RESERVING FOR MATURITY GUARANTEES UNDER UNITISED WITH-PROFITS POLICIES. Wenyi Tong

Transcription

1 RESERVING FOR MATURITY GUARANTEES UNDER UNITISED WITH-PROFITS POLICIES By Wenyi Tong Submitted for the Degree of Doctor of Philosophy at Heriot-Watt University on Completion of Research in the School of Mathematical and Computer Sciences November This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that the copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author or the university (as may be appropriate).

2 I hereby declare that the work presented in this thesis was carried out by myself at Heriot-Watt University, Edinburgh, except where due acknowledgement is made, and has not been submitted for any other degree. Wenyi Tong (Candidate) Professor Angus S. Macdonald (Supervisor) Professor Howard R. Waters (Supervisor) Doctor Mark Willder (Supervisor) Date ii

3 Contents Acknowledgements Abstract xiii xiv Introduction Changes in the Regulatory Environment Thesis Outline GUARANTEES UNDER UNITISED WITH-PROFITS POLI- CIES Operation of UWP Polices Reserving Approaches Literature Review Three Reserving Approaches Models Required Valuation Model Bonus and Asset Allocation Model Real World Asset Model RESERVING FOR A SINGLE UWP POLICY HISTORICALLY Introduction Reserving Approach of Buying Options Notation and Assumptions Mechanism of the Option Approach Results Using the Option Method Dynamic Hedging Mechanism of the Hedging Approach Results Using Discrete Hedging CTE Reserving Mechanism of the CTE Approach Results under the CTE Approach Summary DYNAMIC BONUSES Introduction A Dynamic Bonus Strategy Dynamic Bonuses without Smoothing Dynamic Bonuses with Smoothing iii

4 3.3 Results for the Single Policy with Dynamic Bonuses Case A: Without Smoothing or Allowance for Future Bonuses Case B: Smoothing without Allowance for Future Bonuses Case C: Smoothing with Allowance for Future Bonuses Summary A RISK-FREE RATE CONSISTENT WITH THE WILKIE MODEL Introduction A Yield Curve for the Wilkie Model Results with the Consistent Risk-Free Rate Buying Options Discrete Hedging CTE Reserving Summary A DYNAMIC INVESTMENT STRATEGY Introduction A Dynamic Model Containing Dynamic Investment and Bonus Strategies Results with the Dynamic Model Buying Options Discrete Hedging CTE Reserving Summary SENSITIVITY TESTING FOR THE SINGLE UWP POLICY Introduction Sensitivity to Different Parameters EBRs and Bonus Rates Reserves and Profitability Sensitivity to Different Upper and Lower Probability Boundaries EBRs and Bonus Rates Reserves and Profitability Sensitivity to Different 10-Year Periods Investment Performance of the Two Asset Classes Asset Shares and Guarantees Reserves and Profitability Summary RESERVING FOR A PORTFOLIO OF UWP POLICIES HIS- TORICALLY Introduction Equity Proportions and Regular Bonuses Case A: Without Smoothing or Allowance for Future Bonuses Case B: Smoothing without Allowance for Future Bonuses Case C: Smoothing with Allowance for Future Bonuses Asset Shares and Guarantees in Cases A, B and C iv

5 7.3.1 Asset Shares Guarantees Terminal Bonus Reserves Using the Three Approaches Buying Options Discrete Hedging CTE Reserving Comparison of the Portfolio Reserves Set up by Different Approaches in Case C Profitability of the UWP Policies Buying Options Discrete Hedging CTE Approach Comparison of the Free Estate under Different Reserving Approaches in Case C Sensitivity Testing for the Portfolio in Case C Sensitivity to Different Parameters Sensitivity to Different Probability Boundaries in the Investment Strategy Summary RESERVING FOR THE PORTFOLIO WITHIN THE SIMU- LATED REAL WORLD Introduction A Single 10-Year Policy Average EBR and Average Regular Bonus Rate Portfolio Asset Share and Guarantee Maturing Policies Portfolio Reserves Set up Using the Three Reserving Approaches Option and Hedging Approaches CTE Reserving Free Estate Option Approach Hedging Approach CTE Approach Summary CONCLUSIONS AND FURTHER RESEARCH Conclusions Suggestions for Further Research A Wilkie Model 1995 Version 196 B Investment Data and Derived Initial Conditions 199 C Details of Calculations for the 1991 Policy with a 5% Bonus Rate, a 5% Risk-Free Rate and a 100% EBR 202 v

