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An optimal insurance strategy for an individual under an intertemporal equilibrium
Insurance: Mathematics and Economics 42 (2008) 255–260 www.elsevier.com/locate/ime
An optimal insurance strategy for an individual under an intertemporal equilibrium Chunyang Zhou ∗ , Chongfeng Wu, Shengping Zhang, Xuejun Huang Financial Engineering Research Center for Shanghai Jiaotong University, Shanghai 200052, People’s Republic of China Received November 2006; received in revised form January 2007; accepted 22 February 2007
Abstract In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135–142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem. c 2007 Elsevier B.V. All rights reserved.
Keywords: Optimal insurance strategy; Put option; Expected utility
1. Introduction Risk is essentially related to the uncertainty of loss in the future. To avoid the uncertainty of loss, one individual can transfer the risk to another, who is prepared to take it on. The risk premium is the cost that the individual has to pay to the risk-taker. Before the boom in the derivative market, it was the insurance companies who played the key role of risk-takers. They take on the risks that a risk-averse individual would not take on, and ask for a certain amount of risk premium as a compensation. The problem for the risk-averse individual is how to choose his/her optimal insurance strategy, so as to make him/her well off enough in the future. Previous insurance decision analyses can be divided into two approaches: the equilibrium model and the optimization model. In the equilibrium model, the optimal insurance is determined by the optimal risk sharing between the insured and the insurer. Among the various researchers, Borch (1962) was the first
∗ Corresponding address: Department of Management Science and Engineering, Shanghai Jiaotong University, No. 535, Fahuazhen Road, 200052 Shanghai, People’s Republic of China. Tel.: +86 21 52301087. E-mail address: [email protected] (C. Zhou).
c 2007 Elsevier B.V. All rights reserved. 0167-6687/$ - see front matter doi:10.1016/j.insmatheco.2007.02.005
to use this approach to derive optimal insurance. He sought to characterize a Pareto optimal risk sharing between several risk-averse individuals. Arrow (1970, 1974) and Raviv (1979) use this framework to study optimal insurance problems under different settings. In the optimization model, the optimal insurance is determined by the maximization of the insured’s expected utility or the minimization of their residual risk. Arrow (1963) began this study first. He shows that when premium is given by the expected value principle, the deductible insurance is optimal for a risk-averse von Neumann–Morgenstern insured. Using Arrow’s framework, Deprez and Gerber (1985), Young (1999), Gajek and Zagrodny (2004) and Promislow and Young (2005) use different insurance premium principles and target functions to study the optimal insurance problem. In most of the previous literature, the risk premium is specified exogenously. In this paper, the optimization model is used and the risk premium is specified endogenously by an interequilibrium model. With these settings, we discuss how a risk-averse individual chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It can be proved that the individual’s optimal insurance actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Therefore if the risk
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is traded in the derivative market, the individual can go there and buy a corresponding put option to hedge his/her risk. However, many sources of risk are not traded there and cannot be efficiently transferred. The insurance companies partly do the job that the derivative market cannot, and provide means for the individual to transfer his/her risk. The contract that the insured desires actually is a put option as well. The rest of paper is organized as follows. A two-period optimal problem is constructed in Section 2. Section 3 provides an analytical solution to the optimal problem. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option written on his/her holding asset with a proper strike price. Section 4 concludes the paper.
Proof. Since u 0 > 0, let h(x) = ln g(x) = ln u 0 (x − d) − ln u 0 (x). As the logarithm function is a strictly increasing function, the lemma is valid if h(x) can be proved to be a strictly decreasing function of x. It is easy to prove that the first derivative of h(x) is
2. Model set-up
From Eqs. (1) and (2), we prove the lemma.
We consider a static model over a time period between 0 and T . The individual holds an asset, whose value in time T is X , a random variable on the probability space {Ω , F, P}. Suppose the individual has a liability L in time T ; then the individual’s wealth in time T is Y = X − L, which can be decomposed as
Assumption 3. At time 0, the individual’s consumption is x 0 if they do not buy any insurance. Let β be the time discount factor, and π() be the pricing function. The optimal solution to the problem (3) max u x 0 − ξ · π(I ∗ ) + β Eu Y + ξ · I ∗
Y = Y + − (−Y )+
h 0 (x) =
u 00 (x − d) u 00 (x) − 0 . u 0 (x − d) u (x)
At the same time, Assumption 2 states that the individual has a strictly decreasing degree of absolute risk aversion. So we have for any d > 0, u 00 (x) u 00 (x − d) > − . u 0 (x − d) u 0 (x)
−
(2)
ξ
where = max(Y, 0) = max(X − L , 0) can be regarded as the pure profit in time T , and Z = (−Y )+ = max(L − X, 0) can be looked at as the pure loss. Let x, y, z be the realization of the corresponding random variable. Suppose the individual’s insurance strategy I˜(z) depends on their pure loss z, which means that when a pure loss z occurs to the individual, he/she would get a compensation of I˜(z). Like in the discussion in Young (1999), we have that for any z ≥ 0, I˜(z) satisfies 0 ≤ I˜(z) ≤ z. Since z depends on y, this condition can also be written as:
is ξ ∗ = 1.
Assumption 1. The individual holds an asset and assumes a liability, whose values in time T are X and L respectively. Without any insurance, the individual’s wealth would be Y = X −L. The individual adopts I (y) as their insurance strategy1 to avoid their risk in time T , where I (y) is a continuous function, and satisfies 0 ≤ I (y) ≤ (−y)+ . Let I be the set of insurance strategies which satisfy these conditions.
where
Y+
The assumption as regards the individual’s risk attitude is as follows: Assumption 2. Let u be the individual’s utility function, which satisfies u 0 > 0 and u 00 < 0. Meanwhile, the individual has a strictly decreasing degree of absolute risk aversion.2 Lemma 1. Under Assumption 2, if d > 0, then g(x) = is a strictly decreasing function of x.
(1)
u 0 (x−d) u 0 (x)
1 We here focus on the aggregate position of the insured, instead of their liability. This makes sense if the insured is concerned with his/her total wealth in the future. 2 The DARA utility is an often made and well supported restriction on risk preference. In his seminal paper, Mossin (1968) claims that DARA implies that an increase in wealth leads to a decrease in the optimal insurance. Actually, similar results can be concluded from our model, as can be seen in Proposition 11.
