inverse of the volatility follows a square-root process. In both cases, the optimal portfolio contains a non-zero hedging demand against volatility risk since the Sharpe ratio is modelled directly from the stochastic volatility. Liu (2007) considers the case where the Sharpe ratio is proportional to the volatility, which itself follows the model from Heston (1993). A related strand of the literature has focused on portfolio choice over the life-cycle when the investor earns a stochastic non-financial income (see for example Viceira (2001), Cocco et al. (2005) or Benzoni et al. (2007)). Overall, these papers focus on modelling labour income and thus assume constant investment opportunities. Munk and Sørensen (2010) relax this assumption by adding a mean-reverting short-term interest rate to a model with stochastic labour income. In most of these papers, the fraction of wealth allocated to equities is typically shown to be a decreasing function of time-to-horizon, because of the induced term-structure of equity risk implied by the presence of mean-reversion in equity returns. 28 Regardless of the explanation of why long-horizon investors should hold more stocks than short-horizon investors, these findings seem to provide at best partial justification for the deterministic glide path prescriptions used in the target date fund industry. It should indeed be recognised that the prescription of these models also involves state-dependencies in the optimal allocation strategy, which are not present in current forms of target date funds. In fact, recommending the same allocation for investors with a given time-horizon regardless of market conditions cannot be optimal in the presence of a stochastic 80 An EDHEC-Risk Institute Publication

81 Part II: Introducing the Solvency II Investment Benchmarks 29 - As such, our paper is closely related to Sangvinatos and Wachter (2005). opportunity set, and omitting such statedependencies can lead to severe efficiency cost. As very clearly explained by Viceira (2009), " (...) long-term equity investors should invest more on average in equities than their short-horizon counterparts, but they should also consider periodic revisions of this allocation as market conditions change. It is logically inconsistent to count on reduced long-term risk while ignoring the variation in returns that produces it. This market-sensitive allocation policy is very different from the asset allocation policy of life-cycle funds, whose target mix moves mechanically away from stocks as an inverse function of investment horizon, regardless of market conditions. Thus mean-reversion arguments provide, if anything, only a partial justification for the roll down schedule characteristic of life-cycle funds." A number of recent studies have documented the shortcomings of standard forms of target date funds, including Booth and Yakoubov (2000), Cairns et al. (2006), Lewis and Okunev (2009), Viceira (2009), Bodie et al. (2009), Basu and Brisbane (2009) or Basu and Drew (2009). For example, Cairns et al. (2006) conduct a formal analysis of the welfare loss involved in following a deterministic strategy similar to actual target date fund strategies with respect to the optimal stochastic life-cycle strategy, and they find the associated opportunity cost to be very substantial. In what follows, we propose a comprehensive long-horizon dynamic allocation model in the presence of stochastic inflation and interest rates, mean-reverting equity risk premium and stochastic volatility. Our model extends Munk et al. (2004) by relaxing the somewhat questionable assumption of a perfectly negative correlation between equity returns and risk premium uncertainty. 29 While our setting is rich enough to account for the aforementioned features, we manage to obtain a quasiexplicit representation for the optimal portfolio strategy. Having quasi-analytical expressions for the optimal strategy turns out to be very useful in the numerical and empirical analysis. As mentioned before, implementing such extended forms of life-cycle investing strategies in a real-world context is a serious challenge, which involves providing a parsimonious enough partition of the relevant subjective attributes (mainly duration of the liabilities) and objective attributes (in particular, current estimated level of the risk premium provided by equities) that can cover the overall population of insurance companies and market conditions. Against this backdrop, our results suggest that a parsimonious partition of the set of investors and market conditions can be performed, which allows for a relatively accurate approximation of the true optimal strategy while being consistent with real-world implementation constraints. Our results have important potential implications for the design of improved forms of dynamic investing solutions that could help insurance companies cope with their long-term equity investment problems within the Solvency II regulatory framework. Overall, our results suggest that there is ample room for added value between one-size-fits-all investment products and fully-customised investment solutions, which can only be designed and implemented by the largest insurance companies. The methodology presented in what follows can be used to design An EDHEC-Risk Institute Publication 81

