Preparing for Solvency II Theoretical and Practical issues in Building Internal Economic Capital Models Using Nested Stochastic Projections

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1 Preparing for Solvency II Theoretical and Practical issues in Building Internal Economic Capital Models Using Nested Stochastic Projections Ed Morgan, Italy, Marc Slutzky, USA Milliman Abstract: This paper describes the expected role of internal economic capital models in Solvency II and discusses the case for their use. Practice has been developed by leading insurance companies, which have followed different approaches to building internal capital models, and we give an overview of the key methodological issues. These include, for example, the risk measure to use, the type of liability valuation (statutory or market consistent), the time horizon and the acceptable probability of ruin The authors have developed and implemented such an internal economic capital model for a major US insurer, using nested stochastic projections. Stochastic projections are inherently complex; require long runtimes and results are difficult to analyze. Nested stochastic projections are far more complex because of the additional scenarios run at future durations. Any application which involves projecting forward stochastically a realistic balance sheet (i.e. with liabilities valued on a market consistent basis) will theoretically involve nested stochastic projections. Whilst approximations can be used to avoid nested stochastic modeling, these generally have limitations which can reduce the value of the work. We have encountered challenges and problems in this modeling. The paper describes how we have dealt with them and the insights gained. Challenges include the complex, dynamic interactions required, IT issues (both software and hardware) and the need to compress the model to get manageable performance, while still having results which can be understood and analysed (avoiding black boxes ). Our observations also have applicability to other regulatory and financial reporting developments such as European Embedded Value, UK ICAs and the developing International Financial Reporting Standards. Keywords: Solvency II, EEV, IFRS, ICA, Stochastic, Nested Stochastic, Economic Capital

2 1. Introduction In 2000 the European Commission started the so-called Solvency II process which will lead to a major revision of the solvency margin regime for insurers in the European Union. Solvency II is intended to be much more realistic in assessing the capital requirements for insurers than the current EU solvency basis. The process of developing and agreeing the Solvency II system is complex and beyond the scope of this paper. At the time of writing a draft outline of the Framework Directive has been published and a consultation process is underway with comments having been made by various interested parties. A key body is CEIOPS (the Committee of European Insurance and Occupational Pensions Supervisors), a committee composed of representatives of EU member states supervisors, which advises the European Commission in preparing new measures such as Solvency II. Other interested parties include the CEA (the federation of European insurance company associations), actuarial bodies such as the Groupe Consultatif and the IAA (International Association of Actuaries) and groups such as the CRO Forum representing certain companies. The draft Framework Directive defines two levels of solvency capital, the Solvency Capital Requirement (SCR) and the Minimum Capital Requirement (MCR). The MCR represents an absolute minimum level of capital below which urgent action would be required by the regulator and will be calculated according to a simple formula. The SCR is the level of capital required so that there is a 0.5% probability that assets will not be sufficient to meet liabilities during the following year. This paper mainly focuses on life insurance although many of the comments are also applicable to non-life insurers. The first part of the paper (sections 2-5) describe the role of internal models in calculating the SCR, a comparison with other measures of risk based capital and an overview of some of the main methodological issues involved in internal economic capital models. The second part of the paper (sections 6 to 9) describes some of the practical issues involved in implementing such models. 2. Role of Internal Models The draft Framework Directive envisages regulators will be able to permit insurers to calculate the SCR using internal models provided these have been validated and approved by the regulator. Such models will need to have risk measures, time horizons and scope of risks at least as prudent as those underlying the standard approach 1 to calculating the SCR. It could be questioned whether many regulators are likely to have the resources or expertise to be in a position to validate companies internal models. It is possible that regulators will follow an approach of looking for an independent opinion (e.g. from a consulting actuary or auditor) as to the suitability and accuracy of the model.

