Optimal multi-period insurance contracts

Insurance: Mathematics North-Holland

and Economics

2 (1983) 199-208

199

Optimal multi-period insurance contracts * Itzhak VENEZIA The Hebrew Universi(v, Jerusalem, Israel

Haim LEVY The Hebrew University, Jerusalem, Israel and The Department of Finance, University of Florida, Gainesville,

FL. 3261 I, USA

Received 13 Sepzmber 1982 Revised 7 February 1983

In multi-period insurance contracts (such as automobile insurance contracts). unlike single-period ones. the premiums that the insured must pay increase whenever he Files a claim. Hence, the buyer faces a problem that is absent in one-period models. namely: he must determine for which damages he should file a claim and for which he should not. The optimal claims policy of the buyer is presented for a large class of insurance contracts. It is shown that the buyer will file a claim only if it is larger than some critical value. Based on this it is shown that the buyer prefers a contract that provides full coverage above a deductible for damages that exceed his critical value. In this case the optimal contract is not unique since the buyer is indifferent to the form of the contract for damages below his critical value. It is shown, however. that as in one-period models (Arrow (1963, 1974)) there exists an optimal contract that provides full coverage above a deductible. In multi-period setting. however, the buyer will file a claim only if the damage is sufficiently higher than the deductible. It is also shown that the buyer prefers a strictly positive deductible. Unlike the one-period case (Mossin (1968)). this result holds true even if the premium rates equal the expected payments. Keywork

Premiums, Optimal critical claim size, Deductible.

1. Introduction

In multi-period insurance contracts, the premiums which the insured must pay increase whenever he files a claim. This practice is very popular in automobile insurance, but can also be found in medical and theft insurance. In multi-period insurance contracts, the buyer faces a problem which does not exist in one-period contracts. Namely: the buyer must decide for which damages he should file a claim. and for which he should not, bearing in mind that if he makes a claim, his future rates will increase. This type of insurance, and the optimal claims strategy of the buyer have been analyzed by some authors. Moinar and Rockwell (1966), Lemaire (1976), ( 1977), Seal (1969), Venezia and Levy ( 1980), and Von Lanzenauer (1971), investigated several aspects of dynamic automobile insurance contracts. Keeler, Newhouse, and Phelps (1977) analyzed dynamic models in medical insurance. Most of these authors assumed that the insurance contracts are of the form of full coverage above a deductible (henceforth FCAD), and in practice, indeed, most insurance contracts are of this form. However, whereas it has been proved (by Arrow (1963, 1974)) that optimal one-period contracts provide full coverage above a deductible, it has only been conjectured (see, e.g., Keeler, Newhouse and Phelps (1977), p_ 641) that the same holds true also for dynamic models. Thus, one of the objectives of this paper is to extend .4rrow’s results to the case of dynamic insurance contracts. Our second objective is to examine whether Mossin’s (1968) resuits, showing that the buyer prefers a nonnegative deductible, hold in our case too. The paper is constructed as follows: In Section 2 we describe the market of automobile.insurance. We aiso present the optimal claims strategy of the buyer, assuming that the contracts he holds at any period are l

The authors wish to thank an anonymous

0167-6687/83/503.00

referee for his helpful comments

@ 1983 EIsevier Science Publishers

B.V. (North-Holland)

and suggestions.

I. Venezia, H. Levy / Insurance contracts

200

exogenously given. Based on this we show in Section 3 that the buyer preferes a contract which provides full coverage above a deductible for damages in the relevant range (i.e. damages that exceed his critical value). In this case the optimal contract is not unique since the buyor is indifferent to the form of the contract at damages below his critical value. We show however, that as in one-period models (Arrow (1963, 1974)) there exists an optimal contract that provides full coverage above a deductible. In multi-period setting however, the buyer will file a claim only if the damage is sufficiently higher than the deductible. We also show in this section that the buyer prefers a strictly positive deductible. Unlike the one-period case (Mossin (1968)) this result holds true even if the premium rates are fair (i.e. even if they equal the expected payments).

