Yan Li and Liyan Yang
Prospect theory, the disposition effect, and asset prices
Journal of Financial Economics | Volume 107, Issue 3 (Mar 2013), 715–739

One of the most studied individual trading behaviors is the "disposition effect": investors have a greater tendency to sell assets that have risen in value since purchase than those that have fallen. Because none of the most obvious rational explanations, such as portfolio rebalancing or information story, can entirely account for the disposition effect, an alternative view based on prospect theory has gained favor.

Prospect theory has several salient features: (i) investors evaluate outcomes, not according to final wealth levels, but according to their perception of gains and losses relative to a reference point, typically the purchase price; (ii) investors are more sensitive to losses than to gains of the same magnitude (loss aversion); and (iii) investors are risk-averse for gains and risk-seeking for losses (diminishing sensitivity).

What if everyone acts this way?

It is the diminishing sensitivity feature of prospect theory that researchers often cite as the underlying mechanism of the disposition effect: if a stock is traded at a gain, then the investor is in his risk-averse domain and is inclined to sell the stock; if a stock is traded at a loss, then the investor is in his risk-seeking domain and tends to hold on to the stock. Recent theoretical models, however, suggest that the link from prospect theory and the disposition effect is more nuanced. Specifically, prospect theory will fail to predict the disposition effect when the expected return is high (for an investor to buy a stock, its expected return must be reasonably high, which might encourage the investor to take more risk to continue to hold the stock after a gain), or when returns are positively correlated over time (after a gain, the investor expects another increase in price and would be more likely to hold on to the stock). In all these existing models, stock returns are assumed to follow an exogenous process, and it remains unclear whether prospect theory still predicts a disposition effect when stock returns are affected by the trading of prospect-theory investors.

This question is particularly important in light of two other related asset pricing literatures. First, the loss aversion feature of prospect theory has been used to explain the historical high equity premium. Second, the disposition effect has been employed to generate price momentum, because the disposition effect creates a wedge between a stock's fundamental value and its equilibrium price, leading to price underreaction to information. However, in principle, both a high equity premium and price momentum will make prospect theory less likely to deliver the disposition effect, as suggested by the above-mentioned partial equilibrium models. To examine whether and to what extent prospect theory can simultaneously explain the disposition effect, the momentum effect, and the equity premium puzzle (which are three of the most well-known puzzles and which have been separately investigated using prospect theory), and in particular, to give a definitive answer to whether prospect theory can generate the disposition effect, we need a general equilibrium model in which prospect-theory investors trade stocks and affect stock prices.

Our general equilibrium model

In our JFE paper, we build such a model to examine the implications of prospect theory for trading, pricing, and volumes. We adopt an overlapping-generation (OLG) setup with three generations: age-1, age-2, and age-3. Our OLG setup can be understood as a stylized way of describing how different types of investors in real markets interact with each other. The age-1 investors correspond to new participants to the market; the age-2 investors correspond to the discretionary traders who have been sitting in the market for some time; and the age-3 investors correspond to pure noise investors. All investors trade a risky asset (stock) and a risk-free asset (bond). In each period t, the bond is traded at the constant gross risk-free rate Rf > 1, and the stock's price Pt is determined by investors' trading behaviors.

The stock is a claim to a stream of dividends. The dividend growth rate is independently and identically distributed (i.i.d.) over time, and it can equally likely take either a high value θH or a low value θL (with 0 < θL< θH). However, some investors are more optimistic than others in the sense that they believe the next period dividend growth rate is more likely to be high. This heterogeneity implies that in each period those optimistic investors will hold the stock in equilibrium. Also, for a given investor, he may experience belief changes: it is possible that he was initially optimistic and then becomes pessimistic, and so, he may first buy the stock and later want to sell it, which creates the possibility for prospect theory to affect his selling behaviors.

