Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 457 A Behavioral Economics Exploration into the Volatility Anomaly * The NUCB Graduate School Equity Quantitative Research Department, Nomura Securities Co., Ltd. Abstract Contrary to a commonsense view in traditional finance theories to the effect that expected returns on investments in high-risk securities are higher than those in low-risk investments, in the actual stock market, there are negative correlations, respectively, between the beta value of individual securities measured beforehand and the actual returns realized later, and between the idiosyncratic volatility measured beforehand and the actual returns realized later. Here we, based upon the empirical studies of investor behaviors in the Japanese stock market, present the fact that, behind the beta anomaly, there is a preference for high-beta securities by typical institutional investors whose mandate is to beat a benchmark, and also that, behind the idiosyncratic volatility anomaly, there is a preference for positively skewed securities by individual investors, especially those engaged in margin trading, who overweight low tail probabilities assigned to the state of the world in which they make a lot of money by investing in the positively skewed stock, which could be called a gambling preference. Key words: volatility, anomaly, behavioral bias, institutional investor, individual investor JEL classification: G11, G12, G14 1. Introduction Traditional finance 1 theorists commonly argue that expected return on investment in high-risk securities should be high enough to compensate for the risk, and this is a widely prevailing notion. On the other hand, empirical researchers have found that, in the real financial market, long-run average return of securities with higher risk is actually not higher, and often significantly lower, than that of securities with lower risk. How can we interpret this rather scandalous empirical fact? * We thank Motonari Kurasawa, seminar participants at the Policy Research Institute of the Ministry of Finance, Yasuo Kakuta, Shinichi Hirota, participants at the 5th annual meeting of Association of Behavioral Economics and Finance, and participants at the 20th annual meeting of Nippon Finance Association for extremely helpful comments. 1 We use the term traditional finance to differentiate it from behavioral finance.
458 S Iwasawa, T Uchiyama / Public Policy Review While traditional finance researchers have explored interpretations of this empirical fact while they maintain their belief that taking economic risk in investment should be rewarded with premium, researchers of behavioral finance, a field that emerged as an alternative paradigm to the traditional finance, have proposed convincing interpretations, based on observation of investor behavior in actual financial markets. We begin with reviewing this academic development, and then proceed to our own empirical research using data of investor behavior in the Japanese stock market, where the data supports the behavioral view. 2. Traditional finance theory and volatility anomaly One of the most commonly used measures of a security s risk is standard deviation of its return. However, in the Capital Asset Pricing Model 2 (henceforth CAPM) which is a pillar of the traditional finance theory, an individual security i s risk is measured by β im, which is a function of covariance between security i and the market portfolio, because, in the CAPM equilibrium, rational investors are supposed to own the most diversified portfolio, and only a part of the individual security s risk which cannot be diversified away should be priced. Thus, according to the CAPM, a security i s expected excess return (relative to return on safe assets) is positively correlated with β im, as is shown in equation (1). E[R i R f ] = β im E[R m R f ], β im = Cov (R i, R m ) / Var (R m ) (1) R i, R f, and R m indicate return on an individual security i, a safe asset, and the market portfolio, respectively. In contrast, in the actual stock market, we do not necessarily find positive correlations between the beta value of individual securities measured beforehand and the actual returns realized later, but depending on sample period, we observe zero or negative correlation between them 3. Baker et al. (2011) researched data of all US listed stocks for January 1968 to December 2008. They sorted stocks into five groups for each month according to five-year trailing beta (or at least 24 months trailing beta if the history of data is less than 60 months), and tracked the returns on these portfolios. Then they repeated the task, and calculated the cumulative returns of each group 4. Their result clearly shows the negative correlation between the beta value and the returns. Figure 1 shows that a dollar invested in the lowest-beta portfolio in January 1968 increased 2 Sharpe (1964), Lintner (1965). 3 Several researchers pointed this out in the 1970s. Black et al. (1972) noted that the relationship between risk and return was much flatter than predicted by the CAPM. Haugen and Heins (1972, 1975) pointed out that, over the long run, stock portfolios with lesser variance in monthly returns had experienced greater average returns than their riskier counterparts, and concluded that the conventional hypothesis that a risk generates a special reward cannot be empirically supported. 4 We calculate the return of each portfolio on market capitalization weighted basis.
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 459 Figure 1 U.S. Stock Returns by Beta Quintile, January 1968 December 2008 Notes: For each month, we sorted all publicly traded stocks tracked by CRSP (with at least 24 months of return history) into five equal quintiles according to trailing beta. In January 1968, $1 was invested according to capitalization weights. We estimated beta by using up to 60 months of trailing returns (i.e., return data starting as early as January 1963). At the end of each month, we rebalanced each portfolio, excluding all transaction costs. Source: Baker et al. (2011), Figure 1, Panel C to $60.46 by December 2008. In other words, the investment yielded annual return of 10.5 percent. On the other hand, a dollar invested in the highest-beta portfolio in January 1968 was worth only $3.77 at the end of December 2008. The investment managed to produce a positive return, but the annual return was only 3.3 percent. One can observe a similar pattern in the Japanese stock market. Using data of stocks listed on the TSE First section, we made a similar calculation as Baker et al. (2011) 5. Figure 2 shows that one yen invested in the lowest-beta portfolio in January 1985 increased to 3.61 by June 2012, but that one yen invested in the highest-beta portfolio decreased to 0.69. The annualized return on investment is 4.8 percent for the lowest-beta portfolio and minus 1.4 percent for the highest-beta portfolio. How should one interpret the empirical evidence showing the negative correlation, instead of the positive correlation implied by the CAPM, between the beta value of individual securities measured beforehand and the actual returns realized later? Fama and French (1992), a well known paper as the one proposing that beta is dead, is actually a response 5 Sample period is from January 1985 to June 2012. We estimate beta by using up to 60 months (or at least 36 months) of trailing returns.
