Prospect Theory, developed by Kahneman and Tversky, proposes that people value gains and losses differently, leading to decision-making that deviates from traditional rational models. In terms of stock returns, it suggests investors are more sensitive to losses than to equivalent gains and may make irrational decisions based on this asymmetry. This theory is relevant as it provides insight into investor behavior, especially under uncertainty, and helps in understanding market anomalies and investor decision-making processes.

Elaboration
The main empirical prediction about prospect theory by Barbaris, Mukherjee and Wang is that stocks whose historical distribution have high (low) prospect theory values will have low (high) subsequent returns. This prediction is expected to hold primarily among small cap stocks, in other words, among stocks for which individual investors play a more important role.

Another hypothesis by the article stated that a stock whose future returns are expected to be positively skewed will be 'overpriced' and will earn a lower average return.

• A positively skewed investment return means there were frequent small losses and a few large gains.
• A negatively skewed investment return means there were frequent small gains and a few large losses.

First, we will elaborate on the quantification of the prospect theory (formed by Tversky and Kahneman (TK). It start with the prospect theory value function, given by

(1)  

Typical values for the parameters are and . The value function is plotted in Figure 1, in which we see the investor's risk aversion over gains and the risk seeking over losses. We will elaborate shortly on practical examples.

Furthermore, the weight functions need to be introduced. The main consequences of the weighting functions is that it overweights the tails. Think about a lottery or insurance. With a lottery, you have a small probability p for a big price and with an insurance we pay monthly for a small probability p for a loss. By overweighting the tail probability, the prospect theory can capture both of these choices. The weighting functions (which will be used later on this lesson) are given as follows.

(2)  

(3)  

The parameters in the three equations above have values and .

When thinking about a stock, some of the investors mentally represent it as the distribution of its monthly returns in excess of the market over the past five years. Given a specific stock, we record the stock's return in excess of the market in each of the previous sixty months and then sort these sixty excess returns in increasing order. Suppose that excess returns are negative and are positive. Then, the excess returns are ordered as , which becomes the stock's historical return distribution, with each return having an equal 1/60 probability. Finally, the prospect theory value of this distribution is given as

(4)  

This TK value captures the impression that investors form about a stock, after seeing its historical price fluctuations in a chart.

One important property of TK is that it does not depend on the order in which the sixty past returns occur in time. The justification behind this method is that the TK is capturing an investor's quick, passive reaction to a chart. This reaction is likely to be based on the chart as an integral whole, with the early part of the chart affecting the investor just as much as the later part.

Prospect theory investors are investors that construct their portfolio holdings by taking the tangency portfolio and adjusting it, by increasing their holdings of stocks with high prospect theory values and decreasing their holdings of stocks with low prospect theory values.

The main hypothesis is that TK value of stocks will predict the stock's subsequent return in the cross section. Two methods have been used in order to test this hypothesis:

• The Time Series Test
• The Gama-MacBeth Test

Time Series Test

1. At the start of each month, beginning in July 1931 and ending in December 2010, we sort stocks intro deciles based on their TK-value.
2. Compute the average return of each decile over the entire sample.
3. Report the average return of each decile in excess of the risk-free rate, the four-factor alpha for each decile (the return adjusted by the 3-factor model from Fama & French, 1993, and by momentum), the 5-factor alpha (the 3-factor model and momentum and liquidity) and the characteristics adjusted return.

See the article for an overview of the results. The results support the main hypothesis. The average return on low TK values is significantly larger than the average return for higher TK values.

Fama-MacBeth Test

The Fama-MacBeth test has one advantage over the time series test. This method allows us to examine the predictive power of TK, while controlling for known predictors of the return. Each month, from July 1931 till December 2010, the method runs a cross sectional regression of stocks returns in that month on the TK value that is measured at the start of the month and on variables already known to predict returns.

The results support the main hypothesis. Even after we include the major known predictors of the return, the TK value remains significant.

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