6 D Six Sample Paths in the Simulated Real World 204 References 209 vi

7 List of Tables 4.1 The comparison of the accumulated values of the cashflows for the 1991 policy in Case C with the zero-coupon yield or 5% constant as a risk-free rate The equity backing ratios for the policy issued at the end of 1991 in Case C (%) The comparison of the accumulated values of the cashflows for the 1991 policy in Case C with the dynamic and static EBRs The assumptions for the parameters g, c, TB, σ and τ (%) The EBRs for the 1991 policy in Case C based on the different bases (%) The regular bonus rates declared on the 1991 policy in Case C using the different bases (%) The asset share, guarantee and terminal bonus rate at maturity of the 1991 policy in Case C based on the different bases The reserves using the option method for the 1991 policy in Case C under the different bases The accumulated values of the cashflows incurred using the option method for the 1991 policy in Case C under the different bases The accumulated values of the cashflows incurred by discrete hedging for the 1991 policy in Case C under the different bases The 95th CTE reserves for the 1991 policy in Case C using the different bases The 99th CTE reserves for the 1991 policy in Case C under the different bases The accumulated values of the cashflows incurred to set up CTE reserves for the 1991 policy in Case C using the different bases The EBRs for the 1991 policy in Case C under the standard basis with the different upper and lower probability boundaries (%) The bonus rates for the 1991 policy in Case C under the standard basis with the different probabilities to adjust the EBRs (%) The reserves for the 1991 policy in Case C under the standard basis with the probability boundaries of 95% and 99% to adjust the EBRs The accumulated values of the cashflows for the 1991 policy in Case C under the standard basis with the different probability boundaries The EBRs of the policies in Case C issued at different times under the standard basis with the probability boundaries of 97.5% and 99.5% in the investment strategy (%) vii

8 6.19 The bonus rates declared on the policies in Case C under the standard basis with the probabilities of 97.5% and 99.5% to adjust the EBRs (%) The asset shares, guarantees and terminal bonus rates at maturity for the policies in Case C issued at different times under the standard basis with the probability boundaries of 97.5% and 99.5% The reserves for the three policies in Case C issued at different times under the standard basis with the probabilities of 97.5% and 99.5% The accumulated values of the cashflows incurred for the three policies in Case C under the standard basis with the boundaries of 97.5% and 99.5% The EBRs of each policy in the portfolio in Case A (%) The regular bonus rates declared on each policy in the portfolio in CaseA(%) The EBRs of each policy in the portfolio in Case B (%) The regular bonus rates declared on each policy in the portfolio in CaseB(%) The EBRs of each policy in the portfolio in Case C (%) The regular bonus rates declared on each policy in the portfolio in Case C (%) The asset shares of each policy in the portfolio in Case A The asset shares of each policy in the portfolio in Case B The asset shares of each policy in the portfolio in Case C The guarantees of each policy in the portfolio in Case A The guarantees of each policy in the portfolio in Case B The guarantees of each policy in the portfolio in Case C The asset shares, guarantees and terminal bonus rates at maturity in Cases A, B and C The reserves for each policy in the portfolio using the option method in Case A The reserves for each policy in the portfolio using the option method in Case B The reserves for each policy in the portfolio using the option method in Case C The comparison of the portfolio cashflow and the sum of the individual cashflows incurred by the insurer using the hedging approach in Case C The portfolio asset shares in Case C under the different bases The portfolio guarantees in Case C under the different bases The asset shares and guarantees at maturity in Case C under the different bases The terminal bonus rates declared on the maturing policies in Case C under the different bases (%) The portfolio reserves using the option method in Case C under the different bases The free estate of the insurer using the option method in Case C under the different bases viii

9 7.46 The free estate of the insurer using the hedging approach in Case C under the different bases The 95% portfolio CTE reserves in Case C under the different bases The 99% portfolio CTE reserves in Case C under the different bases The free estate of the insurer who sets up 95% CTE reserves in Case C under the different bases The free estate of the insurer who sets up 99% CTE reserves in Case C under the different bases The portfolio asset shares and guarantees in Case C under the standard basis with the 95% and 99% probability boundaries The asset share, guarantee and terminal bonus rate at maturity in Case C under the standard basis with the 95% and 99% probability boundaries The portfolio reserves in Case C under the standard basis with the 95% and 99% probability boundaries The amount of the free estate in Case C under the standard basis with the 95% and 99% probability boundaries The mean and standard deviation of the simulated EBRs, bonus rates, asset shares and guarantees for the 10-year policy The statistics at the maturity of the 10-year policy The quantiles of the simulated reserves for the 10-year policy The statistics of the accumulated values of the cashflows for the 10- year policy The statistics for the maturing policies during the 30-year extended period B.60 The market indices at 31 December of each year during the period of 1964 to B.61 The derived initial conditions for the 1995 version of the Wilkie model at 31 December of each year during the period of 1964 to C.62 The details of calculations for the 1991 policy under the option approach202 C.63 The details of calculations for the 1991 policy under the hedging approach C.64 The details of calculations for the 1991 policy under the CTE approach203 ix