2. The model Buyers of automobile insurance are usually classified by the sellers into a finite number, J + 1, of risk categories. These categories are deterained by the past history of claims of the buyers ‘. The sellers assume that if the buyer has made many claims in the past, he is more likely to file claims in the future and thus he should be charged higher premium rates 2. A common system of insurance companies for revising premium rates of buyers according to their past claims is the ‘Bonus-Malus System’ (see, e.g., Seal (1969), Lemaire (1976)). According to this system, if a buyer that belongs to categoryj < J at time t makes a claim at that period, he is moved to category j + 1 at (t + 1). If he does not make a claim, he is moved to j - 1 (unless j = 1, in which case he stays at that category). Category J may be interpreted as the highest category for which there is demand for insurance. It is assumed that at category J + 1 the buyer is better off without insurance (alternatively, we could assume that at category J + 1 the buyer is denied insurance). The indexj is a measure of the riskiness of the buyer and thus it determines the premium rates of the buyers at that category. The total premium that the buyer at categoryj pays, depends on the contract Z,(x) that he purchases. The insurance contract Zj(x) is a function that specifies for each claim x, the payment Z,(x) that the buyer will receive (e.g., if the policy gives full coverage above a deductible, say m, then Z,(x) = 0 if x G m and Zj(x) = x - m if x > m). The price of contract Zj is denoted by ‘I( I,). Since a higherj means a higher risk, it is assumed that for any contract I, ‘I( I) > q_ ,( I). 3 The indemnity function Z,(x) is a continuous nondecreasing function of the damage x and it satisfies 0 G Z,(x) Q x and Z,(O)= 0. For ease of exposition we also assume differentiability of Zj(x) except - possibly - at finitely many points. A buyer facing this market has two types of decisions to make: (1) which contract to purchase; and (2) given the contract he has purchased, he must determine a claims policy, i.e. a rule that specifies for which damages he should file a claim, and for which he should not. The second problem, which is unique to multi-period insurance models, is the focus of the rest of this section. Assuming that the contracts the buyer holds are exogenously given. we derive below the optimal claims policy. Consider an insured of category j that holds at time t an insurance contract Z,,(x). Assume that his multi-period utility function is additive of the form m

U(C,,C,+,,. . . . c, ,... ) =

c Pf-Wc,)

(2.1)

‘T=,

where c, is consumption at period 7, u( -) is an increasing, strictly concave and twice differentiable one-period utility, and /3 (/3 < 1) is a discount factor. The insured’s income at time T is denoted by A,, and the damages for a categoryj buyer are denoted by

Risk categoriesalso depend on characteristicssuch as age, derivingexperience,economicstatus, etc. Here we consider these characcteristics as beinggiven. A Bayesianrationalecan be givenfor this behaviorof the sellers,but this is beyond the scopeof this paper. As we shalllater show,the pricer,(1) alsodependsOIIthe claimsstrategyof the buyer. Here we assume that if the claim strategy of a category (j + 1) buyer is the same as that of a categoryj buyer the former will pay more.

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_C,(the time index is suppressed from 2 since it is assumed that the p&ability distributions of damages are independent of time). Denoting the cumulative probability distribution function of .i, by F,(s), we represent the assumption that a category ( j -t- 1) buyer is more accident-prone than a category j buyer by assuming that F,+,(x) G F,(x) for all x, with strict inequality for some values of X. Note. Throughout the paper we shall use the same F,(X) to denote the beliefs of both the buyer and the seller. This assumption is made to facilitate notation. The results of the paper. however. remain the same even if the beliefs of the buyer were different from those of the seller (say the buyer’s were F,“(X), and the seller’s F,(X)). It is also possible that the beliefs of the buyer do not depend onj, but this. again, will not affect the results. It is also assumed that E[u( A - 2,)) is finite. This implies. since the _?I are nonnegative and since F,(x) 2 q+ i(x), that Qu( A - c?,)] are finite for aiij < .Z. Assuming that ail damages are repaired, we obtain that the buyer’s consumption at period 7, F,, is [A - q(Zjr) - zj f Z,,(_C,)] if the buyer suffers a damage k, and files a claim. and [A - ‘;( Z,,) - z,] if the buyer does not file a claim. The purpose of the insured is to choose a claims policy that maximizes the expected value of (2.1). To find this policy the following definitions and notations are required. Let V,,(J+) denote the maximum expected discounted utility of the insured from period I forward, conditional that at t he had a damage J and that he always follows an optimal claim strategy. Denote by CI:, the expectation of V,,(2,). i.e. W,, = SV,,(l?,)], where the expectation E is taken with respect to F,(X). y, thus represents the welfare (infinite period expected discounted utility) of the insured given that at period I he belongs to category j, and assuming that he always pursues an optimal claim strategy. Suppose the insured had a damagey at t, and he has to decide whether or not to file a claim. If he files a claim his utility at t will be u[A, - ‘/ - y -t Z,,(y)]. Starting next period, his risk category will be (j + 1). hence his infinite period expected utility from period (r + 1) forward is, by definition, W, + I.,+ ,. This expected utility is worth PU;, i.,+ ,when discounted to period t. Thus. if the buyer files a claim, his infinite period discounted expected utility zt t will be [u(A, -,r,_-y + Z,,(y))+@W,+,,,+ !], Similarly. it can be shown that if the buyer does not file a claim at z, his infunte period expected utility will be [u( A, - r, -y) + PW,- ,.r+ I 1. It then follows. since V,,(y) represents an optimal policy. that (2.2)