3 From the regulators point of view internal models have the advantage that they encourage insurers to measure and manage their risks. They are more flexible than industry standard models and can be updated as financial markets and the company s business evolve. Furthermore internal models should be able to represent the business of an insurer more closely than a rule based standard approach 2. Internal models are most likely to be suited to large insurers or innovative or niche players for whom the standard formulae are least likely to be representative 3. At the same time it is recognized that the use of internal models will involve additional costs both for the companies operating them and for the regulators who need to approve them. CEIOPS has pointed out that the SCR shares many features with economic capital 4. Economic capital models have been increasingly used by leading insurers as a key tool to manage their risks and capital and to measure their performance. Several leading insurers have developed such models and publicized the results. The development of such economic capital models can therefore be viewed as very much a precursor to the use of internal capital models in the regulatory supervision of companies. The Solvency II is part of a convergence between economic and regulatory management of insurance companies based on a realization that ultimately companies that are profitable and well managed are those which are most likely to remain solvent. Companies are working together through the CRO Forum which groups together 13 of Europe s leading insurance companies and which aims to establish best practice and influence the debate on the shape of Solvency II. Since these large insurers have been leading the development of such models they are viewed as a reference point as to what is practically possible and useful when developing internal models. The CRO Forum has published some proposed Principles for Regulatory Admissibility of Internal Models. These, like the equivalent CFO Forum principles on EEV, establish broad principles but leave some flexibility as to the approach to use on various items. 3. Theoretical Issues in developing Economic Capital Models 3.1 There are several key methodological issues which need to be determined in constructing an economic capital model. This section will summarize some of the key issues in developing an internal economic capital model. 3.2 Liability Valuation Bases - the Total Balance Sheet The IAA and others have referred to the concept of the Total Balance Sheet 5. By this they mean that the true level of financial strength depends not just on the difference between the value of the assets and the liabilities, but also on the basis on which these are valued. The existing EU supervisory systems have tended to rely, in theory at least, on a combination of prudence in both the asset and liability valuation bases and simple formulae to define the required minimum level of solvency. In reality it is not always certain that the combined impact of the asset and liability valuation bases will be prudent. For example, traditional bases have generally ignored the cost of embedded financial guarantees and options,

4 something which has been recognized in the European Embedded Value method used in financial reporting by leading European insurers 6. One of the aims of the European Commission in the Solvency II process was to take into account the work of bodies such as the International Accounting Standards Board (IASB) and to influence their work to try and achieve convergence between market and statutory reporting. 7 The developments of the IASB and others involve moving towards fair value, i.e. to try and have financial reporting standards which give a true and fair picture of the company s business. This generally equates to market consistent valuation, although there are some qualifications to this. Whereas the use of market values for assets is generally uncontroversial (although far from universally in use today) the correct method for liability valuation has been the subject of considerable debate. It is expected that liability valuations should be compatible with IASB methodology 8, but not necessarily identical. In particular the concept which exists in the work of the IASB of taking account of the insurer s own credit worthiness is not appropriate from a regulatory perspective. At least until IASB standards are available, CEIOPS favours the inclusion of risk margins in the liability valuations 9 at least for non-market risks (the treatment of market risks is not yet clear.) In the context of non-life insurance it has said that should equate to a certain level of likelihood (e.g. 75%) that the reserves will be sufficient to cover the liabilities. CEIOPS has suggested that the risk margin in liability valuations is necessary to cover the costs of transferring liabilities to a third party. 10 Those, such as the CEA, in favour of a pure definition of market consistent valuation argue 11 that such risk margins mean that the Total Capital Requirement ( TCR ) is the sum of the SCR and the prudence in the liability valuation basis and that it is this TCR which should be calculated to equate to a 1 in 200 probability of ruin. A few companies using economic capital models have to date based their models on a statutory accounting view of solvency (i.e. on the current rules) 12, but it seems clear that in future all companies will move to some form of economic view of solvency. In any case CEIOPS has said that liability valuations should include an explicit valuation of financial options and guarantees and that this should be done in a manner consistent with market-based values 13. In many cases this would seem to make some form of stochastic calculations inevitable, although, unlike the EEV principles 14 the CEIOPS documents stop short of explicitly stating that a stochastic valuation is required. 3.3 Scope of Risks The work of the IAA 15 broke the risks faced by insurers down into five main categories: underwriting risk; market risk, credit risk, operational risk and liquidity risk. Others have used other classifications of risks or added other subdivisions of these risks. For example the Italian regulator, ISVAP, referred to reputational and legal risks in a recent draft circular on risk management.