Remark 2. Assumption 3 actually is an intertemporal equilibrium model, which is similar to that proposed by Cochrane (2001). Since I ∗ is the optimal insurance, if I ∗ is traded in the market, the individual would buy only one. Lemma 3. Under Assumption 3, the price of I ∗ , or π(I ∗ ) can be written as π(I ∗ ) = E(m I ∗ )
m=β
(4)
u 0 (Y + I ∗ ) u 0 x 0 − π(I ∗ )
(5)
is the stochastic discount factor. Proof. The FOC for optimization problem (3) is −u 0 x 0 − ξ ∗ · π(I ∗ ) π(I ∗ ) + β Eu 0 Y + ξ ∗ · I ∗ I ∗ = 0 or "
# βu 0 (Y + ξ ∗ · I ∗ ) ∗ I . π(I ) = E u 0 x 0 − ξ ∗ · π(I ∗ ) ∗
For ξ ∗ = 1, we have # " βu 0 (Y + I ∗ ) ∗ ∗ I = E(m I ∗ ). π(I ) = E u 0 x 0 − π(I ∗ )
Assumption 4. The stochastic discount factor m is unique. Therefore, the price of an insurance coverage I is π(I ) = E(m I ).
(6)
Assumption 5. The market risk free rate of return between time 0 and T is R f and let α = R −1 f .
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Suppose at time 0 the individual buys an insurance I , with a cost of π(I ). Let H (I ) = R f π(I ); then H (I ) represents the corresponding cost in time T . Therefore the individual’s expected utility of terminal wealth after buying the insurance I in time 0 is U (I ) = Eu (Y + I (Y ) − H (I )) .
(7)
The individual would choose I ∈ I to maximize U (I ). Let I ∗ denote the optimal insurance strategy, and H ∗ be the corresponding terminal cost. 3. The optimal insurance strategy An open δ-neighborhood of y is defined as B(y, δ) = (y − δ, y + δ). We have, like Proposition 3.1 in Young (1999), the following proposition, which provides necessary and sufficient conditions for the optimal insurance I ∗ . Proposition 4. Let δ be a positive sufficiently small value and β˜ = β R f . The optimal insurance I ∗ that maximizes U (I ) satisfies the following necessary and sufficient conditions: (1) I ∗ (y) = 0 for all y ∈ B(y0 , δ) if and only if u 0 (y0 − H ∗ ) 0 0 u (x − α H ∗ ) ≤ E u 0 Y + I ∗ (Y ) − H ∗ . ˜ 0 (y0 ) βu (2) I ∗ (y) = (−y)+ for all y ∈ B(y0 , δ) if and only if u 0 (y0+ − H ∗ ) 0 0 u (x ˜ 0 (y + ) βu 0
− α H ∗ ) ≥ E u 0 Y + I ∗ (Y ) − H ∗ .
(3) 0 < I ∗ (y) < (−y)+ for all y ∈ B(y0 , δ) if and only if u 0 (y0 + I ∗ (y0 ) − H ∗ ) 0 0 u (x − α H ∗ ) ˜ 0 (y0 + I ∗ (y0 )) βu = E u 0 Y + I ∗ (Y ) − H ∗ . Proof can be seen in Appendix A. The conditions in Proposition 4 can be interpreted economically. The left-hand side of each condition is the marginal utility benefit of receiving additional indemnity, while the right-hand side represents the marginal utility cost of paying the corresponding additional premium. More details can be found in Young (1999). Let MEU = E u 0 (Y + I ∗ (Y ) − H ∗ ) ; we can conclude the following corollaries: Corollary 5. If y1 < y2 and I ∗ (y1 ) = 0, then I ∗ (y2 ) = 0. Proof. From Proposition 4 and Lemma 1, we have u 0 (y2
−
H ∗)
˜ 0 (y2 ) βu
u 0 (x 0 − α H ∗ ) <
u 0 (y1
−
H ∗)
˜ 0 (y1 ) βu
u 0 (x 0 − α H ∗ )
≤ MEU. So I ∗ (y2 ) = 0.
a state. At the same time, from I ∗ (0) = 0,3 we have that there would always exist y1 which satisfies I ∗ (y1 ) = 0. The conclusion in Corollary 5 coincides with that drawn by Arrow (1963). Corollary 7. If y1 < y2 and I ∗ (y2 ) = (−y2 )+ , then I ∗ (y1 ) = (−y1 )+ . Proof. From Proposition 4 and Lemma 1, we have u 0 (y1+ − H ∗ ) 0 0 u 0 (y2+ − H ∗ ) 0 0 ∗ u (x − α H ) > u (x − α H ∗ ) ˜ 0 (y + ) ˜ 0 (y + ) βu βu 1 2 ≥ MEU. So I ∗ (y1 ) = (−y1 )+ .
Corollary 8. If for i = 1, 2, 0 < I ∗ (yi ) < (−yi )+ , then I ∗ (y1 ) − I ∗ (y2 ) = y2 − y1 . Proof. From Proposition 4, we have for i = 1, 2, u 0 (yi + I ∗ (yi ) − H ∗ ) 0 0 u (x − α H ∗ ) = MEU. ˜ 0 (y0 + I ∗ (y0 )) βu From Lemma 1, we have y1 + I ∗ (y1 ) = y2 + I ∗ (y2 ) that is I ∗ (y1 ) − I ∗ (y2 ) = y2 − y1 .
Remark 9. Corollary 8 shows that the optimal insurance would try to make the individual equally well off in every state in the future, which is very intuitive from the insured’s view. Note that in Young (1999), Corollary 3.4 states that if z 1 < z 2 , then 0 < I ∗ (z 2 ) − I ∗ (z 1 ) < z 2 − z 1 , which means that the optimal insurance is increasing at a rate more slowly than that at which the losses increase. This conclusion is derived by optimizing the insured individual’s expected utility, but it is intuitive in the insurance company’s eyes. For an analytical solution to the problem, it is worth introducing another assumption. Assumption 6. There exists y0 ∈ R such that I ∗ (y0 ) > 0. Lemma 10. Let f (s) =
u 0 (s − H ∗ ) 0 0 u (x − α H ∗ ). ˜ 0 (s) βu
Then there exists d, which satisfies f (d) = E u 0 Y + I ∗ (Y ) − H ∗ ≡ MEU.
