Risk Measures, Stochastic Orders and Comonotonicity

Transcription

1 Risk Measures, Stochastic Orders and Comonotonicity Jan Dhaene Risk Measures, Stochastic Orders and Comonotonicity p. 1/50

2 Sums of r.v. s Many problems in risk theory involve sums of r.v. s: S = X 1 + X X n. Standard techniques for (approximate) evaluation of the d.f. of S are: convolution, moment-based approximations, De Pril s recursion, Panjer s recursion. Assuming independence of the X i is convenient but not always appropriate. The copula approach: (Frees & Valdez, 1998) Pr [X 1 x 1,,X n x n ]=C [F X1 (x 1 ),...,F Xn (x n )]. Risk Measures, Stochastic Orders and Comonotonicity p. 2/50

3 Sums of r.v. s Problem to solve: Determine risk measures related to S = X 1 + X X n in case F Xi (x) known, C complicated / unknown, positive dependence. How to solve? Derive stochastic lower and upper bounds for S: S l S S u. Approximate R [S] by R [S u ] or R [ S l]. Involves risk measurement, stochastic orderings, choice under risk. Risk Measures, Stochastic Orders and Comonotonicity p. 3/50

4 Example: A life annuity Consider a life annuity a x on (x) with present value S = X 1 + X X n, where X j = { 0 : T (x) j, v j : T (x) >j. All X j are increasing functions of T (x) = (X 1,X 2,,X n ) is comonotonic. Risk Measures, Stochastic Orders and Comonotonicity p. 4/50

5 Example: A risk sharing scheme Consider a loss S 0 that is covered by n parties: S = X 1 + X X n. X j is the layer with deductible a j and maximal payment (a j+1 a j ): X j = with a 0 =0and a n+1 =. The layers X j are comonotonic. 0 : 0 S a j S a j : a j <S a j+1 a j+1 a j : S>a j+1, Many risk sharing schemes lead to partial risks that are comonotonic. Risk Measures, Stochastic Orders and Comonotonicity p. 5/50

6 Example: A portfolio of pure endowments Consider a portfolio (X 1,X 2,,X n ) of m-year pure endowment insurances. X j : claim amount of policy j at time m: X j = { 0 : T (x) m, 1 : T (x) >m. Assumption: all X i are i.i.d. and X i d = X. Risk Measures, Stochastic Orders and Comonotonicity p. 6/50

7 Example: A portfolio of pure endowments Let S =(X 1 + X X n ) Y, where Y is the stochastic discount factor over [0,m]. Assumption: X i and Y are mutually independent. Risk pooling reduces the actuarial risk, not the financial risk: Var [ ] S n = Var[X] n E[Y 2 ] + (E[X]) 2 Var[Y ] (E[X]) 2 Var[Y ]. The terms X i Y are conditionally comonotonic. Risk Measures, Stochastic Orders and Comonotonicity p. 7/50

8 Example: A provision for future liabilities Consider the liability cash-flow stream (α 1, 1), (α 2, 2),...,(α n,n). The provision is invested such that it generates a cumulative log-return Y (i) over the period (0,i). The provision is determined as R [S] with S = n α i e Y (i). i=1 The cumulative returns Y (i) are locally quasi-comonotonic. Illustration: i.i.d. normal yearly returns. Risk Measures, Stochastic Orders and Comonotonicity p. 8/50

9 Cumulative returns accumulated interest rate year Risk Measures, Stochastic Orders and Comonotonicity p. 9/50

10 Cumulative returns accumulated interest rate year Risk Measures, Stochastic Orders and Comonotonicity p. 9/50

11 Comonotonicity Definition: (X 1,,X n ) is comonotonic if there exists a r.v. Z and increasing functions f 1,,f n such that (X 1,,X n ) d =(f 1 (Z),,f n (Z)). Determining the d.f. of (X 1,,X n ) is a one-dimensional problem. Comonotonicity is very strong positive dependency structure. Adding comonotonic r.v. s produces no diversification: If all X i are identically distributed and comonotonic, then X X n n d = X 1. Risk Measures, Stochastic Orders and Comonotonicity p. 10/50

12 An example of comonotonic r.v. s Consider (X, Y, Z) with X Uniform on (0, 1 2 ) ( 1, 3 2 Y Beta (2,2) Z Normal (0,1). (X, Y, Z) mutually independent ) Risk Measures, Stochastic Orders and Comonotonicity p. 11/50