5 For life insurance companies the greatest focus is usually on market risk because equity and interest rates risks are usually the most significant risks faced by life insurers, but other risks can also be material. For example mortality (or longevity) risks can be considerable for many insurers and credit risk can be important both in terms of exposure to reinsurance counterparties and to issuers of corporate bonds. It is also the case that theory and methodology for modeling and measuring market risks is far more evolved than for some of the other risk categories, although the scope and modeling of some other risks such as underwriting risk and credit risk is also fairly advanced. However this is not a reason for ignoring such risks and it will be important to develop credible approaches to these risks if internal models are to be accepted by regulators. For example, the UK Continuous Mortality Investigation (CMI) has published a paper 16 which develops stochastic methodology for projecting mortality. In general the focus is on how to deal with the uncertainty over trends in the level of mortality (particularly with reference to longevity). Normal random fluctuations in the level of mortality experience can generally be fairly easily dealt with in a model. More extreme variations in mortality due to catastrophes, new diseases or major advances in treatment for existing diseases present greater challenges in establishing the likelihood of extreme scenarios. Companies implementing internal models have considered the extent to which longevity and mortality risks in different parts of their portfolio may offset each other causing an implicit hedging 17. Generally the conclusion is that it is difficult to justify taking credit for such effects unless it can be demonstrated that the populations under consideration for the two risks have very similar characteristics. For insurers, operational risk is probably the area where methodology is least developed. Some companies are attempting to develop stochastic operational risk models while others are using a simple formula 18, an approach which has also been favoured by some regulators. Operational risk requires further research and there is a possibility to refer to work done in other related industries. The model which we describe in the second part of this paper involves several of the above risk categories, although some risks are modeled only on a fairly simple basis and some risks (e.g. credit risk) are calculated by separate models and then called by the main economic capital model. In the second part of the paper we concentrate primarily of the issues connected to the modeling of market risks. 3.4 Diversification CEIOPS recognizes 19 that the total capital requirement could be less than the sum of the capital required for individual risks to the extent that these risks are independent. Some mathematical technique such as linear correlation can be used to analyse risk dependencies. However it has been pointed out that risk correlations can behave very differently in extreme scenarios than they do across most of the probability distribution. For example a limited movement in mortality may be largely uncorrelated to economic factors driving market risks, but large

6 movements in mortality would seem intuitively to be more likely to be correlated to market risks. The IAA suggests 20 that through the use of techniques such as copulas a better understanding of tail dependencies can be achieved. Some parties such as the rating agencies have been historically skeptical about giving full credit for diversification, and the UK FSA has indicated that it expects that allowable benefit for diversification of risks across jurisdictions or between life and non-life is likely to be small. The problem of tail dependencies seem to justify this skepticism and suggest that a rigorous approach to understanding risk dependencies should be necessary in order to take full credit for aggregation benefits. Quite a lot of data exists to support the analysis of the correlations between various market risks and quite a lot of analysis and modeling has been done in this area. It is also supported by well established economic theory. Whether and why operating factors such as lapse rates are positively or negatively correlated with the stock market is harder to demonstrate and depends on less well established theory on social trends. As well as the diversification effect of different independent risks, insurance groups with diverse businesses will benefit from group diversification benefits to the extent to which their different businesses have non-correlated risks. Leading European insurers are generally large complex groups operating in both life and non-life insurance and have for many years argued that this diversification benefit helped to smooth earnings. In the light of CAPM, from a shareholder perspective this argument relative to earnings seems spurious because investors could achieve the same smoothing by diversifying their investment portfolios without having to invest in diversified businesses. However the benefits of diversification on capital management seem more concrete since these could not be achieved unless the capital was available to support the different business activities of a company. It is therefore natural that in presenting their economic capital models large insurers companies have emphasized the benefits of diversification on the amount of capital calculated as being required. For example, one of Europe s largest insurers 21 calculated diversification benefits of 46% (the reduction in the group capital required compared to the sum of the capital required for the individual operating companies). Of this, 35% was due to geographic diversification and 17% to segmental diversification. It should be pointed out, however, that from the perspective of local regulators, diversification benefits from a group perspective may not be relevant since they need to ensure the solvency of the individual legal entities which they are supervising. Unless the insurance groups are providing guarantees to provide capital to support potential losses in their operating subsidiaries then the regulation will require an adequate level of capital for each legal entity.

