Jonathan E. Ingersoll, Jr. and Lawrence J. Jin
Realization utility with reference-dependent preferences
Review of Financial Studies | Volume 26, Issue 3 (Mar 2013), 723–767

Why do investors buy and sell stocks? The most natural answer is that investors believe that they have new information and want to act on it to maximize their future returns or limit their risk. Alternatively, investors may trade a stock because selling it at a gain gives them a burst of pleasure by confirming that the original decision to buy was correct. Conversely losses may be avoided as they indicate faulty decisions. In academic terms, we call the former type of explanation belief-based and the latter type preference-based. In our paper, “Realization Utility with Reference-Dependent Preferences” (Ingersoll-Jin (RFS 2013)) and this shorter note, we focus entirely on preference-based explanations for trading.

Profits make me happy, losses make me sad...

Of course, there are many preference-based, as well as belief-based, stories that help explain investors' trading behaviors. Investors may trade stocks after a change in family or employment status or due to new objectives in life. They also may trade when their existing portfolios have become unbalanced. Despite such explanations, quite a few empirical facts remain puzzling. For example, academics have no commonly agreed-upon rationale for why individual investors trade excessively even though they underperform passive indices on average. Our general goal is to develop theoretical models to close this gap. Our particular explanation focuses on realization utility as introduced in a dynamic setting by Barberis-Xiong (JFE 2012). Realization utility is based the idea that whatever pleasure or pain is attained from investing comes only when the investor takes the definite action of closing out a position.

...but the feelings diminish

To provide a realization utility framework to explain investors' trading behaviors, an important question is why do investors sometimes sell stock at a loss? If the purpose of trading stock is to enjoy the good feeling of realizing gains, why don't investors always hold on to losses hoping for a later recovery in the stock price? Again one can think of many belief-based and preference-based explanations, but our goal is to provide a unified explanation of both selling at gains and at losses using the idea of realization utility. In order to do so, we incorporate a key ingredient from the behavioral-economics literature: as the level of overall gains and losses increases, the pleasure and pain brought to investors by an extra dollar of gain and loss tends to decrease. For instance, making an additional one thousand dollar gain on top of a million dollar profit does not bring you as much excitement as does the first one thousand dollars of the profit. This is also true for losing an additional one thousand dollars. Academics commonly call this property diminishing sensitivity, as discussed by Tversky-Kahneman (JRU 1992). Note that this differs from the standard assumption in economic theory that there is increasing sensitivity for larger losses.

In Ingersoll-Jin (RFS 2013), we use a dynamic model to study both realization utility and diminishing sensitivity. Our key finding is that incorporating diminishing sensitivity into a realization utility framework significantly improves the model's match to the empirical facts on trading activities and asset prices. In particular, without diminishing sensitivity, investors never voluntarily sell stocks at a loss. Adding this component, however, generates the model predictions that investors do optimally realize both gains and losses and that the former occurs more frequently.

Our model

In this note, we present a simplified version of the model in our RFS paper to illustrate the key insights. We then discuss how these results help explain some puzzling empirical facts. Finally, we provide thoughts on testing some new predictions.

Consider the following economic setup. An investor purchases a share of a stock. Each period while the share is held, its price increases or decreases by one dollar with probabilities π and 1 – π, respectively. When the investor sells, she gets a burst of realization utility that depends on the size of the gain or loss. She then opens a new trading position by purchasing one share of another stock.

The investor's objective is to time the repeated sales and purchases to maximize her average realization utility per period. Due to the simple nature of this problem, the optimal strategy is to pick two fixed values, L < 0 < G, and sell the first time the stock price increases by G dollars or decreases by L dollars from the original purchase price. For a given strategy, G and L, the average realization utility per period is (1)  frac{ P(G,L) sdot u(G) + [ 1- P(G,L) ]  sdot u(L) }{T(G,L)} ,  ;/var/tmp/iawltxhtml/mathcache//ndisplaymath5a32d9d0169d9e5b97322fad53e06293.svg where P is the probability that the gain is ultimately realized and T is the average duration of each holding period. Because each investment episode is identical ex ante, the sequence of investments is a renewal process. Equation 1 is a statement of the Elementary Renewal Theorem. The function u(⋅) measures the amount of realization utility that the investor receives depending on the size of the gain or loss. We pick the curvature of this function to capture the property of diminishing sensitivity discussed earlier.