82 Part II: Introducing the Solvency II Investment Benchmarks 30 - Munk et al. (2004) assume in fact that the expected excess return on the stock, not its Sharpe ratio, is a mean-reverting process. However, since the volatility of the stock is constant in their model, it is equivalent to assume mean-reversion in the expected excess return or in the Sharpe ratio. new benchmarks for European insurance companies that are representative of a dynamic allocation strategy. The aim of the initiative is to enable all small- or mediumsized European insurance companies that do not have a full internal risk mitigation model to be able to avail of an objective and academically proven external reference. 2. A formal dynamic allocation model in the presence of Solvency II constraints In this section, we solve the long-term optimisation programme of an institutional investor (an insurance company) which faces stochastic investment opportunities and Solvency II-type constraints. We first introduce the setting, and then present the optimal strategy in case of market incompleteness coming from possibly imperfect correlation between stock returns and stock risk premium volatility risk. 2.1 State variables and asset returns We consider an investor with finite horizon T, which can be thought of as the duration of the insurance company s liabilities, and facing several sources of risk. The first of these is interest rate risk. The nominal short-term interest rate is assumed to follow a mean-reverting process, of the type proposed by Vasicek (1977): (2.1) A locally risk-free asset exists, which we shall refer to as cash, whose value is equal to the continuously compounded shortterm rate: Since it has zero duration, the cash does not provide any hedge against interest rate risk. In order to have a hedging instrument against interest rate risk, one would have to use nominal bonds with positive duration. In fact, because the Vasicek model considered here has only one factor, a single constant-maturity bond would be sufficient to replicate the entire term structure. Most academic studies assume that a zero-coupon is traded (see Sorensen (1999) and Munk et al. (2004)). In this paper, we will not assume that bonds are available for trading, because the regulator would impose additional capital requirements on an insurance company that holds assets exposed to interest rate risk. Another reason why we exclude bonds from the investment universe of the Solvency II benchmarks is because these benchmarks are not meant to be regarded as substitutes for the entire allocation policy of the insurance company, but instead merely as substitutes for their equity allocation. The second source of risk is the price of a stock index S : (2.2) As in Kim and Omberg (1996), Wachter (2002) and Munk et al. (2004), the conditional Sharpe ratio of that stock follows a mean-reverting process: 30 (2.3) Empirical evidence that the equity risk premium is time-varying can only be indirect, given that expected returns are not readily observable, and more importantly are extremely difficult to estimate. Since Merton (1980), it has been recognised that 82 An EDHEC-Risk Institute Publication

83 Part II: Introducing the Solvency II Investment Benchmarks 31 - Even if one assumes a constant expected return the task is challenging. As pointed out by Merton (1980), the precision of the naive estimator (the average of past log-returns) is completely insensitive to the sampling frequency. The only way to decrease it is to take longer samples, over which the assumption of a constant expected return becomes less and less realistic We warn the reader that we are using the unit volatility vectors (whose norm is by definition equal to 1), rather than the standard volatility vectors (whose norm is equal to the volatility of the process). statistics are of little help in estimating the equity risk premium. 31 More fruitful approaches are based on exploiting economic relationships, such as the link between risk and return, or the link between dividend yields and future returns (e.g. see Fama and French (1988), Campbell and Shiller (1988), Hodrick (1992), Campbell and Viceira (1999)). As explained in section 4.1, we choose a different approach in the implementation of the model, based on the Solvency II methodology, which incorporates a heuristic counter-cyclical procedure for assessing the capital requirement for equity as a function of a measure of equity cheapness given by the distance between current equity value and the last 3-year average value. Our model also assumes that the conditional variance of stock returns is stochastic. This feature was not present in the related literature (notable exceptions being Kim and Omberg (1996), Liu (2007) and Chacko and Viceira (2005)), but we deem it to be an important component of a long-term investment model, given the large empirical evidence for time-variation in the volatility of stock returns. We follow Heston (1993) by assuming that the conditional variance V=(σ S ) 2 of stock returns follows a mean-reverting square-root process: (2.4) The square root in the diffusion term prevents the process from going negative. The equations that describe the model can also be written in terms of a four-dimensional Brownian motion and of the unit volatility vectors, that are defined by: 32 Not all the risks are spanned by traded assets. Interest rate risk and stock price risk are, but volatility risk and Sharpe ratio risk are not, except for special parameter values. In particular, if one assumes, as in Wachter (2002), that innovations to this state variable are perfectly anti-correlated with the innovations to stock returns, (i.e. if ρ sλ = 1), then Sharpe ratio risk is spanned. In the remainder of section 2, we do not assume such perfectly negative correlations. A market price of risk vector is obtained as: As shown by He and Pearson (1991), the other market prices of risk are those vector processes of the form, where satisfies almost surely for all date t. Each of these prices of risk defines a pricing kernel M, through: There are infinitely many pricing kernels, because there are infinitely many vectors orthogonal to. 2.2 Derivation of the optimal strategy We let ω t denote the weight allocated to the locally risky asset, which is the stock. The weight allocated to the cash is thus 1 ω t. With these notations, we can write the budget constraint of an investor who receives no non-financial income and does not consume as: (2.5) Throughout the paper, we let be the Constant Relative Risk Aversion (CRRA) utility function. The objective of the insurance company is assumed to focus on the maximisation of the risk/return tradeoff in terminal wealth, while risk constraints An EDHEC-Risk Institute Publication 83

