Bounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates

Transcription

1 Bounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates Tom Hoedemakers (K.U.Leuven) Grzegorz Darkiewicz (K.U.Leuven) Griselda Deelstra (ULB) Jan Dhaene (K.U.Leuven) Michèle Vanmaele (Ghent University) Workshop on the Interface between Quantitative Finance and Insurance Edinburgh, April 4-8, 2005

2 Agenda 1. Introduction: problem description and motivation 2. Model assumptions 3. General treatment 4. Convex order and comonotonic risks 5. Comonotonic bounds for a stop-loss premium 6. Upper bounds based on the lower bound plus an error term 7. Decomposition of the stop-loss premium in an exact and an inexact part 8. Application to life annuities with stochastic interest rates 9. Numerical illustration

3 Introduction: problem description and motivation life annuity: a series of periodic payments where each payment will actually be made only if a designated life is alive at the time the payment is due

4 Introduction: problem description and motivation life annuity: a series of periodic payments where each payment will actually be made only if a designated life is alive at the time the payment is due notation: T x : future lifetime of (x) (a person aged x years) G x (t) =Pr[T x t] = t q x,t 0 (G 1 x (1) = ω x) K x = T x : curtate future lifetime of (x) Pr(K x = k) =Pr(k T x <k+1)= k+1 q x k q x = k q x,k=0, 1,... a discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i = ω x I (Tx >i)e Z i e Z i random discount factor over the period [0,i] 1 per annum payable at beginning of each year

5 a continuous whole life annuity: a Tx = Tx 0 e r(τ) dτ see Vanduffel S., Dhaene J. and Valdez E. (2005) Accurate closed-form approximations for constant continuous annuities

6 a continuous whole life annuity: a Tx = Tx 0 e r(τ) dτ see Vanduffel S., Dhaene J. and Valdez E. (2005) Accurate closed-form approximations for constant continuous annuities Why stochastic interest rates? durations of contracts are typically very long (often 30 years or even more) uncertainty about future rates of return becomes very high

7 a continuous whole life annuity: a Tx = Tx 0 e r(τ) dτ see Vanduffel S., Dhaene J. and Valdez E. (2005) Accurate closed-form approximations for constant continuous annuities Why stochastic interest rates? durations of contracts are typically very long (often 30 years or even more) uncertainty about future rates of return becomes very high financial and investment risk unlike the mortality risk cannot be diversified with an increase in the number of policies

8 Actuarial literature on stochastic interest rates calculating first two or three moments of present value function explore properties of random distribution

9 Actuarial literature on stochastic interest rates calculating first two or three moments of present value function explore properties of random distribution criticism: variance is not suitable risk measure to determine solvency requirements for an insurance portfolio two-sided risk measure over- and underestimation of reserve in case of skewed distribution and tail properties of distribution not emphasized

10 Actuarial literature on stochastic interest rates calculating first two or three moments of present value function explore properties of random distribution criticism: variance is not suitable risk measure to determine solvency requirements for an insurance portfolio two-sided risk measure over- and underestimation of reserve in case of skewed distribution and tail properties of distribution not emphasized appropriate risk measures: Value-at-Risk, Tail Value-at-Risk or Expected Short Fall on based stop-loss premium stop-loss premium with retention d: π(x, d) =E[(X d) + ]=E[max(0,X d)]

11 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i mortality process: Makeham s model the number of persons alive at age x: l x = as x g cx with prespecified parameters a>0, 0 <s<1, 0 <g<1 and c>1 respectively for men and women number of newborns: l 0 =

12 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i discounting process: Brownian motion (motivation: Cesari & Cremonini (2003)) e Z i = e Y (i) := e (Y 1+ +Y i ) rv s Y i represent stochastic continuous compounded rate of return over period [i 1,i] and e Y (i) is random discount factor over period [0,i] yearly returns Y i are i.i.d. normally distributed with mean µ and volatility σ Y (i) iµ + σb(i) with B(i) standard Brownian motion independent of K x, µ constant force of interest Y (i) strongly dependent!

13 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i risk measures in particular stop-loss premiums of S policy x analytically cannot be determined

14 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i risk measures in particular stop-loss premiums of S policy x analytically cannot be determined comonotonic approximations (especially the lower bound) has been successfully applied to determine easy to compute and accurate approximations for the problem at hand

15 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i risk measures in particular stop-loss premiums of S policy x analytically cannot be determined comonotonic approximations (especially the lower bound) has been successfully applied to determine easy to compute and accurate approximations for the problem at hand Why is life simple using comonotonic approximations?

