数理解析研究所講究録
第 1857 巻 2013 年 51-57

51

Report on Bounded Insurance Contracts

Yoshinobu Teraoka
Professor Emeritus,
Osaka Prefecture University

Abstract: Insurance contract can be considered from a view point of two person
game between two participants, i.e., the buyer and the seller. This report is therepublication of the results on optimal insurance contracts from the viewpoint ofeach of the two participants, under some plausible conditions, presented by
Teraoka about forty years ago. These were the advancement of the ideas
suggested from Arrow and Miller.
1. Introduction
In this report, we republish to make instruction on the problem of choosing the“optimal bounded insurance contracts” from the view point of each of the twoparticipants, i.e., the buyer and the seller, under some plausible conditions. Arrow[1] and Miller [2] have already described case where the monetaryrefund does nothave an upper bound. In the real world, however, the insurance company does not
pay more than certain amount of money to the beneficiary, so we shall call such acontract“bounded”. We gave four theorems which state that the“optimal boundedcontract” for the buyer is “bounded stop.loss” and one for the seller is “boundedproportional”. These were evidently generalization of the results from Arrow andMiller in [1] and [2]. Those were appeared by Teraoka[3], however, the journal
which printed it has ceased to published more than thirty years ago, consequently
it is very difficult to read the paper now.
We shall also instruct two optimal insurance contracts from the view point ofeach of the two participants, under a kind of duality conditions for the above tworesults. It was shown that the optimal insurance contracts are “minimumtruncated” for the seller and “bounden proportional” for the buyer, and the latter
contract is the common one for the two under disadvantage conditions for eachother. There were also given by Teraoka[4] 1977, however, the paper which
printed them contains the printer’s error in the main result. Thus we shallinstruct our results obtained about forty years ago in this report$\ldots$

52
2. Assumptions
Suppose that buyer faces a positive.valued monetary risk with a cumulative. Thus his expected
money function
distribution $F(x)$ and has$u(\cdot)$$utility\cdot of\cdot$

utihty of facing the risk is$\zeta u(-x)dF(x)$. We also assume that the seller of theis incurred by thein such a way that if the loss
insurance sells a contract
buyer, the seller will pay the buyer an amount $T(x)$ which satisfies
be theassigned positive constant. Let$0 \leq T(x)\leq\min(x,K)$ , where
is a
$x$

$T(\cdot)$

$K$

$\pi$

$pre\cdot$

premium which is usually equal to $\pi=\zeta\tau(x)dF(x)$ . We also assume that the, and that both ofseller has $utility\cdot of$-money function
twice differentiable and concave, that is,$v”(x)\leq 0$ for all. Then the expected utilities for each are$v(\cdot)$$u’(\cdot)\geq 0,$$u(\cdot)$$u”(\cdot)\leq 0,$

and$v(\cdot)$$v’(x)\geq 0$,areand$x$$fu[-\pi-x+T(x)]dF(x)$and$\zeta v[\pi-T(x)]dF(x)$by making the contract.
Let$S_{F}(z)=f(x-z)dF(x)$$=t(1-F(x))dx$

where the expected value

is assumed to exist. For any cumulative distribution function$E(X)=\zeta xdF(x)$$F$

with finite mean $E(X)S_{F}(z)$ is non-negative, convex, and strictly decreasing on, andthe set where it is positive. Furthermore, $S_{F}(z)\geq E(X)-z,$$(0\leq z\leq\infty)$$S_{F}(0)=E(X), \lim_{zarrow\infty}S_{F}(z)=0$. We denote the inverse function of $S_{F}(z)$ by$S_{F}^{-1}(c)$for $0<c\leq E(X)..$
3. The Optimal Bounded Insurance Contracts
Result.1 presents an optimal bounded insurance contract from the viewpoint of
the buyer, and Result.2 offers it for the seller under some condition. We find thatthe optimal contract for the buyer is “bounded stop loss “and one for the seller is“bounded proportional”
Result.1. Let
agreement of

be a positive number, and let$\pi$$T(\cdot)$$K\geq S_{F}^{-1}(E(X)-\pi)$.