460 S Iwasawa, T Uchiyama / Public Policy Review Figure 2 Japanese Stock Returns by Beta Quintile, January 1985 June 2012 Value of 1 Invested in 1985 8 4 Bottom Quintile 2 1 0.5 Top Quintile 0.25 85 87 89 91 93 95 97 99 01 03 05 07 09 11 Notes: For each month, we sort all TSE1 listed stocks into five equal quintiles according to trailing beta. In January 1985, 1 is invested according to market capitalization weights. We estimate beta by using up to 60 months of trailing returns (i.e., return data starting as early as January 1980). At the end of each month, we rebalance each portfolio, excluding all transaction costs. from traditional finance researchers. They began by observing that the relationship between the historical beta and the subsequent return of individual stocks in the US was positive but very weak between years from 1941 to 1990, and negative between years from 1963 to 1990, and claimed that the CAPM was empirically rejected. Then they found that two other characteristics of individual stocks firm size and bookto-market equity have significant correlations respectively with the subsequent return of them, and suggested that each of the factors are related to some economic risks 6. According to their interpretation, stocks with smaller market capitalization produced higher return than those with larger market capitalization, because the former was, in some sense, economically riskier. Similarly, they interpreted that stocks with higher book-to-market equity yielded higher return than those with lower book-to-market equity, because the former was riskier in some economic fundamentals. Here we note that, behind their interpretation, there is a belief that taking economic risk 6 Behavioral finance researchers consider these correlations as an anomaly to traditional finance theory, and an evidence of market inefficiency. In the Japanese stock market, correlation between book-to-market ratio and the stock s subsequent return has been observed for the last decade, although correlation between firm size and the stock s subsequent return has not been significant in the recent market ( 2012b).
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 461 should be rewarded with premium 7. Based on the belief, Fama and French (1993) proposed the three factor model, in which they added firm size and book-to-market equity to the beta as proxies for some economic risks, as a true economic model. Their idea is shown by the equation (2). E[R i R f ] = β im E[R m R f ] + γ i E[SMB] + δ i E[HML] (2) In equation (2), SMB stands for difference in return of small capitalization stocks relative to big capitalization stocks (Small Minus Big), HML stands for difference in return of high book-to-market equity stocks relative to low book-to-market equity stocks (High Minus Low). The paradigm, represented by Fama and French s methodology outlined above, in which researchers interpret the empirical facts observed in the actual financial markets as consistent with beliefs of traditional finance theories is still quite influential in the finance academia. However, in recent years, it has been facing another challenge by empirical evidence proposed by Ang et al. (2006, 2009). They first decomposed realized return of individual security i into the one that is explained by the Fama and French three factor model, and the residual of it, and defined idiosyncratic volatility of the security as standard deviation of the residual. R i = {R f + β i (R m R f ) + γ i SMB + δ i HML} + ε i (3) That is, the idiosyncratic volatility of security i is (Var ε i ]) 1/2. Then they pointed out that stocks with high idiosyncratic volatility measured beforehand have abysmally low subsequent returns on average. Ang et al. (2006, 2009) showed that negative correlation between the idiosyncratic volatility and subsequent return existed in many developed stock markets around the world. Here we show that the relationship is observed in the Japanese stock market. Using data of stocks listed on the TSE First section for January 1985 to June 2012, we sorted stocks into five groups for each month according to five-year idiosyncratic volatility as defined by Ang et al. (2006), and tracked the returns on these portfolios. Then we repeated the task, and calculated the cumulative returns of each group. Figure 3 shows that one yen invested in the lowest-idiosyncratic volatility portfolio in January 1985 increased to 3.59 by June 2012, but that one yen invested in the highest-idiosyncratic volatility portfolio decreased to 0.23. The 7 A traditional finance proposition that taking economic risk should be rewarded with premium should be called a belief, rather than a scientific theory. To empirically prove this proposition, we have to have a particular model of expected return and risk. Here, say that the evidence does not support the proposition. Then the traditional finance researchers would say that the evidence does not necessarily reject the proposition, since the model used for the empirical research may be wrong. But if we carry through this logic, it is impossible to reject the proposition, since we cannot test all models. Karl Popper famously claimed that what is unfalsifiable is unscientific (Popper 2002). Following Popper, we call the proposition a belief.
462 S Iwasawa, T Uchiyama / Public Policy Review annualized return on investment is 4.8 percent for the lowest-idiosyncratic volatility portfolio and minus 5.2 percent for the highest-idiosyncratic volatility portfolio. The difference in annualized return is 995 basis points, and wider than the one between extreme quintile portfolios sorted on beta, 613 basis points. Ang et al. (2006, 2009) showed that the negative correlation between the idiosyncratic volatility and subsequent return was observed in many developed stock markets, and that the correlation was individually significant in each G7 country. The empirical facts identified here should be a surprise for traditional finance theorists who maintain their belief that taking economic risk should be rewarded with premium. First, if we assume that investors have well-diversified portfolios, then the idiosyncratic risk should be diversified away and not be priced, inconsistent with the evidence above. Second, more importantly, if we do not assume that investors do not have well-diversified portfolios, the idiosyncratic risk should be priced, but then investment in high idiosyncratic risk should be rewarded with premium, which is the exact opposite relation to the empirical facts. If we try to interpret these empirical observations while maintaining the belief that taking economic risk in investment should be rewarded with premium, the only way is to propose a new model which shows economic risks in a more proper way, just as Fama and French Figure 3 Japanese Stock Returns by Idiosyncratic Volatility Quintile, January 1985 June 2012 Value of 1 Invested in 1985 8 Bottom Quintile 4 2 1 0.5 Top Quintile 0.25 85 87 89 91 93 95 97 99 01 03 05 07 09 11 Notes: For each month, we sort all TSE1 listed stocks into five equal quintiles according to trailing idiosyncratic volatility, which is defined as standard deviation of residual, or ε it, in equation (3). In January 1985, 1 is invested according to market capitalization weights. We estimate idiosyncratic volatility by using up to 60 months of trailing returns (i.e., return data starting as early as January 1980). At the end of each month, we rebalance each portfolio, excluding all transaction costs.