10 List of Figures 2.1 Maturity payout before and after declaring a bonus at time t, and the increase in the maturity payout after the bonus declaration The asset shares and guarantees for the policy issued at the end of 1991 with static bonus and investment strategies The reserves set up using the option method for the 1991 policy with static bonus and investment strategies and a constant risk-free rate The equity index and exercise price of the options bought for the 1991 policy The hedging error and transaction costs incurred by discrete hedging for the 1991 policy The 95% and 99% CTE reserves for the 1991 policy with static bonus and investment strategies and a constant risk-free rate The unsmoothed bonus rates for the policy issued at the end of 1991 with a static investment strategy The asset share, force of inflation (multiplied by 10,000) and 25% projected maturity asset share for the 1991 policy with a static investment strategy The comparison of the smoothed and unsmoothed bonus rates of the 1991 policy with a static investment strategy The asset shares and guarantees of the 1991 policy with a static investment strategy in Case A The comparison of the reserves using the option method for the 1991 policy in Case A and the case of static bonuses The comparison of the CTE reserves for the 1991 policy in Case A and the case of static bonuses The asset shares and guarantees of the 1991 policy with a static investment strategy in Cases A and B The comparison of the reserves using the option method for the 1991 policy in Cases A and B The comparison of the CTE reserves for the 1991 policy in Cases A andb The asset shares and guarantees of the 1991 policy with a static investment strategy in Cases B and C The comparison of the reserves using the option method for the 1991 policy in Cases B and C The comparison of the CTE reserves for the 1991 policy in Cases B andc x

11 4.19 The yield on the zero-coupon bond with the same maturity date as the 1991 policy The comparison of the reserves using the option method for the 1991 policy in Case C assuming the zero-coupon yield or 5% constant as a risk-free rate over the policy term The comparison of the CTE reserves for the 1991 policy in Case C assuming the zero-coupon yield or 5% constant as a risk-free rate over the policy term The comparison of the bonus rates declared on the 1991 policy in Case C with the static and dynamic EBRs The comparison of the asset shares and guarantees of the 1991 policy in Case C with the static and dynamic investment strategies The comparison of the reserves using the option method for the 1991 policy in Case C with the dynamic and static EBRs The equity index and exercise price of the options bought for the 1991 policy in Case C with the dynamic and static EBRs The comparison of the CTE reserves for the 1991 policy in Case C with the dynamic and static EBRs The asset shares of the 1991 policy in Case C based on the different bases The guarantees of the 1991 policy in Case C based on the different bases The asset shares and guarantees for the 1991 policy in Case C under the standard basis with the different upper and lower probability boundaries The comparison of the equity indices at each policy duration over the different 10-year periods The comparison of the yields on the zero-coupon bonds, each maturing at the end of the policy term, at each policy duration over the different 10-year periods The comparison of the asset shares of the policies in Case C issued at different times under the standard basis with the boundaries of 97.5% and 99.5% The comparison of the guarantees of the policies in Case C issued at different times under the standard basis with the boundaries of 97.5% and 99.5% The comparison of the total asset shares of the portfolio in Cases A, BandC The comparison of the total guarantees of the portfolio in Cases A, BandC The portfolio reserves using the option method in Cases A, B and C The 95% portfolio CTE reserves in Cases A, B and C The 99% portfolio CTE reserves in Cases A, B and C The comparison of the portfolio CTE reserves and the sum of the individual CTE reserves in Case C The comparison of the portfolio reserves set up using different reserving approaches in Case C xi

12 7.41 The free estate of the insurer using the option method in Cases A, B andc The free estate of the insurer using the hedging approach in Cases A, BandC The free estate of the insurer who sets up 95% CTE reserves in Cases A,BandC The free estate of the insurer who sets up 99% CTE reserves in Cases A,BandC The comparison of the free estate under different reserving approaches in Case C The quantiles of the average EBRs over the policy term for each policy in Case C The quantiles of the average regular bonus rates declared over the policy term for each policy in Case C The quantiles of the portfolio asset shares in Case C The quantiles of the portfolio guarantees in Case C The quantiles of the AS/G ratios in Case C The quantiles of the portfolio reserves set up using the option method in Case C The quantiles of the 99% portfolio CTE reserves in Case C The quantiles of the insurer s free estate using the option method in Case C The quantiles of the insurer s free estate by discrete hedging in Case C The quantiles of the insurer s free estate by setting up the 99% portfolio CTE reserves in Case C D.56 Six sample paths of the simulated portfolio asset share and guarantee in Case C D.57 Six sample paths of the simulated reserve required for the portfolio in Case C D.58 Six sample paths of the simulated free estate in Case C xii

