Optimal Allocation of Policy Limits and Deductibles

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1 Optimal Allocation of Policy Limits and Deductibles Ka Chun Cheung Tel: Fax: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada Abstract In this paper, we study the problems of optimal allocation of policy limits and deductibles. Several objective functions are considered: maximizing the expected utility of wealth assuming the losses are independent, minimizing the expected total retained loss, and maximizing the expected utility of wealth when the dependence structure is unknown. Orderings of the optimal allocations are obtained. Key words: Arrangement Increasing Functions; Stochastic order; Dependence Structure; Comonotonicity 1 Introduction Suppose that a policyholder is exposed to n random losses (risks). Through paying premium, (s)he could obtain coverage from an insurer. Two common forms of coverage are (ordinary) deductible and policy limit (c.f. Klugman et al. (2004)). In some situations, the policyholder has the right to allocate the deductibles or the policy limits among the n risks. For example, the compensation package of many big companies includes a commonly called Flexible Spending Account Program, which allows the employees allocate pre-tax dollars toward specific expenses such as health care, medical costs, or dependent care. This is essentially a form of allocating policy limits. This paper addresses the problem of finding the optimal way to allocate policy limits and deductibles. While closed-form expression of the optimal allocation is out of reach in general because of the nonlinear nature of the problem, we could still obtain useful qualitative results concerning the relative size of each allocation. 1

2 The paper is organized as follows. In Section 2, we recall some basic results about arrangement increasing functions and stochastic dominance orders. Section 3 studies the problem of optimal allocation of policy limits. Section 4 gives a parallel treatment for deductibles. Section 5 concludes the paper. 2 Preliminary In this section, we will collect some facts concerning arrangement increasing functions, stochastic dominance orders, and comonotonicity that are useful in the sequel. 2.1 Arrangement Increasing Functions Let τ be any permutation of the set {1, 2,..., n}. For any vector x = (x 1,..., x n ) R n, we use x τ to denote the permuted vector (x τ(1),..., x τ(n) ). Let x [1] x [n] denote the components of x in decreasing order and let x = (x [1],..., x [n] ) denote the decreasing rearrangement of x. Similarly, let x (1) x (n) denote the components of x in increasing order and let x = (x (1),..., x (n) ) denote the increasing rearrangement of x. Definition 1 A real-valued function g(x, λ) defined on R n R n is said to be an arrangement increasing (AI) function if 1) g is permutation invariant, i.e., g(x, λ) = g(x τ, λ τ) for any permutation τ, and 2) g exhibits permutation order, i.e. g(x, λ ) g(x, λ τ ) g(x, λ ) for any permutation τ. Property 2) means that an AI function attains its maximum value when x and λ are similarly ordered, and it attains its minimum value when they are oppositely ordered. The most prominent example of AI function is g(x, λ) = x i λ i. The fact that this is an AI function is indeed the well-known rearrangement inequality. A function φ : R 2 R is called supermodular (or L-superadditive) if φ(r + η, s) φ(r, s) is increasing in s for all r and all η > 0. The next two results are due to Hollander et al. (1977). 2

3 Lemma 1 A function g : R n R n R having the form g(x, λ) = φ(x i, λ i ) is AI if φ is supermodular. Lemma 2 A function g : R n R n R having the form is AI if ψ : R n R is Schur-concave. g(x, λ) = ψ(x λ) For a thorough and detailed account of the above concepts, we refer to Marshall and Olkin (1979). 2.2 Stochastic Dominance Orders Standard references for this subsection include Denuit et al. (2005), Kaas et al. (1994, 2001), Müller and Stoyan (2002), and Shaked and Shanthikumar (1994). Throughout this paper, all the random variables considered are defined on a common probability space (Ω, F, P). We also assume that all the expectations mentioned exist. For any random variable X, F X denotes its distribution function: F X (t) = P(X t). Definition 2 Let X and Y be two random variables. 1. Y is said to be larger than X in the usual stochastic order (resp. increasing convex order, convex order), denoted as Y st X (resp. Y icx X, Y cx X), if E[f(Y )] E[f(X)] for all increasing (resp. increasing convex, convex) function f. 2. Y is said to be larger than X in the hazard rate order, denoted as Y hr X, if 1 F Y (s) 1 F X (s) is non-decreasing in s. The following lemma gives a useful characterization of the usual stochastic order. 3

