How Capital Market assists to solve the Longevity Issues Longevity Risk Modeling and Derivatives Pricing

1. How Capital Market assists to solve the Longevity IssuesLongevity Risk Modeling and Derivatives Pricing Yinglu Deng Assistant Professor in Finance School of Economics and Management Tsinghua University Joint working with Patrick Brockett, Richard MacMinn

2. Longevity Risk • Definition • Longevity risk is the risk that individual lives longer than expected or projected • Dramatic improvements in longevity during the 20th century • In developed countries, average life expectancy has increased by 1.2 months per year • Globally, life expectancy at birth has increased by 4.5 months per year • The impact of the longevity risk • In U.K., as of October 2006 the deficit in FTSE100 Corporations was estimated at £46 billion, however once increased longevity is taken into account, the deficit is more likely £100 billion (Pension Capital Strategies estimates) • In U.S., the new mortality assumptions designated for contribution levels by the IRS for pension funds have increased pension liabilities by 5-10% (Watson Wyatt) • Using up-to-date mortality tables, pension payments for a male born in 1950 increase by 8% Effected Parties • Pension funds • Corporate Sponsored • Government Sponsored • Annuity Providers • Insurance companies • Reinsurance companies • Individuals

3. Longevity Risk underestimation of the life expectancy

4. Why is Financial Management of Longevity Risk Important? • Social Security is going insolvent. If people live longer it will only get worse • Defined benefit pension plans are reserved for an assumed rate of mortality which is wrong. More people living longer will make the pension plan go insolvent • Individuals in defined contribution pension plans (or without plans) encounter the risk of running out of money if they live too long

5. Also A Corresponding Issue of Mortality Risk Effected Parties • Life Insurance Providers • Insurance companies • Reinsurance companies • Government (the insurer of last resort) • Definition • Catastrophic mortality events • 1918 pandemic influenza, more than 675,000 excess deaths from the flu occurred between September 1918 and April 1919 in U.S. alone • H5N1 avian influenza occurred in Hong Kong in 1997, and H1N1 occurred globally in 2009 • Localized natural catastrophic events • Earthquake, Hurricane, Floods, Tsunami • The impact of the mortality risk • The reserves for U.S. life insurance policies stand at around $1 trillion- may be insufficient due to potential catastrophic mortality risk

6. Longevity/Mortality Risk Management • Transferring and Diversifying +Insurance (Mortality Risk) -Life Insurance  +Reinsurance Company +Annuity Provider (Longevity Risk) -Customized Product -Immediate Annuities -Deferred Annuities -Advanced Life Delayed Annuities +Pension funds (Longevity Risk)  +Capital Market -including corporate pensions -Securitization -Structured Product +Reverse Mortgages (Longevity Risk) -Insurance Linked Securities +Life Settlements (Longevity Risk) +Individual 

7. Can the Capital Markets Assist? • Capital Markets have much larger capacity than insurance markets, annuity markets and reinsurance market • Capital Markets provide more diversified approaches to manage the risk. • Derivative instruments can be invented which transfer these risks from those who hold them to those better situated to handle them • Just like other insurance linked securities, or options or bonds built on indices (e.g., weather derivatives, catastrophe derivatives, earthquake bonds, etc.) • Capital Markets provide potential high liquid approach • Capital Markets provide low cost approach • The key point for the emerging of the capital market solution is the model for quantification of the risk and valuation of the derivatives

8. The Potential Longevity Risk Market is Large • Global market: • >$25 trillion according to Swiss Re • UK market (according to Prudential Inc.) • UK government liabilities susceptible to longevity risk exceed £ 2000bn • About £ 1170bn in state pensions • About £770bn are unfunded • About £ 160bn in local authority liabilities • Defined Benefit pension plans are susceptible • Total pension liabilities = about £1000bn • Of which pensions currently in payment = about £ 500bn (so the need is immediate) • Annuity providers are susceptible • Approximately £ 135bn • Defined Contribution plans put the risk on the individual • Total assets = approximately £ 450bn • Of which the share held by those over age 55 = approximately £ 150bn

9. Securitization • Insurance linked securities • The interaction and combination of the insurance industry and the capital market • Load off the non-diversified risk from the insurer or pension balance sheet • An efficient and low-cost way to allocate and diversify risk in the capital market • Enhance the risk capacity of the insurance industry • Examples: • Catastrophe Mortality Bond • Longevity Bond • q-Forwards • Life settlement securitization • Reverse Mortgage Participants • Investment Banks • JP Morgan • Goldman Sachs • Reinsurance Companies • Swiss Re • Munich Re

