A Cautionary Note on Pricing Longevity Index Swaps (Joint work with - PowerPoint PPT Presentation

A Cautionary Note on Pricing Longevity Index Swaps A Cautionary Note on Pricing Longevity Index Swaps (Joint work with Johnny S.H. Li) Rui Zhou Department of Statistics and Actuarial Science University of Waterloo 44th Actuarial Research Conference 2009

A Cautionary Note on Pricing Longevity Index Swaps Objectives ◮ Pricing QxX index swap ◮ Examining the parameter risk and model risk in the pricing ◮ Determining the effect of the uncertainty on the pricing Outline ◮ Mortality derivatives ◮ QxX index Swap ◮ Parameter risk ◮ Model risk ◮ Conclusion

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Mortality Derivatives What are mortality derivatives? ◮ Financial contracts that have payoffs tied to the level of a certain longevity or mortality index ◮ Examples: survivor bond, survivor swap, . . . How to price mortality derivatives? ◮ Mortality model ◮ Wang’s Transform, Q measure, . . .

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Mortality model A two-factor stochastic mortality model (Cairns, Blake and Dowd (2006)) Mathematical Specification: q x , t ln = A 1 ( t ) + A 2 ( t ) x . (1) 1 − q x , t ◮ x → age ◮ t → time ◮ q x , t → realized single-year death probability ◮ { A 1 ( t ) } and { A 2 ( t ) } → discrete-time stochastic processes

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Mortality model A two-factor stochastic mortality model(con’t) q x , t Stochastic Mortality: Recall: ln 1 − q x , t = A 1 ( t ) + A 2 ( t ) x D ( t + 1 ) = A ( t + 1 ) − A ( t ) (2) = µ + CZ ( t + 1 ) ◮ A ( t ) = ( A 1 ( t ) , A 2 ( t )) ′ ◮ µ → constant 2 × 1 vector ◮ C → constant 2 × 2 upper triangular matrix ◮ Z ( t ) → 2-dim standard normal random variable

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Mortality model Model fitting Data ◮ q x , t , x = 65 , 66 , . . . , 109 , t = 1971 , 1972 , . . . , 2005 Model fitting q x , t ln 1 − q x , t = A 1 ( t ) + A 2 ( t ) x D ( t + 1 ) = µ + CZ ( t + 1 ) ◮ First step: Estimate A ( t ) by least square method ◮ Second step: Estimate µ and C through maximum likelihood estimation

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Mortality model Forecasting Steps q x , t ln 1 − q x , t = A 1 ( t ) + A 2 ( t ) x D ( t + 1 ) = µ + CZ ( t + 1 ) ◮ Simulate a set of Z ◮ Obtain corresponding D ( 2005 + k ) , k = 1 , 2 , . . . , 10 ◮ A ( 2005 + k ) = A ( 2005 ) + � k n = 1 D ( 2005 + n ) , k = 1 , 2 , . . . , 10 ◮ Calculate q x , 2005 + k

A Cautionary Note on Pricing Longevity Index Swaps Mortality Derivatives Risk-adjusted probability measure Pricing in Risk-adjusted world Real-world probability measure(P measure) D ( t + 1 ) = µ + CZ ( t + 1 ) (3) Risk-adjusted probability measure(Q measure) µ + C (˜ D ( t + 1 ) = Z ( t + 1 ) − λ ) (4) µ + C ˜ = ˜ Z ( t + 1 ) , where λ is the market price of risk and ˜ µ = µ − C λ .

A Cautionary Note on Pricing Longevity Index Swaps QxX index swap QxX Index “allows market participants to measure, manage and trade exposure to longevity and mortality risks in a standardized, transparent, and real-time manner" ◮ Launched by Goldman Sacs in 2007 ◮ Based on a reference pool consisting of a set of lives underwritten by AVS Underwriting LLC ◮ The index value is the number of lives in the reference pool ◮ Published monthly, providing “real-time" mortality information

A Cautionary Note on Pricing Longevity Index Swaps QxX index swap Payment structure of QxX index swap Payment structure of QxX index swap � � S k − 1 · σ X S 0 12 FIXED PAYER FIXED RECEIVER � � S k − 1 − S k X S 0 ◮ X → nominal amount ◮ S k → index value in the k th month ◮ σ → fixed spread ◮ Goldman Sacs: σ = 500 basis points for 10-year swap

A Cautionary Note on Pricing Longevity Index Swaps QxX index swap Pricing a 10-year QxX index swap 10-year QxX index swap price ◮ QxX index swap is priced by determining the “fair" spread σ Market value of future payments from fixed payer = Market value of future payments from fixed receiver ◮ We need to know the market price of risk λ . In our analysis, ◮ Not enough data to estimate λ for QxX index swaps ◮ Use the estimated market price of risk from BNP/EIB longevity bond

