On systematic mortality risk. and. quadratic hedging with mortality derivatives

Transcription

1 On systematic mortality risk and quadratic hedging with mortality derivatives University of Ulm, December 2, 2006 Thomas Møller, PFA Pension, Copenhagen ( tmoller)

2 Outline. Introduction 2. Systematic mortality risk (With M. Dahl, IME, 2006) 3. Hedging with mortality derivatives (With M. Dahl and M. Melchior, work in progress) Theory and modeling Numerical results

3 Introduction Since 2003, I work in PFA Pension Danish life insurance company/pension fund Mutual company Balance: approximately 27 billion euro participating life insurance contracts (defined contributions) Background: until 2003, Assistant professor, Laboratory of Actuarial Math, Univ. Cph Chief Analyst, Actuarial Innovation market-valuation of liabilities new savings products actuarial support for risk-management and investment depts actuarial research & supervision 2

4 Introduction Brief motivation Systematic mortality risk Large improvements in the mortality in many countries during the last years Risk for life insurers with (guaranteed) annuities (mortality tables were not conservative enough!) Future mortality is difficult to predict (unpredictable!) A new market for mortality derivatives is appearing (mortality/survivor swaps, longevity bonds etc) Necessary to model e.g. the mortality intensity as a stochastic process 3

5 Introduction Some recent literature on systematic mortality risk Marocco/Pitacco (998) Olivieri/Pitacco (2002) Milevsky/Promislow (200) Dahl (2004) Dahl/Møller (2006) Cairns, Blake and Dowd (2004) Miltersen/Persson (2006) Biffis and Millossovich (2006) 4

6 Introduction Long term simulation of number of survivors Example: age 30, simulate number of survivors at age policy-holders.000 policy-holders Expected no. of survivors: 37 Expected no. of survivors: 374 Std.dev without systematic risk: 4.9 Std.dev without systematic risk: 5.4 Std.dev with systematic risk: 7.3 Std.dev with systematic risk:

7 Introduction Simulation for a portfolio of retired Example: start age 75, simulate number of survivors at age policy-holders.000 policy-holders Expected no. of survivors: 43.5 Expected no. of survivors: 435 Std.dev without systematic risk: 4.9 Std.dev without systematic risk: 5.8 Std.dev with systematic risk: 5.2 Std.dev with systematic risk: 2.5 6

8 Systematic mortality risk The mortality intensity is a stochastic process (joint work with M. Dahl) Known at time 0: µ (x + t) is mortality intensity today at all ages x + t Unknown at time 0: ζ(t, x) is relative change in the mortality from 0 to t, age x Mortality intensity: µ(x, t) = µ (x + t)ζ(x, t) (In general, a stochastic process) True survival probability from t to T given information I(t): [ S(x, t, T) = E P e ] T t µ(x,τ)dτ I(t) 7

9 Systematic mortality risk A specific model: Time-inhomogeneous CIR model known from finance: dζ(x, t) = (γ(x, t) δ(x, t)ζ(x, t))dt + σ(x, t) ζ(x, t)dw µ (t) Proposition (Affine mortality structure, Dahl, 2004) The survival probability S(x, t, T) is S(x, t, T) = e Aµ (x,t,t) B µ (x,t,t)µ(x,t) where t Bµ (x, t, T) = δ µ (x, t)b µ (x, t, T) + 2 (σµ (x, t)) 2 (B µ (x, t, T)) 2 t Aµ (x, t, T) = γ µ (x, t)b µ (x, t, T) with B µ (x, T, T) = 0 and A µ (x, T, T) = 0 [ Martingale: S M (x, t, T) = E P e ] T 0 µ(x,τ)dτ I(t) 8

10 Systematic mortality risk Forward mortality intensity f µ (x, t, T) = log S(x, t, T) = µ(x, t) T T Bµ (x, t, T) T Aµ (x, t, T) Survival probability S(x, t, T) = e T t f µ (x,t,u)du e T t µ(x,u)du Change of measure for mortality and financial market Equivalent measure Q Financial market Standard affine model for short rate: dr(t) = (γ r,α δ r,α r(t)) dt + γ r,σ + δ r,σ r(t)dw r (t) 9

