Size, Value & Betting-Against-Beta

요약

CAPM으로 설명되지 않는 시장 이상 현상 중 클래식하게 여겨지는 것으로는 1) 사이즈 효과, 2) 밸류 효과 등이 있다. 그 외에도 다양한 효과에 대한 연구가 이루어지고 있으나, 이 글에서는 Betting-Against-Beta 효과를 살펴보았다. BAB 효과는 변동성이 낮은 자산의 Risk-Adjusted Return이 비교적 높아짐을 의미한다.

먼저 2012년 6월부터 2017년 6월까지 5년간의 횡단면 분석을 실시하였다. 결과적으로 사이즈 및 밸류 효과는 통계적으로 유의미했던 반면, BAB 효과는 그렇지 못하였다.

Summary Statistics for Estimated Coefficient

	Variable	Estimated Coefficient	Standard Error	t-statistics	p-values
	Constant	2.14836	0.18502	11.61	<.0001
	VF	0.27364	0.10433	2.62	0.009
FCTR	SF	0.38217	0.14843	2.57	0.0104
	BF/beta	0.12326	0.09947	1.24	0.216
	ENG	-0.3504	0.52976	-0.66	0.5087
	IND	-0.44625	0.26685	-1.67	0.0952
	MTR	-0.17467	0.26794	-0.65	0.5148
SCTR	IT	0.27535	0.39205	0.7	0.4829
(base: CD)	UTL	-0.60822	0.54648	-1.11	0.2664
	CS	0.73666	0.33174	2.22	0.0269
	TEL	0.35529	0.93259	0.38	0.7034
	HC	2.17689	0.36147	6.02	<.0001

나아가 1983년 1월부터 2017년 10월까지의 시계열 분석에서도 마찬가지로 사이즈 및 밸류효과의 유의성만 확인할 수 있었다.

Summary Statistics for Factor Portfolios’ Performance

	Monthly Return Statistics			Cross Correlation
Factor Portfolio	Monthly Average	Standard Deviation	p-value for Mean = 0	RMRF	SMB	HML	BAB
RMRF	0.0095	0.0785	0.0067		-0.1997	-0.1164	0.1645
SMB	0.0037	0.0394	0.0265	-0.1997		0.3781	0.2134
HML	0.0067	0.0505	0.0032	-0.1164	0.3781		0.2921
BAB	0.0061	0.0666	0.0314	0.1645	0.2134	0.2921

Factor Portfolios’ Performance in Log-Scale, started as 100-point in January 1983

※ 본 리서치는 연세대학교 경제학과 재학 중이며 투자학회(동아리) YIG에 소속된 전현우 학회원과 함께 실시되었습니다. 데이터 핸들링, 회귀분석, 리포트 작성까지 많은 도움이 된 전현우 학회원에게 감사의 말씀드립니다.

본문

Empirical Study:

Cross-Sectional Analysis of Individual Stocks’ Return Attribution

Based on Risk Factors of Size, Valuation, and Betting-Against-Beta

I.       Introduction

 

A.      Asset Pricing Models to Explain Sources of Risk Premium

There have been many attempts to find economically fair expected return for an asset. Despite all those studies, there is no single economic model which can suggest the equilibrium price for an asset until now. Many of empirical studies have found significant market anomalies that can be source of additional (abnormal) return over the market risk premium. Some of them are prevalent so that they are considered as alternative risk factors, while rest bunch of them are still under investigations.

There are literally hundreds of market anomalies captured by empirical studies nowadays. (Zhang et al. (2017)) However, most of them are derived by pure statistical significance rather than based on theoretical explanations for economic risk factors. Some of them can be subject to data mining or can be merely obtained by chance. On the other hand, some of the modern theories for pricing model such as the capital asset pricing model (Sharpe (1964)) is highly theoretical, so that we cannot even construct a hypothesis test.

In this paper, our primary goal is to find risk factors that have significant empirical evidence as well as profound theoretical explanation. Furthermore, we tried to verify the effect on individual stocks’ return distribution by those risk factors.

 

B.       Risk factors we tested

Size factor and value factor were suggested by Fama at al. (1993). Large-sized companies are tend to survive through recession phase in business cycle. In addition, small-sized companies have not enough market power to exploit boom in business cycle. Therefore, investor requires more risk premium for small-sized companies. Valuation factor reflects profitability. Low BTM (Book-To-Market equity value) imply low return on equity. Loading valuation risk factor means the portfolio is tilted towards lack of profitability on invested capital.

