Prospect theory, PT, is the best known descriptive decision theory. Kahneman and Tversky (1979) proved that, when making decisions in a context of risk or uncertainty, most individuals (i) show preferences that depend on gains and losses with respect to a certain reference point, and (ii) form beliefs that do not correspond to the statistical probabilities (their perception of the risks associated with a decision may be biased). The combination of these properties implies a fourfold pattern of risk attitudes that is confirmed by experimental evidence: risk aversion for gains and risk seeking for losses of moderate to high probability; risk seeking for gains and risk aversion for losses of low probability.
We devised our tests to elicit the parameters of two classic specifications for PT. First, the (piecewise) power function by Tversky and Kahneman (1992), were α+ measures risk profile for gains, α- measures risk profile for losses (aversion to a sure loss) and β measures loss aversion. Second, the Prelec-I (Prelec, 1998) weighting function, where γ+ measures the distortion of probabilities in the positive domain and γ- does the same for losses.
The methodology we use for parameter estimation is based on the elicitation of certainty equivalents of prospects with just two outcomes. Following Abdellaoui et al. (2008), the elicitation method consists of three stages, with fifteen questions in total: six questions involving only positive prospects (i.e., a chance to win some positive quantity or zero) to calibrate α+ and γ+, six questions for negative prospects to calibrate α- and γ-, and three questions regarding the acceptability of mixed prospects, in order to estimate β. Figure 1 shows one of the six iterations participants had to answer involving positive domain.
FIGURE 1 – A sample question with positive prospects
In every iteration participants had to choose between a positive prospect (left) and a series of sure positive outcomes (right). To control for response errors we repeated the one outcome at the end of each trial. The certainty equivalent of a prospect was then estimated by the midpoint between the lowest accepted value and the highest rejected value. This procedure allows for the cash equivalent to be derived from observed choices, rather than assessed by the subject.
Participants in the experiment completed the fifteen questions in about 20 minutes and there were no relevant incidents in any of the five sessions. Results evidence our tests resulted largely satisfactory to replicate the main findings of prospect theory. They reiterate the classic evidence of concavity for gains, while risk seeking in the negative domain seems to be more acute. The percentage of individuals with alpha measures below one are 59.5% and 93.7%. We also observe a significant degree of probability weighting in both domains, with distortion being higher in the negative side. Finally, beta estimations are in consonance with classic results in the literature. The percentage of individuals with beta measures above two are 73.0%.
Figure 2 plots the risk attitudes of the average (idealized) participant.
FIGURE 2 – Risk attitudes of the average participant
The fourfold pattern of risk attitudes predicts people tend to be risk seeking for gains of low probability (1% and 5% in our test) and losses of medium and high probability, while we tend to be risk averse for gains of medium and high probability and losses of low probability. The pattern is clearly observable for the average respondent, with the nuance of an about risk neutrality for gains of medium probability.
Finally, we determine the validity of participants’ responses based on two kinds of errors: iteration errors (the reliability of the iterative questions we asked to control for response errors) and fitting errors (the errors obtained in the non-linear regressions for parameter estimation). Results were highly satisfactory in both instances: only 5.6% of responses were contradictory (94.4% reliability), while 80% and 65% of the individual regressions had a coefficient R2 > 90% in the positive and negative domain, respectively.
Results on overconfidence and prospect theory were tested against several priors (gender, age, education, working experience and skills in finance) using several methodological techniques (correlation, regression, anova and factorial analyses). We found these significant (p < 0.05) results:
· Education increases loss aversion
· Women are more overconfident than men in terms of overprecision
· Women are more risk seeking in the positive and negative domains
· Experience reduces objectivity in terms of estimation of self-performance
· Skills in finance increases objectivity reducing probability distortion
· Skills in finance reduces risk aversion
· Overestimation and overplacement come together (E and P correlated)
· Risk seeking comes together in both domains (alpha+ and alpha-)
· Distortion of probabilities come together in both domains (gamma+ and gamma-)
· Loss aversion and risk aversion in the negative domain come together as well
· Regarding the relationship between overconfidence and PT parameters, we find only low statistical evidence (p < 0.1) that more aggressive profiles for losses (alpha- and γ-) are correlated with lower levels of overconfidence (E and P).