(8)
(9)
Proof. From Assumption 6, there exists y0 ∈ R such that I ∗ (y0 ) > 0. If I ∗ (y0 ) < (−y0 )+ , then from Proposition 4(3), we can conclude that f (y0 + I ∗ (y0 )) = MEU. Otherwise, if I ∗ (y0 ) = (−y0 )+ , from Proposition 4(2), we have f y0 + (−y0 )+ = f (0) ≥ MEU. (10)
Remark 6. Corollary 5 means that when the individual’s wealth is beyond a certain level, he/she would not insure such
3 From 0 ≤ I ∗ (y) ≤ (−y)+ , we have 0 ≤ I ∗ (0) ≤ 0, or I ∗ (0) = 0. Actually, for all y ≥ 0, we have 0 ≤ I ∗ (y) ≤ (−y)+ = 0, so I ∗ (y) = 0.
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At the same time, from I ∗ (0) = 0 and Proposition 4(1), we have f (0) ≤ MEU.
(11)
So from Eqs. (10) and (11), we conclude that f (0) = MEU. Proposition 11. Suppose d satisfies Eq. (9); then the individual’s optimal insurance can be written as follows: (1) For a given wealth realization y, the optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0. (2) For a given wealth realization y, the optimal insurance I ∗ (y) = (−y)+ if and only if d ≥ 0. The proof can be seen in Appendix B. Remark 12. (1) Actually, d can be seen as the individual’s wealth target. From Corollary 8, the individual’s optimal insurance strategy is set such that his/her wealth would be the same in every possible state in the future. From Eq. (9), we can see that the wealth target is d when the marginal utility benefit of increasing terminal wealth by insurance is equal to the marginal utility cost of paying the corresponding additional premium. (2) Eq. (14) in Appendix B can be interpreted as follows. The individual’s optimal insurance I ∗ can be seen as trying to make his/her terminal wealth reach d as closely as possible, under the constraint of 0 ≤ I ∗ (y) ≤ (−y)+ . In the state when the individual’s terminal wealth is beyond or equal to d, it is unnecessary for him/her to insure such state; in the state when his/her wealth is below d, but the terminal pure profit can reach d, the individual would take a partial coverage of their loss to make his/her terminal wealth equal to d; if even the terminal pure profit is below the target wealth d, then there is no means but to take full coverage of their loss, so that their terminal wealth could be near to their target. (3) The proof of Proposition 11 does not depend on the specific formulation of the predetermined optimal insurance strategy, so if the insurance strategy I ∗ is optimal, it must satisfy the forms proposed in Proposition 11. There are two types of optimal insurance strategy: when d ≥ 0, I ∗ (Y ) = (−Y )+ ; when d < 0, I ∗ (Y ) = (d − Y )+ . They can be written in a compact formulation as + I ∗ (Y ) = d − − Y where d − = min(d, 0). (4) Actually, the optimal insurance can be written as the function of their holding asset X and their liability L. (a) When d ≥ 0, the individual’s optimal insurance is I ∗ (X, L) = (L − X )+ , and the corresponding cost is π(I ∗ ) = E(m I ∗ ) = E m(L − X )+ , which can be seen as the premium of a put option written on their holding asset X , with a strike price of L. (b) When d < 0, the individ+ ual’s optimal insurance is I ∗ (X, L) = (L + d +− X ) , with a ∗ ∗ cost π(I ) = E(m I ) = E m(L + d − X ) , which can be seen as the premium of a put option written on the asset X , with a strike price of L + d. Therefore, the individual’s optimal insurance strategy can be implemented by buying a put option written on X with a proper strike price.
From Proposition 11, we can conclude the following corollary: Corollary 13. Let H ∗ (d) = R f E m(d − Y )+ ; then the individual’s optimal insurance strategy can be written as follows: 1. The optimal insurance I ∗ (y) = (−y)+ if and only if u 0 (−H ∗ (0)) 0 0 f (0) = u x − α H ∗ (0) ˜ 0 (0) βu 0 + ≥ E u Y − H ∗ (0) . 2. The optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0, where d satisfies u 0 (d − H ∗ (d)) 0 0 u (x − α H ∗ (d)) ˜ 0 (d) βu = E u 0 max(Y, d) − H ∗ (d) .
(12)
At the same time, it is valid that u 0 (−H ∗ (0)) 0 0 u x − α H ∗ (0) f (0) = ˜ 0 (0) βu 0 + < E u Y − H ∗ (0) . Proof. (1) Suppose d satisfies Eq. (9); then from Proposition 11, the individual’s optimal insurance strategy I ∗ (y) = (−y)+ if and only if d ≥ 0. Substitute I ∗ (y) = (−y)+ into Eq. (9); we have u 0 (d − H ∗ (0)) 0 0 f (d) = u x − α H ∗ (0) ˜ 0 (d) βu 0 + = E u Y − H ∗ (0) . (13) Since f (s) is a strictly decreasing function of s, there is a unique solution to Eq. (13): d = f −1 E u 0 Y + − H ∗ (0) . Meanwhile, we have d ≥ 0 if and only if u 0 (−H ∗ (0)) 0 0 f (0) = u x − α H ∗ (0) ˜ 0 (0) βu 0 + ≥ E u Y − H ∗ (0) . (2) From Proposition 11, the optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0, where d satisfies Eq. (12). At the same time, from conclusion (1), we have u 0 (−H ∗ (0)) 0 0 u (x − α H ∗ ) 0 ˜ βu (0) < E u 0 Y + − H ∗ (0) .
f (0) =
To summarize, the process of getting the individual’s optimal insurance strategy is as follows: Step 1: Let I ∗ be the optimal insurance. Step 2: From Proposition 11, we show that if I ∗ is an optimal insurance strategy, it must have a certain formulation, say I ∗ (y) = h(y, d). Step 3: Let I ∗ (y) = h(y, d); the conditions that the parameters should satisfy are determined if I ∗ (y) is the optimal insurance strategy.