In earlier studies, it was assumed that there is no funding constraint (i.e. limited leveraging). An investor can select the optimal portfolio which is efficient and has the highest Sharpe Ratio. Then, the investor can leverage or deleverage to maximize his/her own utility. An investor with strong risk-aversion usually deleverages portfolio, holding both the optimal portfolio and risk free assets. In contrasts, an investor with weak risk-aversion might leverage portfolio, holding leveraged optimal portfolio and short position in risk free assets. This leads to every investor to have same optimal portfolio, the market portfolio (Markowitz (1952)).

Frazzini at al. (2013) suggested that BAB(Betting-Against-Beta) factor generates additional risk premium under the condition that investors have funding constraint. In real world, an investor who has weak risk-aversion is limited in leveraging his/her portfolio. Consequently, investors with weak risk-aversion hold non-optimal portfolios with overweight on high-beta assets and underweight on low-beta assets. This behavior leads to lower returns for high-beta assets and higher returns for low-beta assets. Here, beta refers to market beta from Sharpe (1964). BAB factor portfolio exploits this abnormal return gap.

 

C.       Data Descriptions

The universe is limited by non-financial common stocks of companies listed in Korea Stock Exchange. FICS (Fn-Guide Industry Classification Standard) was used for sector classification.

We obtained all of market price and financial data from FnGuide, except for Treasury bill rates. The data of Treasury bill rates are collected from the Federal Reserve Economic Data, FRED. All returns were calculated including dividend yield.

 

II.    Cross-Sectional Analysis of Return Attribution

 

A.      Return Attribution Model

We adopted cross-sectional approach to detect return contribution from each risk premium to individual stocks. The model is based on ordinary least squares methodology. We took data set of cross-sectional return distribution of individual stocks in investable universe at each time as dependent variable.

We included explanatory variables of classical risk factors such size factor, valuation factor, and beta factor. Also dummy variables for sector classification were implemented as control variables.

Since our model is cross-sectional, it is impossible to impose operating risk factor through market beta. We, instead, put dummy variables for sector classification. Each sector is exposed to each own operating risk. Base case for dummy variables is consumer discretionary(CD) sector

The regression model we designed is formed as follows:

We calculated each SF(Size Factor). SF is basically normalized cross-sectional natural logarithm of ME. (ME at time t refers to the aggregate market value of the equity at the beginning of the holding period.) Logarithm was used to adjust skewness in distribution of naive ME.

 

VF(Value Factor) was calculated from BTM. BTM refers to Book value-to-Market value and calculated based on FYE BE (Financial Year End Book value of the Equity). We used time-lagged BE to consider actual reporting lag. The lagging period was set to be 5 months.

To calculate raw time-series beta, we used sample correlation,  and sample standard deviation of each individual stock’s excess return ( ) and market excess return ( ) over risk free return. Data samples were based on past 3-year monthly data. Based on the raw time-series beta, we derived ex-ante beta. Raw time-series beta of stock i( ), and ex-ante beta of stock i( ) is as follows:

W refers to weighting towards individual stock’s raw time-series beta, while (1-W) refers to cross-sectional mean-reversion multiplier. The cross-sectional mean of beta,  was assumed to be 1 for any time, t. For a raw time-series beta of negative value, we used value of 0.4 for ex-ante beta.  At each time t, we normalized cross-sectional ex-ante betas,  to calculate BF(Beta Factor):

 

All factors were designed both for cross-sectional use and time-series use. Time-series factor data is used to construct factor mimicking portfolios.

 

B.       Statistics

For cross-sectional analysis, we decided to set holding period from June, 2012 to June 2017. Too short holding period may not be enough to exploit risk factors’ premium. However each risk factor exposure can be changed through long-term horizon. We chose 5-year holding period since that can be relatively free from both problems.

 

Table I

Analysis of Variance

	DF	Sum of Squares	Mean Squares	F Value	Pr > F
Model	11	314.5	28.59	8.72	<.0001
Error	408	1337.3	3.28
Corrected Total	419	1651.8

 

Table II

Summary Statistics for Estimated Coefficient

	Variable	Estimated Coefficient	Standard Error	t-statistics	p-values
	Constant	2.14836	0.18502	11.61	<.0001
	VF	0.27364	0.10433	2.62	0.009
FCTR	SF	0.38217	0.14843	2.57	0.0104
	BF/beta	0.12326	0.09947	1.24	0.216
	ENG	-0.3504	0.52976	-0.66	0.5087
	IND	-0.44625	0.26685	-1.67	0.0952
	MTR	-0.17467	0.26794	-0.65	0.5148
SCTR	IT	0.27535	0.39205	0.7	0.4829
(base: CD)	UTL	-0.60822	0.54648	-1.11	0.2664
	CS	0.73666	0.33174	2.22	0.0269
	TEL	0.35529	0.93259	0.38	0.7034
	HC	2.17689	0.36147	6.02	<.0001

Note: (i)Adjusted , (ii)Durbin-Watson test statistic = 2.061

 

III. Interpretation & Alternative Model

 

A.      Interpretation

Assuming significance level of 0.05, beta factor has no significant attribution to return. Although the BAB factor portfolio shows positive risk premium, at least for the selected period, there is no statistical evidence of BAB factor working. On the other hand, size factor and value factor were statistically significant while their mimicking portfolios also had significant realized risk premium. Looking other control variable, except for consumer staples and health care sector, there is no significant difference to the base sector.