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 463 (1993) did by proposing the three-factor model. In the new model, the risk associated with (low) idiosyncratic volatility is considered a proxy for an additional economic risk. But then what exactly is the economic risk associated with investment in low idiosyncratic volatility? Fama and French (1992) suggested that the economic risk associated with investment in stocks with small market capitalization and in stocks with high book-to-market equity is distressed risk. Stocks with high distressed risk would be low-priced, if they are, reflecting their risk, priced properly, and as a result, their expected return would be high. Based on this reasoning, Fama and French argued that average return on stocks with small market capitalization and on stocks with high book-to-market equity is high, because those stocks involve high distressed risk. While the debate on whether their view is justified or not continues even today, it is empirically valid to say that stocks with high book-to-market equity tend to have high distressed risk, and it gives some credence to the three factor model. But if we argue that stocks with low idiosyncratic volatility are fundamentally riskier than those with high idiosyncratic volatility, and therefore tend to yield higher return, what exactly is the economic risk associated with investment in stocks with low idiosyncratic volatility? 8 Ang et al. (2006) considered the hypothesis that exposure to aggregate volatility risk explains the difference in return on low and high idiosyncratic volatility, and tested it. The result, however, was far from satisfying for traditional finance researchers. They concluded that exposure to aggregate volatility risk accounts for very little of the anomalous low returns of stocks with high idiosyncratic volatility. Since Ang et al. (2006, 2009), finance researchers have viewed the negative correlation, respectively, between the idiosyncratic volatility measured beforehand and the actual returns realized later, and between the beta value of individual securities measured beforehand and the actual returns realized later, as anomalies to traditional finance theory 9. More often than not, they sum up both phenomena into one and call it (low) volatility anomaly. However, as we discuss later in detail, we view that different mechanisms operate under the two phenomena, so in this paper we call the former idiosyncratic volatility anomaly, and the latter beta anomaly. One interesting academic development since Ang et al. (2006, 2009) is that researchers of behavioral finance, a field that emerged as an alternative paradigm to the traditional finance, have proposed convincing interpretations, based on observation of investor behavior in actual financial markets, and empirical research that supports such views have accumulated. 8 For example, Huang et al. (2010) added return reversal factor to the three factor model of Fama and French, and found that idiosyncratic volatility anomaly disappears once return reversals are controlled for. While this finding is interesting, traditional finance researchers would need to propose an economic risk associated with large negative return of a stock in the past period, if they want to use this evidence to maintain their belief. 9 Research that explores interpretations of idiosyncratic volatility anomaly while maintaining the belief of traditional finance paradigm are ongoing in various forms. One important research is to estimate time-varying true volatility of an individual stock, as opposed to its historical volatility measured at a particular point. For example, Fu (2009) estimated expected idiosyncratic volatility of an individual stock applying the exponential GARCH model, and argued that there is a positive correlation between the expected volatility and return of the stock.
464 S Iwasawa, T Uchiyama / Public Policy Review In the subsequent sections we review the trend and explore source and mechanism of the anomalies. 3. Behavioral economics interpretation of the stock market anomaly Traditional finance theory generally posits that the market price of an asset is (roughly) equal to its fundamental value, that is, the financial market is efficient. In contrast, behavioral finance researchers view that the price can diverge with the fundamental value, and that the difference between them often remains unexploited for long. Both theoretical and empirical research that validate the behavioral view have much developed recently, and as a result, researchers who take the behavioral views are not the minority any more in finance academia. Once we accept the view that the market price of an asset can diverge with its fundamental value, we can give a straightforward interpretation to anomalies in the stock market; an asset which tends to be overvalued relative to its fundamental value would produce low return on average in the long-run, because the return is suppressed while the overvaluation is corrected. The story that an investment in an overvalued asset can produce poor return in the longrun, even though the investment involves substantial risk, should sound familiar for many people (aside from proponents of traditional finance theory). As an example, average annual return of TOPIX dividend index, a leading Japanese equity index, in 21 years from 1990 to 2011 was minus 2.2 percent with annual risk, measured by its standard deviation, of 24.3 percent. In the meantime, average annual return of Nomura Government Bond Performance Index, a leading index of Japanese government bonds, at the same period was 4.1 percent with annual risk of only 4.8 percent. This simple comparison illustrates that investment in Japanese equity during that period involved high risk, yet produced low return, and not too many people would disagree that one reason behind this result is that Japanese equities at the beginning of the year 1990, with leading indices at historical high, were overvalued relative to their fundamental value. According to the noise trader approach to finance, a leading theory of behavioral finance which was proposed by De Long et al. (1990), two conditions have to be met for an ineffcieint security price to exist in the market. 10 First, there have to be noise traders who trade a security without taking care of its fundamental value (and its relationship with the security price). In order for a stock to be overpriced relative to its fundamental value, there must be an investor who buys it despite its overvaluation. We call such an investor a noise trader. Second, there should not be an unlimited arbitrager who tries to exploit a difference between a security price and its fundamental value without being constrained by anything - neither by amount nor by horizon of investment. In a leading finance textbook authored by traditional theorists, it is said that the financial market is considered efficient because there are many well-informed professionals looking 10 We can find reader-friendly summaries of this theory in Shleifer and Summers (1990) and Shleifer (2000).