C. Zhou et al. / Insurance: Mathematics and Economics 42 (2008) 255–260
4. Conclusion In this paper, we discuss how a risk-averse individual under the intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. The analytical solution is provided as well. The main results in this paper are as follows: (1) The individual would not insure the state where the marginal utility benefit of receiving additional indemnity is less than the corresponding additional premium. The individual takes a partial coverage when the marginal benefit is equal to the marginal cost, and takes a full coverage of their pure loss when the marginal benefit is larger than the marginal cost. (2) There exists y1 ∈ R such that if y > y1 , the individual takes no coverage of his/her wealth, which coincides with the conclusion of Arrow (1963). If the individual takes full coverage of their wealth y2 , he/she will too when his/her terminal wealth is below y2 . If the individual takes partial coverage when his/her terminal wealth is y3 and y4 , he/she will be equally well off in both states. (3) Suppose d satisfies Eq. (9); then d can be seen as the individual’s wealth target in time T . When his/her terminal wealth is d, the marginal utility benefit of increasing terminal wealth by insurance is equal to the marginal utility cost of paying the corresponding additional premium. (4) The individual’s optimal insurance strategy can be seen as trying to make his/her terminal wealth reach d as closely as possible, under the constraint 0 ≤ I (y) ≤ (−y)+ . (5) The individual’s optimal insurance strategy can be implemented by buying a put option written on his/her holding asset with a proper strike price. (6) The cost of avoiding risk can be seen as a risk measure, so the put option premium can be considered as a reasonable risk measure. Jarrow (2002) drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem. Acknowledgement The authors thank the anonymous referee for helpful comments. This work is supported by NSFC(70331001). Appendix A Proof of Proposition 4. (1) Suppose I ∗ (y) = 0 for all y ∈ B(y0 , δ). Let I (y) = I ∗ (y) + V (y), where ≥ 0 and V (y) = 1 B(y0 ,δ) (y) is the indicator function of y defined as 1, y ∈ B(y0 , δ) V (y) = 0, y 6∈ B(y0 , δ). Since I ∗ maximize U (I ), we have 0 ≥ U (I ) − U (I ∗ ) = Eu [Y + I (Y ) − H (I )] − Eu Y + I ∗ (Y ) − H (I ∗ ) 0 = E u Y + I ∗ (Y ) − H (I ∗ ) × I (Y ) − I ∗ (Y ) + H (I ∗ ) − H (I ) + o() = E u 0 Y + I ∗ (Y ) − H (I ∗ ) V − R f E(mV ) + o().
259
Divide both sides by , and let → 0; we have E u 0 Y + I ∗ (Y ) − H (I ∗ ) V ≤ R f E(mV )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . Divide both sides by P (ω ∈ B(y0 , δ)), and let δ → 0+ ; it can be obtained that u 0 y0 + I ∗ (y0 ) − H (I ∗ ) ≤ R f m(y0 )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . Since I ∗ (y0 ) = 0, we can get u 0 y0 − H (I ∗ ) ≤ R f m(y0 )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . From Eq. (5) and β˜ = β R f , we have u 0 (y0 − H ∗ ) 0 0 u (x − α H ∗ ) ≤ E u 0 Y + I ∗ (Y ) − H ∗ . 0 ˜ (y0 ) βu (2), (3) Following the proof as for part (1), we obtain the necessary conditions stated in Proposition 4. From Lemma 1, at most one condition of the three conditions can hold for a given y 0 , so the necessary conditions are also sufficient. Appendix B Proof of Proposition 11. From Assumption 5, H ∗ > 0. Then by Lemma 1, f (s) is a strictly decreasing function of s. For a given wealth realization y, if y+I ∗ (y) = d, then I ∗ (y) = d−y. At the same time, from Proposition 4, we have 0 < I ∗ (y) < (−y)+ , or y < d < y + . Three situations are discussed as follows: (1) If y ≥ d, it can be concluded that I ∗ (y) = 0, which can be proved in the following way. If I ∗ (y) = 0, we have y + I ∗ (y) = y ≥ d. Since f (s) is a strictly decreasing function of s, we have f (y) ≤ f (d) = MEU. Then by Proposition 4, we have I ∗ (y) = 0. (2) If y + ≤ d, then I ∗ (y) = (−y)+ , which can be proved as follows. If I ∗ (y) = (−y)+ , we have y + I ∗ (y) = y + ≤ d. For f (s) a strictly decreasing function of s, we have f (y + ) ≥ f (d) = MEU. Then by Proposition 4, we have I ∗ (y) = (−y)+ . (3) If y < d < y + , then I ∗ (y) = d − y. This can be proved as follows. If I ∗ (y) = d − y, we have f (y + I ∗ (y)) = f (d) = MEU. At the same time, it can be easily verified that 0 < I ∗ (y) < (−y)+ ; so from Proposition 4, we have I ∗ (y) = d − y. I ∗ (y) can formally be written as y≥d 0, I ∗ (y) = d − y, y < d < y + (14) (−y)+ , y + ≤ d. Three cases for d are discussed as follows. (1) If d < 0, from Eq. (14), we can see that the individual would not take full coverage of their loss, and his/her optimal insurance strategy is I ∗ (y) = (d − y)+ . (2) If d = 0, it can be easily verified that I ∗ (y) = (−y)+ . (3) If d > 0, for y + ≤ d ⇔ y ≤ d and y + ≥ d ⇔ y ≥ d, the optimal insurance strategy can be written as
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I ∗ (y) =
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0 (−y)+
y≥d y < d.
Since d > 0, the equation above can be simplified as I ∗ (y) = (−y)+ . Thus the sufficient conditions have been verified. Meanwhile, the necessary conditions are obvious. References Arrow, K.J., 1963. Uncertainty and the welfare economics of medical care. American Economic Review 53, 941–973. Arrow, K.J., 1970. Essays in the Theory of Risk-Bearing. North-Holland Publishing Co., Amsterdam. Arrow, K.J., 1974. Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal 1, 1–42.