such that$\zeta\tau(x)dF(x)=\pi$$\tau_{K}$be the set of all insurance

and $0 \leq T(x)\leq\min(x,K)$ , where

53
Then for any utility function(1)$u(\cdot)$$\max_{T\epsilon r_{K}}\zeta u[-\pi-x+T(x)]dF(x)=\zeta u[-\pi-x+T_{K}(x)]dF(x)$ ,where
(2)

and

$T_{K}^{\cdot}(x)=\{\begin{array}{ll}0, 0\leq x<a_{K}\min(x-a_{K},K) , x\geq a_{K}\end{array}$$a_{K}$

is the unique root ofequation(3)$S_{F}(a_{K})-S(a_{K}+K)=\pi$

and sa tisfies(4)$0\leq a_{K}\leq S_{F}^{-1}(\pi)$Proof. Since(5)$u^{\mathfrak{l}}(x)\leq 0$.

, we have$u[-\pi-x+T(x)]-u[-\pi-x+T_{K}^{\cdot}(x)]\leq\{T(x)-T_{K}^{\cdot}(x)\}u’(-\pi-x+T_{K}^{\cdot}(x))$.

We find that(6)$u’(-\pi-x+T_{K}^{\cdot}(x))=\{\begin{array}{l}u(-\pi-x)\leq u.(-\pi-a_{K})\prime, 0\leq x\leq a_{K}u’(-\pi-a_{K}) , a_{K}<x\leq a_{K}+K.u’(-\pi-x+K)\geq u’(-\pi-a_{K}), x\geq a_{K}+K\end{array}$

The definition of $T_{K}(x)$ gives(7)$T(x)-T_{K}^{\cdot}(x)=\{\begin{array}{ll}T(x)\geq 0, 0\leq x\leq a_{K}T(x)-K\leq 0, x\geq a_{K}\end{array}$Considering (4), (5) and (6), it follows that for any$T(\cdot)\in\tau_{K}$$\zeta u[-\pi-x+T(x)\iota;F(x)-\zeta u[-\pi-x+T_{K}(x)bF(x)$$\leq\zeta^{K}b(x)-T_{K}^{\cdot}(x)b’(-\pi-a_{K})dF(x)+\zeta_{K}^{+K}b(x)-T_{K}^{*}(x)b’(-\pi-a_{K})dF(x)$$+\zeta_{\kappa^{+K}}k(x)-T_{K}(x)\}u’(-\pi-a_{K})dF(x)$$=u\prime(-\pi-a_{K})\zeta k(x)-T_{K}(x)\}fF(x)=0,$

yielding Equation (1).
Since $S_{F}(x)-S_{F}(x+K)$ , forthe condition$E(X)-S_{F}(K)\geq\pi$$x\geq 0$,

is decreasing from $E(X)-S_{F}(K)$ to zero,i.e., $K\geq S_{F}^{-1}(E(X)-\pi)$ assures the unique

54
satisfying Equation (3) and Inequality (4). This completes theexistence of
proof of Result 1.
$a_{K}$Result 2.

Letagreement of

be a positive number and let$\pi$$T(\cdot)$$K\geq S_{F}^{-1}(E(X)-\pi)$such that

Then for any utility function(8)$non\cdot$

be the set ofall insurance

and $0 \leq T(x)\leq\min(x,K)$ , where$\zeta\tau(x)dF(x)=\pi$, and $T(x)/x$ is a$\tau_{K}^{0}$decreasing function of$x$

if $T(x)<K,$$v(\cdot)$$\max_{T\in\tau_{\kappa^{0}}}\zeta u[\pi-T(x)]dF(x)=\zeta u[\pi-T_{K}^{0}(x)]dF(x)$,where(9)

andTK$q_{K}$$0_{(x)=}\{\begin{array}{ll}q_{K}x, 0\leq x<K/q_{K}K, x\geq K/q_{K}\end{array}$is the unique root ofequation(10)$S_{F}(K/q_{K})=E(x)-\pi/q_{K}$and sa tisfies(11)$\pi/E(X)\leq q_{K}\leq\min(1,K/E(X))..$