7 This has led the CRO Forum to propose 22 that there should be a Solo Entity Solvency Test and a Group Solvency Test and that the former should be able to take account of capital within the Group provided that sufficient capital mobility can be demonstrated, the pledge of capital is backed by appropriate formal legal agreements and the credit risk associated with such pledges is allowed for. It is worth noting that restrictions on the movement of capital can exist even within different classes of business in the same legal entity as is the case, for example, for UK with profits business 23. There have already been examples of companies reporting economic capital less than the required statutory solvency capital for some operating units with the implication that this can be offset against excess economic capital elsewhere in the group. As economic and statutory capital measures converge, insurance groups will be looking to ensure that they receive appropriate credit for the benefit of diversification in assessing the SCR, something which is not possible under the current solvency regime. 3.5 Risk Measures VaR, TailVaR and CTE The two risk measures which have generally been viewed as most suitable are VaR (Value at Risk) and TailVaR (Tail Value at Risk) 24. Other measures are also possible such as the standard deviation of the losses which a company may suffer. All these measures are based on a view of the possible outcomes for the future level of solvency as a probability distribution. VaR assesses the probability of ruin at a given quantile of the probability distribution. TailVaR considers both the probability and severity of losses which exceed a given quantile and is defined as the arithmetic average of losses exceeding a given quantile. From a shareholder perspective VaR can be considered adequate because once the net worth has been exhausted, shareholders have lost the value of their shares and are not, in theory at least, interested in the severity of further losses. From a regulatory point of view, however, the magnitude of losses is significant because it will determine the losses to policyholder and hence influence the damage to the reputation of the insurance industry and the regulator. TailVaR is generally considered to deal better with low-frequency high-severity events because it takes more account of the shape of the tail of the distribution. For this reason the IAA has expressed 25 a preference for TailVaR over VaR. Although VaR is commonly used in banking, insurance more commonly involves skewed risk distributions. On the other hand it is often hard to find data to accurately model the tail of the probability distribution. At present the majority of companies economic capital models are using a VaR approach, although some local regulators who have implemented this type of system have favoured one approach and some the other 26. A conditional tail expectation set at the x% level, denoted CTE(x), is the average cost of the highest (100-x)% of the results. It should be noted that CTE(x) is

8 generally greater than a (x + (100-x)) percentile coverage (i.e., CTE(90) is generally greater than the 95th percentile) Time Period and Projection Method Insurers which have implemented internal capital models have tended to follow one of two main approaches, although there are variants on each of them. The first one has a one year time horizon. For example it could involve testing the solvency to a short term shock such as an equity market or interest rate movement. The shock is calibrated to represent a certain probability of event (e.g. a one in every 200 year even) and hence the amount of capital required to survive the shock is the amount needed to ensure continued solvency with this level of probability. This method does not give information about the magnitude of the losses in the tail of the distribution and so could only be used to calculate VaR and not TailVaR. The short term shock method is attractive to companies wishing to avoid stochastic modeling since if solvency can be adequately estimated using deterministic methods then no stochastic modeling is required. Another variant of the one year method could involve projecting stochastically for one year and determining the capital required to have a certainty probability of remaining solvent at the end of this year (e.g. by looking at the worst 0.5% of scenarios). The second method involves a multi-year time horizon where the economic balance sheet of the company is projected for a long period (e.g. 20 years) or until all the liabilities have run-off. A test can either be made that there is adequate capital throughout a certain percentage of these scenarios or at the end of a certain percentage of these scenarios. This multi-year time horizon can give a deeper understanding of the long term risk exposures. If the test of solvency is made at the end of the projection period (i.e. at run-off) then it can ignore the impact of scenarios where there are losses rendering the company insolvent followed by profits (e.g. in the case of market falls followed by recoveries). In a similar way the impact of tax can causes some distortions because profits will be taxed, but losses will normally be suffered gross of tax. On the other hand if the balance sheet is projected stochastically throughout the period and tested for solvency every year this makes the calculations fairly onerous. Another possible weakness of the multi-year or run-off method is that may ignore management actions to some extent. Whereas most stochastic models will allow for some dynamic management actions such as asset rebalancing in the event of market falls, it may be hard to allow realistically for the full range of possible regulatory and management actions on questions such as capital raising and hedging of risks.

9 The IAA has pointed out that 28 there are likely to be various delays due between a solvency assessment being assessed and the regulator taking action due to the need to prepare the reports, regulatory review and decisions on appropriate actions. However this delay is rarely likely to exceed one year which could be taken as implying that a one year horizon for projecting solvency is adequate. Generally most local regulators and the current economic capital models of leading insurers appear to be in favour of adopting a one year time horizon 29. A related decision on the projection method is whether to include new business and if so for how many years. In general companies with economic capital models have only made limited or no allowance for new business 30. Most UK insurers calculating ICAs (internal capital assessments which are intended to be a precursor to Solvency II) have made allowance for one year s new business 31. The model described in the second part of this paper uses a multi-year approach with solvency assessed at the end of the projection period. 3.7 Probability of Ruin The SCR can be defined 32 as the level of capital required to ensure that a company can with a certain degree of certainty for a certain period of time absorb unforeseen losses and continue to meet its liabilities. One way of thinking about this is as the probability of ruin we wish to calibrate to, i.e. are we protecting against a one in every 100 or one in every 200 year event? CEIOPS points out 33 that, in reality, it can not be expected that on an annual basis one in every 200 insurers will therefore fail. Very often the causes of failure of one company will also impact others (e.g. catastrophes or market crashes) and will therefore lead to clusters of failures. It may also be hoped that it proves possible in practice to take corrective action to avert failure quicker than is predicted in the models reducing the real probability of failure. In general there seems to be a more towards using a 99.5% confidence level. Clearly the confidence level will also be related to the time period being looked at and the risk measure being used. A 99.5% TailVaR capital level will be higher than a 99.5% VaR level because the former is intended to provide enough capital to cover the average losses of the worst 0.5% of scenarios. 3.8 Risk Neutral and Realistic or Real World Methods As in other recent developments in financial modeling there is a debate between whether it is more appropriate to use risk neutral or realistic (real world) economic scenarios. Many companies operating economic capital models appear to have used risk neutral approaches and the CRO Forum strongly supports this approach 34. Since the aim of the economic capital model is to determine the amount of capital required to meet the policyholder liabilities with a given degree of certainty a