4 Lemma 3 The following statements are equivalent: (a) X st Y. (b) There exist random variables X and Ỹ, defined on a common probability space ( Ω, F, P), such that X = d X, Ỹ = d Y, and X( ω) Ỹ ( ω) for all ω Ω. Let us remark that both the usual stochastic order and the increasing convex order are closed under convolution: if X st Y (resp. X icx Y ), then X +Z st Y +Z (resp. X +Z icx Y +Z ) whenever Z is independent of X and Y. For further properties on stochastic orders, see the references aforementioned. Finally, we recall the following property of the hazard rate order, which can be founded in Müller and Stoyan (2002): Lemma 4 Let X and Y be two independent random variables. If X hr Y, then g(x, Y ) icx g(y, X) for all g : R 2 R such that both g(x, y) and g(x, y) g(y, x) are increasing in x for all x y. 2.3 Comonotonicity A subset A R n is said to be comonotonic if whenever x = (x 1,..., x n ) and y = (y 1,..., y n ) are elements of A, either x i y i for all i or y i x i for all i. A random vector X = (X 1,..., X n ) in R n is said to be comonotonic if there is a comonotonic subset A of R n such that P(X A) = 1. The intuitive meaning is that comonotonic random variables are always moving together in the same direction. The following lemma provides an equivalent characterization of comonotonicity. Lemma 5 The following statements are equivalent: (a) The random vector X = (X 1,..., X n ) is comonotonic. (b) There exist non-decreasing functions f i (i = 1,..., n), and a random variable Z, such that (X 1,..., X n ) d = (f 1 (Z),..., f n (Z)). In particular, characterization (b) implies that comonotonicity is preserved under non-decreasing transformation of each component. Let F 1,..., F n be n univariate distribution functions. We use R(F 1,..., F n ) to denote the Fréchet space of all the n-dimensional random vectors whose marginal distributions are F 1,..., F n. One of the most important results concerning comonotonic random vectors is that they are maximal within the Fréchet spaces they are lying in with respect to the convex order. 4

5 Lemma 6 If (X c,..., X c ) R(F 1,..., F n ) is comonotonic, then for any (X 1,..., X n ) R(F 1,..., F n ). X X n cx X c + + X c For more information on comonotonicity, we refer to Dhaene and Goovaerts (1996), Dhaene et al. (2000, 2002), and Kaas et al. (2000). 3 Optimal allocation of policy limits Let X 1,..., X n be n risks faced by a policyholder. Through insurance arrangement, (s)he is granted a total of $l > 0 as policy limit with which (s)he can allocate arbitrarily among the n risks. If (l 1,..., l n ) are the allocated policy limits, then we have l i 0 for all i and l l n = l. We call this n-tuple admissible and use S n (l) to denote the class of all such n-tuples. If l = (l 1,..., l n ) S n (l) is chosen, then the benefit obtained from the insurer would be (X i l i ) and hence the total retained loss of the policyholder is X i (X i l i ) = (X i l i ) Maximizing utility - independent case The first problem to be considered is to maximize the expected utility of wealth: max l Sn(l) E [u (w n (X i l i ) + )], where u is increasing and concave Problem L1: w is a fixed constant X 1,..., X n are independent The function u and the constant w represent the utility function and the wealth (after premium is paid) of the policyholder. Let ũ(x) = u(w x). Then ũ is increasing and convex, and problem L1 is equivalent to min l Sn(l) E [ũ ( n (X i l i ) + )], Problem L1 : where ũ is increasing and convex X 1,..., X n are independent 5

6 Proposition 1 Let l = (l 1,..., l n) be the solution to Problem L1, then X i hr X j = l i l j. Proof: Without losing generality, assume that i = 2, j = 1, i.e. X 2 hr X 1. Let l = (l 1,..., l n ) be any admissible allocation with l 1 l 2. Denote the objective function in Problem L1 as G, then it suffices to show that G(l) G( l), where l = (l 2, l 1, l 3,..., l n ) is obtained from l by swapping the first two components. Note that l is also admissible. Consider the following function: g(x 1, x 2 ) = (x 1 l 2 ) + + (x 2 l 1 ) +. It is easy to see that g(x 1, x 2 ) is increasing in x 1, and g(x 1, x 2 ) g(x 2, x 1 ) = (x 1 l 2 ) + + (x 2 l 1 ) + (x 2 l 2 ) + (x 1 l 1 ) + is also increasing in x 1. By Lemma 4, we have g(x 2, X 1 ) icx g(x 1, X 2 ), that is, (X 1 l 1 ) + + (X 2 l 2 ) + icx (X 1 l 2 ) + + (X 2 l 1 ) + and hence (X 1 l 1 ) + + (X 2 l 2 ) + + (X i l i ) + icx (X 1 l 2 ) + + (X 2 l 1 ) + + (X i l i ) + because icx is closed under convolution. Since ũ is increasing and convex, [ ( )] G(l) = E ũ (X 1 l 1 ) + + (X 2 l 2 ) + + (X i l i ) + [ ( E ũ (X 1 l 2 ) + + (X 2 l 1 ) + + )] (X i l i ) + = G( l), which is the desired inequality. This proposition means that if the size of a certain risk is large, then we should allocate a larger policy limit to it, and vice versa. 3.2 Minimizing expected retained loss Instead of maximizing the utility, the policyholder may consider minimizing the expected total retained loss: [ Problem L2: min E l S n(l) 6 (X i l i ) + ]