Proof. First we shall prove Equation (11). Putting $t=K/q$ , Equation (10) is
rewritten by(12)$S_{F}(t)=E(X)-(\pi/K)t.$

From the assumption of$K$, we obtain

and $0<\pi<K.$of (12) exists uniquely and$S_{F}(K)\leq E(X)-\pi$

Hence the root$t^{0}$$\max(E(X),K)\leq t^{0}(=K/q^{0},say)\leq(K/\pi)E(X)$giving$\pi/E(X)\leq q^{0}\leq\min(K/E(X), 1)$

Next we shall prove that

$T_{K}^{0}(x)$.

is an optimal contract for the seller. We

clearly have
(13)

$v[\pi-T(x)]-v[\pi-T_{K}^{0}(x)]\leq\{T_{K}^{0}(x)-T(x)\}n^{!}[\pi-T_{K}^{0}(x)].$

From the definition of TK$0_{(x)}$and $0<q<1$ , we obtain

55
(14)

TK

$0<x\leq K/q$$0_{(x)-T(x)=}\{\begin{array}{l}x[q-\{T(x)/x\}1K-T(x) ,\end{array}$$x\geq K/q$

If $F(K/q)=Pr\{X\leq K/q\}=0$ , then since$rk_{K}^{0}(X)-T(x)\}k_{K}^{0},$(8) is derived from (13).
$=Pr\{X\leq K/q\}>0$(15)

Therefore, we shall prove the case where $F(K/q)$. Suppose that $T(K/q)<K$ , thenfor$T_{K}^{0}(x)-T(x)=x[q-\{T(x)/x\}1$$0<x<K/q,$since $T(x)/x$ is adecreasing function so far as $T(x)<K$ .
obtain from (14) and (15)$non\cdot$

Therefore we$\zeta k_{K}^{0}(x)-T(x)\}iF(x)>0,$

contradicting to(16)$T(x)\in\tau_{K}^{0}$$T(\cdot)=K$ ,Hence we find that for anyfor$T(x)\in\tau_{K}^{0}$$x\geq K/q$

and$\zeta b_{K}^{0_{(X)-T(x)^{)}\mu}}F(x)=\zeta^{/q}\{T_{K}^{0}(x)-T(x)\}iF(x)$$=\zeta_{X\{q-T(x)/x}^{/q}\ltimes F(x)=0.$

From the above results, if $F(K/q)>0$ then there exists a $\gamma\in(0,K/q]$ such that(17)

$T_{K}^{0}(x)=qx\{\begin{array}{l}><\end{array}\}T(x)$if$\{\begin{array}{ll}0< x<\gamma\gamma<x<K/q \end{array}\}.$

Since we obtain$v^{!}[\pi-T_{K}^{0}(x)]=v^{!}(\pi-qx)\{\begin{array}{l}\leq\geq\end{array}\}v^{!}(\pi-q\gamma)$,

if$\{\begin{array}{ll}0< x\leq\gamma\gamma\leq x\leq K/q \end{array}\},$(13), (16) and (17) give$\zeta v[\pi-T(x)]iF(x)-\zeta v[\pi-T_{K}^{0}(x)1tF(x)\leq\zeta k_{K}^{0}(x)-T(x)^{)}fl^{!}[\pi-T_{K}^{0}(x)1fF(x)$$=\zeta^{/q}\{T_{K}^{0}(x)-T(x)^{)}fl^{!}[\pi-T_{K}^{0}(x)\}iF(x)$$\leq\zeta k_{K}^{0_{(X}})-T(x)\}v^{!}(\pi-q\gamma)dF(x)+\zeta^{/q}b_{K}^{0_{(x)-T(x)\}v^{!}(\pi-q\gamma)dF(x)}}$$=v^{!}(\pi-q\gamma)\zeta^{/q}\{T_{K}^{0}(x)-T(x)\}dF(x)=0.$

This complete the proof.