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 465 for mispriced discrepancies between the market prices and the fundamental values of assets 11. However, a growing consensus among economists is that, in real financial markets, no arbitragers are unlimited because of following risks and institutional frictions. First, arbitragers in the real market face a fundamental risk. When they engage in an arbitrage a difference between a security price and its fundamental value, they need to estimate the latter. However, there is a risk that the estimate is simply wrong, or that incoming news will significantly alter the value. Second, they are also exposed to a noise trader risk. When they short-sell a security which they believe is overpriced relative to its fundamental value, there is a risk that many noise traders buy the security for whatever reasons at the same time, overwhelming the arbitragers short-sales. Third, many of the well-informed professionals are institutional investors who manage money of their clients, who are not necessarily patient enough, when their investment return is not as much as anticipated. As a result, institutional arbitragers need to consider a risk that their clients require them to unwind their position when their arbitrage strategy does not work as they expect 12. Once we assume that these risks would limit arbitrage transactions, existence of noise traders can cause an inefficient security price, which diverges from its fundamental value, in the financial market. Applying this logic to interpret the volatility anomaly, we can view that investors who like to buy stocks with high beta and/or high idiosyncratic volatility, even if they are overvalued, cause overpricing of these stocks on average, and low long-run average return of them. Then the questions are what types of investors demand stocks with high volatility even though they are overvalued, and why they do so. We analyze these issues both theoretically and empirically in the following chapters. 4. Institutional investors preference for high-beta stocks and the beta anomaly 4-1. Institutional investors and benchmarking Investors in the stock market are categorized into individual investors who trade their own money and institutional investors who manage and invest their clients money. Among these groups, however, institutional investors have gained their ground for the last half a century. In the US, proportion of public equities held by institutions increased from 20 percent in 1970 to 60 percent today, on the market capitalization weighted basis. A similar trend is observed in Japan, where proportion of listed shares owned by institutions rose from 6 percent in 1980 to 34 percent in 2012, on the market capitalization weighted basis. In addition, 11 See, for example, following excerpt from a finance textbook written by Bodie et al. (2008). for assets that are bought and sold in competitive markets, price is an accurate reflection of value because there are many well-informed professionals looking for mispriced discrepancies between the market prices and the fundamental values of assets. 12 There are other reasons that limit arbitrage. For example, short-selling, which is typically associated with arbitrage transactions, is costly. Also, short-selling is impossible for some assets.
466 S Iwasawa, T Uchiyama / Public Policy Review proportion of shares traded by institutional investors 13 in the Japanese equity market per period has been constantly rising for the last 20 years. In recent years, institutions trading consists of 60 to 80 percent of the trades made in the TSE First section, except for proprietary ones. Generally speaking, institutional investors have the advantage in information and investment knowledge relative to individual investors, because institutions manage larger amount of money than individuals, and as a result, for example, institutions tend to be better serviced by investment brokers than individuals. Therefore, we can view that well-informed professionals looking for mispriced discrepancies between the market prices and the fundamental values of assets are mostly among institutional investors. In fact, we can assume that quite a few professional money managers realize that stocks with low historical volatility tend to outperform stocks with high historical volatility. Then do they overweight stocks with low historical volatility in their portfolios? But if a lot of institutional investors do so, lowvolatility stocks are unlikely undervalued, and the volatility anomaly should weaken, or disappear. This reasoning leads us to conjecture that there are some institutional constraints which limit institutional investors arbitrage transactions to exploit undervaluation of low-volatility stocks. Baker et al. (2011) argue that benchmarking, a typical contract for institutional equity management containing an implicit or explicit mandate to outperform a specific benchmark, is an important reason of why institutional investors do not overweight lowvolatility stocks. It is not easy to appropriately evaluate investment performance of institutional investors. While expected excess return (relative to risk-free rate) and Sharpe ratio, or expected excess return per unit of risk, are measures of investment performance which are recommended by traditional finance theorists, it is not straightforward to judge the skill of a money manager, even though average excess return and Sharpe ratio of her investment performance for the last, say, 10 years are given. The custom of benchmarking, where a fund s performance is expressed in relative terms with a specific benchmark, has developed to ease this problem. It is supposed that investors would easily recognize that a money manager is skillful, once they look at her investment return surpassing the benchmark return for the last 10 consecutive years. In recent years, most institutional investors, except for hedge funds which target absolute return, are benchmarked in some way or other. For example, Sensoy (2009) reported that 61.3 percent of U.S. mutual fund assets are benchmarked to the S&P 500 and 94.6 percent of them are benchmarked to some popular U.S. index, both on the market capitalization weighted basis. There are two types of funds that are benchmarked: passive funds which aim to achieve investment performance that is close to their benchmark, and active funds which try to beat their benchmark. Below we focus on investment behavior of active funds, since they trade 13 Institutional investors here include foreign investors, domestic trust banks, and domestic investment trusts.