7 Notice that we do not make any assumption on the dependence structure of X 1,..., X n (for example, independence) because it plays no role in Problem L2. Problem L2 is a special case of Problem L1 with ũ(x) = x, and hence X i hr X j = l i l j is still valid. However, we can show that the condition involving the hazard rate order can be weakened to the usual stochastic order. Proposition 2 Let l = (l 1,..., l n) be the solution to Problem L2, then X i st X j = l i l j. We first prove a lemma. Lemma 7 The function g : R n R n R defined by g(x, λ) = (x 1 λ i ) + is an AI function. Proof: By the famous Hardy-Littlewood-Pólya theorem (Hardy et al. (1929)), the function R n x ψ(x) = h(x i ) is Schur-concave whenever h is concave. In particular, take h(x) = (x) +, then g(x, λ) = h(x i λ i ) = ψ(x λ) is AI by Lemma 2. Proof of Proposition 2: Again, assume that i = 2, j = 1. Let l and l be the two admissible allocations as defined in the proof of Proposition 1, and H the objective function in Problem L2. We need to show that X 2 st X 1 = H(l) H( l). By Lemma 3, X 2 st X 1 implies that there exist X 1 and X 2 defined on some other probability space ( Ω, F, P) with X 1 d = X 1, X2 d = X 2, and X 2 ( ω) X 1 ( ω) for all ω Ω. Hence H( l) H(l) = E [(X 1 l 2 ) + + (X 2 l 1 ) + (X 1 l 1 ) + (X 2 l 2 ) + ] = Ẽ X 1 l 2 ) + + ( X 2 l 1 ) + ( X 1 l 1 ) + ( X ] 2 l 2 ) + 0, where the last inequality follows from Lemma 7 because X 2 X 1 and l 2 l 1. 7

8 3.3 Maximizing utility - unknown dependence structure In Problem L1, it is assumed that X 1,..., X n are independent. Now we consider the case where the dependence structure is unknown: the only information available is the marginal distribution of each X i. Following Cheung (2006), we employ the maximin formulation: first identify the worst dependence structure, then solve the maximization problem as if this worst dependence structure were the actual one. This formulation represents a conservation attitude toward the uncertainty in the dependence structure. Let F 1,..., F n be the distributions of X 1,..., X n, and R the corresponding Fréchet space. Problem L1 is modified to max l Sn(l) min (X1,...,X n) R E [u (w n (X i l i ) + )], Problem L3: where u is increasing and concave w is a fixed constant Using the transformation ũ(x) = u(w x), Problem L3 is equivalent to min l Sn(l) max (X1,...,X n) R E [ũ( n Problem L3 : (X i l i ) + )], where ũ is increasing and convex We first identify the dependence structure that solves the max part in Problem L3. Lemma 8 Suppose that ũ is increasing and convex. If (X c,..., X c ) R is comonotonic, then [ ] [ ] E ũ( (X i l i ) + ) E ũ( (Xi c l i ) + ) for any (X 1,..., X n ) R and any (l 1,..., l n ) S n (l). Proof: Since the map x (x l i ) + is non-decreasing for each i, the comonotonicity of (X c,..., X c ) implies the comonotonicity of ((X1 c l 1 ) +,..., (Xn l c n ) + ). For any (X 1,..., X n ) R, ((X 1 l 1 ) +,..., (X n l n ) + ) and ((X1 c l 1 ) +,..., (Xn c l n ) + ) belong to the same Fréchet space, and hence (X i l i ) + cx (Xi c l i ) + by Lemma 6. The result follows from the definition of the convex order. This lemma tells us that the worst dependence structure among X 1,..., X n is comonotonicity, i.e. when the risks are moving together in the same direction. When the exact dependence structure is not known, assuming comonotonicity is not totally a wild guess. Very often, the 8