13 An example of comonotonic r.v. s Consider (X, Y, Z) with X Uniform on (0, 1 2 ) ( 1, 3 2 Y Beta (2,2) Z Normal (0,1). (X, Y, Z) comonotonic ) Risk Measures, Stochastic Orders and Comonotonicity p. 11/50

14 Sums of comonotonic r.v. s Notation: (X c 1,...,Xc n) is comonotonic and has same marginals as (X 1,...,X n ). S c = X c 1 + Xc Xc n. Quantiles of S c : F 1 S c (p) = n i=1 F 1 X i (p). Distribution function of S c : n i=1 F 1 X i [F S c(x)] = x. Risk Measures, Stochastic Orders and Comonotonicity p. 12/50

15 Sums of comonotonic r.v. s Stop-loss premiums of S c : (Dhaene, Wang, Young, Goovaerts, 2000) E[S c d] + = n E [ ] (X i d i ) + i=1 with d i = F 1 X i [F S c(d)]. Application: (Jamshidian, 1989) In the Vasicek (1977) model, the price of a European call option on a coupon bond = sum of the prices of European options on zero coupon bonds. Risk Measures, Stochastic Orders and Comonotonicity p. 13/50

16 Theories of choice under risk Wealth level vs. probability level: E[X] = 1 0 F 1 X (1 q) dq. Utility theory: (von Neumann & Morgenstern, 1947) E[u(X)] = 1 0 u [ F 1 X (1 q))] dq with u(x) a utility function. Dual theory of choice under risk: (Yaari, 1987) ρ f [X] = 1 0 F 1 X (1 q) df (q) with f(q) a distortion function. Risk Measures, Stochastic Orders and Comonotonicity p. 14/50

17 Theories of choice under risk Choice under risk: Prefer wealth Y over wealth X if E[u(X)] E[u(Y )]. Prefer wealth Y over wealth X if Additivity: ρ f [X] ρ f [Y ]. If u(0) = 0 and Pr [X 0,Y 0]=0, then E[u(X + Y )] = E [u(x)] + E [u(y )]. If X and Y are comonotonic, then ρ f [X + Y ]=ρ f [X]+ρ f [Y ]. Risk Measures, Stochastic Orders and Comonotonicity p. 15/50

18 Risk measures Definition: = mapping from the set of quantifiable losses to the real line: X R [X]. Have been investigated extensively in the literature: Huber (1981): upper expectations, Goovaerts, De Vylder & Haezendonck (1984): premium principles, Artzner, Delbaen, Eber & Heath (1999): coherent risk measures. Risk Measures, Stochastic Orders and Comonotonicity p. 16/50

19 Construction of risk measures The equivalent expected utility principle: u(w) =E[u(w + R [X] X)]. The equivalent distorted expectation principle: ρ f [w] =ρ f [w + R [X] X]. This leads to distortion risk measures (Wang, 1996): R [X] =ρ g [X] with g(q) =1 f(1 q). Risk Measures, Stochastic Orders and Comonotonicity p. 17/50

20 Distortion risk measures 1 II F (x) X E[X] = I II I 0 x Risk Measures, Stochastic Orders and Comonotonicity p. 18/50

21 Distortion risk measures 1 II II' F (x) X g(f (x)) X E[X] = I (II+II') ρ [X] = (I+I') II E[X] g I I' 0 x Risk Measures, Stochastic Orders and Comonotonicity p. 18/50

22 Examples of distortion risk measures Value-at-Risk: X VaR p [X] =F 1 X (p) =Q p[x]. g(x) =I (x >1 p), 0 x 1. 1 g(x) 0 1 p 1 x Risk Measures, Stochastic Orders and Comonotonicity p. 19/50

23 Examples of distortion risk measures Tail Value-at-Risk: X TVaR p (X) = 1 1 p g(x) =min ( ) x 1 p, 1, 0 x 1. 1 p VaR q [X] dq. 1 g(x) 0 1 p 1 x Risk Measures, Stochastic Orders and Comonotonicity p. 20/50

24 Concave distortion risk measures ρ g [.] is a concave distortion risk measure if g is concave. TVaR p is concave, VaR p not. Concave distortion risk measures are subadditive: ρ g [X + Y ] ρ g [X]+ρ g [Y ]. Optimality of TVaR p : TVaR p [X] =min{ρ g ([X] g is concave and ρ g VaR p }. Optimality of VaR p : (Artzner et al. 1999) VaR p [X] =inf{ρ([x] ρ is coherent and ρ VaR p }. Risk Measures, Stochastic Orders and Comonotonicity p. 21/50