~,(y)=max{u(A,-r.-y)+PW,-,.,,,,u(A,-r,-~+Z,,(v))+PW,+I.,+ij.

The optimal policy of the insured can therefore be inferred from (2.2). A claim will be filed if the damage satisfies ~,t(Y)~u~At-r,-Y)-~(AI-5-Y+z,,~Y))~P~W,~l.,+l

-

w,-,.,+I)*

(2.3)

In order to determine the optimal policy the W,t must be evaluated. For this two alternatives are possible: (1) To assume that the planning horizon is a finite (but arbitrarily large) integer, T. In this case ?.r+ I = 0 for alli and the Wlr can be computed recursively for f G T from (2.2) and using the fact that w,, = wy~)I(2) To assume that all the A, are the same and equal to some A, and that the planning horizon is infinite. In what follows we only consider the second alternative {the stationary case). In this case, the situation of a buyer at categoryi at time ? is the same as that of a buyer belonging to category j at (t + 1). ?hus the index c can be dropped, and the condition (2.3) for the buyer to file a claim can be replaced by

In Appendix 1 we show that the functions h,(y) are continuous and nonincreasing. It hence follows that for eachi. there exists a critical value y, such that a claim will be filed only if the damage Y is larger than or

cyual to_y,. &II Appendix 2 WCshow that the functions W, are finite and unique and suggest a method for their evaluation. On the basis of the W, the optimal claims strategy can be evaluated from (2.4). It is noted

from (2.4) that they! must satisfy (2.5)

No/c ‘f’his value of W, + , is obtained if we assume that once the insured reaches category (J + 1) he stays

there forever. In this case each period his expected utility is E(u(A - ZJ)], and hence his infinite period discounted expected utility is (I - P)-‘E[u(A - T,)]. Alternatively we could assume that the buyer stays at category (J + 1) only one period and then always moves to category J next period (he cannot make a claim if he does not buy insurance). In this case W,, , = E[u(A -a,)] +pWJ. The same results are obtained under these two alternatives. Moreover, the welfare of an insured at category (j - 1) is higher than that of an insured in category (j -I- I ) since the premiums the former has to pay are lower; hence W,_ , z I+$+, . The case wl = W,.+, for some j is trivial since it implies that the bonus-malus system is ineffective for a categoryj driver. Thus we shall assume that W, > W,,. , for all j = I,. .., J. It then follows from (2.5), since I,( .) 2 0, I,(O) = 0, and since M(+) is increasing, that the critical values y, are strictly positive. Since W, = E[L$(y’)] and since for y < _P,the insured does not file a claim and for y > _y,he files a claim, it follows that w,= l”[+

-r,-y)+PW,-,]d
J [( A -

-t

14

r, - .1~ -t

f,(y))

-t-flM;+,]dF,(y)7 j=

i+.--3J.

(2.6)

,‘;

In the next section we remove the asstimption that the contracts are exogenously given and analyze some of the properties of the optima1 (from the insured’s point of view) insurance contracts.

