The Conditional Beta Risk-Return in the Pricing Models of Individual Stocks

This paper finds a significant conditional risk-return relationship even in individual stocks when dual betas based on market conditions are considered in the traditional asset pricing model. The conditional risk-return relationship appears to persist even after adjusting for firm size, total risk, and industry, surpassing the explanatory power of firm size, which has traditionally been identified as a significant variable for returns. Interestingly, this study discovers the so-called November effect instead of the traditional January effect, and in November, beta risk appears to be rewarded in both bull and bear markets regardless of market conditions.

Introduction

Research Motives and Literature Review

The Capital Asset Pricing Model (CAPM), proposed long ago by Sharpe (1964), Lintner (1965), and Black (1972) (hereafter, SLB), suggested that the “expected” return for any risky assets including stock are determined by beta (covariance of asset return and market return per a unit of the variance of market return), the risk-free rate of return, and the expected market return. If an equity security is considered a risky asset here, beta values represent the systematic risk inherent in the individual equity securities. Beta measures the sensitivity of the expected returns of individual stocks to the market returns, serving as a core explanatory variable in the asset pricing model to determine the returns of individual stocks. In the CAPM world, investors are rewarded by a return in excess of the risk-free return in proportion to the systematic risk unique to individual stock securities.

$$\begin{array}{r}E\left(R_i\right)=R_f+{\beta }_i(E(R_m)-R_f)\end{array}$$

where $E(R_i)$, $E(R_m)$, and $R_f$ are the expected return for a stock i, expected market return, and the current risk-free rate of return, respectively.

The commonly echoed phrase among investors related to the CAPM model, “high-risk, high-return,” intuitively resonates because the Security Market Line (SML) described above indicates that the expected returns of individual stock are linearly proportionate to systematic risk, beta and should be compensated accordingly. The CAPM is an ex-ante, or forward-looking, model, based on the assumption that the expected market return is always greater than the risk-free rate ($(E(R_m)-R_f)$ > 0). The word “return” in the phrase of “high-risk, high-return” is unconsciously accepted as a “positive” return by investors.

However, one thing overlooked here is that in ex-post realized returns, the “high-risk, high-return” notion may sometimes result in “high-risk, high-negative return” because there are cases where the realized market return does not surpass the risk-free rate. Therefore, many empirical studies examining the validity of the CAPM using “realized” returns, instead of “ex-ante” data, face inherent limitations. Consequently, the relationship between beta and returns has often been readily contested. Fama and French (1992), a seminal study, showed at most a weak positive relationship between cross-sectional average return and beta over the period 1941 to 1990, and further no relationship over the shorter period 1963 to 1990.

Pettengillet al. (1995) pointed this out well. The traditional single beta CAPM has been tested using “realized” return data given the absence of “expected” data and therefore the risk-return relation was not found and the role of the beta in the model has been negated by many studies. Indeed, it is practically impossible to validate the relationship between risk and returns as described in the CAPM using ex-ante data.

Many studies were conducted to address the theoretical limitations and challenges inherent in validating the CAPM. Some studies report the beta risk-return relation doesn’t appear to be stable over time. Some studies focused on the claim that beta varies with market conditions, such as bull and bear markets, and exhibits asymmetry. Fabozzi and Francis (1977 and 1979) determined the presence of bull and bear markets depending on whether the market return for a month is positive or negative. The conditional dual-beta model was introduced in their study when testing the instability of beta based on the condition of markets, which are bull and bear market. Bhardwaj and Brooks (1993) also found significant differences between systematic risk in bull and bear markets. Chong (2022) reported that a portfolio outperforms the benchmarks for the entire period under study when it is constructed using dual betas based on market conditions.

The empirical evidence clearly shows that there is strong support for a consistent significant relationship between beta and returns conditional on the market condition. Downside beta represents a negative market risk premium, a kind of punishment during negative market excess return periods. Stocks with a higher beta may incur larger losses during market downturns compared to stocks with a lower beta. Therefore, if this early inference holds true, downside beta is expected to show a negative relationship with realized returns.

Research Contributions

This study extends the dual-beta evidence by using more recent and alternative data. This paper reexamines whether the single beta in the traditional SLB’s CAPM explains cross-sectional variation in stock returns over the post-1990 period after the sample period used by Fama and French (1992). In addition, the risk-return relation is explored by introducing dual betas into the traditional CAPM system. Our study differs from most existing literature that investigates the relationship between dual beta and returns based on portfolios (Pettengillet al., 2002; Howton & Peterson, 2005; Chonget al., 2011). In contrast to previous studies that constructed portfolios based on specific criteria for analyzing stocks according to research objectives, our study directly analyzes individual stocks without categorizing them into portfolios. As Blacket al. (1972) argued, the primary reason for analyzing portfolios in previous studies, rather than individual stocks, is to obtain efficient estimates by eliminating noise that could arise from individual stocks. As discussed later, taking this into account, in this study, the noise from individual stocks is controlled by the total risk. Pettengillet al. (2002) concluded their article with a further research question whether or not the conditional risk-return relation presented in their dual-beta model would be held for individual securities. This study is a response to the question.

Furthermore, this study conducts a comparative analysis on how significantly beta risk explains the cross-section of returns in the same period as the beta risk is estimated and in the following period using Fama and MacBeth (1973)’s two-step regression method. One more contribution of this study is its in-depth investigation into how the conditional risk-return relation is influenced by industry, firm size, and seasonal pattern and whether it exists independently of these factors.

The rest of this study is organized as follows: Section 2 provides a description of the data used in the study and outlines the methodology and models employed for analysis. Section 3 presents the cross-sectional regression results of the models. In Section 4, additional analyses and results are presented to explore how other variables, from the perspective of robustness checks, impact the previously presented empirical results. Finally, Section 5 summarizes the key findings and concludes the study.

Data and Methodology

Data

We collect S&P 500 stock securities (as of September 2019, before the COVID Pandemic) and their monthly price/return data from February 1992 to September 2019 (monthly data spanning 28 years). Stocks that were relatively recently added to the S&P 500 as of September 2019 are excluded due to insufficient data. In cases where a company issues multiple classes of stocks, stocks other than the common class are also excluded. After this data filtering process, the sample comprises a total of 461 stocks. One reason for constructing the sample with S&P 500 stocks is the advantage of controlling idiosyncratic price movements more likely to be found in small stocks by focusing on relatively large-cap stocks actively traded in the market. According to Roll (1977), the results of the CAPM vary depending on the chosen market index. Following his criticism, this study considers the stock securities included in the S&P 500 index as a proxy for market returns in the analysis.

