model can be derived from an expected utility model only if a very strong
condition, called risk independence, holds for expected utility. Basically,
this condition requires that lotteries with the same expected payoff can be
preference ranked by their risk measures such that this ranking does not
change when the expected payoff changes.4 This condition holds only for
very few utility functions which, in turn, imply very specific risk measures.
Thus, risk-value models are more general. This is in the spirit of behavioral
research to allow for a wider range of behavior.
3 Efficient Portfolios
We assume a two date-economy with a perfect and complete capital market.
At date 0 investors choose their portfolios which pay off at date 1. A state of
nature at date 1 is defined by the exogenously given aggregate payoff ε; i.e.
the sum of payoffs to all investors.5 ε is a positive variable, ε ∈ (ε,ε) with
the probability density being positive for every ε ∈ (ε,ε). Since we are not
interested in time preferences, the whole analysis is done in forward terms.
Equivalently, the risk-free rate can be assumed to be zero. Define:
pε : = probability density of ε,
eε : = number of claims contingent on state ε, purchased by the investor;
each claim pays off $1 if and only if state ε obtains,
πε : = forward pricing kernel; for every state ε it denotes the price of a
claim contingent on that state divided by the state probability density, πε >
0; E(π) = 1.
Wq : = the investor’s initial forward endowment (wealth), Wq > 0. Since
the investor is endowed with state-contingent claims, Wq equals the forward
4See Bell (1995a, b) for further discussion on this issue.
0The market is said to be complete if for every ε0 ∈ 1R+ there exists a claim which
pays off $ 1 if ε > ε0 and zero otherwise [see Nachman 1988]. We assume the existence of
ε,ε ∈ 1R+ U {∞}such that the state space is identified by (ε,ε).
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