1 Arbitrage risk and the book- to-market anomaly Ali, Hwang and Trombley JFE (2003)

2 I. Introduction Empirical evidence: predictable returns over three to five years for portfolios long in high B/M stocks and short in low B/M stocks. There are two competing explanations. 1. The return to B/M based portfolio strategies represents compensation for risk (Fama and French(1992, 1993)). 2. The return to B/M based portfolio strategies results from systematic mispricing of extreme B/M securities.

3 I. Introduction (con.)  Studies show that market participants underestimate (overestimate) future earnings for high (low) B/M stocks. (La Porta et al., 1997).  Shleifer and Vishny (1997) argue that arbitrage is costly. The risk due to the volatility of arbitrage returns (arbitrage risk) deters arbitrage activity. This study empirically examines the prediction of SV(1997) and provides additional evidence to discriminate between the risk and mispricing

4 I. Introduction (con.) explanations for the B/M effect. 1. If the B/M effect is due to mispricing, it should be greater with higher expected volatility. This study uses historical volatility of the market model residuals as a proxy for idiosyncratic risk. 2. Besides, this study also finds that the B/M effect is greater for stocks with higher transaction costs and stocks with less ownership by sophisticated investors.

5 I. Introduction (con.) Related literature 1. Lakonishok et al., (1994) predict that returns to B/M strategies are smaller for stocks of larger companies (arbitrage costs and investor sophistication affect the existence of mispricing and firm size proxies for these factors). However, La Porta et al., (1997) report that the B/M effect is not much different across small and large firms. 本文的貢獻 : 以更直接的變數來 測度套利成本與投資者的複雜度, 以檢驗 B/M 效果 的價格錯估假說.

6 I. Introduction (con.) 2. Prior empirical studies attempting to explain the existence of systematic mispricing on the basis of arbitrage costs focus mainly on transaction costs, and not much attention is given to arbitrage risk. 本文的貢獻 : 強調套利風險的影響, 並且 examine whether arbitrage risk contributes incrementally to the existence of the B/M effect.

7 II. Data and Variables Returns to B/M based portfolio strategies 1. Samples: the sample consists of all firms on NYSE and AMEX from the period 1976 to B/M is calculated as book value in year t-1 divided by market value of equity at the end of June of year t. 2. Form quintile portfolios with the lowest B/M observations (Q1) and the highest B/M observations (Q5) for each year. Buy-and-hold returns are measured over one-, two-, and three-year holding periods beginning in July of year t.

8 II. Data and Variables (con.) 3. Then compute difference in returns of the extreme B/M portfolios (Q5-Q1) for each sample year and use the time-series of return differences to calculate the mean value and the test statistics. 4. Table 1: B/M based portfolio strategies produce positive returns.  Raw returns (Ret12, Ret24, Ret36)  Size-adjusted returns (SRet12, SRet24, SRet36): raw returns minus the corresponding NYSE/AMEX CRSP size-decile index returns ( 調整規模大小的影響 ).

9 II. Data and Variables (con.) Measures of arbitrage risk, transaction costs and investor sophistication 1. Expected arbitrage risk is Ivolatility: by regressing daily returns on a value-weighted market index over a maximum of 250 days ending on June 30 of year t and then computing the variance of the residuals. 2. Direct transaction costs: (1) Bid-Ask: (bid-ask)/ 0.5*(bid+ask) ( 由 July of year t-1 to June of year t, 共 12 個月之每月最後一個交易日 的最後一小時報價來平均 )(available from 1994 to 1997).

10 II. Data and Variables (con.) (2) Price: the closing price per share in June of year t. 3. Indirect transaction costs Volume: the annual volume of trade from July 1 of year t-1 to June 30 of year t (in million of dollars). 4. Short-selling costs Inst%: the percentage of common stocks owned by institutions at the end of year t-1 (available beginning in 1987)

11 II. Data and Variables (con.) 5. Comprehensive measure of transaction costs Zerofreq: the frequency of zero daily returns over the period July 1 of year t-1 to June 30 of year t. 6. Investor sophistication (1) Analysts: the number of analysts’ estimates included in the database in May of year t (beginning in 1976). (2) Inst#: the numbers of institutional owners at the end of year t-1 (beginning in 1987).