10. Example: A Mortality Bond • Swiss Re Mortality Catastrophe Bond was issued in Dec. 2003 by the Swiss Reinsurance company as the first mortality risk contingent securitization • The bond was issued through a special purpose vehicle (SPV), triggered by a catastrophe evolution of death rates of a certain specified population • The bond had a maturity of three years, a principal of $400m, a coupon rate of LIBOR plus 135 basis points • The amount of principal repayment in the final period declines linearly if the experienced mortality exceeds 130% of a specified rate M0, and is completely eliminated if the mortality is above 150% of M0 . The rate M is based on a weighted mortality index across 5 countries. • The precise payment rate schedule for coupons and the principal is given by the following function:

11. Contract Design of the Mortality Bond LIBOR Interest Rate Swap Counterparty Fixed Return Premium 135bps at t=1,…,T LIBOR+135bps at t=1,…,T Swiss Re Vita Capital Ltd. Debt Investor $400m, at t=0 $400m at t=0 at t=T, Up to $400m, • $0 if no extreme mortality events • at t=T Up to $400m, $0 at the extreme mortality events Mortality Index The Contract Design of the Swiss Re Mortality Bond Issued in 2003

12. Characteristics • The issuer (Swiss Re) gains if the mortality rate is extremely high • The investor gains if the mortality rate is not extremely high (due to higher coupons) • The bond is designed to hedge the portfolio dominated by life insurance/reinsurance policies • The bond is a short-term, standard coupon-plus-principal bond • Coupons float with LIBOR • Principal is at risk from a mortality deterioration • Spread over the LIBOR compensates the investor for bearing the mortality risk • It hedges the issuer (Swiss Re) against an extreme increase in mortality, such as that associated with pandemic influenza.

13. A Longevity Risk Bond • The European Investment Bank (EIB) issued the bond in November 2004, through the structurer /manager BNP Paribas. • Partner Re contracts pay annual floating rate payments (equal to £ 50m* St,realized) to the EIB based on the realized mortality experience of the population of English and Welsh males aged 65 in 2003, where St,realized is the actual survivor rate of a 65 year old for t years • Survivor Rate is defined as the probability a 65 year old will be alive at 65+t. The survivor rate is negative correlated with the mortality rate. So survivor rate is the underlying factor for longevity risk. • Partner Re receive from the EIB annual fixed payments based on a set of mortality forecasts for this cohort, -St,fixed • The mortality forecasts were based on the UK Government Actuary’s Department’s 2002-based central projections of mortality • The bond has an initial value of £ 540 m, an initial coupon of £ 50 m, and a maturity of 25 years. • The net coupon payment structure after the swap is ft (St,realized) =£ 50m*(St,realized -St,fixed) for t = 1; 2; … , T; T = 25:

14. Contract Design of the Longevity Bond The simplified longevity bond structure $50m*St,fixed, at t=1,…,T $540 m, at t=0 Debt Investor Partner Re EIB $50m*St,realized, at t=1,…,T $50m*St,realized, at t=1,…,T +$540m, at t=T If St,realized <St,fixed then Partner Re gains Since the realized survivor rate is less than expected (fixed) and the mortality rate is more than expected If St,realized >St,fixed then Investor gains Since the realized survivor rate is more than expected (fixed) and the mortality rate is less than expected

15. Characteristics • The bond was designed to be a hedge to the holder. • The issuer gains if St,realized is lower than anticipated St,fixed (and conversely, the buyer gains if St,realized is higher than anticipated). • Thus, the bond is a hedge against a portfolio dominated by annuity (rather than life insurance/reinsurance) policies. • The bond is a long-term bond designed to protect the holder against any unanticipated improvement in mortality up to the maturity date of the bond. • St,fixed and St,realizedinvolves a single national survivor index. • The bond is an annuity (or amortizing) bond and all coupon payments are at risk from longevity shocks. More precisely, the payment schedules are directly proportional to the survivor indexes:

16. Contract Design of q-Forward notional*100*qt,fixed at t=1,…,T notional*100*qt,fixed at t=1,…,T • The Characteristics of q-Forwards Pension Funds JP Morgan Life Insurers notional*100*qt,realied at t=1,…,T notional*100*qt,realized at t=1,…,T Longevity Hedge for Pension Funds Mortality Hedge for Life Insurers • Basic building blocks • Standardized contracts for a liquid market • Exchange • realized mortality of a population at some future date, • a fixed mortality rate agreed at inception • Pension funds • hedge against increasing life expectancy of plan members, • Have longevity risk exposure • Life insurers • hedge against the increase in the mortality of policyholders, • Have mortality risk exposure

