Abstract An insurance model, with realistic assumptions about coverage, deductible andpremium, is studied. Insurance is shown to decrease the variance of the cost to the insured,but increase the expected cost, a tradeoff that places our model in the Markowitz mean-variance model.
In a tightening economy, uncertainty plays an ever increasing role. This has led companiesand researchers to a systematic study of risk and its impact, an effort that comes under thetitle of Enterprise Risk Management (ERM). There are several definitions of ERM (e.g.,Casualty Actuarial Society 2003; COSO 2004; Olson and Wu 2010 and Wu et al. 2011),representing a wide spectrum of theory and applications. In this paper we concentrate onone aspect of ERM, namely the mitigation of risk by means of two basic parameters ofinsurance, coverage and deductible.
Insurance, in particular the problem of optimal coverage and deductible, has been widelystudied in the economic literature, mainly through expected-utility (e.g., Arrow 1974;Machina 1995; Pashigian et al. 1966; Raviv 1979; Schlesinger 1981; Smith 1968) andstochastic dominance (Gollier and Schlesinger 1996; Schlesinger 1997). For a detailed ex-position see Dionne (2000) and references therein.
A common feature in the literature is that insurance coverage is unbounded, so the pur-chase of insurance will fully cover any loss above a deductible. Recent works (Cummins and
C. Gaffney (B)Rutgers Center for Operations Research, New Brunswick, NJ 08903-5062, USAe-mail: [email protected]
A. Ben-IsraelRutgers Center for Operations Research and School of Business, New Brunswick, NJ 08903-5062, USAe-mail: [email protected]
Mahul 2004; Zhou et al. 2010) consider insurance problems with upper bounds on coverage(Cummins and Mahul 2004), or reimbursement (Zhou et al. 2010). Both of these papers usethe expected utility criterion.
A purchase of insurance must increase the expected cost to the insurance buyer (the in-sured), for otherwise the insurance seller (the insurer) cannot remain solvent. This expectedcost increase should be compensated by a decrease in risk, that we establish here directlywithout using expected utility.
We consider an insurance model, suggested by automobile insurance, where coverage isbounded, and the problem of the insured is to determine the optimal levels of coverage C anddeductible D. Our model, described in Sect. 2, assumes the premium to be a function of C
and D, and that loss can exceed the coverage. The premium function is based on real-worldinsurance listings. We establish a lower bound (17) on the premium, under the assumptionthat the insurer will not incur an expected loss. The insurance budget, and its implication forthe coverage and deductible, are discussed in Sect. 3.
First-order optimality conditions for the insured are derived in Sect. 4, and show a mono-tone dependence of the optimal deductible on the coverage. In particular, a zero deductiblemay be optimal. The optimality conditions of the insurer are the same, but what is best to theformer is worst for the latter, and consequently the insurer may not offer the policy desiredby the insured, who must then settle for a non-optimal plan.
Our main results, Theorem 1 and Corollary 2 in Sect. 5, show that the variance of theinsured cost, as a function of C and D, is decreasing in C and increasing in D. This tradeoffbetween the expected cost increase and the variance decrease places our model within theframework of the Markowitz mean-variance model, see Sect. 6.
2 The model
Insurance is a way of mitigating the effects of a loss, an adverse event such as acci-dent, fire, illness, etc. We model loss by a random variable X with a distribution functionF(x) = Prob{X ≤ x}, and a density function f (x) = F ′(x). We assume that F(0) = 0 (i.e.the possible losses are nonnegative), and X has finite moments of all orders, in particularexpected value E X, and variance Var X.
A loss x induces a cost L(x), where L(·) is a nondecreasing function. In the uninsuredcase we assume
L(x) = x, x ≥ 0. (1)
Alternatively, the insured (a person, or a company) may buy from the insurer (an in-surance company) an insurance policy in order to control the cost of X, or make it morepredictable. Indeed, the uninsured case (1) does not exist for automobile drivers, who arerequired by law to buy some insurance.
A policy is characterized by the reimbursement I (x) from the insurer for each possibleloss x, and the premium p paid to the insurer. In turn, the reimbursement I (x) and thepremium p depend, as follows, on two parameters, the deductible D ≥ 0 and the coverageC ≥ D.