10 valuation of the liabilities which equates with the market price for another entity to assume these liabilities would be considered appropriate. It would be generally accepted that it is appropriate to discount at rates which reflect the term structure of interest rates and which vary by scenario. The CRO Forum advocates using swap rates which will usually be somewhat higher than the equivalent government bond yields reflecting credit risk in the swap market. However it appears that some companies are using government bond yields for discounting. Some companies are using realistic scenarios and in this case discount rates may be based on corporate bond yields. Supporters of the realistic approach argue that the capital actually available depends on the projected future profits or losses which can only be sensibly projected on realistic bases and that equating insurance liabilities (which are long term, illiquid and with little or no active trading) with apparently similar assets can be misleading. Advocates of market consistency may point to the importance of being able to calibrate values to established market prices, the objectivity of the market consistent approach and the distortions which can be introduced in the realistic approach for example if asset values are in effect projected and discounted at inconsistent interest rates. A full exploration of these issues is clearly beyond the scope of this paper. It is the view of the authors that there are merits in both the risk neutral and the realistic approaches if properly constructed and interpreted.. 4. Parallel Development outside the EU This section notes developments in the USA, Canada and elsewhere similar to Solvency II development in Europe. In the United States, regulators use Risk Based Capital (RBC) as a measure of the sufficiency of surplus. RBC is comprised of 4 elements, referred to as C-1, C-2, C-3 and C-4. C-1 is the capital and surplus necessary to be held for credit fluctuations, C-2 for mortality and morbidity risks and mis-pricing, C-3 for interest rate and equity market fluctuations, and C-4 for general business and operational risks. RBC is generally formulaic, with some use of company specific factors, and with reductions permitted for the use of certain deterministic projections, and stochastic modeling for fixed annuity capital in limited circumstances. The use of internal capital models is being adopted slowly. In one of the first important steps, the NAIC (National Association of Insurance Commissioners), in October 2005, adopted a provision for RBC for variable annuity contracts with Guaranteed Minimum Death Benefits, or living benefits such as guaranteed minimum withdrawal benefits, accumulation benefits or income benefits. Under this provision companies will: calculate their capital requirements using stochastic projections use internally developed models which represent the risks assumed.

11 use company specific assumptions may be used, or prudent best estimate assumptions, and equity and interest return scenarios calibrated to a set of 10,000 scenarios provided by the American Academy of Actuaries be permitted to use projections that reflect the use of hedges, under certain circumstances, if the company has adopted a Clearly Defined Hedging Strategy 35 set RBC at the post-tax, 90 CTE level. This is the average of the additional assets necessary to eliminate the ending surplus deficiency in the 10% of scenarios with the lowest present value of surplus. In addition to using internal models and stochastics, RBC must be calculated using a deterministic projection set by the regulators (referred to as the Standard Scenario), and the actual RBC will be the greater of the amounts produced by the deterministic or stochastic projections. The Standard Scenario was initially proposed as check on the stochastic models, and not a potential floor on RBC. Although results of the calculations for 2005 have yet to be reported, it is believed that the Standard Scenario RBC will in many cases be greater than the modeled RBC. A similar methodology is being considered for the calculation of basic reserves (technical reserves) for the same contracts. Reserves would be calculated using similar models, but with their level established at the pre-tax 65 CTE level of the lowest present value of pre-tax surplus, although higher CTE levels have been proposed. Similar to the Standard Scenario in RBC, there would be a reserve Standard Scenario with less severe assumptions for fixed income and equity returns, and the reserve held would be the greater of the two amounts. In addition the adoption of internal models for Universal Life Insurance is being considered. Both of these are part of a move towards Principles Based Valuation in the US. However, because in the US regulatory capital and surplus requirements are still generally formulaic, companies are developing internal capital models to more accurately reflect their risk profiles and risk management practices. In Canada, principles based supervision and financial reporting have been in effect for several years. Internal models and company specific assumptions, with margins, are used. \ In Switzerland the Swiss Solvency Test (SST) is being introduced in The SST is based on stochastic modeling and extreme scenarios. It is to a large extent formulated in terms of principles and guidelines defined by the supervisory authorities rather than strict formulas. 5. Nested Stochastic Projections 5.1 Circumstances requiring nested stochastic projections