25 Optimality of VaR p Consider a loss X and a solvency capital requirement R [X]. Measuring the insolvency risk: How to choose R[X]? (X R[X]) + E [ (X R[X]) + ]. E [ (X R[X]) + ] should be small choose R[X] large enough. Capital has a cost R[X] should be small enough. Risk Measures, Stochastic Orders and Comonotonicity p. 22/50

26 Optimality of VaR p The optimal capital requirement: R[X] is determined as the minimizer (with respect to d) of E[(X d) + ]+dε, 0 <ε<1. Solution: R[X] =VaR 1 ε [X]. The minimum is given by ε TVaR 1 ε [X]. Geometric proof (for VaR 1 ε [X] > 0): Risk Measures, Stochastic Orders and Comonotonicity p. 23/50

27 Optimality of VaR p E[(X d) + ]+dε with d = Q 1 ε [X] Risk Measures, Stochastic Orders and Comonotonicity p. 24/50

28 Optimality of VaR p E[(X d) + ]+dε with d<q 1 ε [X] Risk Measures, Stochastic Orders and Comonotonicity p. 24/50

29 Optimality of VaR p E[(X d) + ]+dε with d>q 1 ε [X] Risk Measures, Stochastic Orders and Comonotonicity p. 24/50

30 Can a risk measure be too subadditive? (Dhaene, Laeven, Vanduffel, Darkiewicz, Goovaerts, 2005) For losses X and Y, we have that E[(X + Y R[X] R[Y ]) + ] E[(X R[X]) + ]+E[(Y R[Y ]) + ]. Splitting increases the insolvency risk the risk measure used to determine the required solvency capital should be subadditive enough. Merging decreases the insolvency risk subadditivity of the capital requirement is allowed to some extent. the capital requirement can be too subadditive if no constraint is imposed on the subadditivity. Risk Measures, Stochastic Orders and Comonotonicity p. 25/50

31 Can a risk measure be too subadditive? The regulator s condition: E [ (X + Y R[X + Y ]) + ] + ε R[X + Y ] E[(X R[X]) + ]+E[(X R[X]) + ] + ε (R[X]+R[Y ]) VaR 1 ε [ ] fulfills the regulator s condition. Any subadditive R[ ] VaR 1 ε [ ] fulfills the regulator s condition. Mark(ovitz), 1959: We might decide that in one context one basic set of principles is appropriate, while in another context a different set of principles should be used. Risk Measures, Stochastic Orders and Comonotonicity p. 26/50

32 Can a risk measure be too subadditive? The regulator s condition: E [ (X + Y R[X + Y ]) + ] + ε R[X + Y ] E[(X R[X]) + ]+E[(X R[X]) + ]+ε (R[X]+R[Y ]) VaR 1 ε [ ] fulfills the regulator s condition. Any subadditive R[ ] VaR 1 ε [ ] fulfills the regulator s condition. Mark(ovitz), 1959: We might decide that in one context one basic set of principles is appropriate, while in another context a different set of principles should be used. Risk Measures, Stochastic Orders and Comonotonicity p. 26/50

33 Stochastic orderings - Upper and lower tails E[(X d) + ]= surface above the d.f., from d on. E[(d X) + ]= surface below the d.f., from to d. Risk Measures, Stochastic Orders and Comonotonicity p. 27/50

34 Stochastic orderings - Upper and lower tails E[(X d) + ]= surface above the d.f., from d on. E[(d X) + ]= surface below the d.f., from to d F (x ) X d E[(X d ) ] + Risk Measures, Stochastic Orders and Comonotonicity p. 27/50

35 Stochastic orderings - Upper and lower tails E[(X d) + ]= surface above the d.f., from d on. E[(d X) + ]= surface below the d.f., from to d E[(d X ) ] + d F (x ) X Risk Measures, Stochastic Orders and Comonotonicity p. 27/50

36 Convex order Definition: X cx Y any tail of Y exceeds the respective tail of X. Represents common preferences of risk averse decision makers between r.v. s with equal means. Characterization in terms of distortion risk measures: (Wang & Young, 1998) X cx Y E[X] =E[Y ] and ρ g [X] ρ g [Y ] for all concave g. Risk Measures, Stochastic Orders and Comonotonicity p. 28/50