84 Part II: Introducing the Solvency II Investment Benchmarks with respect to the liabilities will be handled separately, as explained below: 33 where g is a function of time and the equity Sharpe ratio given by: (2.6) 33 - It is common in an individual asset management context (see Brennan and Xia (2002), Munk et al. (2004) or Sangvinatsos and Wachter (2005)) to maximise utility from terminal real wealth. It is indeed important to take inflation risk into account in long-term saving plans since an increase in price levels adversely impacts the purchasing power of monetary units. But this aspect of the problem has been extensively discussed in previous work (see e.g. particular Brennan and Xia (2002), and Campbell and Viceira (2001). subject to budget constraint (2.5). (2.6) is a dynamic problem since the control variable is the process of weights allocated to the stock index. In the martingale approach initially developed by Cox and Huang (1989), the dynamic programme is replaced by an equivalent static programme where the control variable is the terminal wealth and the budget constraint is expressed in terms of the present value of the terminal wealth. But the market is incomplete, so there are infinitely many pricing kernels in the economy, hence infinitely many static problems. As shown by He and Pearson (1991), the original dynamic problem (2.6) is equivalent to the static problem obtained with one particular pricing kernel M*, known as the minimax pricing kernel. Formally, the portfolio choice problem is equivalent to: and the optimal portfolio strategy is the one that replicates the optimal payoff. The following proposition gives a full description of this optimal strategy. Proposition 1. Consider the optimisation problem (2.6) subject to budget condition (2.5). The optimal payoff is given by: (2.7) where M* is the minimax pricing kernel, and the optimal wealth process reads: (2.8) The optimal allocation to stock is given by: (2.9) The functions C 1 ( ;γ), C 2 ( ;γ), C 3 ( ;γ) and C 4 ( ;γ) are the solutions to a system of (coupled) ordinary differential equations (ODEs) given in Appendix A.1. Proof. See Appendix A.1. We now discuss the structure of the optimal allocation in more detail. The first term on the right-hand side of equation (2.9) is already present when investment opportunities are constant (see Merton, 1969). It represents the speculative demand for stocks. The only difference from the constant opportunity economy is that the weight allocated to stocks in this term has become a function of the current Sharpe ratio and volatility. The speculative demand increases when the risk-return ratio of the stock is more attractive, and it is decreasing in the volatility. An investor with logarithmic utility (which is obtained for γ = 1) would set the weight allocated to the stock equal to the speculative demand. The second term on the right-hand side of equation (2.9) can be regarded as the hedging demand against interest rate risk of insurance companies 84 An EDHEC-Risk Institute Publication

85 Part II: Introducing the Solvency II Investment Benchmarks 34 - Under quite general assumptions, Wachter (2003) shows that the utility-maximising policy for an extremely risk-averse investor concerned with nominal terminal wealth is to fully invest in a nominal zero-coupon bond that matches his horizon We show in Appendix A.2 that the function C 3 (T t;γ) is positive and increasing. that have an investment horizon of T years. It is clear from the separation formula that for γ > 1, the hedging demand against interest rate risk is increasing in the relative risk aversion γ, which makes sense: more risk-averse investors want to hold more of the portfolio that is risk-free over the long run. In general, this hedging demand rationalises the investment in a pure discount bond with a maturity date matching the investor s horizon. 34 In the context of this analysis, however, we focus on the equity allocation for insurance companies, and recognise that insurance companies manage interest rate risk separately, in the context of their fixed-income portfolio. In the absence of fixed-income securities, some hedging demand can still exist for an equity-only portfolio if equity returns have a non-zero correlation with interest rates. On the other hand, if the correlation ρ Sr is zero, then the hedging demand cancels out, as can be seen from the fact that this hedging demand is proportional to that term. In the empirical calibration exercise, we actually take the correlation between stock returns and interest rates to be zero so that the optimal allocation strategy that we consider in what follows does not contain a hedging demand against interest rate risk. The last term on the right-hand side of equation (2.9) involves the coefficient, which is the beta of the stochastic Sharpe ratio with respect to the stock index. Hence this term represents a hedging demand against Sharpe ratio risk. With reasonable parameter values, the correlation ρ Sλ will be negative and the quantities C 3 (T t;γ) and C 4 (T t;γ) will be positive. 35 Hence the term represents an excess demand for stocks that arise from the uncertainty in the value of future Sharpe ratios. From the ODEs given in Appendix A.1, we see that these quantities depend on both the interest rate and Sharpe ratio models. Hence the non-logarithmic investor (γ 1) must not only know the current opportunity set, (i.e. the risk and return characteristics of the various assets over the next trading interval), but also how these characteristics evolve over time. Overall, the decomposition formula of Proposition 2 is similar to that introduced by Detemple and Rindisbacher (2010). The first term is the speculative demand for the locally risky asset. The second term represents an investment in the portfolio that best replicates the asset that is safe over the long-run, here a zero-coupon bond of maturity T. The third term is a hedging demand against the uncertainty in prices of risk. It is clear from Proposition 2 that the weight allocated to the stock is not only time-dependent, but also state-dependent, in contrast to the heuristic deterministic glide paths of existing life-cycle funds. In particular, the optimal allocation is a function of the stock index Sharpe ratio and the stock volatility. The speculative demand is independent of age, which comes as no surprise given that its purpose is to achieve the best risk-return trade-off over the next trading interval, regardless of the horizon of the investor. By contrast, the hedging demand against Sharpe ratio risk depends explicitly upon age. Moreover, the fact that C 4 is increasing in the time-to-horizon implies that young investors will react more to a rise in the expected return than older ones. An EDHEC-Risk Institute Publication 85

