Asset Pricing John H. Cochrane June 12, 2000

SECTION 3.4 RISK SHARING
Asset prices are set, after all, by investor’s demands for assets, and those demands are set
by investor’s subjective evaluations of the probabilities of various events. We often assume
rational expectations, namely that subjective probabilities are equal to objective frequencies.
But this is an additional assumption that we may not always want to make.
3.4 Risk sharing
Risk sharing: In complete markets, consumption moves together. Only aggregate risk
matters for security markets.
We deduced that the marginal rate of substitution for any individual investor equals the
contingent claim price ratio. But the prices are the same for all investors. Therefore, marginal
utility growth should be the same for all investors
u0 (ci t) = βj u0 (c j
t+1 )
β i u0 (c i t+1)
u 0 (c j
t)
where i and j refer to different investors. If investors have the same homothetic utility function
(for example, power utility), then consumption itself should move in lockstep,
c i t+1
c i t
= cjt+1
c j
t
More generally, shocks to consumption are perfectly correlated across individuals.
This is so radical, it’s easy to misread it at first glance. It doesn’t say that expected
consumptiongrowthshouldbeequal;itsaysthatconsumptiongrowthshouldbeequalexpost.
If my consumption goes up 10%, yours goes up exactly 10% as well, and so does
everyone else’s. In a complete contingent claims market, all investors share all risks, so
when any shock hits, it hits us all equally (after insurance payments). It doesn’t say the
consumption level is the same – this is risk-sharing, not socialism. The rich have higher
levels of consumption, but rich and poor share the shocks equally.
This risk sharing is Pareto-optimal. Suppose a social planner wished to maximize everyone’s
utility given the available resources. For example, with two investors i and j,hewould
maximize
X X
t i t j
max λi β u(ct)+λj β u(ct) s.t. c i t + c j
t = c a t
where c a is the total amount available and λi and λj are i and j’s relative weights in the
59
.
(49)