Borch, K., 1962. Equilibrium in a reinsurance market. Econometrica 30, 424–444. Cochrane, J.H., 2001. Asset Pricing. Princeton University Press. Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Mathematics and Economics 4, 179–189. Gajek, L., Zagrodny, D., 2004. Optimal reinsurance under general risk measures. Insurance: Mathematics and Economics 34, 227–240. Jarrow, R., 2002. Put option premiums and coherent risk measures. Mathematical Finance 12, 135–142. Mossin, J., 1968. Aspects of rational insurance purchasing. Journal of Political Economy 76, 553–568. Promislow, S.D., Young, V.R., 2005. Unifying framework for optimal insurance. Insurance: Mathematics and Economics 36, 347–364. Raviv, A., 1979. The design of an optimal insurance policy. American. Economic Review 69 (1), 84C96. Young, V.R., 1999. Optimal insurance under Wang’s premium principle. Insurance: Mathematics and Economics 25, 109–122.
An optimal insurance strategy for an individual under an intertemporal equilibrium Chunyang Zhou ∗ , Chongfeng Wu, Shengping Zhang, Xuejun Huang Financial Engineering Research Center for Shanghai Jiaotong University, Shanghai 200052, People’s Republic of China Received November 2006; received in revised form January 2007; accepted 22 February 2007
Abstract In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135–142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem. c 2007 Elsevier B.V. All rights reserved.
Keywords: Optimal insurance strategy; Put option; Expected utility
1. Introduction Risk is essentially related to the uncertainty of loss in the future. To avoid the uncertainty of loss, one individual can transfer the risk to another, who is prepared to take it on. The risk premium is the cost that the individual has to pay to the risk-taker. Before the boom in the derivative market, it was the insurance companies who played the key role of risk-takers. They take on the risks that a risk-averse individual would not take on, and ask for a certain amount of risk premium as a compensation. The problem for the risk-averse individual is how to choose his/her optimal insurance strategy, so as to make him/her well off enough in the future. Previous insurance decision analyses can be divided into two approaches: the equilibrium model and the optimization model. In the equilibrium model, the optimal insurance is determined by the optimal risk sharing between the insured and the insurer. Among the various researchers, Borch (1962) was the first
∗ Corresponding address: Department of Management Science and Engineering, Shanghai Jiaotong University, No. 535, Fahuazhen Road, 200052 Shanghai, People’s Republic of China. Tel.: +86 21 52301087. E-mail address: [email protected] (C. Zhou).
c 2007 Elsevier B.V. All rights reserved. 0167-6687/$ - see front matter doi:10.1016/j.insmatheco.2007.02.005
to use this approach to derive optimal insurance. He sought to characterize a Pareto optimal risk sharing between several risk-averse individuals. Arrow (1970, 1974) and Raviv (1979) use this framework to study optimal insurance problems under different settings. In the optimization model, the optimal insurance is determined by the maximization of the insured’s expected utility or the minimization of their residual risk. Arrow (1963) began this study first. He shows that when premium is given by the expected value principle, the deductible insurance is optimal for a risk-averse von Neumann–Morgenstern insured. Using Arrow’s framework, Deprez and Gerber (1985), Young (1999), Gajek and Zagrodny (2004) and Promislow and Young (2005) use different insurance premium principles and target functions to study the optimal insurance problem. In most of the previous literature, the risk premium is specified exogenously. In this paper, the optimization model is used and the risk premium is specified endogenously by an interequilibrium model. With these settings, we discuss how a risk-averse individual chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It can be proved that the individual’s optimal insurance actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Therefore if the risk
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is traded in the derivative market, the individual can go there and buy a corresponding put option to hedge his/her risk. However, many sources of risk are not traded there and cannot be efficiently transferred. The insurance companies partly do the job that the derivative market cannot, and provide means for the individual to transfer his/her risk. The contract that the insured desires actually is a put option as well. The rest of paper is organized as follows. A two-period optimal problem is constructed in Section 2. Section 3 provides an analytical solution to the optimal problem. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option written on his/her holding asset with a proper strike price. Section 4 concludes the paper.
Proof. Since u 0 > 0, let h(x) = ln g(x) = ln u 0 (x − d) − ln u 0 (x). As the logarithm function is a strictly increasing function, the lemma is valid if h(x) can be proved to be a strictly decreasing function of x. It is easy to prove that the first derivative of h(x) is
2. Model set-up
From Eqs. (1) and (2), we prove the lemma.
We consider a static model over a time period between 0 and T . The individual holds an asset, whose value in time T is X , a random variable on the probability space {Ω , F, P}. Suppose the individual has a liability L in time T ; then the individual’s wealth in time T is Y = X − L, which can be decomposed as
Assumption 3. At time 0, the individual’s consumption is x 0 if they do not buy any insurance. Let β be the time discount factor, and π() be the pricing function. The optimal solution to the problem (3) max u x 0 − ξ · π(I ∗ ) + β Eu Y + ξ · I ∗
Y = Y + − (−Y )+
h 0 (x) =
u 00 (x − d) u 00 (x) − 0 . u 0 (x − d) u (x)
At the same time, Assumption 2 states that the individual has a strictly decreasing degree of absolute risk aversion. So we have for any d > 0, u 00 (x) u 00 (x − d) > − . u 0 (x − d) u 0 (x)
−
(2)
ξ
where = max(Y, 0) = max(X − L , 0) can be regarded as the pure profit in time T , and Z = (−Y )+ = max(L − X, 0) can be looked at as the pure loss. Let x, y, z be the realization of the corresponding random variable. Suppose the individual’s insurance strategy I˜(z) depends on their pure loss z, which means that when a pure loss z occurs to the individual, he/she would get a compensation of I˜(z). Like in the discussion in Young (1999), we have that for any z ≥ 0, I˜(z) satisfies 0 ≤ I˜(z) ≤ z. Since z depends on y, this condition can also be written as:
is ξ ∗ = 1.
Assumption 1. The individual holds an asset and assumes a liability, whose values in time T are X and L respectively. Without any insurance, the individual’s wealth would be Y = X −L. The individual adopts I (y) as their insurance strategy1 to avoid their risk in time T , where I (y) is a continuous function, and satisfies 0 ≤ I (y) ≤ (−y)+ . Let I be the set of insurance strategies which satisfy these conditions.
where
Y+
The assumption as regards the individual’s risk attitude is as follows: Assumption 2. Let u be the individual’s utility function, which satisfies u 0 > 0 and u 00 < 0. Meanwhile, the individual has a strictly decreasing degree of absolute risk aversion.2 Lemma 1. Under Assumption 2, if d > 0, then g(x) = is a strictly decreasing function of x.