9 risks a policyholder is facing would exhibit a certain degree of positive dependence. In our Flexible Spending Account example, expenses such as health care, medical costs would by and large move in the same direction: if one of them is large, then most likely the other one would also be large, and vice versa. Modeling the risks to be comonotonic would then be quite realistic. From this lemma, Problem L3 becomes min l Sn(l) E [ũ( n (X i l i ) + )], Problem L3 : where ũ is increasing and convex (X 1,..., X n ) is comonotonic Proposition 3 Let l = (l 1,..., l n) be the solution to Problem L3, then X i st X j = l i l j. Proof: Assume that i = 2, j = 1. Let l and l be the two admissible allocations as defined in the proof of Proposition 1, and I the objective function in Problem L3. We need to show that X 2 st X 1 = I(l) I( l). Since (X 1, X 2 ) is comonotonic, X 2 st X 1 implies that X 2 (ω) X 1 (ω) for all ω (c.f. Cheung (2006)). As g(x, λ) = n (x 1 λ i ) + is an AI function by Lemma 7, we have (X 1 l 1 ) + + (X 2 l 2 ) + + (X i l i ) + (X 1 l 2 ) + + (X 2 l 1 ) + + (X i l i ) + on Ω because we have assumed that l 2 l 1, and hence ( ) ũ (X 1 l 1 ) + + (X 2 l 2 ) + + (X i l i ) + ũ ( (X 1 l 2 ) + + (X 2 l 1 ) + + ) (X i l i ) + because ũ is increasing. Taking expectation on both sides yields the desired result. 4 Optimal allocation of deductibles The situation studied in this section is similar to that we study in Section 3, but the focus is shifted from policy limits to deductibles. Suppose that a policyholder is exposed to n risks 9

10 X 1,..., X n. (S)he needs to decide how to allocate a total deductible of $d > 0 to these risks. If (d 1,..., d n ) are the allocated deductibles, then we have d i 0 for all i and d d n = d. We call this n-tuple admissible and use S n (d) to denote the class of all such n-tuples. If d = (d 1,..., d n ) S n (d) is chosen, then the benefit obtained from the insurer would be (X i d i ) + and hence the total retained loss of the policyholder is X i (X i d i ) + = (X i d i ). 4.1 Maximizing utility - independent case We first consider the problem of maximizing the utility of wealth: max d Sn(d) E [u (w n (X i d i ))], where u is increasing and concave Problem D1: w is a fixed constant X 1,..., X n are independent The interpretations for the function u and the constant w are the same as before. As demonstrated above, problem D1 is equivalent to min d Sn(d) E [ũ ( n (X i d i ))], Problem D1 : where ũ is increasing and convex X 1,..., X n are independent Proposition 4 Let d = (d 1,..., d n) be the solution to Problem D1, then X i hr X j = d i d j. Proof: Assume that i = 2, j = 1, i.e. X 2 hr X 1. Let d = (d 1,..., d n ) be any admissible allocation with d 2 d 1. Denote the objective function in Problem D1 as J, then it suffices to show that J(d) J( d), where d = (d 2, d 1, d 3,..., d n ) is obtained from d by swapping the first two components. Note that d is also admissible. Consider the function g(x 1, x 2 ) = (x 1 d 1 ) + (x 2 d 2 ). It is easy to see that g(x 1, x 2 ) is increasing in x 1, and g(x 1, x 2 ) g(x 2, x 1 ) = (x 1 d 1 ) + (x 2 d 2 ) (x 1 d 2 ) (x 2 d 1 ) 10

11 is also increasing in x 1 becasuse d 2 d 1. By Lemma 4, we have g(x 2, X 1 ) icx g(x 1, X 2 ), that is, (X 1 d 2 ) + (X 2 d 1 ) icx (X 1 d 1 ) + (X 2 d 2 ), and hence (X 1 d 2 ) + (X 2 d 1 ) + (X i d i ) icx (X 1 d 1 ) + (X 2 d 2 ) + because icx is closed under convolution. Since ũ is increasing and convex, [ ( )] J( d) = E ũ (X 1 d 2 ) + (X 2 d 1 ) + (X i d i ) as desired. E [ ũ ( (X 1 d 1 ) + (X 2 d 2 ) + (X i d i ) )] (X i d i ) = J(d), Notice that the conclusion here is exactly opposite to that in Proposition 1 concerning Problem L1: if the size of a risk is large, we should allocate a larger policy limit but a smaller deductible. 4.2 Minimizing expected retained loss Parallel to the development in studying optimal policy limits, here we consider the problem of minimizing the expected total retained loss: [ ] Problem D2: min E (X i d i ). d S n(d) Problem D2 is a special case of Problem D1 with ũ(x) = x,and hence X i hr X j = d i d j is still valid. However, we can weaken the condition to the usual stochastic order. Proposition 5 Let d = (d 1,..., d n) be the solution to Problem D2, then X i st X j = d i d j. Proof: Since (X d) = X (X d) +, Problem D2 is equivalent to [ ] max E (X i d i ) +. d S n(d) The result follows immediately from the proof of Proposition 2. 11