13 Acknowledgements I would like to express my gratitude to all my supervisors, Prof. Angus Macdonald, Prof. Howard Waters, and Dr. Mark Willder for their invaluable guidance, advice and encouragement during the course of my research. I must thank my leading supervisor Mark Willder for his unfailing patience, his understanding in a very difficult period of time, his wide knowledge in the field of life insurance solvency, and his eagerness to share this knowledge. The project carried out in this thesis is sponsored by Standard Life Assurance Company. I would like to thank the sponsor, in particular Dr. David Hare and Douglas Morrison, for the financial and technical support at various stages. This thesis would not be possible without the love and encouragement from my parents. My indebtedness to them for their understanding and respect to all my decisions. I have enjoyed working with my colleagues. Their work in related areas has stimulated my research. No less is my gratitude to my friends in Scotland and China for their great support and wonderful friendship. xiii

14 Abstract As a result of the move by the International Accounting Standards Board (IASB) towards fair value accounting, there is increasing interest in establishing how to value life insurance liabilities, in particular liabilities with embedded options, on a market consistent basis. In the UK, the Financial Services Authority (FSA) is developing a new prudential regulatory regime which moves from the traditional valuation approach to a mark-to-market regime. The recent CP195 proposals (FSA (2003)) develop a twin peaks approach by which the total reserves and capital required are set as the greater of that required under the current statutory basis and that required applying a stress test to a market consistent valuation of assets and liabilities. The aim of this thesis is to investigate the reserves required to meet the maturity guarantees under unitised with-profits (UWP) policies, within the realistic reporting framework. Under the UWP policies, a growth rate in the unit value is promised for the premiums already paid. In addition, the policies allow the holders to participate in the profits of the company through regular and terminal bonus declarations. Thus, the UWP product includes explicit investment guarantees which build up over the policy term. It is to be expected that any guarantee provided will have a cost and hence it should be charged (either to premiums or to asset shares) and reserved for. Three reserving approaches are considered in the thesis. The first two approaches, buying options over-the-counter (OTC) and discrete hedging, apply modern option pricing theory. These approaches are consistent with the realistic peak of the twin peaks approach. The third approach calculates the conditional tail expectation (CTE) reserves by stochastic simulation. The third approach uses the same idea of quantile reserving, recommended by the Maturity Guarantees Working Party xiv

15 in 1980, but with a different risk measure. However, this approach is not favoured by the FSA in CP195 proposals (FSA (2003)). The purpose of applying these three approaches in the thesis is to compare the amount of the reserves required in the current realistic reporting regime with that using traditional stochastic valuation techniques. In the thesis we assume that a fixed 1% (a different percentage is considered in sensitivity testing) of the policyholder s fund is deducted at the end of each policy year as a charge for the guarantees. Reserves are funded by the insurer from its inherited estate. The cashflows incurred by the insurer are calculated, from which profitability of the UWP policies in a 1% stakeholder environment is investigated. We obtain numerical results using both historical data and stochastic simulation. In the historical part, we start from a single UWP policy with a constant bonus rate, a constant risk-free interest rate and a static investment strategy assuming a 100% equity proportion of policyholder s assets. Then we make the model more complicated by adding in a dynamic bonus strategy, a more realistic yield curve and a dynamic investment strategy. We also build up a portfolio which contains different policyholder generations. Under the CTE approach, we can see the benefit of pooling risks when reserving for the fund as a whole instead of setting up reserves separately for each generation. Finally we extend the investigation period and consider the portfolio within the simulated real world. xv

16 Introduction 0.1 Changes in the Regulatory Environment With-profits business has flourished for over a century as a long-term savings vehicle in the UK. The interaction of guaranteed benefits, policyholders reasonable expectations, smoothing, participation in the upside returns and substantial equity exposure makes with-profits a unique investment option. Historically, it has provided an enhanced investment return to many investors. However, recently the need to meet the costs of pensions mis-selling, the closure to new business of Equitable Life, the attribution of AXA s inherited estate, the declining nominal investment returns, lack of transparency and other problems have put with-profits business under increasing criticism. Over the last two years with-profits regulation in the UK has been going through one of the most significant periods of change in its history, to meet the demands for greater transparency and comparability of reporting. Hare et al. (2003) have outlined the development of realistic reporting through the UK regulatory framework. The series of Consultation Papers (CP) published by the FSA, including the recent CP195 proposals (FSA (2003)), demonstrate a move from the traditional valuation approach to a market consistent approach which places values on assets and liabilities consistent with the market values of assets with similar cashflow patterns. CP195 proposes a twin peaks approach for with-profits business to achieve the objective of realistic reporting within the constraints of the EC Third Life Directive. The approach has also been described in Muir and Waller (2003), and Dullaway and Needleman (2003). The regulatory peak is very similar to the existing statutory 1