A Cautionary Note on Pricing Longevity Index Swaps QxX index swap Pricing a 10-year QxX index swap 10-year QxX index swap price (Con’t) Estimates of σ (in basis points) under different choices of λ = ( λ 1 , λ 2 ) λ 1 λ 2 σ 0.375 0 627 0 0.316 619 0.175 0.175 622 Why σ � = 500 bps? ◮ No access to the actual QxX index reference pool ◮ Lack of market data for the swap ◮ Existence of parameter risk and model risk

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Bayesian Method Parameter risk under Bayesian Method ◮ D ( t ) ∼ MVN ( µ, V ) , where V = C ′ C . ◮ Treat µ and C as random variables D ( t ) | µ, V ∼ MVN ( µ, V ) (5) ◮ Use a non-informative prior distribution π ( µ, V ) ∝ | V | − 3 / 2 (6) ◮ Marginal posterior distribution V − 1 | D Wishart ( n − 1 , n − 1 ˆ V − 1 ) , ∼ (7) µ, n − 1 ˆ µ | D ∼ MVN (ˆ V ) ,

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Bayesian Method Estimated marginal posterior density functions for the model parameters 60 3000 800 600 40 2000 Density Density Density 400 20 1000 200 0 0 0 − 0.1 0 0.1 − 1 0 1 0 0.005 0.01 μ 1 μ 2 V(1,1) − 3 x 10 4 4 6 x 10 x 10 x 10 6 6 3 4 4 2 Density Density Density 2 2 1 0 0 0 − 2 − 1 0 − 2 − 1 0 0 1 2 V(1,2) V(2,1) V(2,2) − 4 − 4 − 6 x 10 x 10 x 10 Figure: Simulated marginal posterior parameter distributions. (We denote the i th element in µ by µ i and the ( j , k ) th element in V by V j , k ).

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Impact of parameter risk on pricing Simulated predictive distribution of σ , λ = ( 0 . 375 , 0 ) 150 Without parameter risk With parameter risk 100 Density 50 0 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 σ

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Impact of parameter risk on pricing 95% Confidence Interval for σ λ 1 λ 2 With parameter risk Without parameter risk 0.375 0 (560,693) (574,680) 0 0.316 (553,685) (567,673) 0.175 0.175 (557,686) (571,675) Table: 95% confidence intervals for σ (in basis points) under different choices of λ 1 and λ 2 .

A Cautionary Note on Pricing Longevity Index Swaps Model risk Model risk in pricing − 8.8 0.096 − 9 0.094 − 9.2 0.092 − 9.4 0.09 A 1 − 9.6 A 2 0.088 − 9.8 0.086 − 10 0.084 − 10.2 − 10.4 0.082 1960 1980 2000 2020 1960 1980 2000 2020 Year Year Figure: Estimated values of A 1 ( t ) and A 2 ( t ) , 1971–2005.

A Cautionary Note on Pricing Longevity Index Swaps Model risk Reason for the reverse trend What causes the reverse trend? q x , t Crude mortality curves ln 1 − q x , t = A 1 ( t ) + A 2 ( t ) x 1995−2000 2000−2005 1995 2000 0.08 0.08 Crude mortality rates 1996 Crude mortality rates 2001 1997 2002 0.06 0.06 1998 2003 1999 0.04 0.04 2004 2000 2005 0.02 0.02 0 0 50 60 70 80 90 50 60 70 80 90 Age Age 1995 2000 0.8 0.8 Crude mortality rates 1996 2001 Crude mortality rates 1997 2002 0.6 0.6 1998 2003 1999 0.4 0.4 2004 2000 2005 0.2 0.2 0 0 80 90 100 110 80 90 100 110 Age Age

A Cautionary Note on Pricing Longevity Index Swaps Model risk Reason for the reverse trend What causes the reverse trend? q x , t Life expectancies at age 65 ln 1 − q x , t = A 1 ( t ) + A 2 ( t ) x 18.6 18.4 18.2 18 Life expectancies 17.8 17.6 17.4 17.2 17 16.8 1995 2000 2005 t

A Cautionary Note on Pricing Longevity Index Swaps Model risk Future trends Three possible scenarios −8 0.105 Scenario 1 Scenario 2 Scenario 3 −8.5 0.1 −9 0.095 A 1 (t) A 2 (t) −9.5 0.09 −10 0.085 −10.5 0.08 −11 0.075 1960 1980 2000 2020 1960 1980 2000 2020 Year Year

A Cautionary Note on Pricing Longevity Index Swaps Model risk QxX index swap price under different trends How does the change affect QxX index swap price? Swap spread, σ Scenario 1 Scenario 2 Scenario 3 λ 1 λ 2 0.375 0 627 674 566 0 0.316 619 683 553 0.175 0.175 622 678 558 Table: Swap spread (in basis points) under three different scenarios.

A Cautionary Note on Pricing Longevity Index Swaps Conclusion Conclusion ◮ The swap spread computed from our pricing framework is fairly close to the spread currently offered by Goldman Sachs ◮ The pricing is still very experimental ◮ Parameter risk and model risk are significant in the pricing ◮ No sufficient market price data to estimate market prices of risk ◮ No clear conclusion on how mortality rates may evolve in the future