*gains*, individuals are risk-averse, preferring solutions that lead to a lower expected utility but with a higher certainty (concave value function). On the other hand, faced with a risky choice leading to

*losses*, individuals are risk-seeking, preferring solutions that lead to a lower expected utility as long as it has the potential to avoid losses (convex value function). These examples contradict expected utility theory, which only considers choices with the maximum utility.

^{1}The theory continues with a second concept, based on the observation that people in the tails of the distribution attribute excessive weight to low probability events and insufficient weight to events with high probability. For example, individuals may unconsciously treat an outcome with a probability of 99 percent as if its probability was 95 percent, and an outcome with a probability of 1 percent as if it had a probability of 5 percent. Under- and over-

*weighting*of probabilities is distinct from under- and over-

*estimating*probabilities, a different type of cognitive bias observed in the overconfidence effect.

## New Literature

Nicholas Barberis, Lawrence Jin, and Baolian Wang contribute to the literature on prospect theory and the role it plays in asset pricing with their October 2019 study “Prospect Theory and Stock Market Anomalies.” They begin by noting:Narrow framing results from investors seeking short cuts to simplify complex problems. For example, it may be difficult for an investor to determine how a new risk interacts with pre-existing risks to affect“Prospect theory is often implemented in conjunction with narrow framing, a phenomenon observed in experimental studies whereby, when an individual is thinking about taking on a new risk, he evaluates it to some extent in isolation, separately from his other risks. In the stock market context, this means that, when an investor is thinking about how much money to allocate to a particular stock, he focuses, at least in part, on the potential gains and losses in his holdings of the stock itself.” Nicholas Barberis, Lawrence Jin, and Baolian Wang

*overall*wealth risk. As a consequence, investors evaluate the new risk, at least to some extent, as a stand-alone gamble. The authors go on to explain:

All else equal:“In an economy with prospect theory investors who engage in narrow framing, the price of an asset will depend in part on three asset characteristics: the volatility of the asset’s returns; the skewness of the asset’s returns; and the average prior gain or loss across investors holding the asset, a quantity known as the asset’s ‘capital gain overhang’. Nicholas Barberis, Lawrence Jin, and Baolian Wang

- Investors require a higher average return on more volatile assets: since these investors evaluate each asset to some extent in isolation, and since they are loss averse, they find assets with volatile returns unappealing.
- Investors require a lower average return on assets with more positively-skewed returns: since these investors focus on an asset’s own distribution of potential gains and losses, and since they overweight the tails of this distribution, they find assets with positively-skewed returns attractive.
- Investors require a higher average return on assets where they have larger prior gains.

*higher*average return on small-cap stocks than on large-cap stocks, thereby helping to explain the size anomaly. However, the typical small-cap stock also has more positively-skewed returns, and a more negative capital gain overhang, than the typical large-cap stock. All else equal, these two factors lead prospect theory investors to charge a

*lower*average return on small-cap stocks, thereby hampering the model’s ability to explain the size anomaly. The quantitative approach allows them to discover what the

*overall*prediction of prospect theory for the size anomaly is, once all of these factors are properly combined, allowing one to see which, if any, trait dominates.

## Findings

The study’s data sample covers the period 1963 through 2015. Following is a summary of their findings:- The three characteristics, standard deviation, skewness, and gain overhang, are strongly correlated across anomaly deciles. If the typical stock in decile 1 for some anomaly has more volatile returns than the typical stock in decile 10 for that anomaly, it almost always also has more positively-skewed returns and a more negative capital gain. This holds for 21 of the 22 anomalies, with the only exception being post-earnings announcement drift.
- The model helps explain 13 of the 22 anomalies, in that it predicts a higher CAPM alpha for the extreme anomaly decile portfolio that actually has a higher alpha, empirically. These are the momentum, failure probability, idiosyncratic volatility, gross profitability, expected idiosyncratic skewness, return on equity, maximum daily return, O-score, external finance, composite equity issuance, net stock issuance, post-earnings announcement drift, and difference of opinion anomalies. The strongest results are for the first five.
- For each of the 13 anomalies, the typical stock in the extreme decile with the lower average return is more highly skewed, more volatile, and has a lower gain overhang than the typical stock in the other extreme decile. The greater skewness and lower gain overhang of the former stock leads investors to charge a lower average return on it, while its higher volatility leads investors to charge a higher average return on it. Quantitatively, the first effect dominates.
- There are seven anomalies where the model performs poorly—size, value, long-term reversal, short-term reversal, accrual, asset growth, and investment anomalies. The most notable failures are the size and value anomalies. For example, value stocks are more highly skewed and have a more negative capital gain than growth stocks. This leads prospect theory investors to charge a lower average return on value stocks. However, value stocks are also more volatile, leading investors to charge a higher average return on them. Quantitatively, the first effect dominates. The model, therefore, predicts a lower average return on value stocks, contrary to the empirical facts. In the case of the size anomaly, while stocks in the smallest decile are the most volatile, for which investors require a higher return, they also have positively-skewed returns and trade at a loss, which leads investors to charge a low average return on them. Their analysis shows the second effect overwhelms the first.

## Summary

Barberis, Jin, and Wang make a major contribution to the literature on asset pricing. Their model, based on three elements of prospect theory, takes account of investors’ prior gains and losses and makes quantitative predictions about an asset’s average return based on empirical estimates of its beta, volatility, skewness, and capital gain overhang. The model helps explain 13 of the most prominent anomalies in finance. However, like all models, they are flawed, or wrong. They are not like cameras that provide a perfect picture of the world. If models were perfectly correct, they would be laws, like we have in physics. Instead, they are engines that advance our understanding of how markets work and prices are set. You can benefit from their insights by incorporating their findings into your asset allocation decisions, avoiding stocks, and funds that buy stocks, from the 13 anomalies for which their model predicts low average returns, and buying stocks, and funds that buy stocks, for which their model predicts high average returns. Investors who are subject to behaviors explained by prospect theory, and as a result buy stocks which have low expected returns, should take particular notice.Notes:

- see here for more details https://en.m.wikipedia.org/wiki/Prospect_theory ↩