Reimbursement For each possible loss x the reimbursement I (x|C,D) is given by,
I (x|C,D) = max{x − D,0} − max{x − C,0} =
⎧⎪⎨
⎪⎩
0, if 0 ≤ x < D;x − D, if D ≤ x < C;C − D, if C ≤ x;
see also (Cummins and Mahul 2004, Proposition 1) and (Zhou et al. 2010, Eq. (3)). Such aninsurance policy is called here a {C,D}-policy. The uninsured case I (x) = 0 corresponds toC = D.
Premium The premium p(C,D) is assumed increasing in the coverage C and decreasingin the deductible D (policies with higher deductible are cheaper). The following examplefrom auto-insurance in New Jersey shows the premium as a function of C and D. Note thatnot all values are permissible, in particular there are minimum coverage and deductible.
Example 1 (Auto-insurance) Standard auto insurance policies in the State of New Jerseycontain Personal Injury Protection (PIP), with premiums depending on the deductible D
and coverage C as shown in Table 1. The table lists 6 possible coverages and 5 deductibles;the required minimums are C = $15,000, and D = $250. The standard premium (100 %in Table 1) is for C = $250,000 and D = $250, and changes depending on the insurancecompany. There is also a 20 % co-payment for losses between the deductible selected (bythe buyer) and $5,000. See NJ Auto Insurance Buyer’s Guide (2011) for details.
For example, Jane chose the minimum coverage C = $15,000 (resulting in a reductionof 13 % from the premium for the standard coverage C = $250,000) and a deductible D =$2,500 (getting a 26 % reduction from the standard premium for D = $250). She pays0.87 · 0.74 = 0.64 times the standard premium.
If Jane has an accident resulting in $10,000 of medical expenses, she pays the first $2,500as deductible, and an additional $500 copayment (20 % of the $2,500 that is left of the first$5,000). The insurance pays the remaining $7,000.
Analyzing the data of Table 1, that is typical for car-insurance, we approximate the pre-mium p(C,D) as a product of two functions, each depending on one of the variables,
p(C,D) = α(D)β(C), (3)
where α(D) is a positive, monotonely decreasing function of D, and the function β(C) ispositive and monotonely increasing. The data suggests that the cost of coverage β(C) isaffine in C, say
β(C) = β(0) + mC, for some β(0) ≥ 0, m > 0, (4)
and the premium is therefore
p(C,D) = α(D)(β(0) + mC
). (5)
This is a departure from the common assumption, in the insurance literature, that the pre-mium is proportional to the expected reimbursement, see, e.g., Cummins and Mahul (2004).
Ann Oper Res
Fig. 1 Contour plot of the PIPpremium in Table 1
Zero coverage is typically not allowed, (for example, in New Jersey the minimum cover-age for auto-insurance is $15,000, see Table 1), and therefore the term β(0) in (4) is just adevice for expressing β(C) in its domain.
Example 2 The data of Table 1 is interpolated by (3) with
α(D) = aD−b, a = 1.94, b = 0.123 (250 ≤ D ≤ 2500), (6a)
β(C) = β(0) + mC, β(0) = 0.9, m = 4.5 · 10−7(15,000 ≤ C ≤ 106
), (6b)
up to a multiplicative constant, and plotted in Fig. 1, where darker color indicates higherpremium. The error of the interpolations (6a)–(6b) at the given points is O(10−2).
Example 3 (α(·) exponential) The monotone dependence of the premium on the deductiblecan be modeled by the exponential function,
α(D) = α(0)e−δD, D ≥ 0, (7)
for some δ > 0 and 0 < α(0) < 1.
Example 4 (Life insurance) In life insurance there is no deductible (D = 0), the premiumdepends on the insured’s age and other factors, and is linear in the coverage C, which usu-ally can be bought in multiples of a standard amount, say $10,000. The underlying randomvariable X is discrete with two values (survival, say x = 0, or death, x = 1) and the reim-bursement (2) simplifies to
I (x) ={
0, if x = 0;C, if x = 1.