We utilize individual stocks for our analysis unlike previous studies that used portfolios. Therefore, unlike the traditional three-step process of construction, estimation, and testing adopted in the analysis of the risk-return relationship from portfolios in previous research, our study is conducted in a simpler procedure comprising two steps: estimation and testing stages following the Fama and MacBeth (1973) cross-sectional two-step estimation and testing regression approach. The parameters, including traditional beta and dual betas in the market pricing model (CAPM) and the dual-beta model, respectively, are estimated using the data between February 1992 through January 2012 (referred to as the ‘estimation sample data’). The estimated parameters are then regressed against the realized monthly excess returns during the testing period from February 2012 through September 2019 (referred to as the ‘testing sample data’).

Estimation Methodology

Beta values in the traditional single-beta CAPM are estimated by using standard Ordinary Least Squares (OLS) regression with monthly excess returns during the estimation period, February 1992 through January 2012:

(1) $$\begin{array}{r}{r}_{it}=\widehat{{\alpha }_i}+\widehat{{\beta }_i}{r}_{mt}+{\epsilon }_{it}\end{array}$$

where $r_{it}$ = $R_{it}$ − $R_{ft}$, $r_{mt}=R_{mt}-R_{ft}$, $R_{it}$, $R_{mt}$, and $R_{ft}$ represent stock i’s monthly return, the market return (S&P 500 index), and the monthly 10-year Treasury rate at time t, respectively. Thus, $r_{it}$ and $r_{mt}$ denote the excess returns on stock i and the market returns over the risk-free rate; $\widehat{{\alpha }_i}$ is the intercept, estimating abnormal returns on stock i; β_i is the slope coefficient, indicating the sensitivity of stock i’s return to market return; and ε_it is the error term.

The overall observations from estimation sample data are then segmented into two sub-samples based on whether the monthly market excess return is positive (Up-markets or Bull markets) or negative (Down-markets or Bear markets) to obtain dual betas.

(2) $$\begin{array}{r}{r}_{it}=\widehat{{\alpha }_i}+\widehat{{\beta }_i^{+}}{r}_{mt}\delta +\widehat{{\beta }_i^{-}}{r}_{mt}(1-\delta )+{\epsilon }_{it}\end{array}$$

where δ = 1 is an Up-market condition ((R_mt – R_ft) < 0), δ = 0 is a Down-market condition ((R_mt – R_ft) < 0). $\widehat{{\beta }_i^{+}}$ and $\widehat{{\beta }_i^{-}}$ respectively denote Upside (Bull market) beta and Downside (Bear market) beta. Out of the 240 monthly observations, 140 months and 100 months are used to estimate upside and downside betas. Interestingly, Howton and Peterson (2005) used the median monthly market return to classify a month as either bull or bear, depending on whether the return on the market that month is higher (bull) or lower (bear) than the median market return. However, this approach may not be consistent with the excessive return in the original SBL’s CAPM.

Testing Methodology

Unconditional and conditional relationships between single beta or dual beta and stock realized returns are examined with and without the presence of some other variables (total risk, size, and industry) by using the regression models described below.

Unconditional single-beta models are used to test the relation between beta and stock realized return.

• Unconditional Single-Beta Model:

$$\begin{array}{r}ModelI:r_{it}={\hat{\lambda }}_0+{\hat{\lambda }}_1{\beta }_i+{\theta }_{it}\end{array}$$

• Unconditional Single-Beta Model with Controlling Variables (total risk, size, and industry):

$$\begin{array}{rl}ModelII:r_{it}& =\widehat{{\lambda }_0}+\widehat{{\lambda }_1}{\beta }_i+\widehat{{\lambda }_2}{\sigma }_i^2\\ & +\widehat{{\lambda }_3}\ln \left(Ca{p}_i\right)+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}\end{array}$$

where ${\beta }_i$ and ${\sigma }_i^2$ denote stock i’ beta and total risk. The variable of Ind_i represents a binary variable for the 11 S&P industry sectors, k. When stock i is a company in an industry k (i.e., i $\in$ k), Ind_i = 1; otherwise, Ind_i = 0. ln (Cap_i) is the natural logarithm of stock i’s capitalization (in billion dollars). ${\beta }_i$, ${\sigma }_i^2$, Ind_i, and ln (Cap_i) are all estimated from the estimation sample data (February 1992–January 2012). The realized returns of all 461 stock securities during the testing period (February 2012 through September 2019) are regressed on the previously estimated betas and other variables. If the value for a pricing parameter (i.e., a kind of risk premium), $\widehat{{\lambda }_1}$ is greater than zero, it supports a positive risk-return tradeoff (high-risk, high-positive return), indicating that beta is still relevant and explains the cross-section of stock returns in SBL’s CAPM ($\widehat{{\lambda }_1}>0$).

In Model II, a typical list of control variables such as total risk, firm size, and industry is introduced. Unlike previous related studies that analyze portfolios to examine risk-return relations, our research focuses on individual stocks. Therefore, non-systematic risks related to individual stocks may not be diversified away, and accordingly, they are expected to have a significant impact on the risk-return relation. To control this influence, the individual stock’s variance (standard deviation) during the estimation period is included in the model. Consequently, a significant positive relationship between realized returns and total risk is anticipated ($\widehat{{\lambda }_2}>0$). This implies that holding individual stock is rewarded by bearing not only systematic beta risk but also total risk. As total risk can overlap with beta risk, including this variable in the model may weaken the explanatory power of beta. Therefore, if the model with only beta (Model I) exhibits weak explanatory power for beta, the inclusion of beta in Model II may render it even less meaningful. Fama and French (1992) argued that firm size, measured by market value of equity (ME), is more significant in explaining stock returns than beta. Banz (1981) also reported market equity (ME) to be a significant factor in explaining the cross-section of average returns. Generally, many studies indicate a negative relationship between firm size and average return ($\widehat{{\lambda }_3}<0$).

