be more expensive than suggested by the Black/Scholes model [Christensen
and Prabhala 1998]. Also, our results could help to explain that the pricing
kernels estimated from stock index option prices are leptokurtic and nega-
tively skewed [Longstaff 1995, Brenner and Eom 1996, Jackwerth 2000].
6 Conclusion
The paper considers portfolio choice and asset pricing in a world where in-
vestors’ preferences are modelled by a risk-value approach. We consider risk
functions with an exogenous and with an endogenous benchmark. Both mod-
els yield similar results. In contrast to expected utility, risk-value models do
not constrain the tradeoff between value and risk given a risk measure. This
approach is consistent with the widely observable separation of value and
risk in finance. We have defined properties the risk measure should have.
These properties are verified using empirical findings on risk judgments and
are related to the properties of utility functions.
Looking at efficient individual sharing rules in risk-value models, the first
order conditions display an additional term, the sharing constant. This con-
stant generates a larger variety of sharing rules as compared to expected
utility models. In the risk-value world with the risk function being a nega-
tive HARA-function, these sharing constants determine whether an investor’s
sharing rule is convex, linear or concave relative to that of another investor.
Highly risk sensitive investors tend to buy portfolio insurance from less risk
sensitive investors. Also, the more aggressive investors, i.e. those who de-
mand a higher expected portfolio return, take more risk, but also tend to
buy portfolio insurance.
The pricing kernel is declining and convex in the aggregate payoff. This
is true also in an expected utility equilibrium if the third derivative of the
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