Cost and profit For each possible loss x, the cost to the insured is
L(x|C,D) = p(C,D) + x − I (x|C,D) = p(C,D) +
⎧⎪⎨
⎪⎩
x, if 0 ≤ x < D;D, if D ≤ x < C;x + D − C, if C ≤ x;
(8)
Ann Oper Res
Fig. 2 The costs of the insured and insurer for given C,D
see Fig. 2(a). The corresponding profit of the insurer is
R(x|C,D) = p(C,D) − I (x|C,D) = p(C,D) −
⎧⎪⎨
⎪⎩
0, if x < D;x − D, if D ≤ x < C;C − D, if C ≤ x;
(9)
see Fig. 2(b) which shows the loss −R(x|C,D) of the insurer.The expected insured cost is by (8) and (3),
EL(X|C,D) = α(D)β(C) + E X +∫ C
D
(D − x)f (x) dx + (D − C)
∫ ∞
C
f (x) dx. (10)
Similarly, the expected profit of the {C,D}-insurer is,
ER(X|C,D) = α(D)β(C) −∫ C
D
(x − D)f (x)dx − (C − D)
∫ ∞
C
f (x) dx. (11)
It follows from (8) and (9) that
L(x|C,D) = x + R(x|C,D), for all x ≥ 0,
and consequently, the expected cost of the insured equals the (uninsured) expected loss plusthe expected profit of the insurance company,
EL(X|C,D) = E X + ER(X|C,D), (12)
as can be seen also from (10)–(11).If the insurance company is profitable, as is generally the case, an insurance policy as
above does not reduce the expected cost to the insured—if it did, it would qualify as a “freelunch”.
A {C,D}-policy is called actuarially fair if the expected profit ER(X|C,D) is zero, i.e.if
E I (X|C,D) = p(C,D). (13)
Let F̄ (x) = 1 − F(x) = ∫ ∞x
f (u)du. Then,
∫ C
D
F̄ (x) dx =∫ C
D
(1 − F(x)
)dx
Ann Oper Res
= C − D −∫ C
D
F(x)dx
= C − D +∫ C
D
xf (x) dx − xF(x)]CD (using integration by parts)
=∫ C
D
xf (x) dx + C(1 − F(C)
) − D(1 − F(D)
)
=∫ C
D
(x − D)f (x)dx +∫ ∞
C
(C − D)f (x)dx. (14)
Proposition 1 For any pair (C,D), an insurance plan is actuarially fair if the premiumα(D)β(C) satisfies
α(D)β(C) =∫ C
D
F̄ (x) dx. (15)
Proof For ER(X|C,D) = 0 we need, by (11),
α(D)β(C) =∫ C
D
(x − D)f (x)dx +∫ ∞
C
(C − D)f (x)dx, (16)
and the proof follows by noting that the right sides of (14) and (16) are equal. �
Remark 1 Assuming the expected profit to the insurer is nonnegative, it follows from Propo-sition 1 that, for any pair (C,D), the insurance premium satisfies
α(D)β(C) ≥∫ C
D
F̄ (x) dx, (17)
a condition that the premium must satisfy in order to guarantee an expected profit (or break-even) for the insurer.
3 The insurance budget
The budget B available for buying insurance imposes conditions on the coverage C anddeductible D. Assume first that all the budget is spent. Then, by (3),
α(D)β(C) = B, (18)
an equation that can be solved for D as a function of C,
D = α−1(B/β(C)
), (19)
an increasing function, i.e. buying, with a fixed budget, more coverage makes it necessaryto increase the deductible. Because D is nonnegative, the smallest possible coverage corre-sponds to D = 0 in (18), i.e., C must satisfy,
C ≥ β−1(B/α(0)
), (20)
the right-hand side is the smallest possible coverage (corresponding to zero deductible) forthe given budget B .
Ann Oper Res
Fig. 3 Illustration of Example 5
Example 5 (α(·) exponential) In the exponential case (7), the deductible (19) becomes
D = −1
δlog
(B
α(0)β(C)
)
, (21)
where D ≥ 0 if (20) holds, with D = 0 if B = α(0)β(C).Some of the curves (21) are shown in Fig. 3 for β(C) = C, α(0) = 0.6 and δ = 0.005.
Higher curves (i.e. greater deductible) correspond to lower budgets B .
4 First order optimality conditions
We assume throughout that the support of the distribution F is not contained in [D,C], i.e.,
∫ C
D
f (x) dx < 1. (22)
Indeed if∫ C
Df (x) dx = 1 then by (8) the insured cost L(X|C,D) = p(C,D) + D with
certainty. We also assume the premium is given by (5).The expected value (10) is unchanged if there is a positive probability that no loss occurs
(i.e. x = 0). Indeed, E X requires then a Stieltjes integral, but the contribution of the valuex = 0 is zero.