This study further analyzed the risk-return relationship after classifying individual stocks into the 11 major industry sectors based on the Global Industry Classification System (GICS) and controlling for the potential idiosyncratic characteristics of each industry. This study is the first to introduce industry variables into pricing models to analyze the risk-return relationship using U.S. stocks.

Conditional dual-beta models are developed to investigate the conditional relation between dual beta and stock realized return as follows:

• Conditional Dual-Beta Model:

$$\begin{array}{r}ModelIII:r_{it}=\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )+{\theta }_{it}\end{array}$$

• Conditional Dual-Beta Model with Controlling Variables (total risk, size, and industry):

$$\begin{array}{rl}ModelIV:r_{it}& =\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )\\ & +\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )\\ & +\widehat{{\lambda }_3}\ln \left(Ca{p}_i\right)+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}\end{array}$$

where δ is 1 is an up-market condition ((R_mt – R_ft) < 0), δ = 0 is a down-market condition ((R_mt – R_ft) < 0). Models III and IV replace the single beta in the earlier unconditional single-beta models, Models I and II with dual beta, upside beta (${\beta }_i^{+}$) and downside beta (${\beta }_i^{-}$). Their pricing factors are $\widehat{{\lambda }_1^{+}}$ and $\widehat{{\lambda }_1^{-}}$, respectively. Stock securities with high beta risk should be highly rewarded during bull markets as compensation for large losses during bear markets to hold “high-risk, high-positive or negative return” ($\widehat{{\lambda }_1^{+}}>0$, $\widehat{{\lambda }_1^{-}}<0$).

Like the dual beta, total risk is divided into upside total risk, ${\sigma }_i^{2+}$ and downside total risk, ${\sigma }_i^{2-}$, estimated separately for the bull and bear markets. The pricing factors corresponding to those variables denote $\widehat{{\lambda }_2^{+}}$ and $\widehat{{\lambda }_2^{-}}$. Holding an individual stock cannot diversify away non-systematic risk. Therefore, similar to dual betas, upside and downside total risks are expected to exhibit, respectively, positive and negative relationships with realized returns. In other words, holding more total risk is anticipated to be compensated by greater rewards in bull markets ($\widehat{{\lambda }_2^{+}}$ > 0), but it comes with the compensating factor of potentially facing larger penalties in bear markets ($\widehat{{\lambda }_2^{-}}$ < 0). ${\beta }_i^{+},{\beta }_i^{-},{\sigma }_i^{2+},and{\sigma }_i^{2-}$ are also estimated from the monthly return data during the estimation period (February 1992–January 2012).

Empirical Results

Summary of Estimation Sample Data

The estimation sample comprises 461 large-cap stocks from the S&P 500, representing 11 industry sectors, spanning from February 1992 to January 2012, covering a total of 240 months. As shown in Table I, the average market capitalization of all stocks is $77.47 billion (column (a)). The Communication Services and Technology sectors, dominated by big-tech stocks, have average market capitalizations well over twice the overall average, at $193 billion and $173 billion, respectively, while the Utilities sector is the lowest at $33 billion. The overall average beta value is 1.07, but there is significant variation across industry sectors (column (b)). This indicates that betas vary meaningfully across industries, underscoring the importance of distinguishing them for testing the risk-return relationship. Over the period from 1992 to 2012, the average market betas for up-market, and down-market are 1.027 and 1.085, respectively. This finding implies that the penalty (negative risk premium) in down-markets appears to be greater than the premium for risk in up-markets (columns (d) and (f)).

Table I. Estimation of Unconditional & Conditional Betas and Equality Test of Conditional Betas for 11 Industry Sectors, February 1992 to January 2012

Industry, k (Obs.)	Average	Average	Up-market		Down-market		$H_0:\widehat{{\beta }_k^{+}}=\widehat{{\beta }_k^{-}}$
	capitalization	Beta, β	Avg. excess return	Upside beta,$\widehat{{\beta }_k^{+}}$	Avg. excess return	Downside beta,$\widehat{{\beta }_k^{-}}$
	($Billion)
	(a)	(b)	(c)	(d)	(e)	(f)	(g)
All (461)	77.472	1.068	4.147%	1.027	−2.884%	1.085	(−29.36)**
1. Basic materials (19)	41.397	1.168	3.898%	1.204	−3.177%	1.214	(−1.45)
2. Communication services (18)	193.458	1.127	4.606%	0.841	−3.567%	1.129	(−19.93)**
3. Consumer cyclical (57)	79.641	1.249	5.225%	1.355	−3.229%	1.244	(12.44)**
4. Consumer defensive (33)	78.778	0.578	2.379%	0.602	−1.438%	0.649	(−9.33)**
5. Energy (22)	47.796	1.057	3.960%	0.936	−2.629%	1.177	(−72.6)**
6. Financial (66)	80.152	1.153	3.999%	1.246	−3.373%	1.229	(4.55)**
7. Healthcare (60)	74.562	0.881	3.721%	0.704	−2.031%	1.013	(−60.22)**
8. Industrials (68)	47.322	1.040	3.981%	1.036	−3.002%	1.044	(−1.95)
9. Real estate (29)	33.523	0.996	3.966%	0.765	−2.291%	1.158	(−57.94)**
10. Technology (61)	137.315	1.570	6.483%	1.511	−4.711%	1.294	(38.32)**
11. Utilities (28)	33.059	0.433	1.802%	0.324	−1.192%	0.474	(−37.53)**

However, there are some cases where the opposite holds true. Hawton and Peterson (2005), using data from 1977 to 1993, find average portfolio betas for all (1.112), up-market (1.252), and down-market (0.996) situations. In their study, up-market betas were larger than down-market betas. This difference seems to vary depending on the data sample period. Notably, upside betas and downside betas were found to be statistically significantly different at the 1% level. Specifically, for all industry sectors but one (Basic Materials sector), the null hypothesis of equality of upside and downside betas is rejected at the 1% significance (column (g)).

These results differ from Faff (2001)’s study on Australian stocks. Faff (2001) conducted an equality test of the up-market and down-market betas for a total of 24 industries, and most industries, excluding six (two) at the 1% (5%) significance level, could not reject the null hypothesis. This difference appears to stem from variations in the sampling method. Faff analyzed industry-sorted portfolios’ betas rather than individual stock betas.