86 Part II: Introducing the Solvency II Investment Benchmarks It is unlikely that the allocation to the stock will be a monotonic decreasing function of age as advocated by current financial advice, since the Sharpe ratio and the volatility vary stochastically over time. Nevertheless, both C 3 and C 4 shrink to zero as the investor approaches his horizon date. As a consequence, an investor close to his horizon will only hold the speculative demand and the cash. 2.3 Respecting the Solvency II requirements The strategies that have been introduced in the previous section are designed to maximise expected utility from the terminal wealth at a given horizon. They are optimal in the long-term since they generate the highest possible average asset level for a given uncertainty in the terminal value of the assets. Nonetheless, such optimal strategies may end up violating short-term Solvency II constraints in market conditions with strong bear equity markets. The focus of this section is to design a set of asset allocation strategies that should not only maximise the risk/return trade-off in terminal asset levels, but also satisfy given Solvency II budgets. The idea here is to have a separate control for risk-aversion (i.e., aversion with respect to uncertainty in long-term asset levels), and loss-aversion (i.e., aversion with respect to short-term losses). More precisely, the requirement here is that the wealth generated by the strategy at any time within a year should never fall below a fraction 1 δ of the capital invested at the beginning of the year, where can be interpreted as a given Solvency II equity charge risk budget. This requirement can be mathematically formulated as: A t (1 δ) A 0, almost surely for all t in [0, 1Y] (2.10) Typical values for δ are 5%, 10%, 15%, and 20%. The floor, defined as F = (1 δ) A 0 is the minimum acceptable wealth level, and the outstanding risk budget at time t is the distance of current wealth A t to the floor. In this context, we will consider strategies where the weight allocated to stocks is of the form: (2.11) where ω * is the optimal weight referring to the unconstrained strategy of Proposition 2, m is a constant multiplier and A c is the current value of the strategy. Note that in the limit of a floor converging to zero (no Solvency II capital charges), then the constrained allocation policy matches the unconstrained allocation policy, as it should. Formula (2.11) says that the dollar amount allocated to the unconstrained strategy is an increasing function of the risk budget A c F. On the other hand, when the risk budget shrinks to zero, the portfolio is entirely invested in cash, since ω c is equal to zero. Since the cash investment grows faster than the floor, which is constant, a positive risk budget is recovered just after wealth has landed on the floor. Hence strategy (2.11) guarantees that the wealth is at least always equal to the floor, and if the floor is attained, the portfolio does not remain invested in cash at later dates. 86 An EDHEC-Risk Institute Publication

87 Part II: Introducing the Solvency II Investment Benchmarks 3. From stylised strategies to real world solutions This section discusses the practical implementation of the constrained ω c and unconstrained ω * strategies introduced in the previous part. These strategies are not directly implementable in practice, for the following three main reasons: They assume continuous trading, while trading is possible only in discrete time; They assume that short sales and leverage are permitted; They assume a perfect knowledge of the current Sharpe ratio and volatility of the stock. Note that the last assumption is not highly unrealistic as far as volatility is concerned, since estimation risk is limited for volatility parameter estimates, but it is far too optimistic when it comes to the Sharpe ratio, since it is well-known that expected return estimates are extremely sample-dependent. To address the first two issues, one needs to implement the strategy in discrete time and to resize the weights so as to rule out short positions and leverage. By definition, taking into account such implementation constraints will decrease the expected utility from the terminal wealth. However, it has been shown in Deguest et al. (2011) that the loss in expected utility with respect to the continuous-time optimal strategy remains low compared to deterministic allocations. Addressing the non-observability of the Sharpe ratio is less straightforward. The solution that we propose below is based on a robust estimation of this quantity: rather than guessing its exact value, we pursue the more realistic objective of guessing whether it is low, medium or high. There again, it can be shown that the opportunity cost involved in a parsimonious and robust, albeit imperfect, estimation of the Sharpe ratio is relatively small (see Deguest et al. (2011) for more details). 3.1 Description of Solvency II benchmarks In this section, we describe the asset classes used to build the Solvency II benchmarks. In order to satisfy the Solvency II requirements, we need to impose short-term constraints at each time step. Since profits and losses are computed at the end of each calendar year, it seems natural to design the Solvency II benchmarks as roll-overs of strategies with a given horizon T, stopped after one year, and such that the Solvency II constraints described in section 2.4 are satisfied at each date. In other words, the benchmarks are reset on a yearly basis, with the arbitrary one-year time-frame chosen to be consistent with the Solvency II regulatory focus on short-term losses. We will let the horizon T vary across different values in the empirical illustration, in order to illustrate the properties of the strategy over short to long investment periods. Specifically, we will choose T = 3Y, 5Y, 10Y, 15Y. The unconstrained strategy ω * is of the form given in proposition 2 (where we take the correlation ρ Sr to be zero): leading to the Solvency II strategy given by: An EDHEC-Risk Institute Publication 87