(1)
u 0 (x−d) u 0 (x)
1 We here focus on the aggregate position of the insured, instead of their liability. This makes sense if the insured is concerned with his/her total wealth in the future. 2 The DARA utility is an often made and well supported restriction on risk preference. In his seminal paper, Mossin (1968) claims that DARA implies that an increase in wealth leads to a decrease in the optimal insurance. Actually, similar results can be concluded from our model, as can be seen in Proposition 11.
Remark 2. Assumption 3 actually is an intertemporal equilibrium model, which is similar to that proposed by Cochrane (2001). Since I ∗ is the optimal insurance, if I ∗ is traded in the market, the individual would buy only one. Lemma 3. Under Assumption 3, the price of I ∗ , or π(I ∗ ) can be written as π(I ∗ ) = E(m I ∗ )
m=β
(4)
u 0 (Y + I ∗ ) u 0 x 0 − π(I ∗ )
(5)
is the stochastic discount factor. Proof. The FOC for optimization problem (3) is −u 0 x 0 − ξ ∗ · π(I ∗ ) π(I ∗ ) + β Eu 0 Y + ξ ∗ · I ∗ I ∗ = 0 or "
# βu 0 (Y + ξ ∗ · I ∗ ) ∗ I . π(I ) = E u 0 x 0 − ξ ∗ · π(I ∗ ) ∗
For ξ ∗ = 1, we have # " βu 0 (Y + I ∗ ) ∗ ∗ I = E(m I ∗ ). π(I ) = E u 0 x 0 − π(I ∗ )
Assumption 4. The stochastic discount factor m is unique. Therefore, the price of an insurance coverage I is π(I ) = E(m I ).
(6)
Assumption 5. The market risk free rate of return between time 0 and T is R f and let α = R −1 f .
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Suppose at time 0 the individual buys an insurance I , with a cost of π(I ). Let H (I ) = R f π(I ); then H (I ) represents the corresponding cost in time T . Therefore the individual’s expected utility of terminal wealth after buying the insurance I in time 0 is U (I ) = Eu (Y + I (Y ) − H (I )) .
(7)
The individual would choose I ∈ I to maximize U (I ). Let I ∗ denote the optimal insurance strategy, and H ∗ be the corresponding terminal cost. 3. The optimal insurance strategy An open δ-neighborhood of y is defined as B(y, δ) = (y − δ, y + δ). We have, like Proposition 3.1 in Young (1999), the following proposition, which provides necessary and sufficient conditions for the optimal insurance I ∗ . Proposition 4. Let δ be a positive sufficiently small value and β˜ = β R f . The optimal insurance I ∗ that maximizes U (I ) satisfies the following necessary and sufficient conditions: (1) I ∗ (y) = 0 for all y ∈ B(y0 , δ) if and only if u 0 (y0 − H ∗ ) 0 0 u (x − α H ∗ ) ≤ E u 0 Y + I ∗ (Y ) − H ∗ . ˜ 0 (y0 ) βu (2) I ∗ (y) = (−y)+ for all y ∈ B(y0 , δ) if and only if u 0 (y0+ − H ∗ ) 0 0 u (x ˜ 0 (y + ) βu 0
− α H ∗ ) ≥ E u 0 Y + I ∗ (Y ) − H ∗ .
(3) 0 < I ∗ (y) < (−y)+ for all y ∈ B(y0 , δ) if and only if u 0 (y0 + I ∗ (y0 ) − H ∗ ) 0 0 u (x − α H ∗ ) ˜ 0 (y0 + I ∗ (y0 )) βu = E u 0 Y + I ∗ (Y ) − H ∗ . Proof can be seen in Appendix A. The conditions in Proposition 4 can be interpreted economically. The left-hand side of each condition is the marginal utility benefit of receiving additional indemnity, while the right-hand side represents the marginal utility cost of paying the corresponding additional premium. More details can be found in Young (1999). Let MEU = E u 0 (Y + I ∗ (Y ) − H ∗ ) ; we can conclude the following corollaries: Corollary 5. If y1 < y2 and I ∗ (y1 ) = 0, then I ∗ (y2 ) = 0. Proof. From Proposition 4 and Lemma 1, we have u 0 (y2
−
H ∗)
˜ 0 (y2 ) βu
u 0 (x 0 − α H ∗ ) <
u 0 (y1
−
H ∗)
˜ 0 (y1 ) βu
u 0 (x 0 − α H ∗ )
≤ MEU. So I ∗ (y2 ) = 0.
a state. At the same time, from I ∗ (0) = 0,3 we have that there would always exist y1 which satisfies I ∗ (y1 ) = 0. The conclusion in Corollary 5 coincides with that drawn by Arrow (1963). Corollary 7. If y1 < y2 and I ∗ (y2 ) = (−y2 )+ , then I ∗ (y1 ) = (−y1 )+ . Proof. From Proposition 4 and Lemma 1, we have u 0 (y1+ − H ∗ ) 0 0 u 0 (y2+ − H ∗ ) 0 0 ∗ u (x − α H ) > u (x − α H ∗ ) ˜ 0 (y + ) ˜ 0 (y + ) βu βu 1 2 ≥ MEU. So I ∗ (y1 ) = (−y1 )+ .
Corollary 8. If for i = 1, 2, 0 < I ∗ (yi ) < (−yi )+ , then I ∗ (y1 ) − I ∗ (y2 ) = y2 − y1 . Proof. From Proposition 4, we have for i = 1, 2, u 0 (yi + I ∗ (yi ) − H ∗ ) 0 0 u (x − α H ∗ ) = MEU. ˜ 0 (y0 + I ∗ (y0 )) βu From Lemma 1, we have y1 + I ∗ (y1 ) = y2 + I ∗ (y2 ) that is I ∗ (y1 ) − I ∗ (y2 ) = y2 − y1 .