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 467 more frequently and, as a result, are more influential in the market than passive funds 14. The problem of benchmarked fund managers, in our context, is that they are not likely to overweight low-volatility stocks. Managers benchmarked to an index would overweight high-beta stocks, that is, stocks that are likely to outperform the index, as long as they believe that the index would rise. In fact, if the index indeed increases and if they underweight highbeta stocks, their portfolios are likely to underperform the index. Therefore, it is not in their interest to take such a strategy. Of course, this argument holds only when fund managers expect that the index will rise. We think this assumption generally holds, since it is hard to imagine that equity investment managers run their funds while anticipating a stock index to decline in the long-run, although it is possible that they expect it in the short-run. Investors, however, do not necessarily expect active fund managers to overweight highbeta stocks, but to obtain alpha by selectively investing in stocks that are likely to beat the market in the long-run. If they overweight those stocks with high expected alpha, shouldn t it be the case that they outperform the index without overweighting high-beta stocks? As we show in the simulation below, the answer is no. An active fund manager is commonly evaluated by her fund s excess return relative to her benchmark, and by the information ratio, which is the ratio of the excess return to its tracking error, or its standard deviation. We show below that it is not very likely in her interest for an active fund manager who is obligated to maximize her funds excess return relative to the benchmarked index, and the information ratio, to overweight low-beta stocks, even if their expected alpha is positive. Let an expected return of an individual stock i be expressed by the CAPM. E[R i ] = R f + E[α i ] + β i E[R m R f ] (4) Considering the market portfolio the benchmark applied to fund managers, expected excess return relative to the benchmark and its standard deviation can be expressed in equations (5) and (6). E[R i R m ] = E[α i ] + (β i 1) E[R m R f ] (5) {(β i 1) 2 Var[R m ] + Var[ε i ]} (1/2), ε i = R i {R f + α i + β i E[R m R f ]} (6) A ratio of (5) to (6) is information ratio. Here we simulate the relationship between beta and expected excess return relative to the benchmark, and the relationship between beta and information ratio, assuming that E[R m ] = 10%, R f = 2%, (Var[R m ]) (1/2) = 20%, (Var[ε i ]) (1/2) = 5%, with the cases of E[α i ] = 2, 0, or 2, respectively. 14 If benchmarked investment funds are influential enough in the market, even non-benchmarked funds would be affected. For example, even though some hedge funds consider some stocks with high beta overvalued, if they recognize the risk of additional demand for those high beta stocks arising from benchmarked funds, it would not be easy for them to sell short them.
468 S Iwasawa, T Uchiyama / Public Policy Review Figure 4 Simulation of Expected Excess Return Relative to Benchmark (=E[R i R m ]): The Case for Positive Expected Excess Return of Benchmark (=E[R m R f ]) Expected Excess Return (Relative to Benchmark) 10% 8% 6% 4% 2% 0% -2% -4% -6% -8% -10% 0.35 0.55 0.75 0.95 1.15 1.35 1.55 1.75 i ] +2% ] 0% ] -2% Information Ratio 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6 ] +2% ] 0% ] -2% -0.8 0.35 0.55 0.75 0.95 1.15 1.35 1.55 1.75 Notes: The charts show the relationship between beta-value and expected excess return relative to benchmark (top), and the relationship between beta-value and information ratio (bottom), when we assume E[R m ] = 10%, R f = 2%, σ[r m ] = 20%, and σ[ε i ] = 5%, and when we fix E[α i ] = +2%, 0%, and -2%, respectively, in equations (5) and (6). Let us first examine the relationship between beta-value and expected excess return relative to the benchmark. Given a fixed level of expected alpha, we can observe a positive correlation between beta-value and expected excess return relative to the benchmark. That is, as long as expected return on the market portfolio is positive, institutional investors whose objective is to maximize expected excess return relative to the benchmark are expected to prefer stocks with high-beta to those with low-beta. We should note here that, even though expected alpha is positive, expected excess return relative to the benchmark can be negative when beta-value is low. In our numerical example, even if expected alpha is 2 percent, expected excess return relative to the benchmark is minus 2 percent when beta-value is 0.5. This suggests that active fund managers do not necessarily overweight stocks with positive expected alpha even if they manage to find such stocks. In contrast, even though expected alpha is negative, expected excess return relative
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 469 Figure 5 Simulation of Expected Excess Return Relative to Benchmark (=E[R i R m ]): The Case for Negative Expected Excess Return of Benchmark (=E[R m R f ]) Expected Excess Return (Relative to Benchmark) 10% 8% 6% 4% 2% 0% -2% -4% -6% -8% -10% 0.35 0.55 0.75 0.95 1.15 1.35 1.55 1.75 Information Ratio 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8 0.35 0.55 0.75 0.95 1.15 1.35 1.55 1.75 ] +2% ] 0% ] -2% ] +2% ] 0% ] -2% Notes: The charts show the relationship between beta-value and expected excess return relative to benchmark (top), and the relationship between beta-value and information ratio (bottom), when we assume E[R m ] = 5%, R f = 2%, σ[r m ] = 20%, and σ[ε i ] = 5%, and when we fix E[α i ] = +2%, 0%, and -2%, respectively, in equations (5) and (6). to the benchmark can be positive when beta-value is high. In our numerical example, even if expected alpha is minus 2 percent, expected excess return relative to the benchmark is plus 2 percent when beta-value is 1.5. This simple illustration shows huge motivation for institutional investors who compete with benchmark to overweight high-beta stocks. Next, let us examine the relationship between beta-value and information ratio. Unlike the relationship between beta-value and expected excess return relative to the benchmark, the relationship between beta-value and information ratio is nonlinear, since both numerator and denominator of information ratio are functions of beta-value. Moreover, the relationship between beta-value and information ratio varies depending on size of E[R m ], and on relative magnitude of Var[R m ] to Var[ε i ]. Nevertheless, in a usual case, as shown in our simulation, information ratio is positively correlated with beta-value (the correlation is higher with larger E[R m ] and with smaller Var[R m ] relative to Var[ε i ]), and the implication does not change.
