Insights. Economic capital for life insurers. Welcome... The state of the art an overview. Introduction

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1 January 2013 Insights Economic capital for life insurers The state of the art an overview Welcome......to the first in a planned series of papers examining the various facets of economic capital with a focus on its role and importance in the life insurance industry. We start in this issue with how approaches to modelling economic capital have evolved and the strong influence that Solvency II has exerted, and continues to exert, on what is considered state of the art. Conscious that state of the art can only ever be a temporary label, we conclude by looking at some of the emerging methods and technologies, and s role in developing them, that will shape the future business and financial benefits available from the accurate measure of risk exposure. I trust you will find this and future issues in the series useful and thought provoking. Of course, we would value your insights and the opportunity to inform our own thinking, so please feel free to contact me or one of my colleagues listed at the end of the article. Introduction Economic capital, representing the economic resources required by firms to protect themselves against unforeseen events up to a given risk tolerance, has been used by many insurers as part of a risk management framework their enterprise risk management (ERM) framework for over 10 years. ERM typically starts with a formulation of a firm s risk appetite: that is, an assessment of the risks a firm is willing and unwilling to accept. This leads naturally to a measurement question: how do you measure how much risk you are willing to accept? Economic capital, focusing on extreme outcomes, is part of the answer; with an assessment of less extreme outcomes, such as earnings volatility, forming another part of the assessment. The development of economic capital models in Europe has been given significant impetus by Solvency II. The Directive sets into EU law an approach to regulation based on, amongst other things, management of own funds that include a measure of regulatory capital (Solvency Capital Requirement or SCR) that has a standard economic capital paradigm at its heart. Furthermore, Solvency II provides insurers with the option of defining their regulatory capital requirement using an internal (economic capital) model reflecting a firm s internal assessment of its capital requirements. Solvency II is also fairly prescriptive regarding how it expects firms to develop and validate an internal model and this is influencing internal thinking regarding economic capital too. As a result, for European life firms a standard economic capital paradigm is emerging: the 1-year value at risk (VaR). John Rowland Global Leader Life Capital Modelling

2 The concept is simple: it defines economic capital by following the process below: To what risks is the firm exposed? And how do those risks evolve over a one-year time horizon allowing for the dependent nature of the evolution of risk? Given all possible evolutions of risks over a one-year time frame, what are the possible net asset (or own fund) values? Given all possible net asset values, the SCR (or economic capital) is then measured by considering the 99.5th percentile value. An approach to valuation of the balance sheet: Solvency II bases its valuation on a marketconsistent approach, with notable adjustments. But the 1-year VaR method could equally be defined in terms of IFRS or GAAP earnings as well. Risk measure: the Solvency II definition adopted VaR as opposed to say, tail value at risk, as adopted under the Swiss Solvency Test. The merits of these choices and other core issues are not the focus of this article. We shall focus on how firms have built their models and the state of the art from a modelling perspective. This approach is illustrated in Figure 01 below. When implementing this approach, a number of crucial choices are necessary or will have been made: Time horizon: risks are measured by their evolution over one year, not 10 days (as was standard in banking when VaR was first commonly used) or five years. This choice is somewhat arbitrary but prescribed in Solvency II, and firms should consider risk emergence over all time horizons. Under Solvency II, this requirement is captured under the Own Risk Solvency Assessment (ORSA) requirements. Figure 01. The SCR is defined using a 1-year VaR approach (1) How do the risks we (2) For each possible state, are exposed to evolve what is the value of the over a one-year time frame? balance sheet? (3) Given all possible balance sheets, the SCR is determined by the 99.5th percentile result. Available economic capital Scen 1 Market value assets Market value assets OF RM BEL OF RM BEL Expected Probability of outcome OF Own fund RM Risk margin BEL Best estimate liability Scen 1,000,000 Market value assets OF RM BEL 0.5% Probability T = 0 T = 1 2 towerswatson.com