Contemporaneous Tests of Unconditional and Conditional Risk-Return Relationships

For a comparative purpose, we conducted contemporaneous tests for a risk-return relation and the results are later compared to the findings of predictive tests in Section 3.3. The estimates for all explanatory variables and returns derived from the estimation sample data, spanning from February 1992 to January 2012, are incorporated into all four models (Models I, II, III, and IV).

Table II shows the results of the cross-sectional regressions estimated using the four models. The results of the regression models with single beta only (Model I) and single beta and controlling variables (Model II) are summarized in Panel A. Unlike Fama and French (1992), the results show a robust existence of a positive beta risk-return relation in the traditional single-beta model (SLB’s CAPM), including only one beta as an explanatory variable at the 1% significance level ($\widehat{{\lambda }_1}>0,$ t = 7.93). Theoretically, the parameter ($\widehat{{\lambda }_1}$) in the traditional single-beta model represents the market risk premium. The estimated value of 0.0077 indicates a 9.61% annual return.

Table II. Contemporaneous Return-Beta Relationship, Estimation Period: February 1992 to January 2012

Coefficients		Panel A: UnconditionalRisk-return relationship		Panel B: ConditionalRisk-return relationship
		Without total risk, size, industry	With total risk, size, industry	Without total risk, size, industry	With total risk, size, industry
		Model I	Model II	Model III	Model IV
Intecept	$\widehat{{\lambda }_0}$	0.0039 (3.60)**	0.0003 (−0.11)	0.0115 (14.52)***	0.0036 (1.57)
Single beta	$\widehat{{\lambda }_1}$	0.0077 (7.93)**	0.0010 (0.76)
Upside beta	$\widehat{{\lambda }_1^{+}}$			0.0247 (33.75)**	0.0198 (22.48)**
Downside beta	$\widehat{{\lambda }_1^{-}}$			−0.0329 (−41.93)**	−0.0181 (−16.85)**
Total risk	$\widehat{{\lambda }_2}$		0.0768 (13.38)**
Upside total risk	$\widehat{{\lambda }_2^{+}}$				0.1565 (30.19)**
Downside total risk	$\widehat{{\lambda }_2^{-}}$				−0.1401 (−10.51)**
Firm size	$\widehat{{\lambda }_3}$		0.0005 (1.03)		0.0001 (0.19)
Industry	$\widehat{{\lambda }_4^{+}}:$
Basic materials			−0.0010 (−0.37)		−0.0010 (−0.38)
Communication services			−0.0008 (−0.26)		0.0010 (0.35)
Consumer cyclical			0.0016 (0.88)		0.0001 (0.03)
Consumer defensive			−0.0009 (−0.43)		−0.0007 (−0.33)
Energy			0.0014 (0.58)		0.0025 (1.05)
Financial			−0.0019 (−1.08)		−0.0026 (−1.56)
Healthcare			0.0010 (0.55)		0.0040 (2.3)*
Industrials			0.0020 (0.9)		0.0053 (2.44)*
Technology			0.0024 (1.27)		0.0031 −1.73
Utilities			−0.00205 (−0.89)		−0.0010 (−0.47)
Adj. R2		0.0007	0.0028	0.0583	0.0721
F-Statistic		(62.93)**	(20.62)**	(2,862.40)**	(479.38)**
Pairwise test of equality of risk pricing parameters
$H_0:\widehat{{\lambda }_1^{+}}\widehat{{\lambda }_1^{-}}$				(75.39)**	(32.28)**

Interestingly, when total risk, firm size, and industry variables are added to the models, the coefficient for beta becomes insignificant, and the total risk variable shows a significant association with returns at the 1% significance level, as reported in Model II in Panel A, $\widehat{{\lambda }_2}>0$ (t = 13.38). A noteworthy finding is that the size variable, traditionally seen as important for explaining returns in previous literature, seems to contribute little to explaining the cross-sectional variation in realized returns. This finding contrasts with Lakonishok and Shapiro (1986), which reported that neither the traditional beta nor alternative risk measures (total risk) can explain the return, and only firm size matters. They analyzed portfolios, so the role of total risk in the relationship with returns seems to be insignificant, given that a significant portion of unsystematic risk is diversified away. However, when individual stocks are considered, the compensation for individual stocks’ total risk appears to be highly significant. The SLB’s single-beta pricing model implies that the intercept parameter, ${\lambda }_0$ should be equal to zero. However, the estimate of the parameter is 0.0039 and significantly different from zero (t = 3.60). This result is consistent with Blacket al. (1972) that reported +0.00359.

The results in Panel B report the results of the conditional dual-beta models with and without control variables. The slope coefficients of upside beta ($\widehat{{\lambda }_1^{+}}$) and downside beta ($\widehat{{\lambda }_1^{-}}$) are different from zero at the 1% significance level, and the coefficient values are positive and negative, respectively, as anticipated. Additionally, a significant conditional relationship is found between total risk and returns. These findings provide strong support for a conditional relationship between risk and return.

Predictive Tests of Unconditional and Conditional Risk-Return Relationship

We examined the relationship between risk and return using predictive tests; the estimates of various explanatory variables from the estimation sample data are used to estimate the cross-sectional relationship with realized returns for the subsequent testing period from February 2012 through September 2019. The results of regression models for testing conditional and unconditional risk-return relationships are summarized in Table III. The estimated parameters of binary variables for almost all industries appear to be statistically insignificant and are omitted from Table III. As expected in Lakonishok and Shapiro (1986), the t-statistics for all estimates in the predictive tests reported in Table III are smaller than the t-statistics in the contemporaneous test from Table II. The F-statistics for the fitness of the models also decrease in all models.