470 S Iwasawa, T Uchiyama / Public Policy Review Therefore, institutional investors whose objective is to maximize expected excess return relative to the benchmark, or information ratio, are expected to overweight stocks with highbeta. By the way, in the simulation so far, it is assumed that expected excess return of the market (= E[R m R f ]) is positive. In reality, however, there may be periods when institutional investors expect negative return of the market in the short-run. Given this possibility, we simulate with an assumption that expected return on the market portfolio is negative. In concrete, we simulate with same set of numerical assumptions as above except that we set E[R m ] as minus 5 percent here. As is easily imagined, all conclusions will reverse in this case. That is, given a fixed level of expected alpha, a negative correlation exists between beta-value and expected excess return relative to the benchmark this time. Also, in a usual case as shown in our simulation, information ratio is negatively correlated with beta-value (the correlation is more negative with smaller E[R m ] and with smaller Var[R m ] relative to Var[ε i ]). Institutional investors whose objective is to maximize expected excess return relative to the benchmark, or information ratio, are expected to underweight stocks with high-beta, once they assume negative expected return on the market portfolio. The simulation result implies that active fund managers preference for high-beta stocks dramatically changes, depending on their expectation of the market. When they view that expected return on the market portfolio is positive, they will prefer to buy stocks with highbeta, and, as a result, performance of high-beta stocks relative to low-beta ones should improve. In contrast, when they anticipate negative return on the market portfolio, they will prefer to sell stocks with high-beta, and, as a result, performance of high-beta stocks relative to low-beta ones should deteriorate. Since we can expect higher level of money inflow in the stock market from institutional investors when they expect higher return of the market, we should observe a positive correlation between level of money inflow in the stock market from institutional investors and performance of high-beta stocks relative to low-beta ones. These considerations lead us to following testable hypotheses. First, institutional investors should own high-beta stocks more than low-beta stocks. Second, level of money inflow in the stock market from institutional investors and performance of high-beta stocks relative to low-beta ones should positively correlate. Third, institutional investors should buy more high-beta stocks than low-beta stocks when, on an aggregated basis, they heavily net buy Japanese equities. In contrast, they should sell more high-beta stocks than low-beta stocks when, on an aggregated basis, they heavily net sell Japanese equities. 4-2. Foreign institutional investors preference for high-beta stocks in the Japanese market To our knowledge, little empirical research which directly relates actual behavior of institutional investors with the beta anomaly in the market has been done. In fact, Baker et al. (2011) argued that conducting a direct test of their proposed mechanism was difficult, and
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 471 offered several indirect pieces of evidence. For example, as we saw in chapter 2, the beta anomaly in the US stock market became stronger after the 1970s, compared to the years from 1940 to 1960 (Fama and French, 1992). They interpret that this evidence corresponds to the fact that benchmarked institutional investors have significantly increased for the last 50 years. Here we discuss our own empirical research on the beta anomaly and behavior of institutional investors in the Japanese stock market ( 2012a) 15. In the Japanese equity market, data of both investment flow and share ownership by investor category are available. Therefore, we can analyze investment behavior of institutional investors in terms of both their money flow during a particular period, and their equity share ownership at a particular period of time. This feature makes the hypotheses above testable. Although there are both domestic and foreign institutional investors, we focus solely on foreign investors in the subsequent analysis, observing that the proportion of value of equity traded by domestic institutional investors is minor relative to that of foreign investors. Also, although foreign investors in the data include both institutional and individual investors who trade outside Japan, the latter are considered extremely minor, relative to the former. Let us now proceed to hypothesis testing using our empirical data. First, we test whether foreign investors overweight high-beta stocks relative to low-beta stocks. Our universe here is TSE1 listed companies with March or September ending fiscal years. Using their panel data as of the end of every six months from March 1985 to September 2012, we run a regression, where the dependent variable is a proportion of a company s outstanding shares held by foreign investors and independent variables are historical beta value and other control variables. Independent variables are estimated as of the end of every six months from February 1985 to August 2012. The beta value is estimated by using 60 months of trailing returns 16. The result, shown in Table 1, is consistent with the hypothesis. Regression coefficients on beta value are positive and statistically significant at the 5 percent level, regardless of whether we add other control variables or not. That is, the higher the beta-value, the higher the proportion of outstanding shares held by foreign investors, controlling for market capitalization, forecast return on equity, and analysts investment rating. Second, we test whether there is a positive correlation between foreign investors aggregate money flow into the Japanese equity market and performance of high-beta 15 (2010) is interesting empirical research on the volatility anomaly in the Japanese stock market. They found that volatile stocks tend to be those with high expected earnings growth, and those with high price-to-earnings multiple, and argued that, behind the volatility anomaly, there is investors and analysts excessive optimism on volatile stocks. While this argument is consistent with our hypotheses discussed in 4-1, there remains a question of why volatility causes optimism. In their paper, they mentioned benchmarking as an important driver, but their argument was inconclusive on this point. 16 Because proportion of a company s outstanding shares held by foreign investors is likely to be serially correlated, error terms in our panel regression will have both cross-sectional correlations and time-series correlations. Therefore, we add year dummy variables, and adjust both cross-sectional and time-series correlations to estimate robust standard error for coefficients in our panel regression, following Petersen (2009) and Thompson (2011). We apply the same technique in panel regressions shown in tables 2, 3, and 5.