17 valuation. It is governed by EU rules and based on a comparison of the admissible assets with the sum of the mathematical reserves, the Resilience Capital Requirement (RCR) and the Long Term Insurance Capital Requirement (LTICR). The realistic peak is based on a comparison of assets, including some inadmissibles, with realistic liabilities plus a risk capital margin (RCM). The RCM is required on top of realistic liabilities to provide some resilience to adverse experience. The total reserves and capital required for each with-profits fund are set as the greater of that required under a statutory approach and that required under a realistic, market-consistent approach. Therefore, additional capital, known as the with-profits insurance capital component (WPICC), is required to bring the regulatory surplus down to the level of the realistic surplus if the former is larger. The realistic balance sheet set out by the FSA is the core of the realistic peak. It is essentially split into three important items: the realistic value of assets available to support with-profits business, the with-profits benefit reserve and future policy related liabilities. The realistic balance sheet has also been described in detail in Muir and Waller (2003) and Hare et al. (2003). To determine a value of liabilities, CP195 proposals (FSA (2003)) adopt a put option approach which defines a liability as equal to the value of some underlying asset (asset share) plus an additional option value which reflects the fact that payout is subject to a certain minimum value. Hare et al. (2003) and Dullaway and Needleman (2003) suggest an alternative approach, known as a call option approach, starting with the present value of guarantees to which the cost of options is then added. The addition would be the excess (if any) of asset share over contractual guarantees, which corresponds to a call option. The authors have also described the relative advantages and disadvantages of these two approaches. The FSA sets out three approaches to determine the cost of any guarantees, options and smoothing embedded within the with-profits policies, namely: a stochastic approach using a market consistent asset model the market costs of hedging the guarantees or options a series of deterministic projections with attributed probabilities. 2

18 If the underlying guarantees or options can be hedged in the market then the cost of guarantees or options can be set equal to the market value of the hedge. Under simplifying assumptions direct analytical approaches or closed form solutions might be used. However, to the extent that the value of the guarantees or options is materially affected by management actions taken by the company it is unlikely that a closed form solution can be found which allows for dynamic asset allocation, dynamic bonuses and any cross-subsidy of guarantee costs. Therefore, the FSA states a preference for a stochastic approach using a market consistent asset model. The assets and liabilities of the with-profits fund are projected under a large number of economic scenarios generated by the asset model that has been calibrated to the market prices of financial instruments most relevant to the business being valued. The model should incorporate formulaic rules for the actions that the company s management may take to reduce risks and mitigate costs, for example adopting a more closely matching investment strategy or reducing bonus rates. Describing the characteristics of an asset model for the realistic balance sheet calculation is relatively straightforward, but the key questions are what asset model should be used and how should the model be calibrated? At present, no guidance exists in either of these two areas other than that the results produced should be market consistent. In order to investigate the significance of the choice and calibration of the asset model, Hare et al. (2003) have carried out a survey of option prices produced using six economic models. The authors conclude that market consistent models do not necessarily produce the same result; different models using the same calibration method can produce similar results; and similar models that are calibrated in different ways can produce dissimilar results. The FSA also states that when incorporating management actions into the projection of claims, the company should ensure consistency with its Principles and Practices of Financial Management (PPFM), which indicate to policyholders how an insurance company exercises its discretion in managing with-profits funds, and takes into account its regulatory duty to treat its customers fairly. However, it is not straightforward in practice to reflect the complex interactions between the insurer s financial strength, investment policy and bonus strategy, in a wide range of future 3

19 economic scenarios. In Dullaway and Needleman (2003), three approaches (in ascending order of sophistication) to incorporating management actions are described: A closed form approach. The guarantees or options are valued using a combination of deterministic and closed form solutions (e.g. the Black-Scholes formula) on a market consistent basis, with no or very limited allowance for management actions. A stochastic simulation approach. The guarantees or options are valued using market consistent stochastic projection models, but again with no or very limited allowance for management actions. A dynamic simulation approach. The guarantees or options are valued using market consistent stochastic projection models, with dynamic management actions incorporated. Under the third, i.e. the most complicated approach, if the management actions depend on the insurer s prospective solvency position, there are some practical issues with conducting a nested stochastic investigation. In addition to calculating the realistic values of assets and liabilities, the CP 195 proposals (FSA (2003)) require the calculation of a risk capital margin (RCM). It is defined as the fall in realistic surplus (which is the excess of the realistic value of assets over the realistic value of liabilities plus the risk capital margin) following a specific stress test. The rationale for the RCM calculation is that the insurer should still be in surplus on a realistic basis following an adverse event. The stress test set out in CP195 proposals (FSA (2003)) specifies a scenario including a fall in equity and property values, a widening of credit spreads and a shift in the yield curve in the direction that gives the greatest reduction in the realistic surplus. As commented in Muir and Waller (2003), the prescribed stress scenario only covers the major risks that with-profits funds are likely to be exposed to and ignores other risk factors such as adverse currency movements, operational risk etc. Hare et al. (2003) suggest that in the cases where the with-profits fund holds different assets to the hedge portfolio, a market consistent valuation of liabilities coupled with a short-term quantile based calculation of additional capital requirement is more appropriate from 4