88 Part II: Introducing the Solvency II Investment Benchmarks The process of weights ω c defines a family of Solvency II benchmark strategies for various values of the investment horizon T = 3Y, 5Y, 10Y, 15Y and various values of the Solvency II capital charge budget δ = 5,10, 15 or 20%. As discussed in the introduction to this section, the practical implementation of these benchmarks will be done in discrete time, and will involve short-sale constraints on the unconstrained strategy ω *, and partitions of the state space of volatility and Sharpe ratio. The family of Solvency II benchmarks is parameterised by two subjective parameters: the relative risk aversion γ; the constant multiplier m. The risk aversion γ and multiplier values will be calibrated in such a way that the average allocation to the equity index is sufficiently high to obtain significant exposure to the equity risk premium, but also in such a way that the allocation never gets so high at any point in time that it might lead to violating the Solvency II floor in the empirical analysis. 3.2 Discrete-time trading strategies The optimal strategies of Propositions 2 and 3 were derived under the assumption that trading takes place in continuous time. In practice, however, continuous trading is impossible because it would incur prohibitively high transaction costs. Therefore, all the strategies that we subsequently test are implemented at discrete dates. More formally, let t 0 = 0, t 1,, t n-1,< T denote the trading dates, and let t n = T the investor s horizon. We also define as the realised return on the stock index over the period [t 1, t i+1 ], and as the realised return on the bank account. We consider any strategy, given by a -tuple of weights. Over each time interval [t 1, t i+1 ], the portfolio is left buy-and-hold, hence the return can be expressed as leading to the final wealth In implementation and the numerical analysis in section 5, we take the trading frequency to be monthly, so. It should be noted that discretised optimal strategies are suboptimal, but this opportunity cost remains unavoidable in practice since trading can only occur on a discrete time set. 3.3 Imposing short-sale constraints on asset classes Since no short-sale constraints have been imposed ex-ante, there is no reason why all the optimal constrained weights ω c would fall between 0 and 1. However, short-selling is not a desirable feature in investment solutions for insurance companies. Therefore, this section presents a simple methodology to impose short-sale constraints directly. There are no shortsales if, and only if, the weight allocated to the stock index is between 0 and 1. A simple way to ensure that these conditions are satisfied is to set the weight to zero if it is negative and to one if it exceeds one. The mathematical expression of this transformation is: where (x) + denotes max (0,x). Note that the weights in cash account also need to be recomputed as. 88 An EDHEC-Risk Institute Publication

89 Part II: Introducing the Solvency II Investment Benchmarks 3.4 Introducing a discrete partition of the set of market conditions Since the equity Sharpe ratio λ S and the equity volatility σ S are not directly observable, we will rely on estimation methods in order to estimate these two processes at all times. It is common knowledge that the Sharpe ratio is much harder to estimate than the volatility, therefore we expect to have larger estimation errors coming from the Sharpe ratio estimation. Following Martellini and Milhau (2010), we propose discretising the state space of these two processes, using a finer grid mesh for the volatility in order to take into account the difference in the estimation errors. This discretisation introduces a loss of optimality that has been shown to remain low in Deguest et al. (2011). Let us now describe the partition of the Sharpe ratio process, which is assumed to follow the mean-reverting process (2.3), and the stock index volatility process, which follows the mean-reverting square root diffusion (2.4). We recall that upper and lower bounds have been introduced for these two processes, and therefore, the crudest possible partition of market conditions would involve replacing all realisations of the process λ s by the constant, and all realisations of σ s by. The second partitioning level of the set of market conditions would involve distinguishing between high, moderate and low risk premium levels, and would therefore contain three buckets: and three standard values:, and. More generally, it can be verified that the (k i 0) partitioning level has standard values, where i = 1 for λ S, and i = 2 for σ S. For the upper and lower bounds of λ S and σ S, we propose the following values = 0.25 = 0.75 = 0.05 = 0.40 Note that the volatility bounds are in line with observations of historical volatilities and of the VIX index for most of the available data. However, since the Sharpe ratio is not observable (see section 4.4 for more details on what exact crude proxy will be used for the unobservable Sharpe ratio on the equity index), the choice of the bounds has to be done differently. We therefore propose calibrating them using the parameters of the Sharpe ratio process. Indeed, they roughly correspond to the mean value plus or minus the long-term standard deviation of λ S. Once these bounds are set, we build a partition using a number of standard values (equally spread between the upper and lower bounds) equal to for λ S and to for σ S as previously explained. Since the estimation of the Sharpe ratio is known to be difficult, we will work with very low partitioning levels (k 1 = 2). Nonetheless, the estimation of the volatility is a pretty standard estimation, so we can consider high partitioning levels (k 2 = 5). 4. Calibration of the model In this section, we present the various approaches that we have considered in order to calibrate the model parameters. Table 1 summarises the set of parameters An EDHEC-Risk Institute Publication 89