472 S Iwasawa, T Uchiyama / Public Policy Review Table 1 Panel Data Regression of Proportion of Outstanding Shares Held by Foreign Investors Beta value Log Market Capitalization Book to Market Ratio Forecast Return on Equity Analysts' Investment Rating Number of Observations (1) 0.07 53,936 (3.03) (2) 0.09 0.57 53,936 (4.01) (18.29) (3) 0.08 0.57-0.01 53,936 (3.89) (16.66) (-0.30) (4) 0.10 0.52 0.04-0.05 18,415 (4.85) (20.06) (2.46) (-3.95) Notes: Taking a universe of TSE1 listed companies with March or September ending fiscal years, and their crosssectional data as of the end of every six months from March 1985 to September 2012, we run a panel data regression, where the dependent variable is the proportion of a company s outstanding shares held by foreign investors and independent variables are historical beta value and other control variables. Independent variables are estimated as of the end of every six months from February 1985 to August 2012. The beta value is estimated by using 60 months of trailing returns. Forecast return on equity is calculated as forecast net income divided by book equity as of the end of the most recent fiscal year. Forecast net income is taken from consensus data, where we prioritize those of IFIS, I/B/E/S, and QUICK. When the consensus data are not available, we supplement the forecast data either by a Nomura analyst or by Toyo Keizai. Analysts investment rating is taken from I/B/E/S consensus data, where 1 corresponds to Strong Buy and 5 corresponds to Strong Sell. Although we add year dummies as explanatory variables, we omit to show coefficients on those. Both dependent and independent variables are normalized with an average of zero and standard deviation of one. Inside parentheses are t-statistics adjusted both for serial and cross-sectional correlation (Peterson 2009 and Thompson 2011). portfolios relative to low-beta portfolios in the Japanese equity market. We examine monthly data of foreign investors net purchases of TSE listed stocks, and compare them with the difference in monthly returns between the extreme quintile portfolios sorted on beta value. The result is consistent with our hypothesis. Figure 6, where we show the former by the bar chart and the cumulative difference in monthly returns between the extreme quintile portfolios sorted on beta value by the line chart, gives us an intuition. While performance of high-beta portfolios relative to low-beta portfolios is mostly downward trending, when foreign investors net purchases of Japanese stocks are large positive, the trend either weakens or even reverses. In fact, correlation coefficients between monthly data of foreign investors net purchases of TSE listed stocks and the difference in average annual returns between the extreme quintile portfolios sorted on beta value is 0.39 with t-statistics of 7.69. Third, we examine foreign investors actual investment behavior at the micro level. In particular, we test whether foreign investors buy more high-beta stocks than low-beta stocks
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 473 Figure 6 Foreign Investors Net Purchases of Japanese Equities and the Cumulative Difference in Average Returns between the Extreme Quintile Portfolios Sorted on Beta Value % 0.5 0-0.5-40 -1-1.5-80 trn 3 2 1 0-2 -2.5 Foreign Investors' Net Purchases of Japanese Equities (rhs) -3 Beta Factor Performance (lhs) -120-3.5 85 87 89 91 93 95 97 99 01 03 05 07 09 11-1 -2-3 Notes: Foreign investors net purchases of Japanese equities are the difference in value of purchases and value of sales of TSE listed stocks by foreign investors. Beta factor performance is the cumulative difference in average annual returns between the extreme quintile portfolios sorted on beta value. See notes for Figure 2 for how to calculate portfolio returns. when, on an aggregated basis, they heavily net buy Japanese equities. At the same time, we test whether they sell more high-beta stocks than low-beta stocks when, on an aggregated basis, they heavily net sell Japanese equities. For this purpose, we sort 27 yearly cross-section data from years 1985 to 2011 on size of foreign investors aggregate net purchases of Japanese equities (relative to total market capitalization of Japanese equities). We use nine panels where foreign investors net purchases are the largest, and another nine panels where their net purchases are the smallest, to see their investment behavior in each of the environments. Our universe here is TSE1 listed companies with March ending fiscal years. Using their panel data as of the end of each fiscal year, we run a regression, where the dependent variable is the change in proportion of a company s outstanding shares held by foreign investors during the year, and independent variables are historical beta value and other control variables. Independent variables are estimated as of the end of each fiscal year 17. The beta value is estimated by using 60 months of trailing returns. Panel A of Table 2 shows the result of panels of nine fiscal years when foreign investors aggregate net purchases of Japanese equities were the largest. Regression coefficients on beta 17 This setting causes a partial overlap between measurement period of beta-value which is an explanatory variable, and measurement period of change in proportion of shares held by foreign investors, and a simultaneity problem arises as a result. To avoid the problem, we repeat the regressions shown in Table 2, using trailing beta estimated for 60 months up to the beginning of each fiscal year, but the result basically does not change.
474 S Iwasawa, T Uchiyama / Public Policy Review Table 2 Panel Data Regression of Change in Proportion of Shares Held by Foreign Investors Panel A: Years when foreign investors' net purchases of Japanese equities were the largest Fiscal years 1991, 1993, 1995, 1999, 2003-06, and 2009 Beta value Log Market Capitalization Book to Market Ratio Change in Forecast ROE (1) 0.06 (2.18) (2) 0.08 0.22 (3.14) (6.02) (3) 0.07 0.22-0.03 (3.07) (6.48) (-2.26) (4) 0.08 0.23 0.00 (3.16) (6.04) (1.58) (5) 0.07 0.22-0.03 0.00 (3.23) (6.88) (-1.04) (1.54) Panel B: Years when foreign investors' net purchases of Japanese equities were the smallest Fiscal years 1986-87, 1997-98, 2001-02, 2007-08, and 2011 Beta value Log Market Capitalization Book to Market Ratio Change in Forecast ROE (1) -0.11 (-2.60) (2) -0.11-0.04 (-2.59) (-0.91) (3) -0.11-0.07-0.07 (-2.65) (-1.43) (-2.61) (4) -0.11-0.04 0.01 (-2.67) (-0.85) (0.59) (5) -0.11-0.07-0.08 0.01 (-2.71) (-1.49) (-3.25) (0.71) Notes: Taking a universe of TSE1 listed companies with March ending fiscal years, and their cross-sectional data as of the end of each fiscal year from 1985 to 2011, we run a regression, where the dependent variable is the change in proportion of a company s outstanding shares held by foreign investors during the year, and independent variables are historical beta value and other control variables. Panel A shows the result of panels of nine fiscal years when foreign investors aggregate net purchases of Japanese equities were the largest. Panel B shows the result of panels of nine fiscal years when foreign investors aggregate net purchases of Japanese equities were the smallest. The beta value is estimated by using 60 months of trailing returns. Forecast return on equity is calculated as forecast net income divided by book equity as of the end of the most recent fiscal year. Forecast net income is taken from consensus data, where we prioritize those of IFIS, I/B/E/S, and QUICK. When the consensus data are not available, we supplement the forecast data either by a Nomura analyst or by Toyo Keizai. Although we add year dummies as explanatory variables, we omit to show coefficients on those. Both dependent and independent variables are normalized with an average of zero and standard deviation of one. Inside parentheses are t-statistics adjusted both for serial and cross-sectional correlation (Peterson 2009 and Thompson 2011). value are positive and statistically significant at the 5 percent level, regardless of whether we add other control variables or not. That is, when foreign investors heavily net buy Japanese equities, they buy more high-beta stocks than low-beta stocks. Panel B, showing the result of panels of nine fiscal years when foreign investors aggregate
