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

require F^," < О.³

As proposed by Jia and Dyer (1996), risk functions with F^z < О, F^zz > О,
and F^zzz < О correspond to utility functions with u' > О, u" < О and u"' > О,
i.e. with positive prudence [Kimball 1990].

Standard models of risk perception, e.g. the exponential model, have the
properties noted above. Note that variance neither fulfills property i) nor
property iii). The three properties imply that alternative C (see example
last section) is judged the most risky. Risk judgments found in a number of
empirical studies, see, e.g., Keller, Sarin and Weber (1986), are consistent
with the risk ranking implied by these properties. Property iii) is reflected in
a variety of empirical results that show that people judge alternatives with
potential catastrophic outcomes as being especially risky, see, e.g., Slovic
(1987).

In the following sections we will derive results using properties i) - iii).

The risk function, in general, varies from investor to investor. It is defined on
the range (ê, ê). In order to guarantee optimal internal solutions to portfolio
choice, we add the assumption that F^z(ê) → -∞ for ê → ê and F^z (ê) →
О for ê → ê. At the end, in order to get further results, we need more
detailed information about the risk measure. At that point we will assume
that the risk function belongs to the set of negative HARA-functions. We
consider negative HARA-functions with properties i) - iii). They include

³The necessity of F"' < 0 can be seen as follows:

Let y, z be two states with the same payoff and the same probability p; in both states
the payoff deviates from the expected value by Δ. Their contribution to the total risk of
the portfolio is then given by 2pF (Δ). Now replace the payoff deviation Δ in the states
y and z by a mean preserving spread around Δ, that is: the deviation from the expected
payoff is Δ — a in state y and Δ + a in state z (ɑ > 0). Notice that the expected payoff is
not changed and hence it does not influence the risk contribution of the other states. The
new contribution of the states y, z to the total risk is pF (Δ — ɑ) + pF (Δ + ɑ). The risk
increase, denoted RI_a is then given by: RI_a (Δ) = pF (Δ — ɑ) +pF (Δ + ɑ) — 2pF (Δ)
and the strict convexity of F is equivalent to RI_a > 0 for all Δ and all a > 0. We require
for an increase in Δ that RI'_a (Δ) = pF' (Δ — ɑ) +pF' (Δ + ɑ) — 2pF^, (Δ) < 0. This
holds iff F"' < 0 .

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