1 Prospect theory, mental accounting, and momentum Grinblatt and Han JFE (2005)

2 I. Introduction One of the most well-documented regularities in the financial markets is that investors tend to hold on to their losing stocks too long and sell their winners too soon: “disposition effect” (Shefrin and Statman 1985) The leading explanation for the disposition effect: KT(1979)’s “prospect theory” (PT) and Thaler’s (1980) “mental accounting” (MA). 1. Due to the difference in risk attitude (S-shaped value function), investors subject to PT/MA have

3 I. Introduction (con.) a greater tendency to sell stocks that have gone up in value since purchase. 2. Figure 1: For power utility, the fraction of wealth invested in a stock is unaffected by the (initial wealth) starting point. However, the demand function is shifted by the substitution of a convex utility function to the left of the inflection point. Demand is increased more at point C (as a starting point) than at point D. The critical determinant of demand is the starting position in the value function.

4 I. Introduction (con.) 3. The reference point: when the relevant mental accounts employ the cost basis in a stock as the reference point, the starting positions are dictated by the unrealized capital gains or loss in the stock (for example: extreme “winner” stocks start investors at point D and extreme “loser” stocks start investors at point A). This paper’s focus is on the implication of the “disposition effect” on asset pricing. 1. Price distortion: some investors are subject to PT/MA

5 I. Introduction (con.) behaviors. They have demand functions that are inversely related to the unrealized profits they have experienced on a stock ( 需求受到未實現 利得的反向影響 ). This demand distortion creates a price distortion (with respect to the equilibrium price in the absence of PT/MA investors): tends to generate price under-reaction to public information. 2. Return distortion: as disposition effect trading occurs, the cost bases across investors change as does an appropriate aggregate cost basis for

6 I. Introduction (con.) investors as a whole. On average, the dynamics of this process tend to reduce the absolute spread between the aggregate cost basis and the market price. As a result, the market price in the next trading round reverts towards its fundamental value: stocks with paper capital gains will have higher average returns going forward than stocks with paper capital losses. 3. One implication is that we expect to see momentum in stock returns. Any variable that captures the unrealized capital gains will also be

7 I. Introduction (con.) a predictor of the cross-section of expected returns. This model also distinguishes itself from others that explain momentum in predicting that (one- period) lagged capital gains are sufficient statistics for forecasting the cross-section of returns (see p.314). 1. It is the pattern of past returns, combined with the pattern of past trading volume, that determines whether the stock has experienced an aggregate unrealized gain or loss.

8 I. Introduction (con.) 2. With cross-sectional Fama-MacBeth (1973) regression, this study finds that (unrealized) capital gains variable predicts future returns, even after controlling for the effect of past returns, but the reverse is rarely true.

9 II. The Model The fundamental value and demand function 1. Fundamental value follows a random walk: 2. Rational demand and PT/MA demand ( 後者的需 求會受到利得與損失的影響 ) R t : a reference price relative to which PT/MA investors measure their gains or losses; : a positive constant

10 II. The Model (con.) Equilibrium price and returns 1. Assume the risky stock is in fixed supply (one unit). The equilibrium market price is (eq.4) where < 1 ( : the fraction of PT/MA investors) 此式顯示價格是 fundamental value 與 reference price 的加權平均, 且 market price under-reacts to public information about the fundamental value.

11 II. The Model (con.) 2. Each PT/MA investor is assumed to use a mental account that is separate for each stock. If the relevant reference price is the cost basis for the shares investors acquired of that stock, then (eq.5) where should be related to the stock’s turnover ratio. (P t 是 t 期買入股票者的 reference price) 3. By combing eq.4 and eq.5, the dynamics of the market price can be expressed as follows:

12 II. The Model (con.) 由此式發現 :  In the absence of a mechanism for the reference price to change, there is no expected price change. Heterogeneity in the degree to which investors are subject to PT/MA of any variety induces trades and revises the cost basis of the shares in an investor’s portfolio.

13 II. The Model (con.)  The changes in R are of the same sign as the gain- the difference between the market price and the reference price. This reference price updating leads both the market price and the reference price to revert to the fundamental value. 4. The expected returns is (eq.8)

14 II. The Model (con.)  This equation suggests that a stock’s expected return is monotonically increasing in the percentage unrealized capital gain.  Momentum: Since a stock’s capital gain is likely to be correlated with its past return, the past return is a noise proxy for the unrealized aggregate capital gain.  The portfolio formation horizon over which momentum is likely to be strongest is an intermediate one.

