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

previous periods as suggested by Barberis, Huang and Santos (2000) and by
other factors which have been shown to be important in financial behavior.

An endogenous benchmark depends on the characteristics of the payoff
distribution. The most prominent endogenous benchmark is the expected
value of the payoffs as in variance and other moments. This benchmark im-
plies that the risk measure is location free, i.e. risk does not change if the
return distribution is shifted by adding or subtracting a positive number.
Hence risk is independent of the expected payoff. This is a desirable prop-
erty since the expected payoff (value) is already used as a primitive in the
preference function.

2.2 Properties of a Risk Measure

We now describe the risk measure in more detail. Risk will be measured
according to equation (1) as the expectation of a function of the deviation
of a random variable e from a benchmark e. In the case of an exogenous
benchmark e is a given number, in the case of an endogenous benchmark e
is the expected value E(e).

For simplicity we define the deviation e := e — e.

As a next step we postulate three key properties of the risk function F
which are based on the psychological studies cited before:

i) Outcomes above the benchmark reduce risk and outcomes below in-
crease risk, F(e₁) > F(0) > F(e₂) for e₁ < 0 < e₂. In addition, we require
monotonicity: A higher payoff will contribute less to risk than a lower payoff,
thus F^z < 0.

ii) Mean preserving spreads increase risk, thus F^zz > 0.

iii) The sensitivity to a mean preserving spread is larger in the loss domain
(relative to the benchmark) than in the gain domain. Requiring monotonicity
then implies that the sensitivity decreases if the payoff increases; thus we

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