20 both policyholder protection and regulatory action viewpoints. Similarly, Hibbert and Turnbull (2003) also define the minimum regulatory capital requirement as the capital sufficient to give a 99% probability of meeting the realistic value of guarantees after one year, in an illustrative example to demonstrate possible implications for risk-based capital requirements. This thesis carries out an investigation of the reserves required to meet the maturity guarantees under unitised with-profits (UWP) policies, within the new regulatory realistic reporting framework. The following questions are addressed in our numerical results: How should the maturity guarantees under UWP policies have been reserved for in the past for the realistic balance sheet calculation but using a closed form approach to incorporating management actions? How do these reserves compare with conditional tail expectation (CTE) reserves calculated using traditional stochastic valuation techniques? What amount of reserves will be required in the future under different reserving approaches in the simulated real world? What is the benefit of reserving for the fund as a whole instead of setting up reserves separately for each generation of business? Would those policies issued in the past be sustainable in a 1% stakeholder environment if the reserves had been set up as required? How about the sustainability in the future? What reasonable bonus and investment strategies could have been used in the past? Will these strategies still be reasonable in the future? What is the effect of smoothing regular bonus rates from year to year? What is the effect of reserving for future bonuses as required by the FSA? Following the FSA, this thesis also adopts a put option approach to value liabilities. We assume that all policies survive to maturity. Mortality and lapses are ignored for simplicity. At maturity 100% of unsmoothed asset share, which is the 5

21 accumulated value of the policyholder s fund, is paid to each policyholder. Hence we can take advantage of this approach in that no additional valuation is required for the calculation of the asset share because the market asset value can be used directly. In other words, the thesis concentrates on reserving for the excess of the guaranteed payout over the asset share at maturity. As defined in the realistic balance sheet published by the FSA, we are interested in future costs of financial options less planned deductions for guarantees from with-profits benefits reserve when using the market consistent approaches. 0.2 Thesis Outline Chapter 1 describes the operation of UWP policies and the simplified version considered in this thesis. We review some of the literature on reserving for policies with financial guarantees. Then we briefly discuss the three reserving approaches used in the thesis: buying options over-the-counter (OTC), dynamic hedging internally, and CTE reserving. The chapter also introduces the models used throughout the thesis. Three models with different purposes are mainly considered: valuation model, bonus and asset allocation model, and real world asset model. In Chapters 2 to 5 we build up the overall methodology of reserving for a single UWP policy historically. The results obtained for the 10-year policy issued at the end of 1991 using the three reserving approaches are compared in each chapter. Chapter 2 starts with the simplest case in which we assume a 100% equity backing ratio (EBR) which is the proportion of the asset share invested in equities, a constant bonus rate and a constant risk-free interest rate. In addition, the chapter also introduces the notation and assumptions for the policy. In Chapter 3 a dynamic bonus strategy is introduced. We first assume that regular bonuses are declared according to the bonus strategy without smoothing and that future regular bonuses are ignored when setting reserves. Then we smooth the regular bonus rates from year to year, but again ignore future bonuses. Finally the effect of reserving for the minimum future regular bonuses implied by our smoothing mechanism is investigated. 6

22 In Chapter 4, the yield on the zero-coupon bonds with the same maturity date as the policy is used as a risk-free interest rate. The zero-coupon yield is derived from the consols yield and short-term interest rate using a simple yield curve. We concentrate on the case of smoothing with allowance for future bonuses as it fits better with the current regulatory framework. In Chapter 5 we assume that the policyholder s fund is invested in two asset classes: equities and zero-coupon bonds with the same maturity date as the policy. A dynamic investment strategy is introduced by which the EBRs are adjusted annually. As in Chapter 4 we only consider the case of smoothing with allowance for future bonuses. Chapter 6 investigates the sensitivity of the results on a single policy to different parameters, different probability boundaries used in our dynamic investment strategy and different 10-year periods. In Chapters 7 and 8 we build up a portfolio of single premium 10-year UWP policies which are at different durations. Chapter 7 looks at the portfolio historically with a 20-year investigation period starting at the end of The dynamic bonus and investment strategies are applied to each generation separately. The risk-free interest rate equals the yield on the zero-coupon bonds with the same maturity date as the policy. The effects of smoothing and allowing for future bonuses are both considered. The benefit to the insurer of pooling risks under the CTE reserving approach is investigated, by comparing the amount of the reserves required for the whole portfolio with that set up for each generation of business separately. The portfolio cashflows incurred by the insurer are rolled up at the risk-free interest rate to calculate the amount of the insurer s free estate, from which we can discuss the profitability of the UWP policies over the last 20 years. The chapter also investigates the sensitivity of the portfolio results to different parameters as we do in Chapter 6 for the single policy issued at the end of Chapter 8 extends the investigation period to the end of The real world during the 30-year period starting at the end of 2002 is simulated stochastically. We concentrate on the case of smoothing with allowance for future bonuses. Different 7

