Immunization and Hedging of Longevity Risk
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1 Immunization and Hedging of Longevity Risk Changyu Estelle Liu and Michael Sherris CEPAR and School of Risk and Actuarial Studies UNSW Business School, UNSW Australia 2052 This presentation has been prepared for the Actuaries Institute 2015 ASTIN and AFIR/ERM Colloquium. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions.
2 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
3 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
4 Introduction Motivation Pension funds and life insurers offering annuities hold long term liabilities linked to longevity Risk management of life annuity portfolios should immunize or hedge both interest rate and mortality risks Standard fixed interest duration-convexity hedging must be adapted to allow for both interest rate and longevity risk Our aims To develop an immunization approach along with a delta-gamma based approach allowing for both interest rate and mortality risk. To compare and assess immunization and hedge effectiveness of fixedincome coupon bonds, annuity bonds, and longevity bonds, using simulations of portfolio surplus outcomes for an annuity portfolio.
5 Key insights Fixed-income annuity bonds can more effectively match cash flows but provide limited additional hedge effectiveness over coupon bonds. Longevity bonds, including deferred longevity bonds, reduce risk significantly compared to coupon and annuity bonds, reflecting the long duration of the typical life annuity and the exposure to longevity risk. Longevity bonds are shown to be effective in immunizing surplus over short and long horizons based on Fisher-Weil duration and convexity of cash flows. Delta gamma hedging is only effective over short horizons, requiring more rebalancing than immunization with longevity bonds. Delta gamma hedging is model dependent introducing increased model risk compared to Fisher-Weil immunization.
6 Approach 1. Stochastic Mortality Risk: Schrager (2006) Two-factor Model 2. Stochastic Interest Rate Risk: CIR Model 3. Liabilities: Whole-life annuities (Monthly) for Australian Male current age Assets: Currently Issued Fixed-income Securities (monthly, quarterly, semi-annually): 39 Coupon Bonds and 24 Annuity Bonds (ASX and FIIG) 10 Assumed Longevity Bonds (Yearly) for Australian Male current age 65: 5,10,, 50 years 5. Optimal Asset Allocation: Extend Shiu s Linear Programming Duration-convexity immunization: minimizing assets convexity, matching surplus and duration of assets and liabilities Delta-gamma hedging: minimizing assets gamma, matching surplus and delta of assets and liabilities 6. Hedge Performance: Compare distribution of P&L at future time points (1, 10, and 50 years) Compare risk reduction measures
7 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
8 Risk Models and Measures for Life Annuity and Longevity Bonds
9 Risk Measures for Life Annuity and Longevity Bonds
10 Risk Measures for Life Annuity and Longevity Bonds
11 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
12 Linear Programming: Duration-Convexity Immunization
13 Optimal Bond Portfolio to Immunize a Life Annuity Issued to 65 year olds
14 Optimal Duration-Convexity Bond Portfolio Cash Flows
15 Linear Programming: Delta-Gamma Hedging
16 Optimal Bond Portfolio to Delta-Gamma Hedge a Life Annuity Issued to 65 year olds
17 Optimal Delta-Gamma Hedge Bond Portfolio Cash Flows
18 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
19 Optimal Duration-Convexity Bond Portfolio Surplus Distribution
20 Optimal Delta-Gamma Hedge Bond Portfolio Surplus Distribution
21 Measuring Hedge Effectiveness Risk measures used for portfolio surplus - VaR, Volatility and Expected Shortfall Measures risk reduction as a percentage of the risk
22 Hedge effectiveness-annuity Bonds and Longevity Bonds
23 Hedge effectiveness-delta-gama Hedging compared to Immunization
24 Outline 1. Motivation and Key Insights 2. Interest Rate and Longevity Risk 3. Optimal Bond Portfolios 4. Portfolio Surplus Distribution and Hedge Effectiveness 5. Summary and Wrap Up
25 Summary Fisher-Weil immunization can effectively immunize both interest rate and mortality (longevity) risk. Annuity bonds provide better cash flow matching than coupon bonds, however the hedge effectiveness are similar. Delta-gamma hedging with longevity bonds include both deferred (negative short maturity bond and positive longer maturity bond) and immediate longevity bonds. Longevity bonds alone provide very effective hedging of longevity risk in life annuities - no surprise here but it is surprising that these are not available in government or private bond markets. Raises policy issues as to why the market and government should consider the design of (deferred) longevity bonds to support life insurers offering annuities.
26 Research Contributions Extensions on the current mortality immunization literature Tsai and Chung (2013), Lin and Tsai (2013), and Luciano et al. (2012): Incorporated fixed-income securities and longevity-linked securities to manage both interest rate risk and mortality risk. Developed linear programming methods to solve the optimal asset allocation problems considering any number of assets in the portfolio (not limited by the number of equality conditions). Compared delta-gamma hedging with duration-convexity hedging and used stochastic models to assess both interest rate and mortality risks.
27 Thank you Questions or Discussion Thank you for your attention! Contact: or Find the paper on SSRN
28 Reference
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