17. Six longevity swaps in 2008-09

18. Main Contribution • The first model to give a closed-form solution to the expected mortality rate, and q-forward type products. The closed-form solves the computational complexity problem encountered by most of the complicated structured derivatives • The first model to address the longevity jump and the mortality jump separately in a concise model with only six parameters • The model parameterization is very easy and straightforward, which enables the model implementation very efficient • The model fits the data better than the classical Lee-Carter model and other previous jump models

19. Literature Review • Lee-Carter (1992), benchmark, without jumps, extended by Brouhns, Denuit and Vermunt (2002), Renshaw and Haberman (2003), Denuit, Devolder and Goderniaux (2007), Li and Chan (2007) • Our model incorporates the jump diffusion process • Biffis (2005), Bauer, Borger and Russ (2009), with affine jump-diffusion process, model force of mortality in a continuous-time framework • Our model incorporates the cohort effect • Chen, Cox and Peterson (2009), with compound Poisson normal jump diffusion process • Our model incorporates the asymmetric jump diffusion process • Lin, Cox and Peterson (2009), modeling longevity jump and mortality jump • Our model provides a concise and practical approach

20. Data • HIST290 National Center for Health Statistics, U.S. • Death rates per 100,000 population for selected causes of death • Death rates are tabulated for age group (<1), (1-4), (5-14), (15-24), then every 10 years, to (75-84), and (>85) • Both sex and race categories • Selected causes for death include major conditions such as heart disease, cancer, and stroke

21. Data Figure 1. 1900-2004 Mortality Rate

22. Data Figure 2. Comparison of the Age Group Mortality Rates

23. Quantitative Model Framework • Lee-Carter Framework • Mortality improvement • Different improvement rate for age groups • Dynamic improvement trend • Model Set-up- modeling the force of mortality

24. Quantitative Model Framework • Two-stage procedure Singular Value Decomposition (SVD) method • Regression • Re-estimate

25. Quantitative Model Framework

26. Quantitative Model Framework

27. Quantitative Model Requirement • Stochastic Process • Brownian Motion • Transient Jump • Asymmetric Jump • Non stochastic process • Geometric Brownian Motion • Permanent Jump • Symmetric Jump V.S. • Asymmetric Jump • Phenomenon • Mortality Jump • Short-term intensified effect • Pandemic influenza, like flu 1918 • Longevity Jump • Long-term gentle effect • Pharmaceutical or medical innovation • Compound Poisson- • Double Exponential Jump Diffusion • Positive Jump • Small frequency • Large scale • Negative Jump • Large frequency • Small scale

28. Quantitative Model Requirement • The descriptive statistics of shows asymmetric leptokurtic features. • The skewness of equals to -0.451 • distribution is skewed to the left • distribution has a higher peak and two heavier tails Figure 4. Comparison of actual distribution and normal distribution

29. Quantitative Model Specification Specification • Features • Differentiating positive jumps and negative jumps • Mathematical tractability • Closed-form formula • Concise • Widely implemented

30. Quantitative Model Specification

31. Numerical Method • Parameter calibration • Disentangling jumps from diffusion • Maximum Likelihood Estimation method • The form of the DEJD process satisfies the requirement of the transition density for using MLE • Calibrate parameters • Results indicates • Maximum likelihood value

32. Maximum Likelihood Function

33. Distribution Function of Jumps • (Continued with last page)

34. Model Comparison Figure 4. Comparison of Actual Distribution and Normal Distribution Figure 5. Comparison of Actual Distribution and DEJD Distribution

35. Model Comparison • Compare fitness of DEJD model with Lee-Carter Brownian Motion model and Normal Jump Diffusion model (Chen and Cox, 2009) • Bayesian Information Criterion (BIC) • Allow comparison of more than two models • Penalizes the likelihood for having more parameters (so higher number of parameters is is not always better) • Do not require alternative to be nested • Conservative, heavily penalize over parameterization • The smaller BIC, the better fitness Table 2. Comparison of model fitness

36. Implied Market Price of Risk --Computed with Mortality Bond, recall • Swiss Re Mortality Catastrophe Bond is issued by the Swiss Reinsurance company , as the first mortality risk contingent securitization in Dec. 2003. This issue will be used to obtain the market price of longevity risk for our model. • The bond has a maturity of three years, a principal of $400m, the coupon rate of 135 basis points plus the LIBOR • The precise payment rate schedules are given by the following function:

37. Pricing JP Morgan’s q-Forward Longevity Derivative Recall the structure notional*100*Mt,fixed at t=1,…,T Pension Funds JP Morgan notional*100*Mt,realied at t=1,…,T Longevity Hedge for Pension Funds What is the appropriate market implied fixed rate to use in this derivative?