Remark 9. Corollary 8 shows that the optimal insurance would try to make the individual equally well off in every state in the future, which is very intuitive from the insured’s view. Note that in Young (1999), Corollary 3.4 states that if z 1 < z 2 , then 0 < I ∗ (z 2 ) − I ∗ (z 1 ) < z 2 − z 1 , which means that the optimal insurance is increasing at a rate more slowly than that at which the losses increase. This conclusion is derived by optimizing the insured individual’s expected utility, but it is intuitive in the insurance company’s eyes. For an analytical solution to the problem, it is worth introducing another assumption. Assumption 6. There exists y0 ∈ R such that I ∗ (y0 ) > 0. Lemma 10. Let f (s) =
u 0 (s − H ∗ ) 0 0 u (x − α H ∗ ). ˜ 0 (s) βu
Then there exists d, which satisfies f (d) = E u 0 Y + I ∗ (Y ) − H ∗ ≡ MEU.
(8)
(9)
Proof. From Assumption 6, there exists y0 ∈ R such that I ∗ (y0 ) > 0. If I ∗ (y0 ) < (−y0 )+ , then from Proposition 4(3), we can conclude that f (y0 + I ∗ (y0 )) = MEU. Otherwise, if I ∗ (y0 ) = (−y0 )+ , from Proposition 4(2), we have f y0 + (−y0 )+ = f (0) ≥ MEU. (10)
Remark 6. Corollary 5 means that when the individual’s wealth is beyond a certain level, he/she would not insure such
3 From 0 ≤ I ∗ (y) ≤ (−y)+ , we have 0 ≤ I ∗ (0) ≤ 0, or I ∗ (0) = 0. Actually, for all y ≥ 0, we have 0 ≤ I ∗ (y) ≤ (−y)+ = 0, so I ∗ (y) = 0.
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At the same time, from I ∗ (0) = 0 and Proposition 4(1), we have f (0) ≤ MEU.
(11)
So from Eqs. (10) and (11), we conclude that f (0) = MEU. Proposition 11. Suppose d satisfies Eq. (9); then the individual’s optimal insurance can be written as follows: (1) For a given wealth realization y, the optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0. (2) For a given wealth realization y, the optimal insurance I ∗ (y) = (−y)+ if and only if d ≥ 0. The proof can be seen in Appendix B. Remark 12. (1) Actually, d can be seen as the individual’s wealth target. From Corollary 8, the individual’s optimal insurance strategy is set such that his/her wealth would be the same in every possible state in the future. From Eq. (9), we can see that the wealth target is d when the marginal utility benefit of increasing terminal wealth by insurance is equal to the marginal utility cost of paying the corresponding additional premium. (2) Eq. (14) in Appendix B can be interpreted as follows. The individual’s optimal insurance I ∗ can be seen as trying to make his/her terminal wealth reach d as closely as possible, under the constraint of 0 ≤ I ∗ (y) ≤ (−y)+ . In the state when the individual’s terminal wealth is beyond or equal to d, it is unnecessary for him/her to insure such state; in the state when his/her wealth is below d, but the terminal pure profit can reach d, the individual would take a partial coverage of their loss to make his/her terminal wealth equal to d; if even the terminal pure profit is below the target wealth d, then there is no means but to take full coverage of their loss, so that their terminal wealth could be near to their target. (3) The proof of Proposition 11 does not depend on the specific formulation of the predetermined optimal insurance strategy, so if the insurance strategy I ∗ is optimal, it must satisfy the forms proposed in Proposition 11. There are two types of optimal insurance strategy: when d ≥ 0, I ∗ (Y ) = (−Y )+ ; when d < 0, I ∗ (Y ) = (d − Y )+ . They can be written in a compact formulation as + I ∗ (Y ) = d − − Y where d − = min(d, 0). (4) Actually, the optimal insurance can be written as the function of their holding asset X and their liability L. (a) When d ≥ 0, the individual’s optimal insurance is I ∗ (X, L) = (L − X )+ , and the corresponding cost is π(I ∗ ) = E(m I ∗ ) = E m(L − X )+ , which can be seen as the premium of a put option written on their holding asset X , with a strike price of L. (b) When d < 0, the individ+ ual’s optimal insurance is I ∗ (X, L) = (L + d +− X ) , with a ∗ ∗ cost π(I ) = E(m I ) = E m(L + d − X ) , which can be seen as the premium of a put option written on the asset X , with a strike price of L + d. Therefore, the individual’s optimal insurance strategy can be implemented by buying a put option written on X with a proper strike price.
From Proposition 11, we can conclude the following corollary: Corollary 13. Let H ∗ (d) = R f E m(d − Y )+ ; then the individual’s optimal insurance strategy can be written as follows: 1. The optimal insurance I ∗ (y) = (−y)+ if and only if u 0 (−H ∗ (0)) 0 0 f (0) = u x − α H ∗ (0) ˜ 0 (0) βu 0 + ≥ E u Y − H ∗ (0) . 2. The optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0, where d satisfies u 0 (d − H ∗ (d)) 0 0 u (x − α H ∗ (d)) ˜ 0 (d) βu = E u 0 max(Y, d) − H ∗ (d) .
(12)
At the same time, it is valid that u 0 (−H ∗ (0)) 0 0 u x − α H ∗ (0) f (0) = ˜ 0 (0) βu 0 + < E u Y − H ∗ (0) . Proof. (1) Suppose d satisfies Eq. (9); then from Proposition 11, the individual’s optimal insurance strategy I ∗ (y) = (−y)+ if and only if d ≥ 0. Substitute I ∗ (y) = (−y)+ into Eq. (9); we have u 0 (d − H ∗ (0)) 0 0 f (d) = u x − α H ∗ (0) ˜ 0 (d) βu 0 + = E u Y − H ∗ (0) . (13) Since f (s) is a strictly decreasing function of s, there is a unique solution to Eq. (13): d = f −1 E u 0 Y + − H ∗ (0) . Meanwhile, we have d ≥ 0 if and only if u 0 (−H ∗ (0)) 0 0 f (0) = u x − α H ∗ (0) ˜ 0 (0) βu 0 + ≥ E u Y − H ∗ (0) . (2) From Proposition 11, the optimal insurance I ∗ (y) = (d − y)+ if and only if d < 0, where d satisfies Eq. (12). At the same time, from conclusion (1), we have u 0 (−H ∗ (0)) 0 0 u (x − α H ∗ ) 0 ˜ βu (0) < E u 0 Y + − H ∗ (0) .
f (0) =
To summarize, the process of getting the individual’s optimal insurance strategy is as follows: Step 1: Let I ∗ be the optimal insurance. Step 2: From Proposition 11, we show that if I ∗ is an optimal insurance strategy, it must have a certain formulation, say I ∗ (y) = h(y, d). Step 3: Let I ∗ (y) = h(y, d); the conditions that the parameters should satisfy are determined if I ∗ (y) is the optimal insurance strategy.