12 Nested stochastic projections (sometimes referred to as stochastic-in-stochastic modeling) are projections in which one loop of stochastic projection lies within another one. For example, if the economic balance sheet is projected forward stochastically and for each stochastic scenario in this projection, a stochastic valuation of the liabilities is required then this involves a nested stochastic projection. Usually one stochastic dimension exists when there is a fair (or market consistent) value of liabilities. A second stochastic dimension can be required for a variety of reasons. For example the projection of dynamic hedging within a stochastic projection would give rise to a nested stochastic model. Nested stochastic projections would also be required to calculate a stochastic Embedded Value (EEV or Market Consistent EV) in which the capital or value of liabilities need to be calculated stochastically within the projection. Conceivably under some circumstances three stochastic dimensions can even be required. This could be the case when both future dynamic hedging and a projected economic balance sheet are involved. 5.2 Possible approximations to avoid Nested Stochastic Modeling The practical difficulties involved in making nested stochastic projections will naturally lead insurers to seek approximations which can avoid the necessity for such calculations. However there are limitations with most of the approximations which can be adopted. For example closed form option pricing formulae such as Black Scholes can make calculations much easier and potentially avoid at least one of the stochastic dimensions. On the other hand such formulae do not always approximate closely to the way in the market values the same options and therefore there will be a distortion in the calculated capital. Such distortions can tend to be greater under the more extreme scenarios which are those which influence the result of an economic capital calculation. Another approximation could be to use a single set of scenarios to project the balance sheet and estimate the market value of liabilities. However generally in a nested stochastic model, the economic scenarios for each of the stochastic paths will be generated on the fly using an algorithm. This algorithm may allow for the ruling economic conditions at the point in time on that stochastic path. For example the algorithm for generating interest rate scenarios will normally include an absolute floor on interest rates and hence the scenarios generated will be different depending on the nature of the path. The same could be done for equities either by assuming some form of mean reversion or by assuming a higher level of market volatility following a major stock market fall. If approximate approaches ignore or approximate this dimension (e.g. by approximating the fair value reserves based on a single set of scenarios) they may over- or understate TailVaR because they do not pick up all the effects influencing the available assets and the required future capital at that point. Bad scenarios

13 could potentially have either a double or a compensating effect when looked at in terms of the required and available capital. 5.3 Nested Stochastic Projections will become more common Regulators will normally expect margins for prudence in capital calculations involving approximate methods so companies deciding whether to use approximations or more precise calculations need to balance the cost of more precise calculations against the cost of holding additional capital and the extra management insights which are gained. The UK regulator, the FSA, has said that it encourages the use of nested stochastic approaches in internal capital models when the models and computing capacity permit such an approach 36. Relatively few UK companies have actually implemented such models to date, although significant energy and resources are being dedicated to building such models or finding improved approximations. The authors believe that nested stochastic projections will inevitably become more common place over the next few years. One approach that has been found useful in practice in the case study we consider in the second part of this paper is to base estimates for the risk based capital on the results of a nested stochastic model, but to then convert these into ratios of the reserves in force. This allows the model to be re-run much more easily and allows the model to be used in a wider variety of ways (e.g. for planning) in the knowledge that the accuracy of the estimates can be validated periodically by rerunning the full model. 6. Approaches to the Development of Economic Capital in the US Milliman has worked with large, diversified insurers, based in the United States, to develop and implement internal capital models for their life insurance businesses. Their objectives are to better understand the capital requirements for the business, and be able to explain and justify their views of such capital requirements to rating agencies and other users of company financial statements. Many companies believe that the formulaic approach used by the rating agencies do not appropriately reflect the company s processes and procedures to effectively manage risk and feel confident that their risk management practices would be positively reflected in the determination of economic capital. For example, the companies may want to be able to take credit for the diversification of risks and for the recognition of superior credit management. In order to achieve these objectives, the companies are developing rigorous models of their business designed to calculate required capital on a stochastic basis. Furthermore, since the companies want to be able to project their capital needs into the future, the models are expanded to support a nested stochastic approach. Companies had previously considered other mechanisms to measure capital and earnings. RAROC (Risk Adjusted Return on Capital) has been used, as this is