The two functions, P(G, L) and T(G, L), can be determined by iterated expectations and are  P(G,L) =  frac{1- beta^L}{ beta^G -  beta^L}  qquad T(G,L) = 
     frac{( beta^L-1) sdot G - ( beta^G-1) sdot L}{(1-2  pi) sdot( beta^G- beta^L)} , ;/var/tmp/iawltxhtml/mathcache//udisplaymath86cfe5813d7539199f57ad5d9f646405.svg where β = (1 – π)/π. The u(⋅) function we use for measuring realization utility is a special case of the Tversky-Kahneman (JRU 1992) utility function  u(x)= left {
   begin{array}{cl}
    (x)^{ alpha}  RAWAMP; x  geq 0  RAWBACKBACK;
    -(-x)^{ alpha}  RAWAMP; x   0
   end{array} right. ,
;/var/tmp/iawltxhtml/mathcache//udisplaymathdaa56282b98c1372b364b6aee7778ef7.svg where x is the size of the gain or loss, and α (0 ≤ α ≤ 1) is a parameter that measures the degree of diminishing sensitivity. For α = 1 sensitivity does not diminish at all; the smaller is α the more quickly does sensitivity diminish. Table 1 shows the average realization utility per period for two different specifications.

1: Average realization utility per period
Panel A: Utility parameter: α = 1
L
G –1 –2 –3 –4 –5  – ∞
1 0.20 0.20 0.20 0.20 0.20 0.20
2 0.20 0.20 0.20 0.20 0.20 0.20
3 0.20 0.20 0.20 0.20 0.20 0.20
4 0.20 0.20 0.20 0.20 0.20 0.20
Panel B: Utility parameter: α = 0.5
L
G –1 –2 –3 –4 –5  – ∞
1 0.20 0.27 0.26 0.25 0.24 0.20
2 0.07 0.14 0.16 0.16 0.16 0.14
3 0.04 0.10 0.12 0.12 0.12 0.12
4 0.03 0.08 0.09 0.10 0.10 0.10
The per-period probability of a price increase (π) is 60%. The optimal gain-loss realization strategy is indicated in bold.

Take good news in small doses and bad news in large ones

In Panel A, the utility parameter α is one. In this case, realization utility is linear in the size of gain or loss; that is, there is no diminishing sensitivity. This means that the marginal benefit of a one dollar gain is always the same regardless of how large a gain or loss is. As a result, all strategies yield the same average realization utility per period of 0.20, which is just the average per-period increase in the stock price.

In Panel B, the utility parameter α is 0.5. In this case, the first dollar gained or lost changes utility by one while the tenth dollar of a gain or loss changes utility by only 0.162, so there is a strong degree of diminishing sensitivity. Now there is a unique optimal trading strategy; in every trading episode, the investor waits until the stock price goes up by one dollar or goes down by two dollars before closing it out and opening a new position. The investor is willing to realize losses in this case because a two dollar loss yields a total decrease of 1.4 units of realization utility, whereas two separate one-dollar gains have a total benefit of 2 units of realization utility. Although closing out an underwater position is immediately painful, it frees up cash for future investments that on average generate trading profits and more utility in the future. (Specifically, keeping G = 1 and moving L from –3 to –2 increases the probability of realizing a loss from 12.3% to 21.1%, but it also shortens the average duration of each trading episode from 2.54 to 1.84 periods.) The property of diminishing sensitivity makes a sequence of small gains more beneficial to the investor than an occasional large loss. Together with realization utility, this generates the prediction that the investor will optimally realize small frequent gains and large infrequent losses.