Table III. Realized Return-Beta Relationship, Testing Period: February 2012 to September 2019

Coefficients		Panel A: UnconditionalRisk-return relationship		Panel B: ConditionalRisk-return relationship
		Without total risk, size, industry	Without total risk, size, industry	Without total risk, size, industry	Without total risk, size, industry
		Model I	Model II	Model III	Model IV
Intercept	$\widehat{{\lambda }_0}$	0.0114 (11.87)**	0.0066 (2.97)**	0.0167 (22.18)**	0.0133 (6.37)**
Single beta	$\widehat{{\lambda }_1}$	0.0009 (1.17)	−0.0001 (−0.12)
Upside beta	$\widehat{{\lambda }_1^{+}}$			0.0091 (14.83)**	0.0073 (10.63)**
Downside beta	$\widehat{{\lambda }_1^{-}}$			−0.0136 (−37.27)**	−0.0178 (−13.05)**
Total risk	$\widehat{{\lambda }_2}$		0.0115 (2.00)*
Upside total risk	$\widehat{{\lambda }_2^{+}}$				0.0313 (6.44)**
Downside total risk	$\widehat{{\lambda }_2^{-}}$				−0.1511 (−9.54)**
Firm size	$\widehat{{\lambda }_3}$		0.0015 (3.26)**		0.0009 (2.14)*
Industry	$\widehat{{\lambda }_4^{+}}$:
Adj. R2		0.0000	0.001	0.0636	0.0682
F-Statistic		(1.23)	(4.23)**	(1,428.67)**	(206.92)**
Pairwise test of equality of risk pricing parameters
$H_0:\widehat{{\lambda }_1^{+}}\widehat{{\lambda }_1^{-}}$				(50.74)**	(17.19)**
Pairwise test of symmetry of risk pricing parameters
$H_0:\widehat{{\lambda }_1^{+}}\widehat{{\lambda }_1^{-}}$					(6.46)**

Risk-Return Relation in Unconditional Single-Beta Models

Several conclusions can be drawn from the results presented in Table III. The results presented in Panel A indicate that there is no statistically significant relationship between risk and return in single-beta models. In contrast to the contemporaneous test results (Panel A of Table II), which show a significant relationship between risk and return even without considering controlling variables, the predictive test reveals that the relationship between risk and return is statistically insignificant (t = 1.17), even without incorporating controlling variables. When considering total risk, size, and industry variables, the risk-return relationship remains insignificant, and the parameter estimate of beta shows even a negative sign which contradicts the intuition about the risk-return tradeoff.

However, the coefficient estimates of the total risk variable, which is significant at the 5% level (t = 2.00), indicates a positive relationship between realized returns and total risk (slope coefficient estimate = +0.0115). Stock investments are rewarded not only for taking systematic risk but also for taking total risk. This finding contradicts the claim made by Lakonishok and Shapiro (1986) that the total risk factor is not significant in explaining the cross-sectional returns of stocks.

Risk-Return Relation in Conditional Dual-Beta Models

Panel B of Table III reports the results of conditional dual-beta models with and without controlling explanatory variables. Dual betas, unlike the single beta in the single beta model, exhibit a highly significant and strong relationship with realized returns at the 1% significance level, regardless of the inclusion of other variables. This finding is noteworthy as it indicates that even when upside total risk, downside total risk, and firm size are considered together, dual betas remain statistically significant not only in contemporaneous tests but also in predictive tests. In particular, the t-statistics for the slope estimates of upside and downside betas, 10.63 and −13.05, respectively, are greater than the t-statistics of other variables (upside total risk (t = 6.44), downside total risk (t = −9.54), and size (t = 2.14)). This implies that dual betas dominate other variables-total risk, size, and industry- in explaining the cross-section of realized returns compared to other variables.

As predicted, there is a notably significant relationship between risk and realized returns, with the direction of the relationship changing in different market conditions such as bull and bear markets. In other words, high beta stocks are rewarded with high returns in bull markets (${\lambda }_1^{+}$ > 0), but this comes at the cost of lower returns in bear markets (${\lambda }_1^{-}$< 0), reflecting a significant trade-off between risk and returns. This finding is consistent with Pettengillet al. (1995), Howton and Peterson (2005), and Faff (2001) that report the conditional risk-return relationship based on portfolios. Our finding shows that such a relationship strongly exists even in individual stocks. Some studies (e.g., Hodoshimaet al., 2000) report that the beta-return relationship is significant only in the down-markets. However, this finding implies that high-beta stocks are unfavorably rewarded with low returns in bear markets, but this unfavorable compensation is not meaningfully offset in bull markets, which is difficult to rationalize.

In Panels A and B of Table III, an interesting observation emerges. The estimated coefficients’ t-statistics for single beta ($\widehat{{\lambda }_1}$) and single total risk ($\widehat{{\lambda }_2}$) become more significant when introduced into the dual beta models. The t-value for single beta is only −0.12 (Panel A of Table III), but when upside ($\widehat{{\lambda }_1^{+}}$) and downside ($\widehat{{\lambda }_1^{-}}$) betas are introduced into the model, their t-values become 10.63 and −13.05, respectively. The noteworthy point is that the absolute value of the t-statistic for the downside beta coefficient is larger than that for the upside beta coefficient. This implies that the penalty for taking beta risk in bear markets is greater than the premium for taking risk in bull markets. Beta operates relatively more significantly in bear markets. Similarly, while the t-statistic for total risk is 2.00, the t-statistics for upside total risk ($\widehat{{\lambda }_2^{+}}$) and downside total risk ($\widehat{{\lambda }_2^{-}}$) are 6.44 and −9.54, showing an increased significance of these variables. Covariance and variance in beta risk (${\displaystyle \frac{{\sigma }_{i,m}}{{\sigma }_m^2}}$) and variance of total risk (${\sigma }_m^2$) do not consider the direction of returns. Consequently, positive and negative returns cancel each other out in risk measurement, leading to a weakened risk-return relationship compared to directional dual risk indicators such as dual betas and dual total risk. Pairwise equality of up- and downside betas’ parameters ($H_0:{\lambda }_1^{+}$ = ${\lambda }_1^{-}$) is conducted, and the null hypothesis of equality can be rejected. We also conducted a test to examine whether the two beta parameters exhibit a symmetric relationship with opposite signs ($H_0:{\lambda }_1^{+}$ = $-{\lambda }_1^{-}$). As reported at the bottom of Table III, the null hypothesis of a symmetric relationship is rejected at the 1% significance level (t = 6.46). In summary, our study concluded that a strongly significant conditional risk-return relationship is present, and the strength of this relationship varies between bull and bear markets.

Another result worth noting is about the variable of firm size. As shown in the results of Panels A and B, the relationship between firm size and returns significantly deviates from previous studies, displaying a positive correlation. To understand the reasons behind this outcome, we conducted further analysis.