23 quantiles of the simulated results for the EBRs, regular bonuses, portfolio reserves, and insurer s free estate are given in the chapter. For those policies matured in the following 30 years, we calculate the probability that the guarantees will bite at maturity. Summary statistics of the simulated maturity payouts and terminal bonuses are also calculated. Chapter 9 gives conclusions and some suggestions for further research. 8

24 Chapter 1 GUARANTEES UNDER UNITISED WITH-PROFITS POLICIES 1.1 Operation of UWP Polices With-profits business provides guaranteed and smoothed benefits which protect the policyholders investment value against fluctuations in the financial market. In addition, unitised with-profits (UWP) policyholders can easily work out the value of their investment at any time given the number of units and the current unit price. They can also change their premiums or sum assured when their circumstances change. These characteristics meet today s increasing demand for greater transparency and flexibility, and hence have made UWP contracts very popular in the market in recent years. The operation of UWP policies is different between different insurers, particularly between mutual and proprietary companies. However, basically the product works in the same way as a unit-linked policy. The policyholder pays a premium, from which a charge is deducted via a reduced allocation rate, bid-offer spread and policy fee to cover expenses. The rest is then converted into a certain number of units according to the current unit price. Each year some units are deducted to pay for mortality charges and fund management charges. 9

25 The unit price is set by a mixture of guarantees and bonuses, rather than directly related to the performance of the underlying assets as under the unit-linked policy. The unit price for the premiums already paid grows at a guaranteed rate. It is possible for the guaranteed growth rate to be as low as 0%, and in this case the guarantee still exists as the unit value is not allowed to fall in the future. Usually, the insurer reserves the right to change the guaranteed rate on future premiums. In the same way as conventional with-profits (CWP) policies declare reversionary bonuses, UWP policies apply regular bonuses to increase the unit price beyond the guaranteed rate. The bonus is actually added on a daily basis, but the bonus rate declared is normally x% p.a. until further notice. Nevertheless, the insurer is not forced to add the same bonus throughout the year. If the stock market falls, the insurer may declare a new lower rate applying from that date. Similar to the terminal bonus declared in the CWP product, at maturity the UWP policyholder may receive a payment bigger than his guaranteed unit value obtained by multiplying the number of his units by the current unit price. In order to keep the guarantee at a low level and hence increase investment freedom, the insurer usually promises a low growth rate and declares small regular bonuses to leave room for a large terminal bonus. UWP policies usually include a guaranteed sum assured which is payable on death. Thus, the policyholder on death receives the greater of his unit value, and the guaranteed sum assured. The policyholder can choose the guaranteed sum assured independently of the premium. The insurer can deal with this flexibility by a mortality charge to cover the expected cost of death claims. To protect itself against the risk of financial selection, the insurer retains the right to impose a market adjustment factor to surrenders which may bring the value of their units down to a level close to the asset share (value of the underlying assets) when the stock market falls. The above description shows the general operation of UWP policies in practice. However, in this thesis we look at a simplified version. In order to concentrate on the maturity guarantees, we ignore mortality, lapses and expenses. We also assume that after policy inception the valuation is conducted just after declaring a regular 10

26 bonus. 1.2 Reserving Approaches In this section we first summarise some of the literature on reserving for policies with financial guarantees. Then we describe the reserving approaches used in this thesis for our UWP policies Literature Review Willder (2004) has given a detailed literature review on pricing and reserving for the financial guarantees under unit-linked and participating policies. Here we concentrate on reserving and only review the papers whose reserving approaches are similar to ours. Ford et al. (1980) Ford et al. (1980) produced the report of the Maturity Guarantees Working Party. The working party consider reserving for maturity guarantees under unit-linked policies. The initial reserves are calculated from a large number of stochastically simulated future outcomes, with the assumption that the reserves would be sufficient in 99% of cases. The working party consider both a single policy and a portfolio of policies with different terms. In the single policy case, the authors conclude that smaller reserves are required for longer term policies. The reduction in the portfolio reserves shows the benefit of risk diversification. Collins (1982) Collins (1982) explores an immunization approach to reserving for unit-linked policies with maturity guarantees. A hedge portfolio is constructed and rebalanced discretely to meet the guaranteed maturity benefit. The author finds that the immunization strategy requires the largest rebalancing of assets when the asset share is in the volatile region of the current value of the maturity guarantees discounted at a risk-free interest rate. To reduce the intensity of the asset rebalancing, the author sets up the hedge portfolio as if the policy had a longer term and larger guarantees. 11

