It is similar to v(Xt+ 1 , St, 1)—the middle line in figure 7.1—but is also mildly
concave over gains and convex over losses. This curvature is most relevant
when choosing between prospects that involve onlygains or between
prospects that involve onlylosses.^13 For gambles that can lead to both
gains and losses—such as the one-year investment in stocks that our agent
is evaluating—loss aversion at the kink is far more important than the de-
gree of curvature away from the kink. For simplicity then, we make vlinear
over both gains and losses.
In our framework the “prospective utility” the investor receives from
gains and losses is computed by taking the expected value of v, in other
words by weighting the value of gains and losses by their probabilities. As a
way of understanding Allais-type violations of the expected utility para-
digm, Kahneman and Tversky (1979) suggest weighting the value of gains
and losses not with the probabilities themselves but with a nonlinear trans-
formation of those probabilities. Again, for simplicity, we abstract from
this feature of prospect theory, and have no reason to believe that our re-
sults are qualitatively affected by this simplification.^14
PROSPECT THEORY AND ASSET PRICES 237
Figure 7.2. Kahneman-Tversky value function.
(^13) Indeed, it is by offering subjects gambles over onlylosses or onlygains that Kahneman
and Tversky (1979) deduce the shape of the value function. They propose concavity over gains
because subjects prefer a gamble that offers $2,000 w.p.^1 ⁄ 4 , $4,000 w.p.^1 ⁄ 4 , and $0 w.p.^1 ⁄ 2 to
the mean-preserving spread offering $6,000 w.p.^1 ⁄ 4 and $0 otherwise. Preferences switch
when the signs are flipped, suggesting convexity over losses.
(^14) Two other assumptions inherent in our framework are worth noting. First, we assume
that investors rationally forecast the loss aversion that they will feel in the future. Loewen-
stein, O’Donoghue, and Rabin (1999) suggest that reality is more complex in that people have
trouble forecasting how they will feel about future events. A more complete model would take