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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:

ln qx,t 1 − qx,t = A1(t) + A2(t)x. (1)

◮ x → age ◮ t → time ◮ qx,t → realized single-year death probability ◮ {A1(t)} and {A2(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)

Stochastic Mortality: Recall: ln

qx,t 1−qx,t = A1(t) + A2(t)x

D(t + 1) = A(t + 1) − A(t) (2) = µ + CZ(t + 1)

◮ A(t) = (A1(t), A2(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

◮ qx,t,

x = 65, 66, . . . , 109, t = 1971, 1972, . . . , 2005

Model fitting

ln

qx,t 1−qx,t = A1(t) + A2(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

ln

qx,t 1−qx,t = A1(t) + A2(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 qx,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)

D(t + 1) = µ + C(˜ Z(t + 1) − λ) (4) = ˜ µ + C ˜ Z(t + 1), where λ is the market price of risk and ˜ µ = µ − Cλ.

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

FIXED PAYER FIXED RECEIVER

X

• Sk−1

S0

· σ

12

• X
• Sk−1−Sk

S0

• ◮ X →nominal amount

◮ Sk →index value in the kth 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 = (λ1, λ2) λ1 λ2 σ 0.375 627 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 | D ∼ MVN(ˆ µ, n−1 ˆ V),

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Bayesian Method

Estimated marginal posterior density functions for the model parameters

−0.1 0.1 20 40 60 μ1 Density −1 1 x 10

−3

1000 2000 3000 μ2 Density 0.005 0.01 200 400 600 800 V(1,1) Density −2 −1 x 10

−4

2 4 6 x 10

4

V(1,2) Density −2 −1 x 10

−4

2 4 6 x 10

4

V(2,1) Density 1 2 x 10

−6

1 2 3 x 10

6

V(2,2) Density

Figure: Simulated marginal posterior parameter distributions. (We denote the ith element in µ by µi and the (j, k)th element in V by Vj,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)

0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 50 100 150 σ Density Without parameter risk With parameter risk

A Cautionary Note on Pricing Longevity Index Swaps Parameter risk Impact of parameter risk on pricing

95% Confidence Interval for σ

λ1 λ2 With parameter risk Without parameter risk 0.375 (560,693) (574,680) 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 λ1 and λ2.

A Cautionary Note on Pricing Longevity Index Swaps Model risk

Model risk in pricing

1960 1980 2000 2020 −10.4 −10.2 −10 −9.8 −9.6 −9.4 −9.2 −9 −8.8 Year A1 1960 1980 2000 2020 0.082 0.084 0.086 0.088 0.09 0.092 0.094 0.096 Year A2

Figure: Estimated values of A1(t) and A2(t), 1971–2005.

A Cautionary Note on Pricing Longevity Index Swaps Model risk Reason for the reverse trend

What causes the reverse trend?

Crude mortality curves ln

qx,t 1−qx,t = A1(t) + A2(t)x

50 60 70 80 90 0.02 0.04 0.06 0.08 Age Crude mortality rates 1995−2000 50 60 70 80 90 0.02 0.04 0.06 0.08 Age Crude mortality rates 2000−2005 80 90 100 110 0.2 0.4 0.6 0.8 Age Crude mortality rates 80 90 100 110 0.2 0.4 0.6 0.8 Age Crude mortality rates 1995 1996 1997 1998 1999 2000 1995 1996 1997 1998 1999 2000 2000 2001 2002 2003 2004 2005 2000 2001 2002 2003 2004 2005

A Cautionary Note on Pricing Longevity Index Swaps Model risk Reason for the reverse trend

What causes the reverse trend?

Life expectancies at age 65 ln

qx,t 1−qx,t = A1(t) + A2(t)x

1995 2000 2005 16.8 17 17.2 17.4 17.6 17.8 18 18.2 18.4 18.6 t Life expectancies

A Cautionary Note on Pricing Longevity Index Swaps Model risk Future trends

Three possible scenarios

1960 1980 2000 2020 −11 −10.5 −10 −9.5 −9 −8.5 −8 Year A1(t) 1960 1980 2000 2020 0.075 0.08 0.085 0.09 0.095 0.1 0.105 Year A2(t) Scenario 1 Scenario 2 Scenario 3

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 λ1 λ2 Scenario 1 Scenario 2 Scenario 3 0.375 627 674 566 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