3. Optinlar insurmce contracts Arrow’s result that full cdvcrage above a deductible is the optima1 indemnity contract for the one-period model is modified to show that in the multi-period model the optimal contract provides full coverage above L\ dcductiblc for damages above a critical value (FCADACV). Formally, the contract l,(x) = x - m, for X >s;, where _Y,is the critical value of damage and m, is the deductible. The form of the contract is arbitrary for x <
(1+$)/m+)dr;(x)

(3.1)

Y/

where h, > 0 is some ‘loading factor”. and it is assumed that the X, satisfy X, G X,, ,. This assumption is needed since otherwise it may happen that a category (j + 1) buyer will pay a lower premium than a category j buyer (e.g., if h, is higher enough than X,, , 1. In what follows we show why the optima1 contract I,( .) provides FCADACV. The idea of the proof is similar to that found in Arrow (1963). Arrow showed that, in a one period model, it is optimal to have (l(x) - x) a constant whenever I(x) > 0. Here we show that
It is assumed that the insurance company knows y,. This assumption

is plausible since insurance companies

have many observations

of claims of its policy holders. However,

since y, depends on A and u( .), this implies that the seller should discriminate

between

buyers on the basis of their income (A).

which they seldom do. Our conjecture

between

buyers on these grounds are more political

than economical.

is that the reasons for not discriminating

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proof to the multi-period case there are two additional problems that must be considered, To explain these problems we must briefly review, however, Arrow’s proof. Suppose Z,(s) is an optimal contract. Consider a perturbation of !,(m), namely <(a) = I,(s) + g( B), where g(a) and its derivative g’( *) are small and g(x) = 0 for 0 6 x G y/. Denoting by Li( g) the effect of a perturbation g on the expected utility of the insured and since fi( .) + g( e) is also a feasible contract, then (3.2) should be nonpositive if the actuarial value of g is zero (i.e. if /Omg(x)dF,(x) = 0). Using Taylor’s expansion we have

The idea of the proof is that if l,(*) does not provide FCADACV, then one can construct a function g(e) such that L,(g) > 0. If I/( 4) does not give FCADACV, then there exist x, and x2 such that II, and Z,(x,) are positive and Z,(x,)-x, < Zj(x2)-x2 (or !,(x,) -x, > Z,(x,) -x2, which is a similar case), Construct g(a) to be slightly positive around x,, slightly negative around x *, and zero otherwise, making sure that j$g(x)dq(x) = 0. Since u’ is decreasing,

Consequently, L,(g) > 0, contradicting the optimality of Z,( *). This proof should be modified (the required modifications and the complete proof are given below) in a multi-period model because of the following two reasons: (1) I,( *) and <.( m)need not have the same actuarial value since the critical values corresponding to these contracts need not be the same. Thus the 5 appearing in the first term of (3.2) is not necessarily the same as that appearing in the second one. (2) In the multi-period model one has to maximize w, i = 1,. . . , J, and the 4 are the solutions of the simultaneous equations system (2.6). Now, any perturbation of Z,(a) will change not only w, but also W;, i *j. Thus showing that Lj(g) > 0 is not sufficient to contradict the optimality of lj( e) (e.g., if this perturbation will decrease K, i rj). We note that insurance companies usually offer contracts which specify the deductible, but not the critical values, which are endogenously determined. Thus, one would prefer a definition of the class of optimal contracts which does not refer to these critical limits. Indeed, we show below that there always exists an optimal contract which provides full coverage above a deductible (for all damages). It thus follows that the class of insurance contracts providing FCAD contains an optimal contract for any potential buyer. We summarize the first part of the section by: Theorem 3.1. Zf at any catego
prefers

Proof. Suppose that when at category i, the optimal contract is Z,(x), the optimal critical value is y,, and the corresponding expected utility is W;, i = 1,. . . , J. Suppose that for some category j, the optimal contract t,(s) does not provide FCADACV. In this case there exist x,, x2 >y such that Z,(x,)> 0, !,(x2)> 0 and (Z,(x,)-x,)< (Z,(x,)-x,). Define the contract q(x)= Z,(x)+g(x), where g(x) is defined as follows: P2E

g(x)=

-PIE

i 0

ifx, G x G x, + S, ifx2
(3.3)

otherwise,

where pk denotes the probability are small enough numbers.