11 Systematic mortality risk Change of measure for mortality and financial market Equivalent measure dq = Λ(T) via ( dp dλ(t) = Λ(t ) h r (t)dw r (t) + h µ (t)dw µ (t) + g(t)dm(x, t) ) Require affine under Q. Zero coupon bond prizes P(t, T) = e Ar (t,t) B r (t,t)r(t) where A r (t, T) and B r (t, T) solve t Br (t, T) = δ r,α,q B r (t, T) + 2 δr,σ (B r (t, T)) 2 t Ar (t, T) = γ r,α,q B r (t, T) 2 γr,σ (B r (t, T)) 2 with B r (T, T) = 0 and A r (T, T) = 0 0

12 Systematic mortality risk Two portfolios of insured lives T j,,..., T j,n are i.i.d. given ζ j with P(T j, > t I(T)) = e t 0 µ j(x,s)ds, j =,2 (: own pf, 2: other pf) Counting processes and martingales N j (x, t) = n (Tj,i t) i= t M j (x, t) = N j (x, t) 0 (n N j(x, u ))µ j (x, u)du Insurance payment process (Benefits premiums on pf ) da(t) = (n N (x, T)) A 0 (T)d (t T) + a 0 (t)(n N (x, t))dt + a (t)dn (x, t) (a i, A 0 deterministic functions)

13 Systematic mortality risk Modeling of the mortality in two portfolios dζ j (x, t) = (γ j (x, t) δ j (x, t)ζ j (x, t))dt + σ j (x, t) Here: W µ two-dimensional Brownian motion and σ j (x, t) R 2 ζ j (x, t)dw µ (t) Possibility for correlation between systematic mortality risk in the two portfolios Simple example: ( σ (x, t) σ 2 (x, t) ) = ( σ σ 2 0 σ 22 ) First: focus on risk in portfolio. Hedge with bonds Later: Hedge with bonds and mortality swaps 2

14 Q-martingale: Systematic mortality risk V (t) = E Q [ [0,T] e τ = [0,t] e 0 r(u)du da(τ) F(t) τ 0 r(u)du da(τ) + e t 0 r udu Ṽ Q (t) ] Here, the market reserve is Ṽ Q (t) = E Q [ (t,t] e τ t r(u)du da(τ) F(t) = (n N (x, t))v Q (t, r(t), µ (x, t)) ] where V Q (t, r(t), µ (x, t)) = T t P(t, τ)s Q (x, t, τ) ( a 0 (τ) + a (τ)f µ,q (x, t, τ) ) dτ + P(t, T)S Q (x, t, T) A 0(T) (t<t) 3

15 Systematic mortality risk Risk-minimization (Föllmer/Sondermann, Schweizer) Here: with payment streams (Møller, 200) Market: savings account and long zero coupon bond Discounted price processes: X(t) = P (t, T), Y (t) = Trading strategy: Process ϕ = (ξ, η) with ξ predictable (+ technical conditions) Value process: V (t, ϕ) := ξ(t)x(t) + η(t) 4

16 Systematic mortality risk Payment process A = (A(t)) 0 t T square integrable A(t) A(s) is amount paid by insurer during (s, t] Cost process C(t, ϕ) = V (t, ϕ) t 0 ξ(u) dx(u) + A (t) Risk process R(t, ϕ) = E Q [ (C(T, ϕ) C(t, ϕ)) 2 F(t) ] Criterion of risk-minimization Minimize R(t, ϕ) over ϕ for all t Terminal cost C(T, ϕ) = V (T, ϕ) T 0 ξ(u) dx(u) + A (T) V value after payments. Fix V (T, ϕ) = 0 Payments Unlimited capital A ϕ = (ξ, η) Investments V (ϕ) 5