38. q-Forward Pricing • The fixed rate can be calculated with the closed-form formula directly.

39. Life Settlement Description • A life settlement is a financial arrangement whereby the third party (or investor) purchases a life insurance policy from the person who originally purchased a life insurance policy. • This third party pays the insured an amount greater than the cash surrender value of the policy -- in effect, the trade-in value of the policy as determined by the originating insurance company-- but less than the face value (or the death benefit). • It can be a win-win situation, as the investor can obtain a return on their initial investment and premium payments once the death benefit becomes payable (assuming the insured does not live too much longer than expected when setting the purchase price) and the owner of the policy obtains more money than they otherwise could obtain during their life.

40. Life settlement Pricing • The main factor in the life settlement securities pricing currently is the estimation of the life expectancy of the insured. • We introduce a Whole Life Time Distribution Dynamic Pricing (WLTDDP) method • In the past, the life expectancy is considered in the pricing as the solo variable which represents the expected life time of the insured when he sells his life insurance policy to the third party as a life settlement.

41. Advantage of WLTDDP Method(1) • The net present value of future payment of the life settlement product is contingent on the future life time T of the insured. According to the Jensen’s inequality, using the expected life time E[T] only and pricing the product like one would an E[T]-year bond with a pay off of vE[T] , v being the discount rate, results in incorrect pricing for the traditional method. This traditional price vE[T] is always smaller than the value of the E[vT] which is the true expected net present value. Thus the current method is underpricing the value of the future payoff. • The advantage of WLTDDP method is that it generates a complete life table with the whole distribution of life time so expected net present value can be calculated instead of simply using the expected life time (life expectancy) independent of the underlying mortality distribution.

42. Advantage of WLTDDP Method(2) • The statistical methodology is based upon information theory for adjusting mortality tables to obtain exactly some known individual characteristics, while obtaining a table that is as close as possible to a standard one. • In this way, the method provides more accurate projection and evaluation for the life settlement products, through incorporating more statistical information of the insured’s future life time.

43. Advantage of WLTDDP Method(3) • it incorporates the effect of the dynamic longevity risk through the original life table which is generated from the Double Exponential Jump Diffusion model (DEJD) • The DEJD model incorporates the longevity jump (caused by medical improvement, etc), mortality jump (caused by pandemic influenza, etc) and dynamic main trend of the mortality rate, which provide better explanation and fitness to the historical mortality rate data.

44. Information Theory For distinguishing between two densities on the basis of an observation t, a sufficient statistic is the log odds ratio in favor of the observation having come from in favor of g. It is the amount of information contained in a observation t for discriminating in favor of f over g.  In a long sequence of observations from , the long-run average or expected log odds ratio in favor of f is I(f|g) = This reflects the expected amount of information for discriminating between f and g. Note that I(f|g) ≧0 and = 0 if and only if f=g . Thus, the size is a measure of the closeness of the densities f and g. For a given g, one can minimize I(f|g) over f to find the closest f. If we have constraints, we can do a constrained optimization. E.g., if the mean is give as m, then we have constraints: We do this with the distribution of life given by the mortality table g and the expectation of life m as given by a life settlement medical expert or actuary to find a best fitting mortality table for pricing this individual’s life settlement.

45. Information Theory • To phrase the problem mathematically, we desire to find a vector of probabilities that solves the problem: • Here is the vector of probabilities corresponding to the standard probability distribution. Brockett, Charnes and Cooper (1980) show that the problem has a unique solution, which is: • The parameters can be obtained easily as the dual variables in an unconstrained convex programming problem:

46. An Example of Real Life Policy from State Farm

47. Table 3. Adjusted and Standard Mortality Table for Age 70 Standard table is the fit DEJD cohort age 70, adjusted to have life expectancy of two years

48. Life Settlement Structure

49. Pricing