16 Model assumptions discrete whole life annuity-immediate: a Kx = S policy x = K x e Z i risk measures in particular stop-loss premiums of S policy x analytically cannot be determined comonotonic approximations (especially the lower bound) has been successfully applied to determine easy to compute and accurate approximations for the problem at hand Why is life simple using comonotonic approximations? Are these approximations accurate?

17 General treatment stop-loss premium with retention d: π(x, d) =E[(X d) + ]=E[max(0,X d)] computation: not always straightforward! if only partial information about the claim size distribution is available (e.g. De Vylder and Goovaerts (1982), Jansen et al. (1986), Hürlimann, (1996, 1998) among others) in the case of a sum of random variables S = X X n with unknown dependency structure convex upper and lower bounds for stop-loss premiums of sums of dependent random variables [Kaas et al. (2000), Dhaene et al. (2002) ] ideas of splitting up the expectation in an exact part and the rest [e.g. Curran (1994) ] ideas of conditioning as in Curran (1994) and Rogers and Shi (1995) leads to easy analytical computable bounds

18 Convex order and comonotonic risks Definition: Consider two random variables X and Y.ThenXis said to precede Y in the convex order sense, notation X cx Y, if and only if E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ] d

19 Convex order and comonotonic risks Definition: Consider two random variables X and Y.ThenXis said to precede Y in the convex order sense, notation X cx Y, if and only if E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ] d Property: X cx Y Var[X] Var[Y ]

20 Convex order and comonotonic risks Definition: Consider two random variables X and Y.ThenXis said to precede Y in the convex order sense, notation X cx Y, if and only if E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ] d Property: X cx Y Var[X] Var[Y ] Comonotonicity is a very strong positive dependence structure each two possible outcomes (x 1,,x n ) and (y 1,,y n ) of X =(X 1,,X n ) are ordered componentwise

21 Convex order and comonotonic risks Definition: Consider two random variables X and Y.ThenXis said to precede Y in the convex order sense, notation X cx Y, if and only if E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ] d Property: X cx Y Var[X] Var[Y ] Comonotonicity is a very strong positive dependence structure each two possible outcomes (x 1,...,x n ) and (y 1,...,y n ) of X =(X 1,...,X n ) are ordered componentwise Characterizations: (a) for U uniform(0, 1) we have X d =(F 1 X 1 (U),F 1 X 2 (U),...,F 1 X n (U)),

22 Convex order and comonotonic risks Definition: Consider two random variables X and Y.ThenXis said to precede Y in the convex order sense, notation X cx Y, if and only if E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ] d Property: X cx Y Var[X] Var[Y ] Comonotonicity is a very strong positive dependence structure each two possible outcomes (x 1,,x n ) and (y 1,,y n ) of X =(X 1,,X n ) are ordered componentwise Characterizations: (a) for U uniform(0, 1) we have X d =(F 1 X 1 (U),F 1 X 2 (U),...,F 1 X n (U)), (b) a random variable Z and non-decreasing functions f 1,f 2,...,f n,(or non-increasing functions) such that X d =(f 1 (Z),f 2 (Z),...,f n (Z)).

23 Comonotonic bounds for a stop-loss premium Upper and lower bounds for sums of random variables Kaas, Dhaene and Goovaerts (2000) General result: Let U be a uniform(0,1) random variable. For any random vector X =(X 1,X 2,...,X n ) with marginal cdf s F X1,F X2,...,F Xn,wehave E[X i Λ] cx X i cx F 1 X i Λ (U) cx F 1 X i (U) which we denote by S l cx S cx S u cx S c

24 Comonotonic bounds for a stop-loss premium Upper and lower bounds for sums of random variables Kaas, Dhaene and Goovaerts (2000) General result: Let U be a uniform(0,1) random variable. For any random vector X =(X 1,X 2,...,X n ) with marginal cdf s F X1,F X2,...,F Xn,wehave E[X i Λ] cx X i cx F 1 X i Λ (U) cx F 1 X i (U) which we denote by S l cx S cx S u cx S c S c : keep marginals X i but replace copula by most dangerous comonotonic copula S c is always a comonotonic sum

25 Comonotonic bounds for a stop-loss premium Upper and lower bounds for sums of random variables Kaas, Dhaene and Goovaerts (2000) General result: Let U be a uniform(0,1) random variable. For any random vector X =(X 1,X 2,...,X n ) with marginal cdf s F X1,F X2,...,F Xn,wehave E[X i Λ] cx X i cx F 1 X i Λ (U) cx F 1 X i (U) which we denote by S l cx S cx S u cx S c S l : replace marginals X i by less dangerous E[X i Λ] S l is comonotonic sum if all E[X i Λ] are functions of Λ