12 II. Data and Variables (con.) For each year, all the observations are ranked ( 每 一年根據前述套利風險等指標, 將樣本股票分 成十組, 第一組 G1 代表套利風險與成本最高者, 依序遞減 ). 1. All of the variables exhibit substantial variation between G1 and G10 groups (p.363). 2. Table 2: Spearman correlations among the measure of arbitrage risk, transaction costs, investor sophistication and firm size.

13 II. Data and Variables (con.)  The correlations of ME with other measures are significant different from zero. Also each correlation is also significant less than one. It is not sufficient to examine only the effect of firm size.  The correlations among measures of arbitrage risk, transaction costs, and investor sophistication are significantly less than 1, but they are also not close to zero.

14 III. Empirical results Portfolio tests 1.Independent of B/M rankings, firms are ranked each year in descending order of Ivolatility (G1- G10). This results in 50 portfolios in each year. Q5-Q1 SRet36 are then computed for each of the group (G1~G10).  Table 3 first column: the mean of Q5-Q1 SRet36 for each of the ten groups. The mean of Q5-Q1 Ret36 of group G1 is significantly greater than that of group G10.

15 III. Empirical results (con.)  To examine the relation across all the decile groups, a correlation between group ranks and Q5-Q1 SRet36 of the groups is computed (negative).  Figure 1: annual estimates of Q5-Q1 Ret36 for Ivolatility groups G1 and G10.  Figure 2: cumulated returns for the Q5-Q1 portfolios of groups G1 and G10 at monthly intervals.  These results support the market-mispricing explanation and indicate that arbitrage risk is one of the reasons why arbitrageurs do not trade away the

16 III. Empirical results (con.) systematic B/M-related mispricing. 2. The above analysis is repeated for each of the other measures of arbitrage costs and investor sophistication.  For all the measures, the correlation between Q5- Q1 SRet36 and the group rank for G1 to G10 is negative, consistent with mispricing explanation. Besides, lots of the G1-G10 results are positive and significant. B/M effect increases with arbitrage costs and decreases with investor

17 III. Empirical results (con.) sophistication, consistent with the mispricing explanation. Incremental role of arbitrage risk This study intends to determine the incremental role of arbitrage risk in the existence of the B/M effect (see p.369 regression eq(1)). 1. The coefficients on the interaction terms capture how the B/M effect varies cross-sectionally with the arbitrage cost/investor sophistication measures.

18 III. Empirical results (con.)  A significant negative coefficient on B/M * Ivolatility -1 would suggest that after controlling for other arbitrage cost and investor sophistication measures, Ivolatility explains cross- sectional variation in the B/M effect. 2. Table 4 reports the regression estimates of eq1. and two reduced versions of the model.  Column 1: the presence of B/M effect (consistent with table 1).

19 III. Empirical results (con.)  Column 2: B/M effect increases with the increase in return volatility (-0.138, t=-2.797,consistent with the results in table 3).  Column 3: the results of complete eq1. The coefficient on B/M * Ivolatility -1 remains negative and significant. Among the other interaction terms in eq1, none has a negative and significant coefficient. A likely explanation is that these measures are highly correlated with each other and that more than one measure can capture similar underlying effects.

20 IV. Conclusion The finding is consistent with the Shleifer and Vishny (1997) thesis that risk associated with the volatility of arbitrage returns deters arbitrage activity and is an important reason for the existence of the B/M-related mispricing. The authors also recognize the limitation of their tests in discriminating between the mispricing and risk explanation for the B/M effect. If B/M captures financial distress, then the B/M effect may be stronger for firms with high volatility stocks because the financial distress factor may be more sensitive for such firms.