This simple illustration highlights the results of our model. In the original RFS paper, we also consider many other factors. We examine the more realistic case of proportional rather than absolute price movements. We do not restrict purchases to a single share so the size of the investment affects investors' trading decisions. We consider transaction costs and show that their presence makes investors defer sales until both gains and losses are larger because trading frequently is very costly. Finally, we incorporate positive time discounting, the notion that getting a gain now means more to the investor than getting the same gain a year from now. This accelerates the realization of gains but can either retard or hasten the taking of losses. On the one hand, the investor naturally wants to postpone painful loss taking; on the other hand, the investor has an incentive to accelerate loss taking because by doing so she can realize future gains sooner.

Our model sheds light on a number of puzzling empirical facts. Among these, the most obvious one is something called the disposition effect. This is an empirically robust pattern that investors have higher propensities to sell stocks that have risen in value than stocks that have fallen in value. This is puzzling because empirical research shows that stocks display momentum: stocks that have done well tend to continue to do well in the future, while stocks that have done poorly tend to do poorly in the future. So, if investors do trade based on information, they should exhibit the opposite of the disposition effect. (Odean (JF 1998) examines other explanations such as portfolio rebalancing and tax motives and finds that these cannot explain the disposition effect.)

Under the framework of realization utility with diminishing sensitivity, however, the disposition effect naturally arises. On the one hand, investors tend to sell stocks that have risen in value because by doing so they get positive feelings (positive realization utility). On the other hand, even though selling stock at a loss brings investors immediate negative feelings, closing out losing positions allows investors to take on new investments that on average accelerate future gains and the good feelings when realizing them. It is only when investors have diminishing sensitivity that the good feelings from realizing frequent small gains can more than offset the bad feelings from realizing infrequent large losses.

In our paper, we provide some detailed calibration exercises. Using reasonable parameter values, our model predictions match the magnitudes and frequencies of realized gains and losses and the frequencies of paper gains and losses as observed in the trading data of Odean (JF 1998) and Dhar-Zhu (MS 2006).

Volatility and trading volume

Another empirical pattern that is consistent with our model is the flattening of the security market line. Ang-Hodrick-Xing-Zhang (JF 2006) document that high-beta and high-residual-risk stocks have smaller expected returns than predicted by equilibrium models such as the capital asset pricing model. Our framework gives a novel explanation for this. Stocks with higher volatilities, whether systematic or idiosyncratic, provide more opportunities for investors to earn realization-utility benefits. As a result, these investors tend to hold more of the highly volatile stocks than predicted by other equilibrium models. The excess demand of realization utility investors pushes up the prices of these stocks and decreases their expected returns.

Other findings that our model can help explain include the observations that individual investors trade excessively despite their underperforming market benchmarks even before transaction costs, that the trading volume is higher in rising markets than it is in falling markets, that investors have a higher propensity to sell a stock once its price rises above its recent historical high, that highly valued assets are heavily traded, that there is an empirical V-shaped pattern between the probability of selling a stock and its unrealized paper gain, and that investors hold and trade individual stocks and do not diversify their portfolios as much possible.

Finally, our model generates some new testable predictions particularly if the reference level, the benchmark with which investors divide their trading gains from losses, depends on the history of stock prices. One prediction is that total volume will be higher in a market that rises or falls quickly followed by a slowly trending period than in a market with the opposite pattern. And in both of these markets, the volume should exceed that in a market with a slow steady rise or fall of the same total magnitude.

In summary, our model highlights the important roles that realization utility and diminishing sensitivity play in understanding investors' trading behaviors. It helps explain of the empirical facts that previously seemed puzzling. We hope that future research can provide additional qualitative and quantitative tests of this framework and further compare our model with alternative theories.


02-reymerswael
Marinus van Reymerswaele (1490–1546): Moneychangers. Flemish, 1548.. Reymerswaele is known for his paintings documenting the flourishing economic activity of northern Europe in the 16th century. Does this painting show the age old theme of ridiculing the supposed greed of the affluent money changers or bankers?