26 Comonotonic bounds for a stop-loss premium Upper and lower bounds for sums of random variables Kaas, Dhaene and Goovaerts (2000) General result: Let U be a uniform(0,1) random variable. For any random vector X =(X 1,X 2,...,X n ) with marginal cdf s F X1,F X2,...,F Xn,wehave E[X i Λ] cx X i cx F 1 X i Λ (U) cx F 1 X i (U) which we denote by S l cx S cx S u cx S c F 1 X i Λ (U) stands for the rv f i(u, Λ) with f i (u, λ) =F 1 X i Λ=λ (u)

27 Comonotonic bounds for a stop-loss premium Upper and lower bounds for sums of random variables Kaas, Dhaene and Goovaerts (2000) General result: Let U be a uniform(0,1) random variable. For any random vector X =(X 1,X 2,...,X n ) with marginal cdf s F X1,F X2,...,F Xn,wehave E[X i Λ] cx X i cx F 1 X i Λ (U) cx F 1 X i (U) which we denote by S l cx S cx S u cx S c E[ ( S l d ) + ] E[(S d) + ] E[(Su d) + ] E[(S c d) + ] d (notation: π l (S,d,Λ) π(s,d) π u (S,d,Λ) π c (S,d), d)

28 Kaas et al. (2000): the quantile function is additive for comonotonic risks in case of strictly increasing and continuous marginals, the cdf F S c(x) is uniquely determined by F 1 S c (F S c (x)) = F 1 X i (F S c (x)) = x, ( F 1 S c (0) <x<f 1 S c (1) )

29 Kaas et al. (2000): the quantile function is additive for comonotonic risks in case of strictly increasing and continuous marginals, the cdf F S c(x) is uniquely determined by F 1 S c (F S c (x)) = F 1 X i (F S c (x)) = x, ( F 1 S c (0) <x<f 1 S c (1) ) Dhaene et al. (2002): stop-loss premiums of a sum of comonotonic random variables can easily be obtained from stop-loss premiums of the terms: π c (S,d)= E [( ) ] X i FX 1 i (F S c (d)), + ( F 1 S c (0) <d<f 1 S c (1) )

30 lower bound: S l comonotonic: π l (S,d,Λ) = = E E ( ) F 1 (0) <d< F 1 (1) S l S l [( ) ] E[X i Λ] F 1 E[X i Λ] (F S l (d)) + [ (E[Xi Λ] E [ X i Λ=F 1 Λ (F S l (d)) ]) + ]

31 lower bound: S l comonotonic: π l (S,d,Λ) = = E E ( ) F 1 (0) <d< F 1 (1) S l S l S l not comonotonic: π l (S,d,Λ) = [( ) ] E[X i Λ] F 1 E[X i Λ] (F S l (d)) + [ (E[Xi Λ] E [ X i Λ=F 1 Λ (F S l (d)) ]) + + ( ) E[X i Λ=λ] d + df Λ (λ) ]

32 lower bound: S l comonotonic: π l (S,d,Λ) = = E E ( ) F 1 (0) <d< F 1 (1) S l S l S l not comonotonic: π l (S,d,Λ) = improved upper bound: π u (S,d,Λ) = + E [( ) ] E[X i Λ] F 1 E[X i Λ] (F S l (d)) + [ (E[Xi Λ] E [ X i Λ=F 1 Λ (F S l (d)) ]) + + ( ) F 1 S u Λ=λ (0) <d<f 1 S u Λ=λ (1) ( ) E[X i Λ=λ] d + df Λ (λ) [( ( X i F 1 X i Λ=λ FS u Λ=λ(d) )) ] Λ=λ df Λ (λ) + ]

33 Upper bounds based on the lower bound plus an error term basedontheideasofrogers and Shi (1995) : 0 E [ ] 1 [ ] E[Y + Z] E[Y Z] + 2 E Var(Y Z) Y := S d and Z := Λ 0 E [ ] E[(S d) + Λ] (S l 1 [ ] d) + 2 E Var(S Λ) π eub (S,d,Λ) = π l (S,d,Λ) E [ Var(S Λ) ] [ ] E Var(S Λ) [ ( =E E [ S 2 Λ ] (E [S Λ]) 2) ] 1/2 1/2 =E E[X i X j Λ] ( S l) 2 j=1

34 Decomposition of the stop-loss premium if d Λ such that Λ d Λ S d (or such that Λ d Λ S d) E[(S d) + Λ] = E[S d Λ] = (S l d) +