90 Part II: Introducing the Solvency II Investment Benchmarks that we will consider in our Monte Carlo simulations. where S t is the value at time t of the equity index, and Table 1: Long-term parameter values Nominal Short-Term Rate b 3.06 % a 0.13 σ r 0.98% λ r % Sharpe Ratio κ 0.35 σ λ 23.22% 40% Volatility α 5.07 σ V 48.00% 21.38% Correlations ρ Sλ 1 ρ Sr 0 ρ SV ρ rv 0 This table displays the base case parameter values used in the numerical experiments 4.1 Estimation of the equity Sharpe ratio The current value of the Sharpe ratio is not observable, which makes its estimation difficult. In the empirical results presented in this document, we follow an estimation approach based on the Solvency II methodology used to assign Capital Requirement for Equity (SCRE). Indeed, CEIOPS defines the SCRE as follows (see CEIOPS Solvency II Calibration Paper (2010), paragraph 3.88 page 49): is the moving average index value computed over the last 3Y of daily data. In other words, equation (3.1) means that we compute the return using the average stock index value of the last 3Y, denoted by, and today s value S t, and cap this value to 10% and floor it to -10%, and then add 39% to the final result. If the stock index has increased in value, the expected return looking forward is taken to be lower than before the increase in equity value, and we therefore increase the SCRE by 10% at most. We apply the same rule to estimate whether the stock index Sharpe ratio is high, medium or low. If R t > 10%, then we set, if R t < 10%, then we set, otherwise we set. This arguably crude methodology for estimating the value for the equity Sharpe ratio process has the notable advantage of being internally consistent with the Solvency II prescription. Obviously, other approaches can be used as well. In particular, a standard approach in the academic literature (see for example Viceira, 2009) consists in using the Dividend Yield (or the Price/Earnings Ratio) to make the Sharpe ratio observable. In order to do that, one performs a linear regression of the stock index excess log-returns with respect to the log-dividend yield, which can be written as: (3.1) This result gives us the value of the two parameters m and p, and then, an estimate of the expected excess log-return 90 An EDHEC-Risk Institute Publication

91 Part II: Introducing the Solvency II Investment Benchmarks 33 - Even if the focus of the paper is on equity investments, having a stochastic interest rate model that generate reasonable bond prices is an attractive feature. X t over the period [t, t = h], can be obtained as follows: The equation above relies on the fact that h is a short horizon, so the expected excess log-returns X t can be approximated by. The result of the regression is given in figures 3 and 4 for the US and European markets respectively. One can notice that the two expected excess log-returns oscillate around 5% with higher regime (around 15%) and lower regime (around -5%). From the expected excess log-return, we can easily extract the Sharpe ratio by using the h=3m historical volatility. 4.2 Interest rate parameters: σ r, λ r, a and b In order to calibrate the interest rate parameters, we downloaded the yield curve of the US Treasury Zero-Coupon bond, and of the French Government from Bloomberg. Then, we considered two techniques to estimate the model parameters: (i) Maximum Likelihood; (ii) Moment Matching. Both of the approaches are superior to using short-term interest rates time-series, which do not give consistent values for bond prices. 33 The approach (i) has been extensively used in previous academic and practitioner studies (see for example by Fisher and Gilles, 1996). The advantage of this method is that even for a single-factor Vasicek model, several yield maturities can be used in the calibration procedure. This method has been implemented in Duffee (2002) to calibrate more sophisticated models called "essentially affine". The main assumption is that we observe a zero-coupon bond yield without noise (here we actually need as many yields as the number of factors in the term structure model): (3.2) Then, we assume that all the other observations with different maturities ι 1,, ι m will contain an additional measurement error like in the Kalman filter framework: where r t is equal to using equation (3.2). Since the measurement errors added to the other observations are assumed to be independent of each other and of the short rate r t, the log-likelihood of the system is simply the sum of the log-likelihoods from the pure time series (equation (3.2)) and the pure cross-sectional estimations. If the distribution of the measurement errors is not Gaussian, then a quasi maximum likelihood method can be used to compute an approximation of the objective function to maximise. Note that this approach has also been used to calibrate a dynamic asset allocation model in Sangvinatsos and Wachter (2005). Approach (ii) relies on matching the expected excess log-return of two bonds with long maturities over a bond with a very short maturity, say 3M. The main advantage of this method is that a closed-form equation can be derived for the expected excess log-return: An EDHEC-Risk Institute Publication 91