14 consistent with the measures that are being used for the P&C and banking businesses that are also often part of the diversified financial services company. However, the risks of the life insurance and annuity businesses are longer term, while the other businesses have a shorter term focus. While RAROC is based on Economic Capital, which is a long term measure, the return and the solvency standard is measured over a term of only one year, which some consider to be too short Therefore, companies are beginning to implement a new modeling process, on a new modeling platform. The modeling process has a very broad scope, in that it includes the development of models for all lines of business and the ability to consolidate those models at a corporate level. We believe that while some companies have models of all their businesses modeled in various ways, few if any have them on a single software platform, which allows the models to interact with each other efficiently. A diversified company s life insurance business often includes both fixed and variable universal life insurance, traditional ordinary life and term insurance, fixed and variable annuities, various types of payout annuities, and various types of pension coverages. All of these businesses must be modeled. By developing a model on a single software platform that covers a variety of risks, correlations and interactions, diversification benefits will be implicitly allowed for. These comprehensive nested stochastic models may be used to develop the Economic Capital necessary to support the credit risks and interest rate and equity market risks. Other means and other models may be used to estimate the capital necessary to support the mortality and morbidity fluctuation risks and the operational and general business risks. 7. Challenges to Building Nested Stochastic Internal Capital Models: Our experience is that the challenge in implementing internal capital modeling is to build a model for each business unit suitable for both Business Planning and for modeling the necessary stochastic and nested stochastic scenarios and paths. These may include both existing business and one or more years of new sales. In the following discussion, the term scenario is used to refer to an array of future yield curves and equity returns starting at the valuation date. The term path refers to additional arrays of yield curves and equity returns which begin at defined future time steps along each of the scenarios. The term node refers to a future point in time at which models are run using nested stochastic paths. The next major challenge in developing the capital models is to reduce the size of the business planning models so they may be run over a large set of nested stochastic scenarios in a reasonable amount of time. The various line of business planning models may range in size from 10,000 model points and several thousand assets up to 150,000 or 200,000 model points and 10,000 or more assets. There may also be 8 or more nodes, i.e. at durations 1 2,3,4,5,10,15,20. The aggregate of all of the models could be 250,000 or more model points and 10,000 to 15,000 assets. If the models are run without compression, the smallest model

15 would have about 90 billion calculations assuming 10,000 model points, 1000 scenarios and 8 nodes at which 1000 nested stochastic scenarios were run over a 30 year projection period, i.e.10,000x1,000x9x1,000. In the aggregate there would be 2,250 billion cell calculations, assuming 250,000 cells. Therefore, the models may be compressed down to 3,000 to 4,000 model points per line of business, by combining contracts with various issue ages, gender, issue years or other underwriting characteristics. It is unnecessary to compress the assets. To validate the compression techniques and models to be used for running the nested stochastics, a company can run a stochastic projection with a limited number of scenarios in both the compressed (small) and uncompressed (large models). Cashflows from each model may be compared, focusing on the present value of average cashflows across all scenarios on a Year by Year basis, or other parameters. 8. Solutions to the Challenges: A solution for many of these challenges is to develop the new models on a new software platform with robust, standard capabilities. After the models are developed, support for nested stochastic projections, the calculation of economic capital, and stochastic reporting capabilities can be added. A company may implement its internal capital modeling by building a deterministic model for each business unit suitable for Business Planning. Then the company may layer on existing assets and investment and reinvestment strategies. Finally, dynamic policyholder behavior functions and interaction between the assets and the liabilities may be added. Then the models may be thoroughly validated until management is comfortable with their predictive accuracy. Once management became comfortable with models for use in business planning, they felt they had good models that could then become the basis for stochastic projections. The challenging next steps involved refining what the details of the capital models would be and building in the nested stochastics. The primary challenge was to reduce the size of the models to allow running nested stochastics, and how to run the nested stochastics in a reasonable amount of time. As noted above, billions of cell projections are necessary. Software Issues - Important factors in the selection of a software platform rather than building models from scratch include the robust asset-liability projection support provided by the platform, and the existence of powerful validation reports and tools. One line of business may serve as an example of how the internal capital model process may work. A major line of business may have 120,000 liability cells in its business planning model. For the development of Economic Capital using nested stochastics, this model may be compressed into 3,000 liability cells. Using the compressed model, projections using 1,000 stochastic scenarios are run, and at