that the damage will lie in the interval (x,, xk + a), k = 1, 2, and E and S

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Itwrrrcmce conlructs

The function g(x) was constructed so that for a fixed critical limit y,, the actuarial values of Z,(x) and 4(x) would be the same. Suppose now that when at category i *.j the buyer purchases f,(x) and when atj 5 purchases J(x) and denote the maximal obtainable expected utilities corresponding to this strategy by W;. In what follows we show that F& W, for i =z I ,. . , , .I and i?$ > W, for some k, thus contradicting the assumption that the r,(x) arc optimal and implying that no contract can be optimal unless it provides FCADACV. Lcl * I” I ,...,d, denote the expected utilities of the buyer if he holds f,(x) when at i *j and J(x) when at j, and if he maintains the nonoptimal critical values y, (the same critical values he optimally determines if he purchases Z,(x) when at i = I,. . ., J). It then follows from the nonoptimality of they, that W;> R, i- I,..., J. Since we show in Lemma I that R > lJ$, i = 1,. . . , J, and ak > ‘wk for some k, it follows that q> U: for i= l,..., J and wk > W, for some k, contradicting the optimality of Zi( .). To show that there always exists an optimal contract which provides FCAD, we note that the form of the optimal contract Z,(x) is arbitrary for x G y,. Thus, if I,(x)= x - m, for x a?,, we may choose an optimal FCAD contract f:(x) as follows: x - nz, IT(x) =

for x )_ “,,

i 0 for x < n2,.

Z,!(x) is equivalent to Z,(x) for both the buyer and the seller, hence it is optimal too. Lemmal.

@,i/,q,i=l

EI

, . . . , J, and l%‘L> W, for some k.

Proof. From (2.6) it follows that W, satisfies ~=JU~‘(N(~,(X))+P~-,)~~(~)+~~(U(~,(~))+~~+,)~F,(X) J,, where+,(x)=A-q(Z,)-~and#~(x)=A--_q(Z,)-x+Z,(x). Likewise, if the buyer purchases contract Z,(x) and maintains the nonoptimal satisfies

critical value _Y,,then ei

(3.4) where s,(x) = n -Y,(J)-x and $,(x)=A -v,(q)-x+<(x). We note from the definition of ~(a) in (3.3) that G,(x) and #,(x) are the same outside the intervals [x,, xI, + 61, k = 1, 2, and q(x)= $(x) for all x. We therefore obtain, by adding and subtracting [A+s”[$j(x)]d.~(x),

k = 192,

from the second term on the right-hand side of (3.3) that (3.5) where (3.6) From the concavity of II, and since Z,( x,) - x, < I,( x2) - x2, it follows that K, > 0. For i *j the e satisfy (since at i *j the buyer purchases the contract Z,(x)) ti

=~“b[@~,(x)] +P@-,}dF,(x) +/m{u[q,(x)] +fHi’., ,)de(x) 0

where K, = 0 for i fj.

.,;

+ K,

(3.7)

Defining 0, = @ - q, i = I,. . . , J, D, = D,, and DJ+ , = 0, we can rewrite (3.5) and

I. Veneziu, H. Levy / Insurance contracts

205

(3.7) as D, + W, = (Jb;[u(lPi(x))

+PF-IId&

+/m[ll(4i(x) .I’,

+pDj._IF;.(yi)+pD,+IGi(yi)+Ki,

+BF+l]dC(x)}

i=l,*..,J,

(3.8)

where Gi( y,) = 1 - &(y,). However, since the term in braces in (3.8) is by definition D, satisfy Di=/3Di_,1;1(yi)+/?Di+,Gj(yi)+Ki,

i= l,...,

w, it follows that the

J.

