STOCHASTIC

RISK AVERSION 125
If z is actuarially neutral, the risk premium and insurance premium coincide.
The results of this paper will be stated in terms of the risk premium JI, but could
equally easily and meaningfully be stated in terms of the cash equivalent or insurance
premium.
3. LOCAL RISK AVERSION
To measure a decision maker's local aversion to risk, it is natural to consider his
risk premium for a small, actuarially neutral risk z. We therefore consider n(x, z) for
a risk z with E(z)=Q and small variance 0. We assume the third absolute central moment of z is of smaller
order than of. (Ordinarily it is of order of.) Expanding u around x on both sides
of (1), we obtain under suitable regularity conditions 3
(4a) u(x - n) = u(x) - nu'(x) + 0{n 2 ),
(4b) E{u(x + z)} = E{u(x) + zu\x) + iz V(x)+0(2 3 )}
= u(x) + iofu"(x) + o(of) .
Setting these expressions equal, as required by (1), then gives
(5) Tz(x,z) = i