92 Part II: Introducing the Solvency II Investment Benchmarks Approach (ii) is less standard but can be used to verify that we retrieve the results obtained with approach (i). The results of these two calibration approaches are given in tables 2 and 3, and appear to be very close, except for λ r, which is known for being a difficult parameter to calibrate. Table 2: Calibration of Interest Rate Parameters with Moment Matching US ZONE (since Mar 1989) Calibrated Parameters EURO ZONE (since Dec 1994) a σ r 1.06 % 0.96 % λ r Model Implied Quantities 3.91 % 4.01 % 5.97 % 6.78 % 7.63 % % 1.59 % 1.42 % 2.43 % 2.41 % 3.11 % 3.56 % 2.09 % 2.49 % This table displays the results of the calibration of the interest rate parameters obtained by matching the average excess return of 5Y and 10Y bonds over the bond with a maturity equal to 3M. The long-term rate is calibrated separately using the average of the 3M yield, leading to b = 3.06% for the EURO ZONE and b = 3.82 % for the US ZONE. Both datasets come from the French and the US yield curves obtained from Bloomberg. Table 3: Calibration of Interest Rate Parameters with Maximum Likelihood US ZONE (since Mar 1989) Calibrated Parameters EURO ZONE (since Dec 1994) a σ r 1.08 % 0.98 % λ r Model Implied Quantities 3.97 % 3.61 % 6.05 % 5.51 % 7.70 % 7.02 % 2.54 % 1.93 % 3.86 % 2.95 % 4.92 % 3.76 % 2.12 % 1.93 % This table displays the results of the calibration of the interest rate parameters obtained by running a Maximum Likelihood approach to fit a partial representation of the Yield Curve given by the 3M, 1Y, 3Y, 5Y and 10Y yields. The long-term rate is calibrated separately using the average of the 3M yield, leading to b = 3.06% for the French YC and b = 3.82% for the US YC. Both datasets come from the French and the US yield curves obtained from Bloomberg. 4.3 Volatility parameters: σ V, α and The current value of volatility can be computed from daily returns of a stock index, either using the historical volatility formula or by fitting an EGARCH model. It can also be obtained from volatility indexes such as the VIX index for the US data or the VSTOXX index for European data. We have favoured the historical volatility because it is a simple estimator, and it can be computed from any stock returns, whereas existing volatility indexes only reflect the volatility of a given pool of stocks. Concerning the parameters of the volatility dynamics, there are two main approaches for the calibration of Heston s model: (iii) Filtering Approach; (iv) Joint-calibration of Stock-Return together with a Volatility Proxy. One key difference is that approach (iii) focuses on estimating the vector of parameters using only historical data while approach (iv) extends the information in historical data with the information available in option prices. The general advantage of (iii) is that the only data needed is the time series of the stocks S t. However, the variance of the log-return of S t is not observable, which implies that the statistical estimation is a filtering 92 An EDHEC-Risk Institute Publication

93 Part II: Introducing the Solvency II Investment Benchmarks problem. The advantage of (iv) is that the level of volatility can be extracted from option prices, or from previous returns (with the possibility of using a GARCH model, or simply the historical volatility). Further on, (iv) provides the only way to estimate the market price of risk. A reliable implementation, however, relies on the observation of the volatility (through volatility indices for example), which is the case for the European and US markets, but may be a challenging requirement for the Asian markets. This limitation can be overcome by adopting an approach in which the instantaneous volatility can be proxied through implied volatility. This approach is called the integrated volatility proxy method (see Ait-Sahalia and Kimmel, 2007). Concerning (iii), there are several estimation methods discussed in the academic literature. Moment-based techniques are described by Chacko and Viceira (2003). The approach is based on the characteristic function of In S t which can be derived in closed form. The parameters of the model are estimated through the generalised methods of moments by minimising the error between the real and the imaginary parts of the theoretical and the empirical characteristic functions. Chacko and Viceira (2003) report parameter estimates for the S&P 500 based on daily, weekly and monthly returns. A similar technique is discussed by Jiang and Knight (2002). Other methods include approximating the conditional moments of integrated volatility through high-frequency data (see Bollerslev and Zhou, 2002), filtering techniques based on the characteristic function (see Bates, 2006), and a Bayesian approach (see Eraker, 2001). Different statistical methods can be used in approach (iv). Eraker (2001) uses a Bayesian approach, whereas Ait-Sahalia and Kimmel (2007) apply a maximum likelihood method. In order to follow the latter approach, one has to determine the joint likelihood function of the observed data, as opposed to, for example, conditional moments. Since, in general, the transition likelihood function for a stochastic volatility model is not known in closed-form, one can apply the closed form approximation method of Ait-Sahalia (2008). From there, a gradient-descent algorithm can be used to compute the maximum of the joint log-likelihood of (S t, V t ). Note that there are several empirical studies in the academic literature that report parameter estimates of Heston s model for the S&P 500. It is however difficult to compare the values across papers because different authors chose different samples and also different frequencies. Another reason, as reported by Chacko and Viceira (2003), is the interplay between the model parameters. In tables 4 and 5, we display the calibration results of the various papers that have been cited above. As we can observe, the GMM approach does not seem to be robust enough for the estimation of the model parameters. Moreover, a direct computation of the expected value of the VIX index (resp. the VSTOXX index) and of its volatility (using the adjusted volatility daily returns described below): corroborates the results of Ait-Sahalia and Kimmel (2007) (see table 6). Therefore, we will use the calibration results obtained by Ait-Sahalia and Kimmel (2007) in our simulation. An EDHEC-Risk Institute Publication 93