17 Systematic mortality risk Kunita-Watanabe decomposition: V (t) := E Q [A (T) F(t)] = V (0) + 0 ξa,q (u) dx(u) + L A,Q (t) where ξ A,Q is predictable L A,Q is a square integrable martingale X and L A,Q are orthogonal t Theorem.! risk-minimizing strategy ϕ = (ξ, η) with V (T, ϕ) = 0: ξ(t) = ξ A,Q (t) The minimum risk process η(t) = V (t) A (t) ξ A,Q (t)x(t) R(t, ϕ) = E Q [ (L A,Q (T) L A,Q (t)) 2 F(t) ] 6

18 Systematic mortality risk Intrinsic value process: dv,q (t) = ν V,Q (t)dm Q where (x, t) + ηv,q (t)dw r,q (t) + ρ V,Q (t)dw µ,q (t) (t) = B(t) a d (t) Ṽ p,q (t) η V,Q (t) = γ r,σ r Ṽ,Q (t) ν V,Q ρ V,Q,j (t) = σµ,j (x, t) µ (x, t) µ Ṽ,Q (t) Risk-minimizing strategy determined from Galtchouk-Kunita-Watanabe decomposition: dv (t) = ξ A,Q (t)dx(t) + dl A,Q (t) 7

19 Systematic mortality risk Risk-minimizing strategy in a pure bond market (ξ B (t), η B (t)) = (ξa,q (t), Ṽ,Q (t) ξ A,Q (t)p (t, T)) where ξ A,Q (t) = The unhedgeable risk η V,Q (t) γ rσ B r (t, T)P (t, T) dl Q (t) = ν V,Q (τ)dm Q µ,q (x, τ) + ρv,q, (τ)dw (τ) + ρ V,Q,2 (τ)dw µ,q 2 (τ) (See Dahl/Møller (2006)) 8

20 Systematic mortality risk Sources of risk from GKW-decomposition: dv (t) = ξ A,Q (t)dx(t) + ν Q (t)dm Q (t) + ρv,q (t)dw µ,q (t) Financial risk: ξ A,Q dx Unsystematic mortality risk: ν Q dm Q Systematic mortality risk: ρ V,Q dw µ,q Properties of the optimal strategy: ξ = ξ A,Q eliminates the financial risk is unable to deal with other risks 9

21 Hedging with mortality derivatives Extending the market with mortality swaps (joint work with M. Dahl and M. Melchior) Underlying payment processes: da swap j (x, t) = (n j N j (x, t))dt n j j tp xdt (Defined for portfolios j =,2) Traded price process: Z,Q j (x, t) = E Q [ T τ 0 e 0 r(u)du da swap j (x, τ) F(t) We assume this process is traded on extended market (B, P, Z j ) ] j = : same portfolio (same systematic and unsystematic risk) j = 2: another portfolio (systematic risk correlated) 20

22 Hedging with mortality derivatives Motivation/idea Mortality swaps are available in the reinsurance markets The mortality swap contains systematic and unsystematic risk If we use Z, we hedge with process driven by 3 sources of risk (M, W µ,, W µ,2 ) Can use this process to balance the systematic and unsystematic risks in the insurance portfolio Using a swap on another portfolio introduces a new unsystematic risk M 2, but eliminates part of the systematic risk 2

23 Hedging with mortality derivatives Dynamics for the traded process dz,q (t) = νz,q (t)dm Q where ν Z,Q (t) = η Z,Q (t) = ρ Z,Q + T t (x, t)+ηz,q (t)dw r,q (t)+ρ Z,Q (t)dw µ,q (t) P (t, τ)s Q (x, t, τ)dτ γ rσ (n N (x, t)) γ rσ T t T t B r (t, τ)p (t, τ)s Q B r (t, τ)p (t, τ) τ p x n dτ,j (t) = σµ,j (x, t) µ (x, t)(n N (x, t))( + g (t)) T t B µ,q (t, τ)p (t, τ)s Q (x, t, τ)dτ Useful for finding the risk-minimzing strategy (x, t, τ)dτ 22