The goal of this study was to verify statistical significance of risk factors: size, valuation, and betting-against-beta in Korean stock market. We can conclude that the traditional size and valuation factors are still effective in Korean stock market whereas betting-against-beta does not hold. BAB factor has fine theoretical backgrounds, so this result of rejecting the BAB factor seems to be interesting. One explanation can be suggested out of many; Korean market has not been well diversified and efficient enough to regard market beta as sole considerable risk.

 

B.       Alternative Model

Since, the model above contains lots of statistically unreliable dependent variable, we would like to suggest alternatives. Below show the top 10 ranked alternative models based on Bayesian information criterion(BIC).

 

Table IV

Top-10 Alternative Models based on BIC

Ranking	Adjusted R-Square	R-Square	AIC	BIC	Variables in Model
1	0.171	0.1809	503.3304	505.5396	VF SF IND CS HC
2	0.1725	0.1843	503.5579	505.862	VF SF BF IND CS HC
3	0.1691	0.179	504.29	506.4715	VF SF BF CS HC
4	0.1668	0.1748	504.459	506.5582	VF SF CS HC
5	0.1708	0.1827	504.391	506.6669	VF SF BF IT CS HC
6	0.1708	0.1827	504.3969	506.6726	VF SF IND IT CS HC
7	0.1707	0.1825	504.4818	506.7546	VF SF IND UTL CS HC
8	0.1728	0.1866	504.3739	506.7675	VF SF BF IND IT CS HC
9	0.1727	0.1865	504.4316	506.823	VF SF BF IND UTL CS HC
10	0.1703	0.1822	504.6665	506.9331	VF SF ENG IND CS HC

 

The Best alternative model based on BIC is the one with VF, SF, IND, CS, and HC. So we took regression analysis one more time for this model. Resulting coefficients of value factor and size factor are same as 0.3 approximately.

 

Table V

Analysis of Variance

Source	DF	Sum of Squares	Mean Square	F-statistic	Pr > F
Model	5	298.7874	59.75748	18.28	<.0001
Error	414	1353.01239	3.26815
Corrected Total	419	1651.7998

 

 

Table VI

Summary Statistics for Estimated Coefficient

Variable	Estimated Coefficient	Standard Error	t-statistic	p-value
Intercept	2.02799	0.11639	17.42	<.0001
VF	0.30176	0.1008	2.99	0.0029
SF	0.31748	0.14319	2.22	0.0272
IND	-0.38804	0.22056	-1.76	0.0792
CS	0.86537	0.29975	2.89	0.0041
HC	2.26806	0.33445	6.78	<.0001

Note: (i)Adjusted , (ii)Durbin-Watson test statistic = 2.07

 

IV. Factor Mimicking Portfolios

 

A.      Constructing Factor Mimicking Portfolios

We designed our own factor portfolios in this study. Market return proxy,  was calculated by value-weighted sum of every stocks’ returns in time t. RMRF then constructed by .  is return of risk free assets. Consequently, our RMRF excludes return contribution from financial sector companies. Our returns for market proxy did not adjusted for any company specific events, and also calculated monthly basis. For these reasons returns for our market proxy may be different to total return of KOPSI.

Studies of market anomalies and multi-factor models generally construct a factor portfolio using both value-weighting approach and equal-weighting approach together to adjust correlation with other risk factors. Value-weighted approach has its advantage in better corresponding to real investment opportunities and in considering wealth effect of total investors. Also Fama pointed out in his paper (1998) that if value-weighted factor portfolios are used to capture evidence of anomalies instead of equal-weighted portfolio, then the anomalies shrink in long-term analysis and often disappear. This means that anomalies detected by equal-weighted portfolio can be difficult to replicate in the real world. For these reasons, HML and BAB factor portfolios are constructed using value-weighting approach. They were not adjusted to exhibit low correlations with other factor portfolios.

However, in SMB factor portfolio under value-weighting approach, an asset with the smallest size among the large-cap category have smaller return contribution to SMB factor portfolio than an asset with the biggest size among the small-cap category. This makes it difficult to impose value-weighting approach in SMB factor portfolio. To avoid this problem, we took equal-weighting approach for SMB factor portfolio. We excluded micro-capitalization companies which are below 20-percentile from the universe.