Firm Size in Conditional Risk-Return Relation

Contrary to many other studies, this study indicates that firm size (ln (Cap)) is related positively with returns (t = 2.14, significant at the 5% level). The relationship between dual beta and returns appears to be valid ($\widehat{{\lambda }_1^{+}}$) > 0 and $\widehat{{\lambda }_1^{-}}$ < 0) and significant even after controlling for firm size. We hypothesized that the pricing parameter of size ($\widehat{{\lambda }_3})$ might likely to vary depending on market conditions, such as bull markets and bear markets. To test this possibility, Model IV has been modified as follows:

$$\begin{array}{rl}ModelI{V}^{\prime }:r_{it}& =\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )+\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta \\ & +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )+\widehat{{\lambda }_3^{+}}\widehat{\ln (Ca{p}_i{)}^{+}}\delta \\ & +\widehat{{\lambda }_3^{-}}\widehat{\ln (Ca{p}_i{)}^{-}}(1-\delta )+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}\end{array}$$

Just like the dual betas, dual variables, $\widehat{\ln (Ca{p}_i{)}^{+}}$ and $\widehat{\ln (Ca{p}_i{)}^{+}}$ of firm size (upside size and downside size) were obtained based on bull and bear markets, Then, they were substituted for the single variable of firm size in Model IV. The results of the model are summarized in Panel B of Table IV. Panel A re-presents the results from a part of Panel B of Table III for the sake of comparison convenience.

Table IV. “Firm Size” Parameters in Dual-Beta Models

Panel A:	$rit=\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )+\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )+\widehat{\hat{{\lambda }_3}ln(Ca{p}_i)}+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}$–Model IV
$\widehat{{\lambda }_1^{+}}$	t-statistics	$\widehat{{\lambda }_1^{+}}$	t-statistics	$\widehat{{\lambda }_3}$	t-statistics
0.0073	(10.63)**	−0.0178	(−13.05)**	0.0009	(2.14)*
Panel B:	$r_{it}=\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )+\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )+\widehat{{\lambda }_3^{+}}\widehat{\ln (Ca{p}_i{)}^{+}}\delta +$ $\widehat{{\lambda }_3^{-}}\widehat{\ln (Ca{p}_i{)}^{-}}(1-\delta )+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}$–Model IV′
$\widehat{{\lambda }_1^{+}}$	t-statistics	$\widehat{{\lambda }_1}$	t-statistics	$\widehat{{\lambda }_3^{+}}$	t-statistics	$\widehat{{\lambda }_3^{-}}$	t-statistics
0.0049	(6.85)**	−0.0136	(−9.66)**	0.0033	(6.95)**	−0.0031	(−5.87)**

The parameters of the dual variables of firm size have greater magnitudes of t-statistics (upside size’s t = 6.95, downside size’s t = −5.87, significant at the 1% level) compared to the t-statistic (t = 2.14, significant at the 5% level) of the single size variable. It appears that the size variable better explains the returns when introduced as dual variables. This finding presents that a single variable of size without considering market conditions (e.g., bull and bear markets) does not properly explain the relationship between size and returns. This finding suggests that the variable of firm size should be differentiated by market conditions and incorporated into pricing models. The upside size variable is positively related to the returns $(\widehat{{\lambda }_3^{+}}>0)$ while the downside size variable shows a negative relationship with the returns $(\widehat{{\lambda }_3^{-}}<0)$. This finding presents that larger-cap stocks tend to provide greater returns in bull markets but may conversely exert less favorable returns compared to smaller stocks in bear markets. These results contrast with previous studies reporting a negative relationship between firm size and returns. Again, this is attributed to a sample bias towards large-cap stocks. Our finding implies that many prior studies claiming a negative relationship between firm size return based on just a single variable of size might have reflected only a part of the whole picture. This issue seems to warrant more comprehensive follow-up research. Also, the conditional risk-return relationship remains valid and significant even after dual size variables are introduced in the models.

Robustness Checks

Test for Conditional Risk-Return Relationship in Industry Sectors

In this section, we examined whether the unconditional and conditional risk-return relationship exists at the industry level and, if so, to what extent it holds significance. Table V shows the outcome of this analysis. The models used here are those applied in the previous predictive test, specifically the unconditional single beta model with controlling variables (Model II) and the conditional dual-beta model with controlling variables (Model IV), excluding only the industry variable, $In{d}_i$.

$$\begin{array}{r}ModelII-1:r_{it}=\widehat{{\lambda }_0}+\widehat{{\lambda }_1}{\beta }_i+\widehat{{\lambda }_2}{\sigma }_i^2+\widehat{{\lambda }_3}\ln \left(Ca{p}_i\right)+{\theta }_{it}\end{array}$$

$$\begin{array}{rl}ModelIV-1:r_{it}& =\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )\\ & +\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )\\ & +\widehat{{\lambda }_3}\ln \left(Ca{p}_i\right)+{\theta }_{it}\end{array}$$

Table V. Test Results of Conditional Risk-Return Relationship in Industry Sectors

Industries	Model II-1	Model IV-1		$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$	$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$
	$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_1^{+}}$	$\widehat{{\lambda }_1^{-}}$
All	0.00004 (0.05)	0.0073 (11.49)**	−0.0184 (−13.81)**	(18.15)**	(7.26)**
Basic materials	0.0086 (1.52)	0.0063 (1.73)	−0.0048 (−0.83)	(1.64)	(0.22)
Communication services	−0.0088 (−1.45)	−0.0034 (−1.02)	−0.0128 (−1.88)	(1.31)	(4.13)*
Consumer cyclical	−0.0030 (−1.54)	0.0045 (3.60)**	−0.0068 (−2.16)*	(3.30)**	(0.48)
Consumer defensive	−0.0026 (−0.5)	0.0064 (1.66)	−0.0080 (−1.12)	(1.84)	(0.03)
Energy	−0.0019 (−0.25)	-0.0034 (−0.68)	−0.0010 (−0.16)	(0.32)	(0.03)
Financial	−0.0059 (−2.29)*	0.0051 (3.37)**	−0.0168 (−6.06)**	(6.88)**	(13.88)**
Healthcare	0.0051 (1.39)	0.0060 (2.35)*	−0.0102 (−2.35)*	(3.32)**	(0.66)
Industrials	−0.0045 (−1.77)	0.0062 (3.49)**	−0.0194 (−9.77)**	(10.94)**	(19.88)**
Real estate	0.0046 (1.48)	0.0060 (2.93)*	−0.0188 (−4.78)**	(5.58)**	(8.40)**
Technology	0.0011 (0.38)	0.0011 (0.62)	−0.0149 (−3.37)**	(3.38)**	(8.45)**
Utilities	0.0301 (0.59)	−0.0047 (−0.2)	−0.0122 (−0.35)	(0.17)	(0.15)

The models are applied to all individual industries. The results of Model II-1 show no significant evidence that the single beta explains the cross-section of returns across all industry sectors. In fact, the beta parameters in more than half of the 11 industry sectors show negative signs. In contrast, for dual-betas (Model IV-1), there is evidence of a significant risk-return relationship in about half of the 11 industry sectors. There are some industries where the estimated coefficients of both upside beta and downside betas are negative. It means that even in bull markets, there may not be proper compensation for taking risk. From an investment trading perspective, this could imply a disadvantage for these industries.