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4. Conclusion In this paper, we discuss how a risk-averse individual under the intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. The analytical solution is provided as well. The main results in this paper are as follows: (1) The individual would not insure the state where the marginal utility benefit of receiving additional indemnity is less than the corresponding additional premium. The individual takes a partial coverage when the marginal benefit is equal to the marginal cost, and takes a full coverage of their pure loss when the marginal benefit is larger than the marginal cost. (2) There exists y1 ∈ R such that if y > y1 , the individual takes no coverage of his/her wealth, which coincides with the conclusion of Arrow (1963). If the individual takes full coverage of their wealth y2 , he/she will too when his/her terminal wealth is below y2 . If the individual takes partial coverage when his/her terminal wealth is y3 and y4 , he/she will be equally well off in both states. (3) Suppose d satisfies Eq. (9); then d can be seen as the individual’s wealth target in time T . When his/her terminal wealth is d, the marginal utility benefit of increasing terminal wealth by insurance is equal to the marginal utility cost of paying the corresponding additional premium. (4) The individual’s optimal insurance strategy can be seen as trying to make his/her terminal wealth reach d as closely as possible, under the constraint 0 ≤ I (y) ≤ (−y)+ . (5) The individual’s optimal insurance strategy can be implemented by buying a put option written on his/her holding asset with a proper strike price. (6) The cost of avoiding risk can be seen as a risk measure, so the put option premium can be considered as a reasonable risk measure. Jarrow (2002) drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem. Acknowledgement The authors thank the anonymous referee for helpful comments. This work is supported by NSFC(70331001). Appendix A Proof of Proposition 4. (1) Suppose I ∗ (y) = 0 for all y ∈ B(y0 , δ). Let I (y) = I ∗ (y) + V (y), where ≥ 0 and V (y) = 1 B(y0 ,δ) (y) is the indicator function of y defined as 1, y ∈ B(y0 , δ) V (y) = 0, y 6∈ B(y0 , δ). Since I ∗ maximize U (I ), we have 0 ≥ U (I ) − U (I ∗ ) = Eu [Y + I (Y ) − H (I )] − Eu Y + I ∗ (Y ) − H (I ∗ ) 0 = E u Y + I ∗ (Y ) − H (I ∗ ) × I (Y ) − I ∗ (Y ) + H (I ∗ ) − H (I ) + o() = E u 0 Y + I ∗ (Y ) − H (I ∗ ) V − R f E(mV ) + o().
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Divide both sides by , and let → 0; we have E u 0 Y + I ∗ (Y ) − H (I ∗ ) V ≤ R f E(mV )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . Divide both sides by P (ω ∈ B(y0 , δ)), and let δ → 0+ ; it can be obtained that u 0 y0 + I ∗ (y0 ) − H (I ∗ ) ≤ R f m(y0 )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . Since I ∗ (y0 ) = 0, we can get u 0 y0 − H (I ∗ ) ≤ R f m(y0 )E u 0 Y + I ∗ (Y ) − H (I ∗ ) . From Eq. (5) and β˜ = β R f , we have u 0 (y0 − H ∗ ) 0 0 u (x − α H ∗ ) ≤ E u 0 Y + I ∗ (Y ) − H ∗ . 0 ˜ (y0 ) βu (2), (3) Following the proof as for part (1), we obtain the necessary conditions stated in Proposition 4. From Lemma 1, at most one condition of the three conditions can hold for a given y 0 , so the necessary conditions are also sufficient. Appendix B Proof of Proposition 11. From Assumption 5, H ∗ > 0. Then by Lemma 1, f (s) is a strictly decreasing function of s. For a given wealth realization y, if y+I ∗ (y) = d, then I ∗ (y) = d−y. At the same time, from Proposition 4, we have 0 < I ∗ (y) < (−y)+ , or y < d < y + . Three situations are discussed as follows: (1) If y ≥ d, it can be concluded that I ∗ (y) = 0, which can be proved in the following way. If I ∗ (y) = 0, we have y + I ∗ (y) = y ≥ d. Since f (s) is a strictly decreasing function of s, we have f (y) ≤ f (d) = MEU. Then by Proposition 4, we have I ∗ (y) = 0. (2) If y + ≤ d, then I ∗ (y) = (−y)+ , which can be proved as follows. If I ∗ (y) = (−y)+ , we have y + I ∗ (y) = y + ≤ d. For f (s) a strictly decreasing function of s, we have f (y + ) ≥ f (d) = MEU. Then by Proposition 4, we have I ∗ (y) = (−y)+ . (3) If y < d < y + , then I ∗ (y) = d − y. This can be proved as follows. If I ∗ (y) = d − y, we have f (y + I ∗ (y)) = f (d) = MEU. At the same time, it can be easily verified that 0 < I ∗ (y) < (−y)+ ; so from Proposition 4, we have I ∗ (y) = d − y. I ∗ (y) can formally be written as y≥d 0, I ∗ (y) = d − y, y < d < y + (14) (−y)+ , y + ≤ d. Three cases for d are discussed as follows. (1) If d < 0, from Eq. (14), we can see that the individual would not take full coverage of their loss, and his/her optimal insurance strategy is I ∗ (y) = (d − y)+ . (2) If d = 0, it can be easily verified that I ∗ (y) = (−y)+ . (3) If d > 0, for y + ≤ d ⇔ y ≤ d and y + ≥ d ⇔ y ≥ d, the optimal insurance strategy can be written as
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0 (−y)+
y≥d y < d.
Since d > 0, the equation above can be simplified as I ∗ (y) = (−y)+ . Thus the sufficient conditions have been verified. Meanwhile, the necessary conditions are obvious. References Arrow, K.J., 1963. Uncertainty and the welfare economics of medical care. American Economic Review 53, 941–973. Arrow, K.J., 1970. Essays in the Theory of Risk-Bearing. North-Holland Publishing Co., Amsterdam. Arrow, K.J., 1974. Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal 1, 1–42.
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