37 Stochastic order bounds for sums of dependent r.v. s Theorem: (Kaas et al., 2000) For any (X 1,,X n ) and any Λ, we have that n i=1 E[X i Λ] cx n i=1 X i cx n i=1 X c i Notation: S l cx S cx S c. Assume that all E[X i Λ] are functions of Λ S l is a comonotonic sum. Why use these comonotonic bounds? One-dimensional stochasticity. ρ g [ S l ] and ρ g [S c ] are easy to calculate. If g is concave, then ρ g [ S l ] ρ g [S] ρ g [S c ]. Risk Measures, Stochastic Orders and Comonotonicity p. 29/50

38 On the choice of Λ (Vanduffel et al., 2004) Let n S = α i e Y (i) and S l = i=1 n α i E i=1 [ ] e Y (i) Λ with α i > 0 and (Y 1,,Y n ) normal. First order approximation for Var [ S l] : [ n ] [ n ] Var[S l ] Corr 2 α i E[e Y (i) ]Y (i), Λ Var α i E[e Y (i) ]Y (i). i=1 Optimal choice for Λ: Λ= n α i E i=1 [ e Y (i)] Y (i). i=1 Risk Measures, Stochastic Orders and Comonotonicity p. 30/50

39 The continuous perpetuity Local comonotonicity: Let B(τ) be a standard Wiener process. The accumulated returns exp [µτ + σb(τ)] and exp [µ (τ + τ)+σb(τ + τ)] are almost comonotonic. The continuous perpetuity: (Dufresne, 1989; Milevsky, 1997) S = 0 exp [ µτ σb(τ)] dτ has a reciprocal Gamma distribution. Risk Measures, Stochastic Orders and Comonotonicity p. 31/50

40 The continuous perpetuity Numerical illustration: µ = 0.07 and σ = Squares = (Q p [S],Q p [S c ]), Circles = (Q p [S],Q p [S l ]). Risk Measures, Stochastic Orders and Comonotonicity p. 32/50

41 An allocation problem Problem description: Consider the loss portfolio (X 1,...,X n ). How to allocate a given amount d among the n losses? Allocation rule: minimize the expected aggregate shortfall: min n i=1 d i=d E ( n i=1 [ (Xi d i ) + ] ). Risk Measures, Stochastic Orders and Comonotonicity p. 33/50

42 An allocation problem Solution of the minimization problem: Let S = X X n and S c = X1 c + + Xc n. For all d i with n i=1 d i = d, wehave E [ (S c d) + ] n E [ ] (X i d i ) +. i=1 As E [ (S c d) + ] = n E i=1 [ (Xi F 1 X i [F S c(d)] ) ], + the optimal allocation rule is given by d i = F 1 X i [F S c(d)]. Risk Measures, Stochastic Orders and Comonotonicity p. 34/50

43 Asian options (Dhaene, Denuit, Goovaerts, Kaas & Vyncke, 2002) A European style arithmetic Asian call option: {A t } = price process of underlying asset, T = exercise date, n = number of averaging dates, K = exercise price. Pay-off at T = Arbitrage-free time-0 price: ( 1 n n 1 i=0 A T i K ) + AC (n, K, T )=e δt ( E 1 n n 1 i=0 A T i K ) +, where δ = risk free interest rate and E is evaluated wrt Q. Risk Measures, Stochastic Orders and Comonotonicity p. 35/50

44 Asian options The comonotonic upper bound: AC (n, K, T ) e δt = e δt n ( E n 1 i=0 E 1 n n 1 i=0 A c T i K ) [ ( ) ] A T i F 1 A T i (F S c (nk)) + + The upper bound in terms of European calls: AC(n, K, T ) n 1 i=0 with K i = F 1 A T i (F S c (nk)). e δi n EC (K i,t i) Risk Measures, Stochastic Orders and Comonotonicity p. 36/50

45 Asian options Static super-replicating strategies: (Albrecher et al., 2005) At time 0, for i =1,...,n,,buy e δi n n 1 i=0 K i = K. European calls EC(K i,t i) with 1 n Hold these European calls until expiration. Invest their payoffs at expiration at the risk-free rate. Payoff at T : 1 n n 1 i=0 Price at time 0: 1 n (A T i K i ) + n 1 i=0 ( 1 n n 1 i=0 A T i K e δi EC(K i,t i) AC(n, K, T ) ) + Risk Measures, Stochastic Orders and Comonotonicity p. 37/50