When investor i enters the market, he is endowed with W1,i units of consumption good. He can trade at ages 1 and 2, leaving his final wealth as W3,i and his capital gains/losses as X3,i. His time t utility, Uti, is then given by Uti = Eti[ v(X3,i) ] , where Eti[⋅] is his expectation at time t, and X3,i = W3,i – Rf2⋅ W1,i measures the capital gains/losses, and where  v(x) =  left {  begin{array}{cl}
    x^{ alpha} ,  RAWAMP;  text{if } x geq 0  RAWBACKBACK;
    - lambda sdot(-x)^{ alpha} ,  RAWAMP;  text{if } x  0
  end{array}  right.
;/var/tmp/iawltxhtml/mathcache//udisplaymath136e46912908b605219432997675303f.svg is the standard prospect-theory value function that is used to evaluate the gains/losses.

Parameter α ∈ (0, 1] governs the value function's concavity/convexity and parameter λ ∈ [1,∞) controls loss aversion. When α = λ = 1, both diminishing sensitivity and loss aversion vanish, reducing the preferences to a standard risk-neutral utility representation. Figure 1 plots v(x) for the cases of α = 0.5 and α = 2.25: it is concave over gains and convex over losses, and it has a kink at the origin.

1: The value function of prospect theory
The parameters to the v(x) function here are α = 0.5 and λ = 2.25.

Our financial market works as follows. In each period, before trading occurs, the stocks are initially held by the age-2 and age-3 investors who were initially optimistic and already purchased stocks in previous periods. All those age-3 investors sell stocks because they will exit the economy at the end of the period. Whether age-2 investors continue to hold the stock depends on their expectations of future dividend growth rate, and the sufficient pessimistic ones end up selling the stock. The (reversed) disposition effect concerns the different behaviors of those age-2 investors as a group in good versus bad dividend news. Their state-dependent behaviors will influence prices by shifting the aggregate demand function, generating momentum or reversal. The age-1 investors who just enter the market and the remaining age-2 investors who didn't purchase the stock last period decide whether to buy the stock by comparing utilities from buying with those from not buying; those optimistic among them will end up buying the stock.

We analytically solve the equilibrium in the case of risk-neutral utility (α = λ = 1), which serves as our benchmark economy. In the case of prospect-theory utility, we use numerical method to solve the equilibrium. In the numerical analysis, we take one period to be six months, and set θH = 1.19 and θL = 0.83, to match the empirical values of the mean and volatility of the net annual dividend growth rate. The risk-free rate is set at Rf = 1.0191.

1: Trading and pricing implications of prospect theory
Panel A: Implications of diminishing sensitivity (by varying α and fixing λ = 1)
Variables α = 0.3 α = 0.5 α = 0.88 α = 1
DispEffect 1.93 1.61 1.13 1.00
MomEffect, in % 5.62 3.28 0.76 0.00
EqPremium, in % 1.57 1.54 0.93 0.00
Panel B: Implications of loss aversion (by varying λ and fixing α = 1)
Variables λ = 1 λ = 2.25 λ = 2 λ = 4
DispEffect 1.00 0.95 0.89 0.81
MomEffect, in % 0.00 –0.25 –0.56 –1.03
EqPremium, in % 0.00 3.77 5.61 7.91
Panel C: Quantitative analysis (by varying α and fixing λ = 2.25)
Variables α = 0.37 α = 0.48 α = 0.52 α = 0.61 α = 0.88 Empirical
DispEffect 2.15 1.79 1.68 1.49 1.07 2.24
MomEffect, in % 4.97 3.59 3.16 2.32 0.34 5.27
EqPremium, in % 5.09 5.08 5.02 4.99 4.63 3.84
This table uses simulations to examine the implications of prospect theory for the disposition effect, the momentum effect, and the equity premium. In the simulation, we take one period to be six months. In each period, a good dividend shock and a poor dividend shock are equally likely. Parameters θH and θL are calibrated at 1.1913 and 0.8310, to match the mean and volatility of the annualized dividend growth rate. The risk-free rate is set at Rf= 1.0191. The empirical values in Panel C are borrowed from previous studies or are computed based on NYSE/Amex data from 1926–2009. Parameter α determines diminishing sensitivity, and parameter λ controls loss aversion. When α < 1, diminishing sensitivity is active, and a smaller α means that investors are more risk averse over gains and more risk loving over losses. When λ > 1, loss aversion is active, and a larger λ means that investors are more loss averse.

