Heterogeneity of Investors and Asset Pricing in a Risk-Value World

Compare the solution of the exogenous benchmark model with that of
the endogenous benchmark model. From equation (12), raising the payoff
in some state always reduces risk in the case of an exogenous benchmark.
From equation (8), for an endogenous benchmark an increase in some payoff
lowers [raises] risk if the payoff e_ε is lower [higher] than f~ ¹(E[f (ê)]). This
follows from the impact of the payoff increase on the benchmark. Yet, in
the case of an exogenous benchmark, the risk reduction is higher the lower
the payoff. Therefore, the risk reduction induced by raising the payoff in
some state minus the expected risk reduction induced by raising the payoff
in every state, —f (e_ε) + E[f (ê)], in the exogenous benchmark model has the
same properties as the risk reduction in the endogenous benchmark model.
This explains why the first order conditions (11) and (14) look precisely the
same although the optimal portfolios are different because of the different
benchmarks.

In spite of the formal identity of (11) and (14) the optimal solutions to
both problems differ substantially. If the payoff constraint (6) is not binding,
then the optimal solution to the endogenous benchmark-problem is to buy
only the risk free asset. The exogenous benchmark-problem without the
payoff constraint is formally the same as the traditional state preference -EU-
choice problem Max E[u(e)] subject to the budget constraint (5). This follows
since minimizing risk with the risk function being a negative utility function
is formally the same as maximizing expected utility. Hence the investor
minimizing risk with an exogenous benchmark chooses a risky portfolio e^
even if the payoff constraint (6) is not binding. This implies E(e(o)) > Wq
or, equivalently, an expected gross portfolio return E(R) > 1. The payoff
constraint becomes binding only if the required expected payoff is raised
above this expectation, i.e. if WqR* > E(e(₀)); (see figure 1).

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