(3.9)

This system of J equations in J unknowns (the 4,) can be written in matrix form as B.D-K

(3.10)

where D==(D ,,.,.,

DJ)‘, K=(K

(1-PC)

-46

-PF2

1

B=

,,...,

K,)’ and the (J X J)

matrix B is defined by

0

-PG $5

1 *

I

lo

-PC,

-PFJ

1

where I;; = &(yi) and Gi = Gi( yi). It is easy to verify that all diagonal elements of B and all of its row sums are positive, and that its off-diagonal elements are nonpositive. It then follows the B- ’ contains only nonnegative elements and hence D = B- 'K contains nonnegative elements only. ’ Since Kj > 0 it follows from thej-th equation in (3.9) that Dk > 0 for some k. 0 Having established that there always exists an optimal contract which provides full coverage above a deductible, we proceed to investigate the determination of the optimal deductible (and thus of the optimal constract). We show: Theorem 3.2. Suppose the buyer purchses contracts I,(x) the optimal deductibles m,* satisfy rn; > 0, j = 1,. . . , J.

which provide full coverage above a deductible. Then

Proof. Let my, ~7, and Wj* denote the optimal deductible, critical value and resulting expected utility, respectively, j = 1,. . . , J. These values are obtained by solving the following maximization problems ? W, = max max /‘(u(A

“I

y,

-r,--x)+PW,_,)dF,(x)

0

+ J”(u(A-r,-m,)+pu;+,)dF,(x), Yi

j=l,...,J,

subject to y, = (1 + Xi)J,?(x - mi)dr’;(x). It is easy then to verilfy that rn: is the solution of the following maximization

(3.11)

problem ‘:

m~x~(m)=~*[u(A-r,-x)+~~~,]dF,(x)+~~[~(A-~-m)-I-~W,*,,]dF,(x) y,* ’ See, e.g., Bellman (1970), p. 305. 6 Differentiating each of the J equations in (3.11) with respect to m, and y,, and equating the derivatives to zero we obtain 25 equations. These equations together with the J equations in (3.1 I) yield a system of 3J equations in 3J unknowns. We assume that the solution of this system yields the optimal critical values and deductibles and hence also the maximal values for the W,. ’ We employ the fact that if some function g( Z,, . . . , Z,,) is maximized by (Zt.. . . , Z,*), then Zf maximizes g( Z,, Zy, . . . . Z,:).

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206

subject to r, = (1 -t-X,)/,:(x - nz)de(x), Noting that ar,/arn = -G,(y;)-h,G,(yF)

Wam=G,(y:)

Insurance

cowacts

and given u/;*, y,*, i = 1,. . . ,J, and mT, i *j. where G,(y)= 1 -F,(y), one obtains that

(Jo”[u’(A

- ,;-m)]d+)

-q-x)-u’&4

I’*

’ ll’( +h, J 0

A -

yl -.\-)d);(x)+/mu~(~

-Y,--nt)dF,(x)

.r;’

From the concavity of u and since U’ > 0, it follows that aH/am deductible nt; must be strictly positive. 0

.

(3.12)

> 0 at m = 0, and thus the optimal

We note in (3.12) that LflH/aml,=, > 0 even if Xj = 0. * That is, the buyer prefers a strictly positive deductible even if the premium is fair. This is different from Mossin’s (( 1968), p. 562) result showing that in one-period models a strictly po!;itive deductible is necessary only if X > 0. The reason for this difference is that in one-period models, filing a claim involves no cost, hence, if the premium is fair, the insured prefers as much insurance as possible, and thus a zero deductible. In multi-period insurance, filing a claim is Gostly, since it moves the buyer to a higher risk category. Thus, even if the premiums are fair, it does not pay to insure against small risks and the buyer therefore prefers a positive deductible.

Appendix 1 Lemma A.I.

The function

h(y)

defined by

h(y)=@-r,-y)-u(A-r,-y+I,(y)) is continuous and nonincreasing.

Continuity and differentiability (except, possibly, at a finite number of points) follow from our assumptions on u and I,(.). Differentiating h we obtain:

Proof.

h’(y)=

-u’(A

= -u’(A

(1 -Z;(y))z/(A

-r,-y)+

-r,-y)+u’(A

-r,-y+l,(y))

-r,-y+I,(y))--I,‘(y)u’(A

-_YI-y-l-l,(y)).

Concavity of u and I,(y) > 0 imply that -u’(A

-r,-y)+u’(A

-r,-y+I,(y))
Also, nonnegativity of I’ and U’ implies that -I;(y)u’(A

-r,-y+Z,(y))
and hence the result. A similar argument hoids when left and right derivatives replace I’ and h’.