We report the results in Table 1. The variable DispEffect is the ratio of “proportion of gains realized” (PGR) to “proportion of losses realized” (PLR). If DispEffect > 1, then investors exhibit a disposition effect in our model, and if DispEffect < 1, they exhibit a reversed disposition effect. We measure momentum as MomEffect = E (Rt + 1 | θt = θH) – E (Rt + 1 | θt = θL), where Rt + 1 is the gross return on the stock between time t and t + 1. That is, MomEffect is the difference in the expected return following positive dividend news and following negative dividend news. If MomEffect > 0, then there is momentum in stock returns, and if MomEffect < 0, then there is reversal. The variable EqPremium = E(Rt – Rf) is the equity premium, i.e., the average stock return in excess of the risk-free rate.

Diminishing sensitivity, momentum, and the equity premium

Panel A examines the implications of diminishing sensitivity by conducting comparative static analysis with respect to α. We here set parameter λ at 1 to remove the loss aversion feature of the preference. In particular, when α = 1 (i.e., when investors are risk-neutral), investors do not exhibit the (reversed) disposition effect, and returns do not exhibit momentum(reversal) and have a mean equal to the risk-free rate. However, as long as α<1, the diminishing sensitivity feature of prospect theory drives a disposition effect, a momentum effect, and a positive equity premium.

The intuition is as follows. When a stock experiences good news and increases in value relative to its purchase price, investors who previously purchased it are in the concave, risk-averse region of the value function of prospect theory. Conversely, when a stock experiences bad news, its investors face capital losses, and they are in the convex, risk-seeking region. So, facing good news, they are keen to sell the stock (i.e., a disposition effect), and their selling makes the stock price underreact to the initial good news in equilibrium, leading to subsequent higher returns (i.e., a momentum effect). Facing bad news, they are reluctant to sell, absent a premium; the price underreacts to the initial bad news, giving rise to subsequent lower returns. The intuition for the positive equity premium is subtle and it is driven by the behavior of age-1 investors who are more likely to wait to buy the stock in the future when their value function is more curved.

Loss aversion, reverse disposition, and price reversal

Panel B examines the implications of loss aversion by conducting comparative static analysis with respect to the parameter λ and by setting parameter α = 1 to remove the diminishing sensitivity feature. We find that, loss aversion drives a reversed disposition effect, a reversal in stock returns, and a positive equity premium. The intuition is the following. Loss aversion means that the prospect-theory value function has a kink at the origin. Investors are afraid of holding stocks if they are close to the kink. As is well known in the literature, loss aversion produces positive equity premiums in equilibrium, and thus, good news will push investors far from the kink and bad news will push them close to the kink. As a result, when facing gains, investors are more likely to hold stocks (i.e., a reversed disposition effect); the increased demand resulting from the reversed disposition effect causes the stock price to “overreact” to the initial good news, pushing the current price even higher and leading to lower subsequent stock returns (i.e., a reversal in stock returns). This effect is symmetric, so that when a stock experiences bad news, the opposite happens.

Diminishing sensitivity is more important than loss aversion

Given that different components of prospect theory often make opposite predictions regarding investors' trading and asset prices, it is nontrivial to examine whether prospect theory can explain the data in economies with preference parameters α and λ set at their empirical values. Panel C conducts such a quantitative analysis. Specifically, abundant empirical studies estimate λ to be close to 2, and hence we fix λ at 2.25. Previous empirical studies have obtained different estimates for the value of α, and we report results for all the possible estimates of α: 0.37, 0.48, 0.52, 0.61, and 0.88. We also report in Panel C the historical values for the variables of interest, which are either borrowed from previous studies or computed based on NYSE/Amex data from 1926–2009. We find that, for all five possible values of α, the diminishing sensitivity component of prospect theory dominates the loss aversion component, which implies that prospect theory indeed helps to explain the disposition effect, the momentum effect, and the high equity premium. In particular, for the case of α = 0.37, the model-generated variables are quite close to their empirical values.

In our JFE paper, we also use our model to explore the volume and pricing implications of dividend volatility and skewness and suggest new testable empirical predictions. Our analysis highlights the importance of using prospect theory to advance our understanding of individual trading behavior and salient asset-pricing phenomena.

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