We summarize the differences between risk-value models and expected
utility models as follows.
1. The risk-value model with an exogenous benchmark is formally the
same as the expected utility model if the payoff constraint (6) is not
binding. If the payoff constraint is binding, then the investor has to
take more risk so as to satisfy the payoff constraint. Hence the payoff
constraint adds a new element to the optimization. For risk functions
with an endogenous benchmark the investor takes risk if and only if
the payoff constraint is binding.
2. If the benchmark is endogenous, then risk measurement depends on
this endogenous benchmark which conflicts with the axioms of von
Neumann/Morgenstern.7
7A third difference between risk-value models with an endogenous benchmark and
EU-models relates to satiation. In the EU-model, marginal utility is always positive, by
assumption. In the risk-value model, the investor may be worse off if she receives an
additional payoff in some state with a high portfolio return. Consider as an example the
preference function F(e, risk) with a > 0 and ë = E(e), P(e,risk) = aë —risk, with risk
as defined in equation (1). Differentiate the preference function with respect to eε . This
yields:
∂ P
=pε[a + f(eε)-E[f (ê)]].
o' ¾
If the portfolio payoff is random, there must exist a state ε with∕(⅛) < E[∕(e)] (for
example, the state with the lowest τrε). Then, given a sufficiently small a, we have ∂
P ∕∂eε < 0. Hence an increase in the state ε-portfolio return may reduce the investor’s
welfare which contradicts the usual assumption of non-satiation. If, however, all prices for
state-contingent claims, τrε, are positive, then the investor always chooses his/her optimal
portfolio such that he∕she never reaches or crosses satiation. This follows since a risk
free-asset exists and the investor can always purchase fewer claims contingent on these
critical states, invest the saved money in the risk-free asset and, thereby, increase his/her
welfare.
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