3 Implementation the challenges Clearly economic capital pre-dates Solvency II, and before Solvency II firms had used the 1-year VaR approach to measure their economic capital. Few, however, had attempted to implement the nested stochastic approach displayed in Figure 01. Some quick mathematics shows that, if assumptions are made about the distribution of risks, the dependency structure between risks and the behaviour of the balance sheet, a simple stress test/correlation matrix calculation can be adopted. For many firms, economic capital started with such a calculation and the Solvency II Standard Formula is a formula that uses the simplicity of this approach. Similar to Basel II, Solvency II permits simple methods the Standard Formula implementable by any insurance entity alongside an option for complex methods an internal model suitable for firms with more complex risk exposures. As noted above, simplifying assumptions are necessary to set up a stress test/correlation matrix-style economic capital calculation. As a consequence, the Standard Formula is appropriate for simpler organisations, where it can provide a reasonable assessment of capital requirements. However, it does not provide much detail about the risk profile of an organisation. Stress test approaches have been in use for a number of years and the limitations of such an approach are well known. Solvency II, with an explicit focus on management of risk as well as measurement of the SCR, seeks to address these by permitting the use of internal economic capital models. This leads to a wish list for an internal economic capital model at least if the model is to prove useful. This wish list consists of both technical and business considerations. Technical considerations Address the shortcuts implicit in the Standard Formula/stress test correlation matrix approach allow for complex risk distributions and more general dependency structures. Allow for non-linear and non-separable behaviour of the balance sheet. Allocate capital across the legal and management hierarchy by risk and product. Explicitly model capital fungibility and transferability restrictions, re-insurance and tax. Business considerations Allow quick/daily estimates of economic capital requirements. Effectively this implies that the model needs to be able to estimate (accurately) the economic capital requirement without requiring underlying asset and liability matching (ALM) models to be run. For most firms, this is equivalent to separating a calibration process from the risk reporting process. A methodology and system that is simple to understand by management and to implement. Be flexible enough to enable what if analysis for example, what if a restructure happened; what if a new product was launched; what if a new re-insurance arrangement was adopted. Multiple controlled user access with auditability, repeatability and workflow capability. The technical challenges led firms to go back to the original 1-year VaR definition. Adopting this implementing a full Monte Carlo simulation enables an economic capital model to explicitly address the technical challenges. However, for all bar the simplest liabilities, full Monte Carlo simulation fails to meet the business requirements due to run-time issues associated with the production of all possible balance sheets. Thus, firms have found that to implement the full Monte Carlo simulation, methods that efficiently estimate the balance sheet are required. These methods or approximate models are known as proxy models. Stress test approaches have been in use for a number of years and the limitations of such an approach are well known. Solvency II...seeks to address these by permitting the use of internal economic capital models. towerswatson.com 3

4 Proxy models replicating portfolios, loss functions and Least Squares Monte Carlo In the present context, proxy models started in the mid-2000s with firms seeking to construct replicating portfolios of their liabilities. A replicating portfolio is a portfolio of assets with the same risk return characteristics as the liabilities being replicated. They are typically constructed using optimisation techniques with either present values of liabilities or individual simulated liabilities (cash flows) being targeted. was at the forefront of developments and implemented a number of economic capital models using replicating portfolios, and replicating portfolio-based models form the basis of a number of internal models across Europe. Replicating portfolios, by construction, focus on market and credit risk. When using a replicating portfolio, capital for insurance and operational risk is usually modelled separately and aggregated to generate the SCR. The separation, for certain types of liabilities, can be a material shortcoming. For example, liabilities such as pay-out annuities have material non-linear interaction between longevity/mortality and market/credit risk, and with participating business there is often significant interaction between lapse risk and market/credit risk. To address this issue, replicating portfolios were generalised, first to include non-traded asset functions of insurance risk and then polynomials in all risk factors directly. In a sense the polynomial can be thought of as a Taylor series expansion of the replicating portfolio. In time, the connection between replicating portfolios and the polynomial has dropped and many firms just construct the polynomial without reference to its replicating portfolio origin. The polynomials have become known as loss functions. Two methods, illustrated in Figure 02, are used to generate loss functions: curve fitting or Least Squares Monte Carlo (LSMC). Figure 02. Loss function generation curve fitting and LSMC (1) Curve fitting (2) LSMC A small number of realisations of risk factors for example, For each realisation, value the balance sheet accurately (for example, 1,000 5,000 simulations). A large number of realisations of risk factors for example, 1,000 10,000. For each realisation, value the balance sheet approximately (for example, 2 10 simulations). Portfolio value Portfolio value Portfolio value in base case and sensitivities Loss function fitted to base + sensitivities Portfolio value from simulations Loss function based on LSMC 4 towerswatson.com

