Original Research
Frontiers in Applied Mathematics and Statistics ·
Volume 12 - 2026 | https://doi.org/10.3389/fams.2026.1953346
Capital structure risk and portfolio returns: evidence from Indian stock market
- Jayashree Somu1
- Nirakar Barik1*corresponding author
- Department of Commerce, School of Social Sciences and Languages, Vellore Institute of Technology, Vellore, India
Abstract
Capital structure is an important determinant of investment decision as it influences share price as well as the overall earnings of the firm. Hence, this study tries to examine the risk exposure of capital structure factor in explaining the excess portfolio returns. Further, it also compares the explanatory power of Fama French three factor model (FF3fm) with the Modified Fama French three factor model (MFF3fm) which replaces the value factor with two capital structure measures such as Long-term debt/Total Assets (LD/TA) and Total Liabilities to Total Assets (TL/TA). The study has employed three models such as Fama French three factor model (FF3fm), Modified Fama French three factor model (MFF3fm) by replacing the value factor with capital structure factors, and Extended Fama French three factor model (EFF3fm) which incorporates both the factors for analysis purpose. The results of the study reveal a strong size, value, and capital structure effect in the Indian stock market, with a significant capital structure premium indicating that firms with high (low) capital structure earn higher (lower) returns. Further, MFF3fm is found to be a better descriptor of excess portfolio returns compared to both FF3fm and Extended Fama–French three factor model (EFF3fm). The lower GRS F statistics, coupled with higher adjusted R 2, further substantiate the superiority of MFF3fm and underscore the importance of capital structure as a relevant factor in explaining portfolio returns. These findings contribute to the asset-pricing literature by providing empirical evidence on the role of capital structure factors in explaining portfolio returns, with particular reference to the Indian stock market.
Keywords
Introduction
In securities markets, investors always look for a strategy which will help them to earn abnormal returns. One common approach is to construct portfolios based on firm characteristics that are believed to influence stock returns, and testing such characteristic-based portfolios has generated substantial evidence of market anomalies. For instance, evidence from earlier studies document that the portfolios based on firm size (market capitalization) show small firms consistently outperform large firms. Similarly, the growth stocks are continuously outperformed by value stocks if portfolios are constructed using book-to-market ratio (BM) which exhibits value effect (1, 2). Later these findings were formalized by Fama and French (3) stating that “market risk premium, size of the firm, and book-to-market ratio” are the three factors which can explain the market returns. The model including these three factors also known as Fama French three factor model (FF3fm) became a cornerstone of a modern finance that helps to predict the stock return better than the traditional “capital asset pricing model (CAPM)”.
Based on the framework introduced by Fama and French (3), there are numerous studies in developed market which tested the robustness of FF3fm. Capaul et al. (4) tested the FF3fm across the six major developed equity markets considering portfolio returns from 1981 to 1992 and found that value firms consistently yielded higher returns than the growth firms across all the markets by confirming the presence of value premium in the international markets. Subsequently, in 1998, Fama and French (5) extended this analysis to thirteen developed and emerging markets and found that the variations in equity returns are mostly driven by size and value factors beyond the market beta. Griffin (6), demonstrated that domestic version of FF factors explain portfolio returns better than the global factors, which confirms the model adaptability to local market conditions. Other studies such as, Drew and Veeraraghavan (7), and Clive Gaunt (8) also subsequently validated the FF3fm across various international markets including Malaysia and Australia. These findings collectively validate the FF3fm as a cross-market framework and motivate its application across the global market.
In Indian context, the FF3fm has attracted considerable researcher attention given the unique structural characteristics of Indian stock market. Connor and Sehgal (9) were among the first to test FF3fm in Indian stock market and observed that returns were influenced by both the firm size and the book-to-market characteristics. In line with the above findings, Kumar and Sehgal (10) further confirmed the influence of size premium across different portfolios and the value effects found to be weaker when “P/E ratio” is considered as proxy to measure the company value. Sehgal and Balakrishnan (11) re-examined the explanatory power of CAPM and FF3fm to account for excess portfolio returns in the Indian equity market. The findings of the study reveals that the FF3fm provides a superior explanation than the traditional CAPM. Balakrishnan (12) examined various asset pricing frameworks in capturing the stock return variation on portfolios and found that FF3fm performed better than the CAPM in estimating excess returns for size–value sorted portfolios. Balakrishnan (13) further confirmed the presence of size and value effect in Indian stock market.