94 Part II: Introducing the Solvency II Investment Benchmarks Table 4: Calibration of Heston Model Parameters with Maximum Likelihood. S&P 500 (Jan Sept 2003) S&P 500 (Feb Jun 2010) % % α σ V % 9.55 % ρ SV % % 3.22 % 2.58 % This table reports the results of two papers: Ait-Sahalia and Kimmel (2007) and Martellini and Stoyanov (2011) using S&P 500 data to calibrate the Heston model by running a Maximum Likelihood algorithm. Daily observations were used by Ait-Sahalia and Kimmel (2007) whereas Martellini and Stoyanov (2011) used monthly observations. Table 5: Calibration of Heston Model Parameters with the Generalised Method of Moments (GMM) S&P 500 ( ) CRSP VW ( ) % % α σ V % % ρ SV % % Table 6: Naive Estimates of the Long-Term and Volatility of the Equity Volatility VIX (Jan Mar 2011) VSTOXX (Jan Apr 2011) % % σ V % % 1M Hist Vol (Jan Mar 2011) 1M Hist Vol (Jan Apr 2011) % % σ V % % This table displays the estimates of the average volatility and the volatility of volatility using the VIX and VSTOXX index data in the upper half of the table, and using historical volatility in the lower half. Daily observations were used for both indices. Note that figures 7 and 8 illustrate that if volatility indices such as VIX or VSTOXX are not available, then the simple computation of the historical volatility or the use of a EGARCH model to extract information from the stock returns can be good volatility proxies for the calibration of the Heston model parameters % % This table reports the results of two papers: on the left, Chacko and Viceira (2003) and on the right, Jiang and Knight (2002) using S&P 500 data (resp. CRSP Value Weighted) to calibrate the Heston model by running a Generalised Method of Moments algorithm. Daily observations were used by both papers. Figure 7: VIX Index, and other Volatility Proxies for the S&P 500 This figure displays the VIX index from inception (2 January 1990) to 31 March 2011, together with the historical volatility of the S&P 500 computed with a lag of one month, and a proxy for the volatility using the EGARCH(1,1) model. 94 An EDHEC-Risk Institute Publication

95 Part II: Introducing the Solvency II Investment Benchmarks Figure 8: VSTOXX Index, and other Volatility Proxies for the EUROSTOXX 50 This figure displays the VSTOXX index from inception (4 January 1999) to 8 April 2011, together with the historical volatility of the EUROSTOXX 50 computed with a lag of one month, and a proxy for the volatility using the EGARCH(1,1) model. 4.4 Sharpe ratio parameters: σ λ, κ and Regarding the parameters of the Sharpe ratio process, there is no clear consensus in the literature, since the Sharpe ratio is not observable. One reference paper using a mean-reverting process for the Sharpe ratio is Munk et al. (2004). The authors used a Kalman filter to calibrate their model (since they do not have a stochastic volatility component, their model remains Gaussian, and the Kalman filter can be used). The speed of mean reversion found in Munk et al. (2004) is equal to 6.08%, but their estimate comes with a high variance, which shows that this parameter is rather difficult to calibrate. The long-term mean of the Sharpe ratio is 44.10%, which is of the same order as the historical Sharpe ratio on the S&P500. Finally, they obtain a volatility for the Sharpe ratio equal to 4.70%. Figure 9: Sharpe Ratio of the S&P 500 This figure displays the Sharpe Ratio of the S&P 500 computed from the quarterly time-series of the expected excess log-return together with the 1M historical volatility. An EDHEC-Risk Institute Publication 95