**Prospect theory value as investment factor**

Prospect theory value is a valid investment factor, particularly in episodes of apparent market inefficiency. Prospect theory is a popular model of irrational decision making. It emphasizes a realistic mental representation of expected gains and losses and an individual’s evaluation of such representations. Prospect theory explains asymmetric loss aversion (view post here) and gambling preferences (view post here). Since mental representations of expected returns and volatility are often driven by price charts, prospect theory value can be estimated based on historic asset return distributions. Assets with a high prospect theory value should have low subsequent returns and vice versa. This proposition holds even if part of the market is fully rational as long as there are balance sheet and risk limits. Empirical academic papers have confirmed the prospect theory value in international equity, corporate bond and foreign exchange markets.

The below are mostly quotes from three empirical papers referenced at the bottom of the post. Cursive text and text in brackets have been added for clarity.

The post ties up with this site’s summary on implicit subsidies in financial markets.

### What is prospect theory?

“Despite the popularity of the expected utility theory, experimental evidence…shows that investor behavior in the real world…deviates from what expected utility theory predicts…Prospect theory introduced by Kahneman is a __more realistic decision-making framework to evaluate risk__.” [Xu, Kozhan and Taylor]

“Prospect theory [is] widely considered to be the most successful descriptive theory for decision making.” [Zhong and Wang]

“Decision-making under prospect theory [involves] __two steps: ‘representation’ and ‘valuation.’__ First, for any risk that an agent is considering, he forms a mental representation of that risk. More precisely…the agent forms a mental representation of the gains and losses he associates with taking the risk. Second, the agent evaluates this representation – this distribution of gains and losses – to see if it is appealing…Tversky and Kahneman (1992) provide detailed formulas that specify the value that a prospect theory agent would assign to any given distribution of gains and losses.” [Barberis, Mukherjee and Wang]

“Prospect theory __differs from expected utility__ theory in two important respects.

- First, within the expected utility framework, the utility function is assumed to be continuously differentiable, and it is a concave function of terminal wealth. In contrast,
__the prospect theory value function is a function of gains and losses relative to a__. The function is kinked at zero and is concave at gains [decreasing pleasure] but convex at losses [increasing pain]. Therefore, prospect theory value function*reference point*__better captures real-world investor’s loss aversion and the focus on the incremental change of wealth__rather than the terminal wealth level. - Second, while the relation between probabilities of events and weights is linear in the expected utility framework, prospect theory introduces a non-linear probability weighting function. Such a weighting function
__captures the gambling preference of investors in the real world to overweight extreme tail events__. The theory also models investors’ perception of gain probabilities differently from loss probabilities.

Collectively, these properties ensure that prospect theory offers a more realistic description of the way investors evaluate risk compared to the rational expected utility framework.” [Xu, Kozhan and Taylor]

“For many investors, their mental representation of a stock is given by the distribution of the stock’s past returns… The most obvious way that __investors can learn about a stock’s past return distribution is by looking at a chart of the stock’s past price movements__ – specifically, at the chart that usually appears, front and center, when they look up information about the stock…They believe the past return distribution to be a good and easily accessible proxy for the object they are truly interested in, namely the distribution of the stock’s future returns…This belief may be mistaken.” [Barberis, Mukherjee and Wang]

“Individual investors have a strong preference for the lottery-type demand, while institutional investors lean toward insurance-type demand.” [Zhong and Wang]

### Prospect theory value and asset returns

“The model makes a simple prediction…__stocks with high prospect theory values will have low subsequent returns, on average, while stocks with low prospect theory values will have high subsequent returns__. The intuition is clear: stocks with high prospect theory values are appealing to some investors; these investors tilt toward these stocks in their portfolios, causing them to become overvalued and to earn low subsequent returns.” [Barberis, Mukherjee and Wang]

“Assuming the…market consists of two types of investors: rational investors thinking in line with the expected utility framework, and irrational investors evaluate risk in the way described by prospect theory. Then these __irrational investors are more willing to hold high prospect theory value [assets]__…If these investors account for a non-trivial proportion…their trading activities will shift demand…and therefore affect expected returns in the equilibrium. Specifically, they will bid high prospect theory value [assets] to be temporarily appreciated and overvalued. [Xu, Kozhan and Taylor]

“Arbitrageurs, being risk-averse, do not aggressively exploit existing mispricing because of fear that the mispricing gap widens in the nearest future. Therefore, even if the predictive pattern is not fully captured by conventional risk factors, it may still be generated by exposure to noise trade risk.” [Xu, Kozhan and Taylor]

### Empirical evidence for stocks

“Each month we compute, for each stock in the cross-section, the stock’s prospect theory value…Based on a review of…sources, we suggest that a natural mental representation of a stock’s past return distribution is the distribution of its monthly returns over the previous five years.” [Barberis, Mukherjee and Wang]

“Consistent with our hypothesis__, we find that the coefficient on the stock’s prospect theory value, averaged across all the monthly regressions, is significantly negative: stocks with higher prospect theory values have lower subsequent returns__. We also find, again consistent with our framework, that this result is particularly strong among small-cap stocks… the predictive power of prospect theory value for subsequent stock returns is stronger among stocks that are less subject to arbitrage – for example, among illiquid stocks and stocks with high idiosyncratic volatility.” [Barberis, Mukherjee and Wang]