Denote EL(X|C,D) by U(C,D). Its respective derivatives give rise to the first orderoptimality conditions
∂U
∂C= α(D)β ′(C) −
∫ ∞
C
f (x) dx = 0
= mα(D) −∫ ∞
C
f (x) dx = 0, by (4), (23a)
∂U
∂D= α′(D)β(C) +
∫ ∞
D
f (x)dx = 0, if D > 0, (23b)
∂U
∂D= α′(D)β(C) +
∫ ∞
D
f (x)dx ≥ 0, if D = 0. (23c)
Ann Oper Res
Remark 2 Writing (23a) as∫ ∞
C
f (x) dx = mα(D), (24)
we note that the left side of (24) is decreasing in C, and for fixed C, the right side is decreas-ing in D, by the assumption on α(·). It follows that C and D move in the same direction, ahigher deductible D corresponds to a higher coverage C.
If D is a differentiable function of C we get from (23a)
dD
dC= −
∂2U
∂C2
∂2U∂D∂C
= − f (C)
mα′(D), (25)
which is positive by the assumptions on the premium function α(·).
Remark 3 The first-order optimality conditions (23a)–(23b),
Prob{X ≥ C} =∫ ∞
C
f (x) dx = mα(D) (26a)
Prob{X ≥ D} =∫ ∞
D
f (x)dx = −α′(D)β(C) (26b)
can be solved for D and C. For D = 0 the optimality condition (23c) gives the inequality
α′(0)β(C) + 1 ≥ 0, or by (4),
C(0) ≤ − 1
m
(1
α′(0)+ β(0)
)
, (27)
an upper bound on the coverage, corresponding to a zero deductible. There may be externalupper bound Cmax on coverage, say
C ≤ Cmax. (28)
For example, in the auto-insurance data of Example 1, Cmax = $1 million.
Corollary 1 A sufficient condition for a positive deductible is∫ ∞
− 1m ( 1
α′(0)+β(0))
f (x) dx > mα(0). (29)
Proof It follows from Remark 2 that the lowest possibly optimal coverage C(0) correspondsto D = 0. C(0) is determined from (26a),
∫ ∞
C(0)
f (x) dx = mα(0),
and in addition must satisfy (27). These two conditions are satisfied only if∫ ∞
− 1m ( 1
α′(0)+β(0))
f (x) dx ≤ mα(0), (30)
which is then a necessary condition for D = 0. The reverse inequality, (29), is thereforesufficient for a positive deductible. �
Ann Oper Res
Optimality conditions for the insurer It follows from (12) that the first order optimalityconditions are the same as those of the insured, which is to be expected since this is a zero-sum game, and the best outcome for one player is the worst for the other. It follows that theinsurer may not offer the optimal policy desired by the insured, who must then settle for anon-optimal plan.
The next example illustrates, for the commonly occurring Gamma distribution (see Bor-toluzzo 2011; Hewitt and Lefkowitz 1979; Jorgensen and De Souza 1994, and Smyth andJorgensen 2002), the calculation of C and D satisfying the necessary optimality conditions.
Example 6 (The Gamma Distribution) Let the random variable X have the Gamma distri-bution with scale 1/λ and shape k ≥ 1,
f (x) = λk
Γ (k)xk−1e−λx, x ≥ 0, (31)
in particular,
Γ (k) = (k − 1)!, (32)
if k is integer, in which case (31) is the Erlang distribution. Let α(D) be given by (7),
α(D) = α(0)e−δD, D ≥ 0.
Zero deductible For D = 0 to be optimal we must have, by (30),
∫ ∞
S
f (x) dx = λk
Γ (k)
∫ ∞
S
xk−1e−λx dx = e−λS
k−1∑
i=0
(i!)−1(λS)i ≤ mα(0),
where
S = − 1
m
(1
α′(0)+ β(0)
)
. (33)
Positive deductible Equations (26a)–(26b) become
e−λC
k−1∑
i=0
(i!)−1(λC)i = mα(0)e−δD, (34a)
e−λD
k−1∑
i=0
(i!)−1(λD)i = α(0)β(C)δe−δD (34b)
or,
eδD−λC = mα(0)∑k−1
i=0 (i!)−1(λC)i, (35a)
e(δ−λ)D = α(0)β(C)δ∑k−1
i=0 (i!)−1(λD)i. (35b)
Ann Oper Res
From (35a) we get D as a function of C,
D = 1
δ
(
log
(mα(0)
∑k−1i=0 (i!)−1(λC)i
)
+ λC
)
(36)
which can be substituted in (35b) to give an equation for C.Figure 4 gives some contour lines of the expected cost (10) for the Gamma distribution
with k = 2 and λ = 10−3. For the premium (7) we use δ = 10−2 and α(0) = 0.5, and for (4)we use β(0) = 50. In Fig. 4(a) the slope m in (4) is m = 0.25, and in Fig. 4(b), m = 0.125.