35 Decomposition of the stop-loss premium if d Λ such that Λ d Λ S d (or such that Λ d Λ S d) E[(S d) + Λ] = E[S d Λ] = (S l d) + π(s,d) = dλ not = I 1 + I 2 E[(S d) + Λ=λ]dF Λ (λ)+ + d Λ E[S d Λ =λ]df Λ (λ) I 2 = + d Λ E[X i Λ=λ]dF Λ (λ) d(1 F Λ (d Λ ))

36 Decomposition of the stop-loss premium if d Λ such that Λ d Λ S d (or such that Λ d Λ S d) E[(S d) + Λ] = E[S d Λ] = (S l d) + π(s,d) = dλ not = I 1 + I 2 E[(S d) + Λ=λ]dF Λ (λ)+ + d Λ E[S d Λ =λ]df Λ (λ) I 2 = + d Λ I 1 I upp 1 := E[X i Λ=λ]dF Λ (λ) d(1 F Λ (d Λ )) dλ E[(S u d) + Λ=λ]dF Λ (λ) partially exact/comonotonic upper bound: π pecub (S,d,Λ) = I upp 1 + I 2

37 I 1 I low 1 := = dλ dλ (E[S Λ=λ] d) + df Λ (λ) ( ) E[X i Λ=λ] d df Λ (λ) upper bound based on lower bound dependent on retention 0 I 1 I low = 1 2 dλ dλ 1 ( E[(S d)+ Λ=λ] (E [S Λ=λ] d) + ) dfλ (λ) (Var (S Λ=λ)) 1 2 df Λ (λ) (1) 1 2 ( E [ Var (S Λ) 1{Λ<dΛ }])1 2 ( E [ 1{Λ<dΛ }])1 2 ε(d Λ ) + π deub (S,d,Λ) = π l (S,d,Λ) + ε(d Λ ) Note: π deub (S,d,Λ) = π eub (S,d) for d Λ =+ in (1)

38 Application to life annuities with stochastic interest rates stop-loss premium of a sum of lognormal random variables with a stochastic time K x horizon K x : S x = α i e Z i with S k = π(s x,d)=e [ (S x d) + ] =EKx [E = = [( Kx α i e Z i d ) K x ]] + [( k ) Pr[K x = k]e α i e Z i d k=1 ω x k=1 k q x π(s k,d), k α i e Z i (deterministic time horizon!) + ]

39 π(s x,d)=e [ ω x ] (S x d) + = k=1 k q x π(s k,d) π bound (S x,d,(λ)) = ω x k=1 π emub (S x,d,λ) =π l (S x,d,λ)+ π min (S,d,Λ) = ω x k=1 k q x π bound (S k,d,(λ k )) ω x k=1 k q x min ( 1 [ ] ) 2 E Var(Sk Λ k ),ε(d Λk ) k q x min ( π c (S k,d),π u (S k,d,λ k ),π pecub (S k,d,λ k ),π emub (S k,d,λ k ) )

40 Lemma: LetX be a lognormal random variable of the form αe Z with Z N(E[Z],σ Z ) and α R. Then the stop-loss premium with retention d equals for αd > 0 where π(x, d) = sign(α)e µ+σ2 2 Φ(sign(α)b 1 ) dφ(sign(α)b 2 ), µ =ln α +E[Z] σ = σ Z b 1 = µ + σ2 ln d σ b 2 = b 1 σ

41 Lemma: LetX be a lognormal random variable of the form αe Z with Z N(E[Z],σ Z ) and α R. Then the stop-loss premium with retention d equals for αd > 0 where π(x, d) = sign(α)e µ+σ2 2 Φ(sign(α)b 1 ) dφ(sign(α)b 2 ), µ =ln α +E[Z] σ = σ Z b 1 = µ + σ2 ln d σ b 2 = b 1 σ comonotonic upper bound: π c (S k,d)= k α i e E[Z i]+ σ2 Zi [ ] 2 Φ sign(α i )σ Zi Φ 1 (F S c k (d)) d ( ) 1 F S c k (d) where F S c k (d) can be obtained by solving: k α i e E[Z i]+sign(α i )σ Zi Φ 1 (F S c (x)) k = x

42 Choice of the conditioning and decomposition variable conditioning random variable: Λ k = k γ i Z i

43 Choice of the conditioning and decomposition variable conditioning random variable: Λ k = (a) Taylor-based : γ i = α i e E[Z i] k γ i Z i Λ k linear transformation of first order approximation to S k Λ k d Λk = S k d d Λk = d k α i e E[Zi] (1 E[Z i ])