46 Asian options The cheapest super-replicating strategy: The price 1 n n 1 i=0 e δi EC(K i,t i) of the super-replicating strategy is minimized for K i = F 1 A T i (F S c(nk)). The optimal strategy corresponds to the comonotonic upper bound. Remarks: Similarly, comonotonic bounds can be derived for basket options (Deelstra et al., 2004). The Ki can be determined from the European call prices observed in the market. The model-free approach can be generalized to the case of a finite number of exercise prices (Hobson et al., 2005). Risk Measures, Stochastic Orders and Comonotonicity p. 38/50

47 Asian options Numerical illustration in a Black & Scholes setting: Risk-free interest rate = e δ 1=9%per year, {A t } : geometric Brownian motion with A 0 = 100 and volatility per year σ =0.2, n = 10 days, T = day 120. K LB MC (s.e ) UB (0.85) (0.81) (0.75) (0.59) (0.33) Risk Measures, Stochastic Orders and Comonotonicity p. 39/50

48 Strategic portfolio selection (Dhaene, Vanduffel, Goovaerts, Kaas & Vyncke, 2005) Provisions for future liabilities: α 1,α 2,..., α n : positive payments, due at times 1, 2,..., n. R = initial provision to be established at time 0. Risk Measures, Stochastic Orders and Comonotonicity p. 40/50

49 Strategic portfolio selection (Dhaene, Vanduffel, Goovaerts, Kaas & Vyncke, 2005) Provisions for future liabilities: α 1,α 2,..., α n : positive payments, due at times 1, 2,..., n. R = initial provision to be established at time 0. R 0 reserve at time 0 Risk Measures, Stochastic Orders and Comonotonicity p. 40/50

50 Strategic portfolio selection (Dhaene, Vanduffel, Goovaerts, Kaas & Vyncke, 2005) Provisions for future liabilities: α 1,α 2,..., α n : positive payments, due at times 1, 2,..., n. R = initial provision to be established at time 0. R α 1 α 2 α consumptions at times 1, 2,... Risk Measures, Stochastic Orders and Comonotonicity p. 40/50

51 Strategic portfolio selection Investment strategy i, (i =1,,n): Yearly returns: (Y (i) 1,,Y(i) n ). The stochastic provision: S (i) = n j=1 α j e (Y (i) 1 +Y (i) 2 + +Y (i) j ). The provision principle: Available provision at time j: R (i) 0 = ρ g [S (i)]. R (i) j = R (i) j 1 ey (i) j α j. Risk Measures, Stochastic Orders and Comonotonicity p. 41/50

52 Strategic portfolio selection The optimal investment strategy: (i,r0 ) follows from R 0 =min i [ R (i) 0 =minρ g S (i)]. i Avoid simulation by considering comonotonic approximations for S (i). Example: the quantile provision principle: R (i) 0 = Q p [S (i)] =inf { ( ) R 0 Pr R n (i) 0 } p. Risk Measures, Stochastic Orders and Comonotonicity p. 42/50

53 Strategic portfolio selection: numerical example The Black-Scholes framework: 1 riskfree asset: δ = risky assets: ( µ (1),σ (1)) =(0.06, 0.10) ( µ (2),σ (2)) =(0.10, 0.20) with Corr [ ] Y (1) k,y (2) k =0.5 Constant mix strategies: π =(π 1,π 2 ) π i = (time-independent) fraction invested in risky asset i, 1 2 i=1 π i = fraction invested in riskfree asset. Risk Measures, Stochastic Orders and Comonotonicity p. 43/50

54 Strategic portfolio selection: numerical example Yearly consumptions: α 1 =...= α 40 =1. Stochastic provision: S (π) = 40 i=1 e (Y 1(π)+Y 2 (π)+ +Y i (π)). Optimal investment strategy: R 0 =min π Q p [S (π)]. Approximation: [ R 0 =minq p S α ( απ (t))] [ ( min Q p S l απ (t))]. α with π (t) = ( 5 9, 4 9) and α = proportion invested in π (t). Risk Measures, Stochastic Orders and Comonotonicity p. 44/50

55 Strategic portfolio selection: numerical example minimal reserve optimal risky proportion p Solid line (left scale): minimal initial provision R0 l as a function of p. Dashed line (right scale): optimal proportion invested in π (t), as a function of p. Risk Measures, Stochastic Orders and Comonotonicity p. 45/50

56 Generalizations Provisions for random future liabilities: Goovaerts et al. (2000), Hoedemakers et al. (2003, 2005), Ahcan et al. (2004). The final wealth problem : Dhaene et al. (2005). Stochastic sums: Hoedemakers et al. (2005). Positive and negative payments: Vanduffel et al. (2005). Other distributions: Albrecher et al. (2005), Valdez et al. (2005). Risk Measures, Stochastic Orders and Comonotonicity p. 46/50