Classically factor portfolios are constructed to be rebalanced annually. However we designed our factor portfolios to have monthly rebalancing.

 

                       i.              Constructing SMB factor portfolio

An asset i at the beginning of the month which had SF bigger than 0 was considered small ME and put into ‘small’ portfolio. Assets that have SF smaller than 0 were put into ‘big’ portfolio. Then SMB portfolio consisted of holding long in the small portfolio and short-selling the big portfolio at the same time. SP refers to return of the small portfolio, and BP refers to that of the big portfolio.

                     ii.              Constructing HML factor portfolio

We put assets with VF over 0 into ‘high portfolio’. Assets with VF under 0 were put into ‘low portfolio’ Then, HML factor portfolio was built by holding long in high portfolio and short-selling low portfolio simultaneously. HP stands for return of high portfolio while LP means return of low portfolio.

                   iii.              Constructing BAB factor portfolio

We went long in stocks with positive BF values, while short-selling stocks with negative BF values. In each long and short position, they held counter position in risk free assets to take only the excess returns of each portfolio. In addition, long-portfolio was levered up and short-portfolio was de-levered so that the two portfolios could have same ex-ante beta exposure of 1. LB(Low Beta) refers to return of long-portfolio, and HB(High Beta) means return of short-portfolio.

B.       Summary Statistics for Factor Portfolios’ Performance

Our cross-sectional regression model includes several factors. These factors should promise risk premium, i.e., they should provide long-term positive returns.

 

Table Ⅶ

Summary Statistics for Factor Portfolios’ Performance

	Monthly Return Statistics			Cross Correlation
Factor Portfolio	Monthly Average	Standard Deviation	p-value for Mean = 0	RMRF	SMB	HML	BAB
RMRF	0.0095	0.0785	0.0067		-0.1997	-0.1164	0.1645
SMB	0.0037	0.0394	0.0265	-0.1997		0.3781	0.2134
HML	0.0067	0.0505	0.0032	-0.1164	0.3781		0.2921
BAB	0.0061	0.0666	0.0314	0.1645	0.2134	0.2921

 

Figure Ⅰ

Factor Portfolios’ Performance in Log-Scale, started as 100-point in January 1983

 

Factor mimicking portfolios show the evidence of long-term risk premium for those risk factors. However, it is interesting that BAB factor portfolio can have different interpretation depending on the investment period. Nevertheless, recent 5 years in which our cross-sectional analysis was based on the data have shown all of those mimicking portfolios performed well.

 

Ⅳ. Conclusion

 

Cross sectional analysis cannot impose market risk factor consisting of operating risk and financial risk. Yet other alternative risk factors such as size and value can be captured with cross-sectional analysis. In fact, alternative risk factors’ return attribution is rarely captured in simple time-series regression analysis. In order verify return attributions from the factors, we chose to use cross-sectional regression model.

In our study, size factor and value factor suggested by Fama at al. were turned out to have meaningful effect on individual stocks’ return distribution. While, BAB factor was not significant. This can be understood as either (i) lack of BAB risk premium or (ii) misuse of risk estimator (market beta).

With long-term investment horizon, factor mimicking portfolios performed well except for BAB factor portfolio again. The BAB factor portfolio’s return was not significantly over 0 with significance level of 0.05.

A portfolio can be exposed to certain risk factors. Tilting toward certain risk factors leads to levered return and risk. Since a portfolio manager has to control the risk exposure to meet the target return and risk, the manager has to decide based on relevant risk factors. This study investigated what factors could be relevant and what could be irrelevant to provide a right guide for portfolio management in Korean stock market. We hope the results of the study in this paper can be used to identify right risk factors by portfolio managers.

 

Ⅵ. References

 

Fama, E.F. (1998), “Market efficiency, long-term returns, and behavioral finance”, Journal of Financial Economics 49, 283-306

Fama, E.F. and K.R. French (1993), "Common risk factors in the returns on stocks and bonds", Journal of Financial Economics 33, 3–56.

Frazzini, A. and L. H. Pedersen (2013), “Betting Against Beta”, Journal of Financial Economics, Forthcoming.

Kim, T. (2015), “An empirical study on the BAB factor in Korean stock market”, M.D. Dissertation

Lee, C., J. Jang (2015), “Size, Book-to-Market, and Momentum Effects across Economic States: Evidence from the Korean Stock Market”, Korean Journal of Financial Management 32

Markowitz, H. (1952), “Portfolio Selection”, Journal of Finance 7, 77-91

Sharpe, W.F. (1964) “Capital Asset Prices: a Theory of market equilibrium under conditions of risk”, Journal of Finance 19, 425-442

Zhang, L., C. Xue and K. Hou (2017) “Replicating Anomalies”