24 Hedging with mortality derivatives GKW-decomposition of V,Q in the extended market (B, P, Z ) where and ξ Q dv,q (t) = ξ Q (t)dp (t, T) + ϑ Q (t)dz,q (x, t) + dlq (t) dl Q (t) = ( ν V,Q + + (t) ϑ Q (t)νz,q ηv,q (t) = (t) ϑ Q (t)ηz,q (t) γ r,σ B r (t, T)P (t, T) ϑ Q (t) = Here: κ Q,j ) (t) dm Q (x, t) ( ρ V,Q ), (t) ϑq (t)ρz,q, (t) dw µ,q (t) ( ρ V,Q ),2 (t) ϑq (t)ρz,q,2 (t) dw µ,q 2 (t) νv,q (t) + ρ V,Q, (t)(κq, (t)) ρ V,Q,2 (t)(κq,2 (t)) ν Z,Q (t) + ρ Z,Q, (t)(κq, (t)) + ρ Z,Q,2 (t)(κq,2 (t)) (t) = νz,q (t)λ Q (x,t) ρ Z,Q,j (t) 23

25 Hedging with mortality derivatives Interpretation: Optimal number of swaps: The strategy balances the three sources of risk: the unsystematic mortality risk and the two factors driving the systematic risk ϑ Q νv,q (t) + ρ V,Q, (t) = (t)(κq, (t)) + ρ V,Q,2 (t)(κq,2 (t)) ν Z,Q (t) + ρ Z,Q, (t)(κq, (t)) + ρ Z,Q,2 (t)(κq,2 (t)) Optimal position in bonds: Identical to the previous position (without swaps) added a position which eliminates the new interest rate risk in the swaps ξ Q (t) = ϑq (t)ηz,q (t) η V,Q (t) γ rσ B r (t, T)P (t, T) 24

26 Hedging with mortality derivatives GKW-decomposition of V,Q in the extended market (B, P, Z 2 ) where dv,q (t) = ξ Q 2 (t)dp (t, T) + ϑ Q 2 (t)dz,q 2 (x, t) + LQ 2 (t) and dl Q 2 (t) = νv,q (t)dm Q (x, t) ϑq 2 (t)νz,q 2 (t)dm Q 2 (x, t) ( + ρ V,Q ), (t) ϑq 2 (t)ρz,q 2, (t) dw µ,q (t) ( + ρ V,Q ),2 (t) ϑq 2 (t)ρz,q 2,2 (t) dw µ,q 2 (t) ξ Q ηv,q 2 (t) = (t) ϑ Q 2 (t)ηz,q 2 (t) γ r,σ B r (t, T)P (t, T) ϑ Q 2 (t) = ρ V,Q, (t)(κq 2, (t)) + ρ V,Q,2 (t)(κq 2,2 (t)) ν Z,Q 2 (t) + ρ Z,Q 2, (t)(κq 2, (t)) + ρ Z,Q 2,2 (t)(κq 2,2 (t)) Here: κ Q 2,j (t) = νz,q 2 (t)λ Q 2 (x,t) ρ Z,Q 2,j (t) 25

27 Interpretation: Hedging with mortality derivatives Optimal number of swaps: (Similar interpretation). The strategy balances the three sources of risk: the unsystematic mortality risk and the two factors driving the systematic risk Optimal position on bonds: Similar intepretation as in previous model Note: The investment in the alternative swap introduces new unsystematic risk related to the insurance portfolio dl Q 2 (t) = νv,q (t)dm Q (x, t) ϑq 2 (t)νz,q 2 (t)dm Q 2 (x, t) ( + ρ V,Q ), (t) ϑq 2 (t)ρz,q 2, (t) dw µ,q (t) ( + ρ V,Q ),2 (t) ϑq 2 (t)ρz,q 2,2 (t) dw µ,q 2 (t) 26