56, we have the, then letting
If $F(x)>0$ for any finite
unbounded cases (Arrow[l] and Miller[2]), in which the optimal contract for the$Karrow\infty$$x\geq 0$

buyer is (from Result 1) of stop-loss type with stop-loss point$a_{\infty}=S_{F}^{-1}(\pi)$, andthe optimal contract for the seller is (from Result 2) of proportional type with therate $q_{\infty}=\pi/E(X)$ .
Here we consider the optimal insurance contracts under a kind of duality
conditions for the above two resuts. Result 3 shows an optimal bounded
insurance contract from the view point of the seller under generous conditions,and Result 4 suggests it for the buyer under disadvantage conditions. It is foundthat the optimal contract for the seller is “minimum truncated” and one for trhebuyer is “bounded proportional” it is the very same contract as one for the seller
under disadvantage conditions given by Result 2.

Result 3. Let
agreement of$\pi$such that$T(\cdot)$$K\geq S_{F}^{-1}(E(X)-\pi)$be a positive number and let$\zeta\tau(x)dF(x)=\pi$

be the set ofall insurance$\tau_{K}$and $0 \leq T(x)\leq\min(x,K)$ , where.

Then for any utility function$v(\cdot)$$\max_{T\in\tau_{K}}\zeta v[\pi-T(x)]dF(x)=\rfloor^{\infty}v[\pi-T.(x)]dF(x)$

,

where
$T,(x)=\{\begin{array}{ll}x, 0\leq x<a_{K}b, x\geq a_{K}\end{array}$

and

$b$

is the unique root ofequation $S_{F}(b)=E(X)-\pi$ , that is$b=S_{F}^{-1}(E(X)-\pi)..$(We omit the proof since it can be found in Teaoka[4]).

Result 4.
agreement ofLet$T(\cdot)$$\pi$

be a positive number, and letsuch that$\zeta\tau(x)dF(x)=\pi$’$\tau_{K}$be the set ofall insurance

and $0 \leq T(x)\leq\min(x,K)$ , where57$K\geq S_{F}^{-1}(E(X)-\pi)$, and $T(x)/x$

Then for any utility function

is a no increasing $fu$ ction of$n^{}$$x$

if $T(x)<K.$$u(\cdot)$$\max_{T\in\tau_{K}}, \zeta u[-\pi-x+T(x)]dF(x)=\zeta u[-\pi-x+T_{K}^{0}(x)]dF(x)$ ,where
TK

and$q_{K}$$0_{(x)=}\{\begin{array}{ll}q_{K}x, 0\leq x<K/q_{K}K, x\geq K/q_{K}\end{array}$is the unique root ofequation$S_{F}(K/q_{K})=E(x)-\pi/q_{K}$and sa tisfies$\pi/E(X)\leq q_{K}\leq\min(1,K/E(X))$.

(We also omit the proof since we can refer to Teraoka[4].)Note that TK$0_{(x)}$is a common contract under disadvantage conditions for thetwo participants, the buyer and the seller of the insurance. Furthermore,in contrast with
$T_{K^{+}}(x)$$T_{K^{+}}(x)$and$T\kappa^{0}(x)$$T_{*}(x)$istakes a compromised position between

and $T.(x)$ .

As a simple example of our results we examined the case of automobile
physical damage insurance for private passenger automobile $(small\cdot size)$ in [4].

REFERENCES[1] Arrow, K. (1963): “Uncertainty and the welfare economics of medical care”,

Amer. Econ. Rev.. 33, 942.973.
[2] Miller, R. B. (1972):“Insurance contract as a two person game”, Manag. Sci. 18,444.447.
[3] Teraoka, Y. (1972): “Bounded insurance contracts”, Rept. Statist. Appl. Res.

JUSE 19, 110.115.
[4] Teraoka, Y. (1977): Some remarks bounded insurance contracts”, Journ. Japan
Statiist. Soc., 7, 35-42.