STOCHASTIC

Let the random variables Og^Sl and S £ 0 represent the percentage loss to be incurred
on an insurance contract and the size (liability) of this contract. The total dollar loss incurred
on a contract selected randomly from the company's portfolio is xS.
Let us consider the following three reinsurance schemes. Using "pro rata quota share"
a percentage 100(1 — a), where 0 < a < 1, of the ceding company's liabilities is given to a
reinsurance company. Hence axS is the retained loss incurred on a contract selected randomly
from the insurance company's portfolio.
With "excess of loss," a part of all losses exceeding a cutoff point M is ceded to the reinsurance
company. Hence the retained loss is min{xS,M} = XSAM. When using "sliding
quota share" no liability is ceded unless the contract size S exceeds the cutoff point C in
which case 100(S— C)/S% of all liability is ceded. Hence the retained loss is X(SA C).
Let the expected value of a nondecreasing convex function of the retained loss denote
the "risk level." It is clear that the choice of the parameters a, M, and C largely determines
the risk level" of these three schemes. To make a valid comparison let us fix a and choose
M = M, and C = C, so that E{txxS) = £(XSA AQ = E(X(SAC,)), that is, each scheme
has equal mean retained loss.
(a) Show that C, > M,. [Hint: Suppose the contrary.]
(b) Show that xS A M« is less risky that either x(S A C,) or ocxS. [Hint: Utilize Theorem
3 in the Hanoch-Levy paper.]
(c) Interpret the results.
(d) What can one show concerning the variances of the three schemes ?
23. (An unorthodox utility function with desirable properties) If z=tan/, then
/ = tan"' z is called the arctangent, denoted arctanz. For z e E 1 ,
and
|arctanz| g 1
d 1
— arctan z = .
dz 1 + z 2
Consider the utility function u(w) = arctan(l + w) over wealth w.
(a) Show that for w > 0, u is positive, bounded, strictly increasing, strictly concave,
has positive strictly decreasing absolute risk aversion, and positive increasing relative
risk aversion.
(b) Show that the nonstrict versions of the properties just given obtain when w £ 0.
(c) Show that the limiting value of the relative risk-aversion index is 2.
Exercise Source Notes
Portions of Exercise 2 were based on Klevorick (1969); Exercises 3 and 4 were adapted
from Leland (1971); Exercise 5 was based in part on Bessler and Veinott (1966); portions
of Exercise 6 were adapted from Whitmore (1970); Exercise 7 is due to Professor S. L.
198 PART II. QUALITATIVE ECONOMIC RESULTS