In these figures, lighter shades correspond to lower costs, and the optimal pair {C,D} isindicated by the brightest spot. Figure 4(b) illustrates that a zero deductible may be optimal,since the unconstrained minimum gives a negative deductible. The optimal deductible D = 0lies on the closest contour (where D ≥ 0) to the theoretical optimum.
Fig. 4 Contour lines of (10), theexpected cost
Ann Oper Res
5 Variance
The insured cannot expect to find an insurance plan that will lower his expected cost, becauseby (12) this would entail an expected loss for the insurer. In this section we establish thatinsurance lowers the variance of the cost, which is a rationale for buying insurance.
We compute the variance of loss when insured, and compare to the variance when thereis no insurance. Let X be the loss when there is no insurance, so we have
Var X = E X2 − (E X)2. (37)
For convenience we write (8) as
L(x|C,D) = p(C,D) + x + φ(x|C,D), (38)
where
φ(x|C,D) =
⎧⎪⎨
⎪⎩
0, if 0 ≤ x < D;D − x, if D ≤ x < C;D − C, if C ≤ x.
(39)
In particular, if C = D,
φ(x|C,C) = 0, x ≥ 0. (40)
We assume that C,D are given, and do not write them if they are not needed explicitly, thusp(C,D) is abbreviated p, φ(X|C,D) abbreviated φ(X), L(X|C,D) is written L(X), etc.
The variance of L(X) is therefore
VarL(X) = Var X + Varφ(X) + 2 Cov(X, φ(X)
)
= Var X + Varφ(X) + 2[E(Xφ(X)
) − E X Eφ(X)]. (41)
The following theorem relates the variance VarL(X|C,D) to the coverage C and de-ductible D.
Theorem 1 Let F,C,D satisfy (22),
∫ C
D
f (x) dx < 1.
Then
∂
∂CVarL(X|C,D) < 0, (42)
and
∂
∂DVarL(X|C,D) > 0. (43)
The proof is given in Appendix.The variance VarL(X|C,D) is thus a decreasing function of C (for fixed D) and an
increasing function of D (for fixed C):
Ann Oper Res
Corollary 2
(a) Let 0 < D < C1 < C2, and let
∫ C2
D
f (x)dx < 1.
Then
VarL(X|C2,D) < VarL(X|C1,D) < Var X. (44)
(b) Let 0 < D1 < D2 < C, and let
∫ C
D1
f (x)dx < 1.
Then
VarL(X|C,D1) < VarL(X|C,D2) < Var X. (45)
Proof It follows from (40) that for C = D,
VarL(X|C,C) = Var X. (46)
(a) The left inequality in (44) follows from (42) and the right inequality from (46), writingVar X as VarL(X|D,D).
(b) is similarly proved. �
Using (58) in Appendix,
1
2
∂
∂CVarL(X)
=∫ ∞
C
f (x) dx
[
C
∫ C
0f (x)dx − D
∫ D
0f (x)dx +
∫ D
0xf (x) dx
+∫ ∞
C
xf (x) dx
]
−∫ ∞
C
xf (x) dx,
approaching zero as C → ∞ provided
limC→∞
∫ ∞
C
xf (x) dx = 0 (47)
holds. Our assumption that E X is finite precludes heavy-tailed distributions such as the log-Cauchy distribution, where (47) does not hold.
Example 7 Consider the Gamma distribution (31) with k = 2 and λ = 10−3. Figure 5 showscontour lines of VarL(X) in the (C,D)-plane. For convenience we represent these contourlines as the curves β − VarL(X) = 0 for several values of β . The red line is D = C, corre-sponding to the uninsured case (where the variance is Var X). Lower curves correspond tolower values of VarL(X). These contour lines illustrate the flattening out of the variance,and the fact that above a certain threshold value, increasing the coverage C has a negligibleeffect on the variance.
Ann Oper Res
Fig. 5 Contour lines of (41), thevariance
6 Mean-variance analysis
A {C,D}-insurance plan was shown to reduce the variance of the costs incurred by theinsured, while increasing their expected value. This leads naturally to the Markowitz mean-variance model (Markowitz 1952, 1959), that is adapted to give the objective
minC,D
(E Z + λVar Z), (48)
where Z is the random cost in question. The parameter λ expresses the tradeoff betweenthe mean (expected cost) and variance, and represents the decision maker’s attitude towardsrisk.