Appendix

2

Theorem. The functions w/, j = 1, . . . , J, are finite approximations method presented in (A.2).

s Except

0

and unique

and can be computed

for the trivial cases y,* = co where a claim is never made, or F,( y,*) = 0, where a claim is always

by the successive

made.

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207

Proof. Let D denote the set (1,. . . , J}, and let C( 0) denote the set of all functions + : D + R where R is the real line. Let 0 = (0 ,,.. ., oJ) E c= x;= ,C(O). Consider the transformation jJ u) defined by ~(u)=~[max{~(A-r,-9)+~~j_,,~(A-~-~+~~(~))+p~i,,}].

l,...,J.

(A-1) In what follows we show that the transformationJ( u) is a contraction mapping, hence from Ross (1970), p. 125 it follows that there exist unique functions UT,j = 1,. . . , J, satisfying j=

uT=E[max{u(A-r,-p)+&,,u(A-r;-Y-I-I,(p))+&?+,}]. These functions u,? are by definition the ?$. Ross (1970) has also shown that q=uDi*=

lim fi”(uO) *--Leo where o” is an arbitrary element of ?? and A” is the n-th application of the transformation fi. To show that 4 is a contraction mapping we need to demonstrate that for any II’, u2 E C maxIf, i

-fi(u’)I

64.2)

Q pmaxluf - $1 i

where p is some parameter satisfying 0 < p < 1. For this we note that ~(o,)-f;(v2)=E[max{u(A-r,-y’)+~~~_,,u(A-r,-9+Zj(~))+~~~+,}] -E[max{u(A

-~-9)+r~u~_,,~(~

-~-~+I,(~))+P~~+,}].

is continuous and nondecreasing, SincefromLemmaA.1 [u(A-r,-y’)-u(A-ri-y’+Zj(~))J (similarly to the way (2.6) has been derived) that (A.3) may be written as .&(u,)-f,(u2)=~[a(A

-r,-Y)+~ff:-,]d~(Y)+J~[u(A

(A-3) it foilows

-Q-Y+I,(Y))+&,+,]d~(Y) YJ

-t*[u(A -

-r,-Y)+Z3$.,]d~(Y)

00 u A-q-

S[(*

(A.4)

Y+I,(y))+PU$?+I]dfj(Y)

YJ

where Y, and yj2 are the critical values corresponding to &( u’) and fJ( u2), respectively. Since the critical value yj’ is such that it maximizes the second expectation at the right-hand side of (A.3), it follows that if we substitute yj for yj2 in (A.4) then the difference between the integrals will (weakly) increase, and hence

+ /m[~(~A-~-y+Zj(y))+Bu~+,]d~(y) Y’

4 -

‘: 0

[

u

(

A-r,-y)+pu;_,]d<.(y)

J~[~I(A-~,-Y+~,(Y))+~U:,,]~~(Y) Y/

=p[l;;(y;)(u:_,-uj+,)+(l wheree(y)=j{dF;(y), Likewise

and of courseO
+j(v:))(u;-I

-uf+J]

(A.5)

1.

(A-6)

I. Veneziu, H. Levy / Insurunce contructs

208

From (AS) and (A.6) it thus follows that maxJf,( u2) -h(v’)I proof. q

=Gp max,luf - $1, and this completes the

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Lemaire, J. (1976). Driver vs. company: Optimal behavior of the policyholder. Scund. Actuarial J. 8, 209-219. Lemaire, J. (1977). La soif du bonus. ASTIN Bull. 9. 18 I- 190. Molnar, D. and T. Rockwell (1966). Analysis of policy movement in a merit-rating program: An application of Markov processes, J. of Risk and Insurance, 265-276.

Mossin, J. (1968). Aspects of rational insurance purchasing. J. of Political Economy 76, 553-68. Ross, S. (1970). Applied Probability Models with Optimizatioti Applicutions. Holden-Day, San Francisco. Seal, H. (1969). Stochastic Theory of a Risk Business. John Wiley & Sons, New York. Venezia, 1. and H. Levy (1980). Optimal claims in automobile insurance. Rev. Econom. S&d. 47, 539-549.