Moreover, the estimates of downside beta parameters consistently show negative signs ($\widehat{{\lambda }_1^{-}}<0$) across all industries. The test results for the equality of dual betas indicate that, overall, the null hypothesis ($H_0:{\lambda }_1^{+}=$ ${\lambda }_1^{-}$) that upside and downside betas are the same is rejected at the 1% significance level. In industry-specific tests, except for communication services, energy, utilities, and basic materials, where dual-betas are statistically similar, all other industries show statistically different dual-betas. A test ($H_0:{\lambda }_1^{+}$$=-{\lambda }_1^{-}$) for the same magnitude but opposite direction (i.e., if the dual betas’ signs only change) of dual betas yield varying results across industries, but overall, the rejection of the null hypothesis at the 1% significance level indicates that there is no symmetry between the two betas. Our findings lead to the conclusion that the conditional risk-return relationship is valid regardless of industry sectors. Since there are no prior studies on conditional risk-return relationships at the level of individual industry sectors, a comparison with our findings is not available.

Test for Conditional Risk-Return Relationship in Firm Sizes

Here, we examined whether the relationship between dual beta and returns still exists within each size portfolio when stocks are distinguished into deciles based on firm size (capitalization). We applied the Models II-2 and IV-2, which are modified versions of Models II and IV with the variable of firm size omitted:

$$\begin{array}{r}ModelII-2:r_{it}=\widehat{{\lambda }_0}+\widehat{{\lambda }_1}{\beta }_i+\widehat{{\lambda }_2}{\sigma }_i^2+\widehat{{\lambda }_4^k}In{d}_i+{\theta }_{it}\end{array}$$

$$\begin{array}{rl}ModelIV-2:r_{it}& =\widehat{{\lambda }_0}+\widehat{{\lambda }_1^{+}}{\beta }_i^{+}\delta +\widehat{{\lambda }_1^{-}}{\beta }_i^{-}(1-\delta )\\ & +\widehat{{\lambda }_2^{+}}{\sigma }_i^{2+}\delta +\widehat{{\lambda }_2^{-}}{\sigma }_i^{2-}(1-\delta )+\widehat{{\lambda }_4^k}In{d}_i\\ & +{\theta }_{it}\end{array}$$

As shown in Model II-2 of Table VI, there is no significant relationship between single beta and returns within the size-sorted decile portfolios based on firm size. Similar to the industry analysis discussed earlier, the signs of some single beta coefficients are even negative in a few decile portfolios, contrary to theory and intuition. In contrast, for dual betas, the risk-return relationships of all decile groups and their signs are clearly aligned with expectations. The coefficients on dual betas in Model IV-2 are statistically significant, being positive ($\widehat{{\lambda }_1^{+}}$ > 0) in up-markets and negative ($\widehat{{\lambda }_1^{-}}<0$) in down-markets. The results indicate the significance of the dual beta in explaining stock returns remains, unlike the single beta even after controlling for the influence of firm size.

Table VI. Test Results of Conditional Risk-Return Relationship in Firm Sizes

Size decile	Model II-2	Model IV-2		$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$	$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$
	$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_1^{+}}$	$\widehat{{\lambda }_1^{-}}$
All	−0.00008 (−0.08)	0.0074 (10.76)**	−0.0173 (−12.71)**	(17.36)**	(6.10)**
1	−0.0070 (−1.53)	−0.0023 (−0.94)	−0.0144 (−3.16)**	(2.39)*	(3.14)**
2	−0.0040 (−1.14)	0.0086 (3.11)**	−0.0247 (−8.16)**	(8.73)**	(3.69)**
3	0.0000 (0)	0.0131 (6.22)**	−0.0125 (−4.23)**	(8.05)**	(0.17)
4	0.0057 (1.24)	0.0045 (1.71)	−0.0127 (−2.45)*	(3.23)**	(1.32)
5	−0.00003 (0)	0.0043 (0.73)	−0.0142 (−1.61)	(1.82)	(0.90)
6	−0.0022 (−0.45)	0.0072 (2.65)**	−0.0195 (−5.43)**	(7.07)**	(2.40)*
7	0.0011 (0.32)	0.0032 (1.53)	−0.0162 (−4.22)**	(4.82)**	(2.76)**
8	0.0000 (0)	0.0056 (2.83)**	−0.0332 (−7.77)**	(8.93)**	(5.48)**
9	−0.0037 (−0.93)	0.0065 (3.11)**	−0.0310 (−6.46)**	(7.38)**	(4.56)**
10	−0.0059 (−1.56)	−0.0017 (−0.67)	−0.0028 (−0.95)	(1.18)	(0.26)

Another interesting finding is that the absolute values of the estimated coefficients and t-statistics for downside beta are consistently larger than those of upside beta in all decile portfolios, except for the third decile. This finding implies that high beta stocks are rewarded with high returns in bull markets (${\lambda }_1^{+}$ > 0), but this comes at the cost of much lower returns in bear markets (${\lambda }_1^{-}$ < 0), reflecting a significant trade-off between risk and returns. To determine whether the beta-return trade-off effect is more pronounced in downside markets compared to upside markets, regardless of firm size, we test the null hypothesis ($H_0:{\lambda }_1^{+}=$$-{\lambda }_1^{-}$) that the absolute values of the upside beta and downside beta are equal. As shown in the last column of Table VI, the null hypothesis is rejected for more than half of the size-sorted decile portfolios.