In (48), Z = L(X|C,D) if insurance is purchased, and Z = X in the uninsured case.There is a critical value λ∗ where a Markowitzian decision maker would be indifferent be-tween buying or not buying insurance, and that value is given by
EL(X|C,D) + λ∗ VarL(X|C,D) = E X + λ∗ Var X
or
λ∗ = EL(X|C,D) − E XVar X − VarL(X|C,D)
. (49)
For λ > λ∗ it is optimal to buy insurance, and for λ < λ∗ insurance cannot be justified in theMarkowitz model, although—in the case of automobile insurance—it is required by law.
7 Conclusion
We studied an insurance model, suggested by automobile insurance, and showed that thevariance of the cost to the insured is a decreasing function of the coverage and an increasingfunction of the deductible. Since insurance increases the expected cost, there is a trade-off between the cost increase and the risk decrease, suggesting a Markowitzian approachto modeling the problem of optimal insurance coverage—minimizing a weighted sum ofexpectation and variance.
Ann Oper Res
Our model considered a single (“typical”) customer, but in practice an insurer has manycustomers with different loss distributions, and their totality determines the expected valueand variance of the insurer profit. The insurer thus has an analogous trade-off between ex-pected profit and variance, and he can aim for the “right” customers by designing policiesthat will attract them. We defer this for future study.
Appendix: Proof of Theorem 1
We prove (42)
∂
∂CVarL(X|C,D) < 0
given that (22) holds,∫ C
D
f (x) dx < 1,
which implies that either∫ ∞
C
f (x) dx > 0, (50)
or∫ D
0f (x)dx > 0, (51)
hold.From (39),
Eφ(X|C,D) =∫ C
D
(D − x)f (x) dx + (D − C)
∫ ∞
C
f (x) dx, (52)
and
Varφ(X|C,D) = Eφ2(X|C,D) − (Eφ(X|C,D)
)2
=∫ C
D
(D − x)2f (x)dx + (D − C)2∫ ∞
C
f (x) dx
−(∫ C
D
(D − x)f (x) dx + (D − C)
∫ ∞
C
f (x) dx
)2
. (53)
By (41) the partial derivative of VarL(X) w.r.t. C is
∂
∂CVarL(X) = ∂
∂CVar X + ∂
∂CVarφ(X) + 2
∂
∂C
(E[Xφ(X)
] − E X Eφ(X)). (54)
We calculate next the three derivatives on the right side.
(a) Clearly,
∂
∂CVar X = 0. (55)
Ann Oper Res
(b)
∂
∂CVarφ(X) = (2C − 2D)
∫ ∞
C
f (x) dx + 2∫ C
D
(D − x)f (x) dx
∫ ∞
C
f (x) dx
− 2(C − D)
(∫ ∞
C
f (x) dx
)2
= 2∫ ∞
C
f (x) dx
[
C
∫ C
0f (x)dx
− D
∫ D
0f (x)dx −
∫ C
D
xf (x) dx
]
(56)
(c) From (39),
E(Xφ(X)
) − E X Eφ(X) = D
[∫ ∞
D
xf (x)dx − (E X)
∫ ∞
D
f (x)dx
]
− C
[∫ ∞
C
xf (x) dx − (E X)
∫ ∞
C
f (x) dx
]
−[∫ C
D
x2f (x)dx − (E X)
∫ C
D
xf (x) dx
]
.
Therefore
2∂
∂C
(E[Xφ(X)
] − E X Eφ(X)) = 2
[
E X∫ ∞
C
f (x) dx −∫ ∞
C
xf (x) dx
]
. (57)
Substituting (55), (56), and (57) in (54) we get
1
2
∂
∂CVarL(X) =
∫ ∞
C
f (x) dx
[
C
∫ C
0f (x)dx − D
∫ D
0f (x)dx −
∫ C
D
xf (x) dx + E X]
−∫ ∞
C
xf (x) dx
=∫ ∞
C
f (x) dx
[
C
∫ C
0f (x)dx − D
∫ D
0f (x)dx
+∫ D
0xf (x) dx +
∫ ∞
C
xf (x) dx
]
−∫ ∞
C
xf (x) dx (58)
Rearrange (58) to get
1
2
∂
∂CVarL(X) = C
∫ ∞
C
f (x) dx
∫ C
0f (x)dx +
∫ ∞
C
f (x) dx
∫ ∞
C
xf (x) dx (59)
−∫ ∞
C
xf (x) dx +∫ ∞
C
f (x) dx
(∫ D
0xf (x) dx − D
∫ D
0f (x)dx
)
.