28 Hedging with mortality derivatives Have also derived GKW-decomposition of V,Q in the extended market (B, P, Z, Z 2 ) dv,q (t) = ξ Q (t)dp (t, T)+ϑ Q (t)dz,q (x, t)+ψq (t)dz,q 2 (x, t)+lq (t) More involved expressions. Now use both mortality swaps to hedge dynamically the risk inherent in the life insurance portfolio 27

29 Hedging with mortality derivatives: Numerical results t Realization of the short rate over a period of 60 years in two different stochastic scenarios (red and blue line) 28

30 Hedging with mortality derivatives: Numerical results t Price of a zero coupon bond with maturity T = 60 years in two different stochastic scenarios (red and blue line). 29

31 Hedging with mortality derivatives: Numerical results Portfolio (j) µ j (x,0) γ j (x, t) δ j (x, t) σ j, (x, t) σ j,2 (x, t) µ 0 (x) µ 0 2 (x) Parameters for mortality intensities. We consider two portfolios, n = 00, n 2 =,

32 Hedging with mortality derivatives: Numerical results age at time of death age at time of death Deaths in the insurance portfolio (left plot) and deaths in the population (right plot) in two stochastic scenarios (red lines and blue lines) 3

33 Hedging with mortality derivatives: Numerical results t t The new hedging instruments Intrinsic value processes for survivor swap on the insurance portfolio (left plot) and survivor swap on the population (right plot) in two scenarios (red and blue lines) 32

34 Hedging with mortality derivatives: Numerical results t The liability - to be hedged! Intrinsic value processes for the insurance contract in two different stochastic scenarios (red and blue line) 33

35 Hedging with mortality derivatives: Numerical results t Picture: Number of survivor swaps on the insurance portfolio held at time t in the market (B, P, Z ) (in scenario ) 34

36 Hedging with mortality derivatives: Numerical results t Picture: Number of survivor swaps on the the population held at time t in the market (B, P, Z 2 ) (in scenario ) 35

37 Hedging with mortality derivatives: Numerical results t t Left plot: Black line is number of survivor swaps on the insurance portfolio in the (B, P, Z, Z 2 ) market. Red line is number of survivor swaps on the insurance portfolio in the (B, P, Z ) market The right plot: Black line is number of survivor swaps on the population in the (B, P, Z, Z 2 ) market. Grey line is the difference between the investments in the survivor swap on the insurance portfolio from the (B, P, Z ) market and the (B, P, Z, Z 2 ) market scaled by a factor 0 36

38 Hedging with mortality derivatives: Numerical results 0 e+00 2 e+05 4 e+05 6 e+05 ξ in the (B,P) market ξ in the (B,P,Z ) market ξ in the (B,P,Z 2 ) market ξ in the (B,P,Z Z 2 ) market t Number of zero coupon bonds held. Hedge for the interest rate risk inherent in the insurance portfolio and for the interest rate risk in the mortality swaps 37

39 Hedging with mortality derivatives: Numerical results n n 2 R(0,Ψ V ) n R(0,Ψ B ) 00, , ,000 0, ,000 00, ,000 00, n n n 2 R(0,Ψ ) R(0,Ψ 2 ) R(0,Ψ ) n n 00, , ,000 0, ,000 00, ,000 00, n The minimum obtainable risk in the various markets 38

40 Hedging with mortality derivatives: Status and future research We have also studied Strategies in discrete time for the mortality swap combined with continuous time hedging for the bond (have derived optimality result) We are currently Finishing the paper Extending the numerical work further Comparing with alternative mortality derivatives 39

41 Hedging with mortality derivatives: References Dahl (2004). Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts, Insurance: Mathematics and Economics 35, 3 36 Dahl, Melchior and Møller (2006). On systematic mortality risk and Quadratic hedging with mortality swaps, in progress Dahl and Møller (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: Mathematics and Economics 39, Møller (200). Risk-minimizing hedging strategies for insurance payment processes, Finance and Stochastics 5, Møller (2002). On valuation and risk management at the interface of insurance and finance, British Actuarial Journal 8, Møller and Steffensen (2007). Market-valuation methods in life and pension insurance, forthcoming, Cambridge University Press 40