44 Choice of the conditioning and decomposition variable conditioning random variable: Λ k = (a) Taylor-based : γ i = α i e E[Z i] k γ i Z i Λ k linear transformation of first order approximation to S k Λ k d Λk = S k d d Λk = d k α i e E[Zi] (1 E[Z i ]) (b) Maximum variance : γ i = α i E [ e i] Z = αi e E[Z i]+ 1 2 σ2 Z i maximization of first order approximation for Var(S l k ): ( 2 k k Var(S l k) Corr[ α i E[e Z i ]Z i, Λ k ]) Var[ α i E[e Z i ]Z i ]

45 Choice of the conditioning and decomposition variable conditioning random variable: Λ k = (a) Taylor-based : γ i = α i e E[Z i] k γ i Z i Λ k linear transformation of first order approximation to S k Λ k d Λk = S k d d Λk = d k α i e E[Zi] (1 E[Z i ]) (b) Maximum variance : γ i = α i E [ e i] Z = αi e E[Z i]+ 1 2 σ2 Z i maximization of first order approximation for Var(S l k ): ( 2 k k Var(S l k) Corr[ α i E[e Z i ]Z i, Λ k ]) Var[ α i E[e Z i ]Z i ] k Λ k d Λk = S k d d Λk = d α i E[e Z i ] (1 E[Z i ] 12 ) σ2zi

46 Choice of the conditioning and decomposition variable conditioning random variable: Λ k = k γ i Z i basedonthestandardized logarithm of the geometric average G =( k S k) 1/k as in Nielsen and Sandman (2002) Λ k = ln G E[ln G] Var[ln G] = k (Z i E[Z i ]) Var( k Z i) Λ k d Λk = S k d d Λk = k ln ( d k) k (E[Z i]+ln(α i )) Var( k Z i)

47 Numerical illustration mortality process: Makeham s model the number of persons alive at age x l x = as x g cx (a >0, 0 <s<1, 0 <g<1 and c>1) with parameters (l 0 = ) for men: a = , s = , g = , c = discounting process: Brownian motion e Z i = e Y (i) := e (Y 1+ +Y i ) yearly returns Y i are i.i.d. normally distributed with mean µ =0.07 and volatility σ =0.1 Y (i) 0.07i +0.1B(i)

48 consider a whole life annuity on a life (65) (male) with yearly unit payments E [( ) ] S policy 65 d + = ω 65 k=1 k q 65 E [ ( Sk d ) + S k = S k with α i =1and Z i = Y (i) (i =1,...,k) Monte Carlo estimates based on simulations ] d LB MC (s.e ) MIN EMUB PECUB ICUB CUB (8.49) (5.48) (0.51) (0.19) (0.01) (0.002)

49 consider a portfolio of N 0 homogeneous life annuity contracts for which the future lifetimes of the insureds T x (1),T x (2),...,T (N 0) x are assumed to be independent for sufficiently large N 0 (Law of Large Numbers) ω x ω x E N i e Y (i) d = E N 0 N i e Y (i) d N 0 N ω x N 0 E ip x e Y (i) d, N 0 where N i denotes the number of survivals after the i-th year for large portfolios of life annuities it suffices to compute stop-loss premiums of an average portfolio S average x : α i = i p x S average x = (i =1,..., ω x ) ω x Y (i) ip x e +

50 consider S average 65 (male persons) Monte Carlo estimates based on simulations d LB MC (s.e ) EMUB PECUB ICUB CUB (0.37) (0.13) (0.035) Remark: E[S average x ]=E[S policy x ] S average x cx S policy x E[(S average x d) + ] E[(S policy x d) + ] d >0

51 Conclusions methodology of estimating stop-loss premiums of strongly dependent random variables

52 Conclusions methodology of estimating stop-loss premiums of strongly dependent random variables efficient way to obtain precise price intervals of the stop-loss premiums of life annuities

53 Conclusions methodology of estimating stop-loss premiums of strongly dependent random variables efficient way to obtain precise price intervals of the stop-loss premiums of life annuities application to an average portfolio when mortality risk is assumed to be fully diversified

54 Conclusions methodology of estimating stop-loss premiums of strongly dependent random variables efficient way to obtain precise price intervals of the stop-loss premiums of life annuities application to an average portfolio when mortality risk is assumed to be fully diversified application to single life annuity decompose value of stop-loss premium by conditioning apply best (smallest) upper bound on each of components separately

55 Conclusions methodology of estimating stop-loss premiums of strongly dependent random variables efficient way to obtain precise price intervals of the stop-loss premiums of life annuities application to an average portfolio when mortality risk is assumed to be fully diversified application to single life annuity decompose value of stop-loss premium by conditioning apply best (smallest) upper bound on each component separately numerical illustrations: LB very accurate decomposition significantly improves UB

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