16 each of a number of future points in time (nodes), on each scenario, 1,000 Nested Stochastic projections can be run to determine future capital needs. These nodes may be at the end of each of the first several years and every five or 10 years thereafter. Generally, a model with about 10,000 to 12,000 model points and 10 nodes at which nested paths are run takes about 12 hours to run one scenario with one engine. Based on such projections, Economic Capital factors may be developed for each future year as a percentage of Account Value or Reserves. These factor results are then stored in a table. The process above is repeated for each line of business: Nested stochastic projections are run, factors are developed using the ratio of Economic Capital to an appropriate parameter of reserves or Account Value, and these factor results are stored. In addition to developing the factors using the Nested Stochastic projections, deterministic projections for each line of business are run using the larger models. The Economic Capital factors determined using the nested stochastic projections are applied to the appropriate future year drivers developed from each larger model. Then the results of multiplying the parameters by the capital factors are summed for each scenario, and capital at the appropriate desired level is determined by ranking the aggregated results for the 1,000 scenarios. If all of the risks are modeled, aggregating these results across all product lines give full benefit for diversification, as within any particular scenario favorable results for some lines of business may offset large capital requirements for others. This could lead to the benefit of diversification being taken at the aggregate, or Corporate, level, while the benefits of diversification may not be allocated down to the line of business level. Conversely, companies may also recognize that generally, rating agencies will not give full credit for reduction of Economic Capital requirements as a result of diversification. Therefore some adjustment must be made to the factors determined. Hardware Issues Because of the complexity of the stochastic modeling process, a means must be developed to run the multiple projections required in a reasonable time frame. One part of the solution that companies have developed is the ability to run the projections on a grid. Companies are running actuarial applications and projection systems on grids as large as 500 engines. (One engine means one processor, thus a dual processor computer is considered to be two engines.) Some but not all of the engines used may be dedicated to Economic Capital calculations. It is not necessary to have all of the engines dedicated solely to actuarial projections, and machines that would otherwise be idle overnight, such as those used by clerical staff, may be used. Companies have noted that often processing projections for the Variable Annuity line of business is the constraining factor on time. Running one scenario can be an overnight process for one engine. Path Reduction - Another part of the solution to reduce the time necessary to complete the projections is path reduction. Before a projection is initiated a path reduction parameter may be selected; based on that parameter only a proportion p of scenarios will be selected to run. The selection is based upon the results of

17 the running all of the scenarios and all of the paths at the end of the first year. In order to save runtime, only scenarios with the worst proportion p of scenarios will be run fully with all of their paths. However, the ability of the software to operate in a grid environment may avoid the need to reduce the number of paths modeled. Validation - Basic stochastic liability models should be validated for each line of business. This can be aided by software that can generate audit reports at a number of different levels. A report that shows the cashflows for a cell and path can be particularly useful. A significant amount of validation should be performed on cash flow generated by path, and may be done using spreadsheets or other methods. Cash flows should be reviewed on several paths, at several points in time, and for all lines of business. A model platform that has the ability to specify what scenario/path to run is particularly useful, so that if it desired to review the results of a particular path, a detailed audit report is available by rerunning the path. Third Party Review and Auditability - Another consideration in developing capital models would be that of developing model processes and output formats so that the insurance company can satisfy the 3rd party reviewers and/or the auditors whose opinions are valued by the regulators. The greater the reliance the insurer places on the internal capital model, the greater the thoroughness of the review. The insurer will want to run the models more frequently, and will want the models checked more frequently. This could lead to a greater reliance on models with nicely formatted output, and the ability to drill down into the stochastic and nested stochastic to understand what is driving the model results. 9. Use of the Economic Scenario Generator: The modeling platform includes an internal Economic Scenario Generator which may be parameterized by the user. The Economic Scenario Generator provides both interest rates and equity returns. For nested stochastic projections, this allows the paths to be generated as needed when the projections at each node of the scenarios are run, as discussed in Section 5.2 above. A discussion of the choice of Economic Scenario Generator is beyond the scope of this paper. In addition to using the internal Economic Scenario Generator, it is useful to have the ability to import externally projected scenarios and paths into the model. 10. Conclusion Stochastic and nested stochastic modeling are certain to be crucial tools in life insurance company management. As well as for Solvency II and NAIC RBC, this methodology may be applied to other developments in financial reporting such as International Financial Reporting Standards, UK Individual Capital Assessment, and stochastic embedded values. This type of projection presents considerable practical challenges and requires new approaches to a variety of software, hardware, and modeling issues. We