5 Curve fitting refers to the process of constructing a loss function by interpolation or regression from a relatively small number of full balance sheet valuations. Curve fitting can be seen as a natural generalisation of the stress test approach, as stress tests can be thought of as linear loss functions. Curve fitting has strong benefits, not least its simplicity you run the ALM model and fit the curve. There are challenges, however, if the loss function is truly to be a generalisation of the stress test approach. Liabilities are multidimensional, so that the best-fit polynomial (now thought of as the Taylor series expansion of the liabilities) may be a function of risk factors with 100 or so non-zero terms, and the process of curve fitting does not in itself prescribe the structure of the polynomial. However, many firms have adopted a curve fitting approach as a valuable extension of the stress test/correlation matrix approach and have utilised grids to ensure efficient production of the 100 or 200+ balance sheets required. Whereas curve fitting adopts a brute force approach to the construction of loss functions, LSMC offers a more efficient alternative by exploiting mathematical methods. LSMC originated in work undertaken to estimate American option prices using Monte Carlo simulation. In insurance terms, LSMC builds on the early replicating portfolio work that fitted replicating portfolios to Monte Carlo simulations of liabilities. Essentially with LSMC, the polynomial is constructed as a regression of very approximate values of the liabilities against values of risk factors. Done well, using what we now call advanced LSMC, from a single projection of 20, ,000 simulations, the method constructs and fits an optimal polynomial without requiring foreknowledge of the optimal structure of the polynomial. Done well... advanced LSMC...constructs and fits an optimal polynomial without requiring foreknowledge of the optimal structure of the polynomial. RiskAgility EC From a business perspective, a workable methodology is only part of the solution. Life firms need software to deploy the methodology. developed RiskAgility EC specifically to deliver the Monte Carlo simulation of the 1-year VaR definition of economic capital (using proxy models) required by Solvency II. The system is constructed around the risk factor loss function paradigm outlined above and was designed to address both the technical and business challenges described earlier. The software, now in version 2.2, has matured so that firms selecting this software are able to implement an internal model from inception to deployment in a business-as-usual (BAU) context in three to four months. A specific goal with RiskAgility EC was to create a system that broke the link between the underlying ALM systems and management reporting. ALM models are used as calibration tools, with most firms choosing to move towards a hard close calibration cycle of either two or four calibrations per year. Once calibrated, management reporting of risk and capital metrics is available daily as RiskAgility EC allows users to monitor risk exposures and update values for variations in financial markets and volumes of business in force. With RiskAgility EC monthly, weekly or even daily monitoring of economic capital positions are possible. Quick solvency monitoring is not new, but RiskAgility EC industrialises the process and provides a tool that enables much more robust monitoring of solvency across the business. Another key facet of RiskAgility EC is its enterprise nature. Installation models vary, but it is not desktop software; rather, it is a solution designed to allow multiple users in multiple locations access in a controlled way to the same model. Consequently, the solution was developed with embedded audit and governance features users check out elements of the model they wish to review or change; runs are managed via job scheduling tools; and versions of models and assumptions are strictly controlled last year s run is repeatable as the system captures it automatically. Experience shows that the governance features initially make some actuaries nervous but the IT infrastructure meets firms IT departments needs, as it fits in with the typical emerging strategy for systemisation and automation of core actuarial models. towerswatson.com 5