(60)
Ann Oper Res
The right side of (59) can be written as
= C
∫ ∞
C
f (x) dx
∫ C
0f (x)dx +
∫ ∞
C
xf (x) dx
(∫ ∞
C
f (x) dx − 1
)
= C
∫ ∞
C
f (x) dx
∫ C
0f (x)dx −
∫ ∞
C
xf (x) dx
∫ C
0f (x)dx
=∫ C
0f (x)dx
(
C
∫ ∞
C
f (x) dx −∫ ∞
C
xf (x) dx
)
< 0 if (50) holds, because then∫ ∞
C
xf (x) dx > C
∫ ∞
C
f (x) dx.
Similarly, (60) is negative if (51) holds, because then
D
∫ D
0f (x)dx >
∫ D
0xf (x) dx.
This completes the proof of (42) if any of the conditions (50)–(51) holds.The inequality (43) is similarly proved.
References
Arrow, K. (1974). Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal, 1, 1–42.Bortoluzzo, A. (2011). Estimating total claim size in the auto insurance industry: a comparison between
Tweedie and zero-adjusted inverse Gaussian distribution. BAR, Brazilian Administration Review, Cu-ritiba, 8(1). Available from http://www.scielo.br/scielo.php, access on 09 Dec. 2012.
Casualty Actuarial Society, Enterprise Risk Management Committee (2003). Overview of Enterprise RiskManagement, May 2003.
COSO (2004). Enterprise Risk Management-Integrated Framework Executive Summary, Committee of Spon-soring Organizations of the Treadway Commission.
Cummins, J. D., & Mahul, O. (2004). The demand for insurance with an upper limit on coverage. The Journalof Risk and Insurance, 71, 253–264.
Dionne, G. (Ed.) (2000). Handbook of insurance. Boston: Kluwer Academic.Gollier, C., & Schlesinger, H. (1996). Arrow’s theorem on the optimality of deductibles: A stochastic domi-
nance approach. Economic Theory, 7, 359–363.Hewitt, C. C. Jr., & Lefkowitz, B. (1979). Methods for fitting distributions to insurance loss data, in Proceed-
ings of the casualty actuarial society, LXVI, pp. 139–160.Jorgensen, B., & De Souza, M. C. P. (1994). Fitting Tweedie’s compound Poisson model to insurance claims
data. Scandinavian Actuarial Journal, 69–93.Machina, M. J. (1995). Non-expected utility and the robustness of the classical insurance paradigm. The
Geneva Papers on Risk and Insurance Theory, 20, 9–50.Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7, 77–91.Markowitz, H. M. (1959). Portfolio selection: efficient diversification of investments. New York: Wiley.New Jersey Auto Insurance Buyer’s Guide A-705(4/10) (2011)Olson, D. L., & Wu, D. (2010). Enterprise risk management models. Berlin: Springer.Pashigian, B. P., Schkade, L. L., & Menefee, G. H. (1966). The selection of an optimal deductible for a given
insurance policy. The Journal of Business, 39, 35–44.Raviv, A. (1979). The design of an optimal insurance policy. The American Economic Review, 69, 84–96.Schlesinger, H. (1981). The optimal level of deductibility in insurance contracts. The Journal of Risk and
Insurance, 48, 465–481.Schlesinger, H. (1997). Insurance demand without the expected–utility paradigm. The Journal of Risk and
Insurance, 64, 19–39.Smith, V. (1968). Optimal insurance coverage. Journal of Political Economy, 76, 68–77.
Smyth, G. K., & Jorgensen, B. (2002). Fitting Tweedie’s compound Poisson model to insurance claims data:dispersion modeling. ASTIN Bulletin, 32, 143–157.
Wu, D., Olson, D., & Birge, J. (2011). Introduction to special issue on “Enterprise risk management in oper-ations”. International Journal of Production Economics, 134, 1–2.
Zhou, C., Wu, W., & Wu, C. (2010). Optimal insurance in the presence of insurer’s loss limit. Insurance.Mathematics & Economics, 46, 300–307.