Test for Conditional Risk-Return Relationship in Seasonality

We employed Models II and IV to examine the existence of the beta risk-return relationship on individual months throughout years and identify any seasonal patterns in the relationship. As reported in Model II of Table VII, the relationship between risk-return is neither consistent nor significant across most months, showing significance only in November. The beta coefficient ($\widehat{{\lambda }_1}$) in the month of May appears to be significant but exhibits a negative sign contrary to typical empirical expectation. “November” only as a significant month in our study contradicts Tinic and West (1984) and Pettengillet al. (1995) that find a significant positive relation in January only. Our finding appears to align with recent empirical research results indicating that the stock market tends to exhibit a strong upward seasonal effect (so called “November Effect” or “Turkey Rally”) in November, rather than in January since the 1986 Tax Reform Act. There is ample empirical literature supporting our finding including studies by Bharaet al.(1999), He and He (2011), etc. In fact, according to Smith (2023), the month of November has been the strongest month for stocks since 1950. As shown in Table VII, the average excess return for the month of November is 2.69%, which is the highest among all twelve months. Our finding clearly confirms the November effect.

Table VII. Test Results of Conditional Risk-Return Relationship in Seasonality

Months	Excess return	Model II	Model IV		$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$	$H_0:{\lambda }_1^{+}={\lambda }_1^{-}$
		$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_1^{+}}$	$\widehat{{\lambda }_1^{-}}$
All	1.24%	−0.0001 (−0.12)	0.0073 (10.63)**	−0.0178 (−13.05)**	(17.64)**	(6.46)**
January	2.01%	0.0021 (0.59)	0.0222 (9.77)**	−0.0287 (−7.59)**	(12.58)**	(1.37)
February	2.42%	0.0045 (1.76)	0.0056 (3.42)**	−0.0191 (−4.99)**	(6.28)**	(3.07)**
March	1.91%	0.0001 (0.04)	0.0050 (3.09)**	−0.0129 (−4.4)**	(5.78)**	(2.21)*
April	1.14%	0.0004 (0.05)	0.0023 (0.49)	−0.0220 (−1.36)	(1.49)	(1.13)
May	−0.15%	−0.0060 (−2.01)*	0.0083 (4.59)**	−0.0447 (−10.72)**	(12.50)**	(7.63)**
June	1.57%	−0.0033 (−1.36)	0.0061 (3.68)**	−0.0114 (−3.79)**	(5.51)**	(1.45)
July	1.99%	0.0018 (0.70)	0.0029 (−1.77)	−0.0173 (−3.19)**	(3.67)**	(2.48)*
August	0.01%	−0.0047 (−1.49)	0.0129 (5.07)**	−0.0108 (−3.44)**	(6.53)**	(0.48)
September	0.84%	0.0029 (1.19)	0.0069 (4.27)**	−0.0110 (−3.75)**	(5.78)**	(1.15)
October	1.05%	−0.0029 (−0.87)	0.0080 (3.58)**	−0.0155 (−4.18)**	(5.92)**	(1.61)
November	2.69%	0.0057 (2.09)*	0.0027 (1.54)	0.0104 (−1.88)	(1.37)	(2.18)*
December	−0.51%	−0.0015 (−0.49)	0.0123 (6.08)**	−0.0222 (−6.56)**	(9.55)**	(2.31)*

The results from the dual-beta model (Model IV) are similar to the earlier analyses based on industry sectors and firm sizes, but they exhibit a stronger and more significant risk-return relationship. The signs of the coefficients for upside and downside betas are consistently positive and negative, respectively, in all months except for November. The dual betas’ parameters of the month of January are large among all twelve months. It means stocks with high beta enjoy significant compensation for the high risk in bull markets but also are penalized by the risk in bear markets. The magnitudes of the beta parameters suggest that the pricing of beta varies by month and market condition. Interestingly, in November, contrary to theoretical expectations, the downside beta shows a significantly positive relationship with returns at the 10% significance level (t = 1.88). This implies that stocks with high downside beta are compensated not only in bear markets but also in bull markets during November. These results do not align with the theory and intuition and may be attributed to some anomaly or the so-called November effect.

Conclusion

Most studies on the relationship between beta risk and returns in the Capital Asset Pricing Model (CAPM) have been conducted using 10–20 portfolios comprising stock securities constructed based on estimated betas for the efficiency of empirical validation. Such studies have generally failed to find a significant relationship between beta risk and returns. In contrast, our study finds that even for individual stocks, the single beta in the traditional asset pricing models fails to significantly explain the cross-section of returns.

In both bull and bear markets, distinguishing upside and downside dual betas during the beta estimation period, we observe a significant risk-return relationship in contemporaneous tests. Moreover, the directions of the relationships between upside/downside betas and returns align well with theoretical predictions and intuition. A predictive test is also conducted to assess the extent to which the estimated betas and other variables significantly explain the returns realized in the subsequent period. In this test, while the single beta still fails to show a significant relationship with returns, dual betas continue to exhibit a meaningful relationship with realized returns, consistent with expectations. Overall, our findings indicate the dual betas really matter.

The evidence for dual betas appears to persist even after considering firm size, total risk, and industry variables, surpassing the explanatory power of firm size, which has traditionally been identified as a significant variable for returns. Contrary to conventional results, our empirical results indicate that the variable of “firm size” shows a weak relationship with returns. We also discover that, similar to dual betas, firm size exhibits varying relationships with returns depending on market conditions. When grouping stocks by their firm size, the relationship between firm size and returns is not consistent across each group, showing varying directions and magnitudes of the impact of firm size on realized returns depending on market conditions. Therefore, instead of uniformly considering the relationship between size and returns as negative, it may be more reasonable to say that this relationship is determined by the sensitivity of returns to firm size and its direction and magnitude in down or bear markets. Further research appears necessary to explore how and why firm size influences returns differently in bull and bear markets.

Unlike previous related studies (Hodoshimaet al., 2000; Pettengillet al., 1995), our study reports that the single beta does not show a significant relationship with returns in the month of January, exhibiting a meaningful relationship only in the month of November (at the 5% significance level). Surprisingly in November, which demonstrates the highest average excess returns (2.69%), both upside and downside betas show a positive significant relationship with returns, leading to the November effect.