12 4.3 Maximizing utility - unknown dependence structure Recall that F 1,..., F n are the distributions of X 1,..., X n, and R is the corresponding Fréchet space. As in Section 3.3, we consider the case where the dependence structure is unknown. Employing the maximin formulation, we have max d Sn(d) min (X1,...,X n) R E [u (w n (X i d i ))], Problem D3: where u is increasing and concave w is a fixed constant which is equivalent to Problem D3 : min d Sn(d) max (X1,...,X n) R E [ũ( n (X i d i ))], where ũ is increasing and convex The first step is to identify the dependence structure that solves the max part in Problem D3. Lemma 9 Suppose that ũ is increasing and convex. If (X c,..., X c ) R is comonotonic, then [ ] [ ] E ũ( (X i d i )) E ũ( (Xi c d i )) for any (X 1,..., X n ) R(F 1,..., F n ) and any (d 1,..., d n ) S n (d). Proof: The proof is exactly the same as in Lemma 8, except replacing the non-decreasing map x (x l i ) + there by the non-decreasing map x (x d i ). Problem D3 now becomes Problem D3 : min d Sn(d) E [ũ( n (X i d i ))], where ũ is increasing and convex (X 1,..., X n ) is comonotonic Proposition 6 Let d = (d 1,..., d n) be the solution to Problem D3, then X i st X j = d i d j. To prove this proposition, we need the next lemma. 12

13 Lemma 10 The function g : R n R n R defined by g(x, d) = (x 1 d i ) is an AI function. Proof: Consider the real-valued function φ(x, d) = x d defined on R 2. Let η > 0. Then for any r, the difference φ(r + η, s) φ(r, s) = (r + η) s r s is increasing in s, hence φ is supermodular. Therefore, g(x, d) = (x 1 d i ) = φ(x i, d i ) is an AI function by Lemma 1. In fact, this lemma can also be proved from Lemma 7 by noting that (x i d i ) = x i (x i d i ) +. The proof given above is chosen since it can illustrate yet another method of verifying that a given function is AI. Proof of Proposition 6: Assume that i = 2, j = 1. Let d and d be two admissible allocations as defined in the proof of Proposition 4, and K the objective function in Problem D3. We need to show that X 2 st X 1 = K(d) K( d). Since (X 1, X 2 ) is comonotonic, X 2 st X 1 implies that X 2 X 1 on Ω. As g(x, d) = n (x 1 d i ) is an AI function by Lemma 10, we have (X 1 d 2 ) + (X 2 d 1 ) + (X i d i ) (X 1 d 1 ) + (X 2 d 2 ) + (X i d i ) on Ω because we have assumed that d 2 d 1, and hence ( ) ũ (X 1 d 2 ) + (X 2 d 1 ) + (X i d i ) ũ ( (X 1 d 1 ) + (X 2 d 2 ) + ) (X i d i ) because ũ is increasing. Taking expectation on both sides yields the desired result. 13

14 5 Conclusion In this paper, the problems of optimal allocation of policy limits and deductibles are studied. For each case, three problems are considered: maximizing the expected utility of wealth assuming independence, minimizing the expected total retained loss, and maximizing the expected utility of wealth when the dependence structure is unknown. Because of the simple relationship x = (x d) + (x d) +, the conclusions for the optimal policy limits problem is exactly opposite to the optimal deductibles problem: it is optimal to allocate a larger policy limit and a smaller deductible to a risk if its size of potential loss is relative large. As a possible further line of research, it would be interesting to investigate whether we can shift our basis of comparison of the risks from size to variability, by means of convex-type orders instead of hazard rate order or the usual stochastic order. We conjecture that the policyholder should allocate a larger policy limit and a smaller deductible to a risk if its variability is relatively large. 6 Acknowledgments This work was supported by a grant from the Natural Sciences and Engineering research Council of Canada. The author would like the thank the anonymous referees for helpful comments and suggestions. References [1] Cheung, K.C., Optimal portfolio problem with unknown dependency structure. Insurance: Mathematics and Economics 38, [2] Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., Actuarial Theory for Dependent Risks. Wiley. [3] Dhaene, J., Goovaerts, M.J., Dependency of risks and stop-loss order. ASTIN Bulletin 26, [4] Dhaene, J., Wang, S., Young, V.R., Goovaerts, M., Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries 2000, [5] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics 31, [6] Hardy, G.H., Littlewood, J.E., Póyla, G., Some simple inequalities satisfied by convex functions, Messenger Math. 58,

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