15 III. Empirical analysis Basic idea: test the theoretical model’s price dynamics (eq.8) by analyzing the relation between aggregate capital gains and the cross-section of expected returns. 1. The relevant reference price: estimating a proxy for the market’s cost basis in a stock (eq.9). 此式中 t 期 的參考價格為落後期價格的加權平均, 權數為該 股票於 t-n 期最後一次被交易, 而後到 t-1 期皆未被 交易的機率. 2. Data: weekly returns, turnover (weekly share trading volume divided by the number of outstanding shares)

16 III. Empirical analysis (con.) and market capitalization data from the MiniCRSP. The sample period, from July 1962 to December 1996, consists of 1,799 weeks. Regression description ( 迴歸變數 ) The regression is eq.10 ( 式中省略 t 與 j 的標示 ) r t j : week-t return of stock j ; : stock j’s cumulated return from week t-t 2 to t-t 1 (the prior cumulated returns over short, intermediate, and long horizons are used as control regressors for the return effects)

17 III. Empirical analysis (con.) : the logarithm of firm j’s market capitalization at the end of week t-1 (control for the return premium effect of firm size) : stock j’s average weekly turnover over the 52 weeks prior to week t (control for the possible effects of volume) : capital gains-related proxy (eq.11) ( 實際上在根據第九式估計 reference price 時, 是 取落後五年來計算 )

18 III. Empirical analysis (con.) Summary statistics 1. 圖 1 為每一週橫斷面股票之 capital gains overhang ( 即 g) 的 weekly time series ( 三條線分別 代表 10%, 50%, 90%): 顯示 there is wide cross- sectional dispersion and a fair amount of time- series variation in this regressor g. 2. Table 1  Panel A: 迴歸式中各變數的基本敘述統計量 ( 包 含 time-series means and standard deviations

19 III. Empirical analysis (con.) of the cross-sectional averages of the independent variables, along with time-series means of their 10 th, 50 th, 90 th percentile).  Panel B: 檢驗 capital gains g 與其他解釋變數之 間的關係. Regress g (cross-sectional) on stock j’s cumulated returns and average weekly turnover for three past periods. 發現 g 的橫斷面差異有 59% 可以被 past returns( 正 ), past turnover( 負 ) 與 firm size ( 正 ) 所解釋 ( 請參考 p.321).

20 III. Empirical analysis (con.) Double sorts The basic idea is that past returns also predict risk- adjusted returns, but should be noisier predictors. 故 作者先透過以 past one-year return 與 capital gains overhang 來進行雙重排序, 檢驗所形成之投組的平 均報酬. 1. Table 2  Panel A: the average capital gains (left) and the average past returns (right)  Panel B: first sort on past return (then sort on capital gains) 時, 25 個投組的 average weekly returns

21 III. Empirical analysis (con.)  Panel C: first sort on capital gains (then sort on past returns) 時, 25 投組的 average weekly returns 2. 由表 2, 發現  During non-January months, for each R quintiles, the average returns of portfolios increase monotonically with their capital gains overhang quintiles ( 且 G5-G1 is significant for each of the past return quintile). However, the reverse is not true. This result is consistent with the model’s prediction.

22 III. Empirical analysis (con.)  During January months, the results are not consistent with the model’s prediction. Within panel B’s past return quintiles, the January returns of G5 stocks tend to be below those of G1 stocks. This may reflect a December tax-loss selling effect or a size effect. Regression results ( 迴歸結果 ) Table 3 presents the average coefficients and time-series t-statistics for the regression eq.10 and its variations.

23 III. Empirical analysis (con.) 1. Panel A and B 的結果一如預期 : when g is excluded from the regression, there is reversal of returns at both the very short and long horizons, but persistence in returns over intermediate horizon. Panel B indicates there is a volume effect. 2. Panel C: when g is included in the regression, there is no longer an intermediate horizon momentum effect. Except for January, there is a remarkably strong cross-sectional relation between g variable and future returns, with a

24 III. Empirical analysis (con.) sign (+) predicted by the model ( 由於後 10% 與 10% 投組的 g 值差距約 60% ( table 1 ), 這表示輸 家與贏家投組的平均報酬差異約為 12.5% / year). 3. Seasonalities  Momentum strategies that form portfolios from past returns over intermediate horizons appear to be most effective in December and there is a strong reversal in January.  : drift downward in December and revert to it normal positive value in early January. Since the

25 III. Empirical analysis (con.) downward shift in in December implies that market prices move closer to fundamental values, it implies that for stocks with capital losses, the decline in represents an added force that makes the stock’s market price decline even further. Similarly, the increase in in early January would make the prices of these same stocks with capital losses deviate again from their fair values, leading to a January reversal (p.328).