A SURVEY OF BEHAVIORAL FINANCE
local curvature to reject E over a wide range of wealth levels, it must be an extraordinarily concave function, making the investor extremely risk averse over large stakes gambles.
The final piece of prospect theory is the nonlinear probability transfor- mation. Small probabilities are overweighted, so that π (p) p. This is de- duced from KT’s finding that
(5000, 0.001) f (5, 1),
(5, 1) f (5000, 0.001),
together with the earlier assumption that υ is concave (convex) in the do- main of gains (losses). Moreover, people are more sensitive to differences in probabilities at higher probability levels. For example, the following pair of choices,
(3000, 1) f (4000, 0.8; 0, 0.2),
(4000, 0.2; 0, 0.8) f (3000, 0.25),
which violate EU theory, imply
π(0.25) π (0.2)
π (1) π (0.8)
The intuition is that the 20 percent jump in probability from 0.8 to 1 is more striking to people than the 20 percent jump from 0.2 to 0.25. In par- ticular, people place much more weight on outcomes that are certain rela- tive to outcomes that are merely probable, a feature sometimes known as the “certainty effect.”
Along with capturing experimental evidence, prospect theory also simulta- neously explains preferences for insurance and for buying lottery tickets. Although the concavity of υ in the region of gains generally produces risk aversion, for lotteries which offer a small chance of a large gain, the over- weighting of small probabilities in figure 1.2 dominates, leading to risk- seeking. Along the same lines, while the convexity of υ in the region of losses typically leads to risk-seeking, the same overweighting of small probabilities induces risk aversion over gambles which have a small chance of a large loss.
Based on additional evidence, Tversky and Kahneman (1992) propose a generalization of prospect theory which can be applied to gambles with more than two outcomes. Specifically, if a gamble promises outcome xi with probability pi, Tversky and Kahneman (1992) propose that people as- sign the gamble the value
π υ ( ) , i x i