“Prior research has tested the implications of prospect theory for stock returns under the assumption that investors use a forward-looking representation of stock returns…In this paper, we test prospect theory under the assumption that investors use a backward-looking representation. Both… present evidence that supports their respective assumptions. This __suggests not only that prospect theory is helpful for understanding the cross-section of returns, but also that both forward- and backward-looking representations of stocks may be commonly used by investors__.” [Barberis, Mukherjee and Wang]

“We find that a significant part of prospect theory value’s predictive power for returns comes from the ‘probability weighting’ component of prospect theory. Under probability weighting, the agent overweights the tails of a return distribution, a device that, among other things, captures the widespread preference for lottery-like gambles. The fact that probability weighting plays an important role in our results suggests – and we confirm this in the data – that a high prospect theory value stock is a stock whose past returns are positively skewed. Part of what may be driving our results, then, is that __when investors observe the stock’s past return distribution, perhaps by looking at a price chart, they see the skewness, which, in turn, leads them to think of the stock as a lottery-like gamble and hence to find it appealing__.” [Barberis, Mukherjee and Wang]

### Empirical evidence for bonds

“Our main empirical prediction is that __a bond’s prospect theory value based on its historical return distributions has predictive power for its future returns with a negative sign__. This predictive power should be stronger for junk bonds in which individual investors play a more important role.” [Zhong and Wang]

“Both portfolio and regression analyses show that bonds with higher (lower) prospect theory value will earn lower (higher) future returns. In other words, prospect theory values can predict future returns. Our findings are robust to various specifications of prospect theory value and model specifications.” [Zhong and Wang]

“Due to fear of being less respected by their peers, __institutional investors are generally more loss-averse than individual investors__. The loss aversion of institutional investors explains why the loss aversion accounts for most predictive power of prospect theory in bond market.” [Zhong and Wang]

### Empirical evidence for foreign exchange

“Our paper…empirically examines the role of prospect theory in explaining the cross-section of currency returns… If a fraction of currency investors indeed deviate from the expected utility in their decision-making processes, and assess the risk of currency in line with prospect theory, then their trading activities following prospect theory should affect expected currency returns in the equilibrium.” [Xu, Kozhan and Taylor]

“Spot and one-month forward exchange rates at the daily frequency from January 1, 1985 to February 28, 2018 are collected from Barclays and Reuters through Datastream… Our main empirical analysis focuses on a sample consisting of fifteen exchange rates of developed economy currencies against the US dollar.” [Xu, Kozhan and Taylor]

“We need to construct a measure of prospect theory value at the currency level. The empirical use of prospect theory requires two steps.

- First, investors need to form a mental representation of a risk…investors mentally represent the risk of a currency using the past distribution of exchange rate returns. The historical exchange rate price chart is perhaps the first piece of information that will appear to investors when searching for information about currency. Hence,
__investors are very likely to use price charts (and therefore the exchange rate past distributions) to form a mental representation__about how risky the currency is. - The second step is that investors need to value whether such a representation is appealing or not. We apply the Tversky & Kahneman formula to the past distribution of exchange rate changes and construct currency-level prospect theory value, which reflects how appealing a currency to a prospect theory investor.” [Xu, Kozhan and Taylor]

“[We provide] empirical evidence that the prospect theory value is an important…driver for the cross-sectional variations of currency excess returns.” [Xu, Kozhan and Taylor]

“We find that the __prospect theory value, derived from the historical distribution of exchange rate changes, negatively and significantly forecasts the cross-section of future currency excess returns__. The predictive relation is not only statistically significant but also economically meaningful. A one standard deviation increase in prospect theory value is associated with a 3.6% per annum drop of currency returns in the following month. The predictive power remains strong when controlling for other currency characteristics.” [Xu, Kozhan and Taylor]

“Moreover, sorting currencies into five portfolios based on prospect theory values, we find that __high prospect theory value currencies significantly underperform their lower value pairs__ by about 5% per annum. A long-short strategy buying (shorting) low (high) prospect theory value currencies has only moderate correlations with other currency risk factors, equity risk factors, and hedge fund factors. Abnormal returns (alphas) after controlling for these factors remain statistically significant at 1%…Our __findings speak in favor of the existence of mispricing at the individual currency level rather than the explanation of systematic risk exposure__.” [Xu, Kozhan and Taylor]

“To further test the mispricing hypothesis, we interact the prospect theory value variable with proxies of limits to arbitrage, speculative demand, and investors’ attention. We find that the __predictive power is strengthened when the FX market volatility, the global risk aversion, and the financial market stress are high, namely when arbitragers are more difficult__ to correct for mispricing. Moreover, the predictive relation is also stronger when the global investor sentiment is high, and the relation is weaker when investors pay more attention to macro-fundamentals and hence pay less attention to the historical currency performance, due to the limited attention capacity. Collectively, both the difficulty for rational arbitragers to remove mispricing and the propensity of irrational traders to trade speculatively contribute to the predictive pattern.” [Xu, Kozhan and Taylor]

### References

Barberis, Nicholas and Abhiroop Mukherjee and Baolian Wang (2016), “Prospect theory and stock returns: An empirical test”, Review of Financial Studies 29(11), https://ssrn.com/abstract=2528149

Xu, Qi, Roman Kozhan and Mark Taylor (2020), “Prospect Theory and Currency Returns: Empirical Evidence” https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3629061

Zhong, Xiaoling and Junbo Wang (2018), “Prospect theory and corporate bond returns: An empirical study,” Journal of Empirical Finance, Elsevier, vol. 47(C), https://ideas.repec.org/a/eee/empfin/v47y2018icp25-48.html