57 References ( Ahcan, A.; Darkiewicz, G.; Goovaerts, M.J.; Hoedemakers, T. (2004). Computation of convex bounds for present value functions of random payments. To appear. Albrecher, H.; Dhaene, J.; Goovaerts, M.J.; Schoutens, W. (2005). Static hedging of Asian options under Lévy models: The comonotonicity approach. The Journal of Derivatives 12(3), Artzner, Ph.; Delbaen, F.; Eber, J.M.; Heath, D. (1999). Coherent measures of risk, Mathematical Finance 9, Deelstra, G.; Liinev, J.; Vanmaele, M. (2004). Pricing of arithmetic basket options by conditioning. Insurance: Mathematics & Economics 34, Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vyncke, D.(2002). Comonotonicity in actuarial science and finance: Applications. Insurance: Mathematics & Economics 31(2), Dhaene, J.; Laeven, R.; Vanduffel, S.; Darkiewicz, G. Goovaerts, M.J. (2005). Can a coherent risk measure be too subadditive? Submitted. Dhaene, J.; Vanduffel, S.; Goovaerts, M.J.; Kaas, R.; Vyncke, D. (2005). Comonotonic approximations for optimal portfolio selection problems. Journal of Risk and Insurance 72(2), Risk Measures, Stochastic Orders and Comonotonicity p. 47/50

58 References ( Dhaene,J.; Wang, S.; Young, V.; Goovaerts M.J. (2000). Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries 2000(2), Dufresne, D. (1990). The distribution of a perpetuity with applications to risk theory and pension funding. Scandinavian Actuarial Journal, 9, Embrechts, P.; Mc.Neil, A.; Straumann, D. (2001). Correlation and dependency in risk management: properties and pitfalls, in Risk Management: Value at Risk and Beyond, edited by Dempster, M. and Moffatt, H.K., Cambridge University Press. Frees, E.W.; Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), Goovaerts, M.J.; De Vylder, F.; Haezendonck, J. (1984). Insurance Premiums, North-Holland, Amsterdam. Goovaerts, M.J.; Dhaene, J.; De Schepper, A. (2000). Stochastic upper bounds for present value functions. Journal of Risk and Insurance, 67(1), Hobson, D.; Laurence, P.; Wang, T. (2005). Static-arbitrage upper bounds for the prices of basket options. Risk Measures, Stochastic Orders and Comonotonicity p. 48/50

59 References ( Hoedemakers, T.; Darkiewicz, G.; Goovaerts, M.J. (2005). Approximations for life annuity contracts in a stochastic financial environment. Hoedemakers, T.; Beirlant, J.; Goovaerts, M.J.; Dhaene, J. (2003). Confidence bounds for discounted loss reserves. Insurance: Mathematics & Economics, 33(2), Hoedemakers, T.; Beirlant, J.; Goovaerts, M.J.; Dhaene, J. (2005). On the distribution of discounted loss reserves using generalized linear models. Scandinavian Actuarial Journal 2005(1), Jamshidian, F. (1989). An exact bond option formula. The Journal of Finance XLIV(1), Huber, P.J. (1981). Robust Statistics, Wiley, New York. Kaas, R.; Dhaene, J.; Goovaerts, M.J. (2000). Upper and lower bounds for sums of random variables. Insurance: Mathematics & Economics 27(2), Milevsky, M.A. (1997). The present value of a stochastic perpetuity and the Gamma distribution, Insurance: Mathematics & Economics, 20(3), Valdez, E.; Dhaene, J. (2005). Convex order bounds for sums of dependent log-elliptical random variables. To appear. Risk Measures, Stochastic Orders and Comonotonicity p. 49/50

60 References ( Vanduffel, S; Dhaene, J.; Goovaerts, M.J. (2005). On the evaluation of saving-consumption plans. Journal of Pension Economics and Finance 4(1), Vanduffel, S.; Hoedemakers, T.; Dhaene, J. (2005). Comparing approximations for risk measures of sums of non-independent lognormal random variables. North American Actuarial Journal, to be published. Wang, S.; Young, V. (1998). Ordering risks: expected utility theory versus Yaari s dual theory of risk. Insurance: Mathematics & Economics 22(1), Risk Measures, Stochastic Orders and Comonotonicity p. 50/50