27 Boyle and Hardy (1997) Boyle and Hardy (1997) consider a stochastic simulation approach and an option pricing approach to reserving for Canadian segregated funds which are unit-linked policies with maturity guarantees. Under the stochastic simulation approach the authors simulate the performance of the unit fund using the Wilkie model. Quantile reserves are set up at a chosen probability level. Under the option pricing approach they construct a hedge portfolio derived from the Black-Scholes equation. As continuous hedging is not possible in practice, the authors consider time-based and move-based hedging strategies. The comparison of the two strategies is conducted using a simulation approach. The authors conclude that the move-based strategy provides superior hedging performance because the tracking errors are smaller for the same expected hedging costs. The authors also consider buying corresponding options externally from a bank or another financial institution. The reserves required are the cost of buying these options. This method provides a 100% probability that the guarantee will be honoured assuming that there is no counter-party risk, which is the risk that the option provider defaults. Hardy (1999) Hardy (1999) compares the quantile reserves calculated using different investment models. A lognormal model and a regime switching lognormal (RSLN) are considered and both of them are calibrated to Toronto Stock Exchange data. The author demonstrates a danger of being under-reserved if using an insufficiently fat tailed distribution to model the investment performance. Hardy (2000) As in Boyle and Hardy (1997), Hardy (2000) also calculates reserves for segregated funds using the stochastic simulation and option pricing approaches. Under the simulation approach the author compares the initial quantile reserves obtained using the Wilkie Model and the lognormal model, with and without allowance for fund management charges. To calculate the additional capital required to increase the reserves or the capital released from the reserve fund, the author adopts a corridor approach whereby the reserves are strengthened if the probability of sufficiency falls below 92.5% and are weakened if the probability rises above 99.8%. Under the option 12

28 pricing approach the author considers both discrete hedging and buying options. Hardy (2001) Hardy (2001) compares the conditional tail expectation (CTE) reserves calculated using the lognormal model and the RSLN model. As in Hardy (1999), the author demonstrates a danger of being under-reserved if the investment performance is simulated by an insufficiently fat tailed distribution. Hare et al. (2000) Hare et al. (2000) consider maturity guarantees under conventional and unitised with-profits policies at different durations. The authors first calculate reserves on the current UK statutory reserving basis including the minimum solvency margin and resilience reserve. Then they find that the statutory reserves are mostly inadequate if reserves are required at a 99% probability level using the stochastic simulation approach. The investment performance is simulated by the Wilkie model with adjusted low inflation parameters. The authors also calculate the ratio of the 99% quantile reserve to the statutory minimum reserve at each duration. Yang (2001) Yang (2001) considers guaranteed annuity options (GAOs) attached to pensions policies. The initial reserves are calculated using a stochastic simulation approach and a hedging approach. Under the simulation approach, both quantile and CTE reserves are calculated. Under the hedging approach, the author derives closed form solutions for the hedging price. The tracking error and transaction costs are compared under the hedging strategies with annual and monthly rebalancing. Wilkie et al. (2003) Wilkie et al. (2003) extend the work carried out in Yang (2001). The authors consider reserving for a pensions-type contract with GAOs both historically and in the stochastically simulated real world. The stochastic simulation and option pricing approaches are used. Under the simulation approach, the quantile and CTE reserves are both calculated. The authors also compare the initial reserves calculated for the whole portfolio with different terms to maturity with those calculated for each individual policy separately. Then the reserves required each year are calculated using both marking-to-market and corridor approaches. 13

29 Hibbert and Turnbull (2003) Hibbert and Turnbull (2003) mainly calculate the fair value of guarantees under a single conventional with-profits policy using a market-consistent asset model. The authors also investigate the sensitivity of this fair value to a set of decision rules for bonuses, equity backing ratio and policyholder behaviour. The authors suggest that reserves should be set up so that the fund would be sufficient to meet the realistic value of guarantees after one year with a 99% probability. Haberman et al. (2003) Haberman et al. (2003) consider reserving for unitised with-profits contracts using stochastic simulation techniques. Assets are modelled with a Brownian motion. The authors consider three smoothing schemes for regular bonuses. The initial reserves are calculated for the guaranteed benefit, future regular bonuses and terminal bonus. Then the authors investigate the sensitivity of the required amount of reserves to changes in the model parameters. Summing up, Collins (1982), Boyle and Hardy (1997), Hardy (2000), Yang (2001), Wilkie et al. (2003) and Hibbert and Turnbull (2003) set up reserves based on option pricing theory. Ford et al. (1980), Boyle and Hardy (1997), Hardy (1999), Hardy (2000) and Hare et al. (2000) calculate quantile reserves using stochastic simulation techniques. CTE reserves are considered in Hardy (2001). Yang (2001) and Wilkie et al. (2003) calculate both quantile and CTE reserves Three Reserving Approaches Having reviewed some of the literature on approaches to reserving for policies with financial guarantees, here in this section we briefly describe our reserving approaches used throughout the thesis. The overall methodology is similar to that of Boyle and Hardy (1997), though they considered unit-linked policies. The maturity payout under a UWP policy corresponds to that of a combination of shares and European put options. Thus, the first approach adopted in this thesis is to buy the corresponding options from a third party so that the guarantees can always be met at maturity. We only look at the part of the reserve fund held in addition to the asset share, so the amount of the reserves calculated in the thesis 14