Policy Research Institute, Ministry of Finance, Japan, Public Policy Review, Vol.9, No.3, September 2013 475 net purchases of Japanese equities were the smallest, exhibits their contrasting behavior. Regression coefficients on beta value are negative and statistically significant at the 5 percent level, regardless of whether we add other control variables or not. That is, when foreign investors heavily net sell Japanese equities, they sell more high-beta stocks than low-beta stocks. Our results are consistent with the hypotheses implied by Baker et al. (2011). First, foreign investors overweight high-beta stocks relative to low-beta stocks. Second, foreign investors net purchases of Japanese equities and performance of high-beta portfolios relative to low-beta portfolios are positively correlated. Third, when foreign investors heavily net buy Japanese equities, they buy more high-beta stocks than low-beta stocks. In contrast, when foreign investors heavily net sell Japanese equities, they sell more high-beta stocks than lowbeta stocks. Our results suggest that high-beta stocks underperform relative to low-beta stocks on average over the long-run, because high-beta stocks tend to be overvalued during when foreign investors heavily net buy Japanese equities, and as a result they suffer from sharp corrections during when foreign investors heavily net sell Japanese equities. 5. Individual investors gambling preference and the volatility anomaly 5-1. Individual investors gambling preference People sometimes buy a lottery ticket. This fact is explained by Kahneman and Tversky (1979) who find that people become risk taking when they are offered a small chance of winning a very large payoff. Based on this observation, Barberis and Huang (2008) build a model of a stock market in which investors who have the gambling preference overvalue lottery-like stocks 18. In general, because people are risk averse, they do not participate in a gamble, even though its expected payoff is positive. For example, most people would not take a gamble with a 50 percent chance of losing 100,000 versus a 50 percent of winning 110,000. However, many would take a gamble with a 99.9 percent chance of losing 100 versus a 0.01 percent of winning 500,000, even if its expected payoff is negative 19. This shows that people avoid risk when they face with a medium chance of winning a medium payoff, but they take risk when they face with a small chance of winning a very large payoff. The latter preference drives people to buy a lottery ticket, even though its negative expected payoff is well-known. There is no surprise that there are individual investors who are driven by such a gambling preference. To be more precise, those investors with a gambling preference would be interested in a 18 Baker et al. (2011) mentioned representativeness bias (Tversky and Kahneman 1983) and overconfidence (Fischhoff et al. 1977, Alpert and Raiffa 1982) in addition to the gambling preference, as psychological biases that lead investors to prefer volatile stocks. To our knowledge, however, little research has related individual investors representativeness bias and/or overconfidence that are orthogonal to the gambling preference to their demand for volatile stocks. Cornell (2009) argued that overconfidence leads investors to demand stocks with high volatility, and stocks with high skewness. 19 A standard expected utility theory assuming people s risk aversion cannot explain this observation.
476 S Iwasawa, T Uchiyama / Public Policy Review possibility of winning a saliently positive payoff, rather than in a large volatility. Because this nature should be more appropriately expressed by skewness, which measures difference between probability of extremely positive return and that of extremely negative return, it can be said that they prefer stocks with high skewness. Skewness of stock i is defined in equation (7). SKEW i = E[(R i μ i ) 3 ] / σ i 3 (7) In equation (7), R i is return of security i, and μ i and σ i are average and standard deviation of security i s return, respectively. There is no statistical relationship between skewness and volatility. However, it is an empirical fact that, in the cross-section, stocks with high-skewness tend to be volatile 20. Boyer et al. (2010) summarized the reasons behind this correlation as follows. First, limited liability of equity implies that stocks with large volatility tend to be those with high skewness. Second, idiosyncratic volatility is positively related to corporate growth options, and the presence of growth options implies greater skewness in returns. Third, higher idiosyncratic volatility may be related to technological revolutions, and these revolutions may lead to industry shake-outs which in turn imply greater skewness in returns as a few winners emerge and other firms fail. In a famous cumulative prospect theory, Tversky and Kahneman (1992) developed a nonexpeced utility model where people avoid risks facing profits but take risks facing losses, and apply a probability weighting function in which they overweigh small chances with large positive payoffs but they underweigh small chances with large negative payoffs. Barberis and Huang (2008) studied the asset pricing implications of the cumulative prospect theory, In their model, investors under the cumulative prospect theory own non-diversified portfolios, and overweight stocks that are lottery-like in other words, stocks with high skewness. In equilibrium, high-skew stocks become overvalued, and produce low average return. While people show gambling preferences in experiment, it is not obvious whether they exhibit the same preferences in their portfolio decision-making, and whether asset prices are affected by them or not. Thus we test the following hypotheses implied by Barberis and Huang (2008). First, positively-skewed stocks yield higher return than negatively-skewed stocks in the long run. Second, domestic individual investors should overweight positivelyskewed stocks relative to negatively-skewed stocks. Third, a positive correlation should be observed between domestic individual investors money flow into the Japanese equity market and performance of positively-skewed portfolios relative to negatively-skewed portfolios. Fourth, domestic individual investors should buy more positively-skewed stocks than negative-skew stocks when, on an aggregated basis, they heavily net buy Japanese equities. In contrast, they should sell more positively-skewed stocks than negatively-skewed stocks when, on an aggregated basis, they heavily net sell Japanese equities. In the model of Barberis and Huang (2008), security prices are affected by the difference 20 Chen et al. (2000) also pointed out that volatile stocks tend to have high skewness in the U.S. market.