6 Validation of proxy models advanced LSMC and the direct method Having developed an internal model, either with RiskAgility EC or other aggregation software, the final challenge under Solvency II is validation of the model either for internal use or as part of an application for use as an approved internal model. Solvency II has an exhaustive set of validation requirements, and firms undertaking a formal Solvency II validation need to map validation actions against specific Solvency II requirements. More broadly, validation falls into two categories: Technical does the model generate materially accurate estimates of economic capital? In addition to the general methodology, this would be expected to cover: modelling of technical provisions; modelling of risks including risk dependency; use of proxy models; and convergence of the Monte Carlo simulation itself (given that a 99.5 th percentile risk metric is used). Use is the model being used to assist with the management of the business; is use of the model feeding back into future improvements of the model. A key challenge in the technical validation is validation of the proxy model. As noted above, these can be constructed using either replicating portfolios or loss functions. Until recently, proxy model validation relied on ad hoc methods because, despite widespread use of grids, direct validation against a full nested stochastic simulation of the balance sheet is still not possible and mathematical methods to prove accuracy appeared not to exist. Recent work by has changed this, by explicitly addressing two issues: 1. Efficient construction of loss functions that can be demonstrated to be robust, in the sense that they fit the underlying balance sheet and produce an error in the SCR that can be measured. 2. Derivation of an estimate of the error in the SCR when using a proxy model. The first we have described before in this article: advanced LSMC. The second is called the direct method of estimating the SCR. The advances rely on: Specific (nested) scenario sets that are used to create a very approximate value of the liabilities across several thousand realisations of risk factors. Analysis of the sources of breakdown and error arising in the SCR from methods used to date to construct replicating portfolios or loss functions (either using curve fitting or LSMC). The advances have been tested with clients in Europe and we expect them to become standard as economic capital modelling continues to evolve. In test cases we have seen replicating portfolios giving rise to material errors in the SCR, which the new techniques can measure and correct. Critically, these techniques have a mathematical foundation and hence can be used to prove accuracy of the SCR. Given that some European regulators are challenging the use of proxy models without proper formal validation, these are significant developments. Both will be subject to further articles in this series. Conclusion the state of the art So, after 10 or more years of development of economic capital models in the life insurance industry, what is the state of the art? In Europe, it is fair to say that Solvency II dictates much of what would be considered best practice, but what of state of the art. The answer here has both business and technical facets. The following, by highlighting use, captures the business requirements: All models are wrong; some are useful. George Box Following George Box s lead, we argue that the best economic capital models are those that are useful and used to manage the business. A well-understood (approximate) model used in day-to-day management of a life insurer, is of more use to management, and would give more comfort to a Board of Directors and also regulators, than a slow to update, highly accurate and overly complex model. The best economic capital models are models such as RiskAgility EC that are able to provide firms with daily solvency reports, aggregating and allocating risk and capital across a legal and management hierarchy and allowing flexible analysis quickly addressing the questions asked by management as they manage the business. The state of the art will push boundaries and go further than this. Technology and methodology does not stand still and armed with new tools, actuaries and management can ask new questions and seek new levers in the battle for understanding and competitive advantage. Approximations inherent in the use of proxy models, even if these approximations are understood, create demand for new improved techniques. The new methods developed by addressing the accuracy of proxy models will enable actuaries to give management more confidence in the accuracy of these models, thus enabling more complex analysis and decisions to be taken with greater confidence. We expect to see use of the new techniques described above becoming standard and a model such as RiskAgility EC incorporating the new analysis will remain, temporarily at least, the state of the art. 6 towerswatson.com

7 Financial life modelling software Global contacts Clients in more than 30 countries leading P&C and life insurance companies, multinationals, pension funds, mutual funds and asset managers use our systems for enhanced risk and capital management. RiskAgility RiskAgility EC RiskAgility SF Star ESG MoSes UK John Rowland William Machin Peter Murphy Tim Wilkins Adam Koursaris Paris Jean-Francois Cartier jean-francois.cartier@towerswatson.com Jonathan Farrant jonathan.farrant@towerswatson.com Guillaume Beneteau guillaume.beneteau@towerswatson.com Zurich Bernhard Gose bernhard.gose@towerswatson.com Stockholm Simon Stronkhorst simon.stronkhorst@towerswatson.com Spain Rosa Salas rosa.salas@towerswatson.com US Mark Scanlon mark.scanlon@towerswatson.com Owen Stein owen.stein@towerswatson.com Asia-Pacific Penny Fosker penny.fosker@towerswatson.com Cologne Aleksander Rejman aleksander.rejman@towerswatson.com Alexey Ivanov alexey.ivanov@towerswatson.com towerswatson.com 7

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