Capital structure is a specific mix of debt and equity including both financial obligations and operating liabilities that firm uses to finance its operations and sustain long-term growth. There is strong debate among researchers about, how debt financing of firm influences the firm value since the seminal work of Modigliani and Miller (14). Subsequently theories such as the “trade-off theory by Kraus and Litzenberger (15)”, “pecking order theory by Myers and Majluf (16),” and “market timing theory Baker and Wurgler (17)” collectively argue that capital structure decisions of a firm directly affect the cost of capital, financial risk and its overall market worth. A large body of empirical literature has examined the relationship between leverage and expected stock returns. Early studies confirmed positive association between firms’ leverage and predicted stock returns, with rise in firms’ debt level increases risk borne by shareholders and generating a strong return premium (Basu, (18); Banz, (1); Bhandari, (19)). Subsequent research extended this work across different markets and time periods documenting that leverage possess a substantial predictive information regarding the anticipated stock returns and firms with different debt level significantly generates different return patterns (Welch, (20); Baturevich and Muradoglu, (21); Yang et al., (22)). This mechanism is further refined by recent studies which shows that the market leverage acts as a driving force behind the value premium through investment irreversibility. Ozdagli (23), Chong and Kim, (24) states that the capital structure volatility as distinct pricing factor. Friewald et al. (25) document that the debt maturity plays a very crucial role with short-term leverage yields a substantial positive equity premium due to its systematic risk related to debt rollover.
Despite these evidences, traditional asset pricing models such as CAPM and FF3fm fails to explain the stock return variation fully which motivated the researchers to explore other factors to get better explanation about the excess portfolio returns. Leverage is used as an explicit risk factor in asset pricing frameworks across various stock market around the globe. Muradoglu and Sivaprasad (26) examined the effect of leverage mimicking portfolios by modifying the traditional four factor model and found that the leverage mimicking portfolio explains the variations in stock return better than the Carhart four factor model indicating that leverage is a powerful pricing factor. Similarly, Koseoglu (27) revealed that addition of leverage factor enhances the explanatory ability of asset pricing models by exploring the Istanbul stock exchange. Boubaker et al. (28) constructed portfolios based on size, book-to-market, and leverage in the French stock market and found highly levered firms earn a positive leverage premium. Maiti and Balakrishnan (29) revealed that leverage-based factors explain stock return variation more effectively than the size and value factors in Indian stock market. Following this Jain et al. (30) confirmed, model that incorporates size, value, leverage and liquidity performs better than the traditional asset pricing models, indicating that leverage is an important dimension of risk in stock return. Notably, all of these studies operationalise capital structure through the debt-to-equity ratio, a narrow measure of financial leverage.
It is clearly evident from existing literature that debt/equity ratio (D/E) is widely used in asset pricing (Bhandari, (19); Fama and French, (2)) as a measure of leverage that proxies for capital structure of firms. But there are also ample evidences, which recognise debt/equity as narrow measure of financial leverage that captures only formal borrowings of firms and ignores broader financing obligation of the firms. According to the empirical findings of Rajan and Zingales (31) capital structure is highly sensitive to the choice of leverage definition, and the measure of total liabilities to total assets (TL/TA) constitutes a broader and often more informative measure than the debt-equity. Following this, Bevan and Danbolt (32) extended this to UK market and confirmed the definitional dependence, showing that trade credit and other operating liabilities account for majority of total liabilities which makes the D/E ratio particularly incomplete. Adding to this Nissim and Penman (33) made theoretical distinction explicit: usually a firms balance sheet consist both the financial liabilities and operating liabilities, and the two components carries a different risk implication and also priced differently by the markets. Similarly, Welch (34) argues that financial debt-to-asset ratio which actually underlies D/E based measure is fundamentally flawed because it implicitly treats non-financial liabilities as equity and thereby recommends a broader liability-based measures instead.
Recent evidences further confirm that capital structure shapes broader return dynamics beyond the traditional leverage-return relationship. Li et al. (35) employed both long-term debt to total assets (LD/TA) and total liabilities to total assets (TL/TA) as capital structure measures in the US market and documented the differential return patterns across portfolios sorted on two measures confirming that capital structure is an independent dimension that cannot be captured by size and value proxies alone. Duong Le (36) extended this idea to Australian market and confirmed that capital structure plays major role in shaping stock return behaviour.
Despite the growing body of literature, critical measurement gap persists in the Indian context. Most of the existing studies that considers leverage and firm characteristics to construct portfolios, rely on narrow financial leverage measurement such as debt-equity ratio which captures only the financial borrowing of the firm’s and which completely ignores the operating liabilities such as trade credit, accounts payable. Furthermore, limited attention has been given to portfolio level analysis based on capital structure, particularly in combination with other important firm characteristics. While anomalies like size-value or size-momentum have been widely studied, but their interaction with capital structure remains largely unexplored in India.
Considering these gaps, the present study examines whether the asset-scaled measures of capital structure yield a different asset pricing inference in Indian stock market. The primary objective of the study is to test whether or not the portfolios constructed based on capital structure can generate abnormal returns. Second, the study tries to explore if inclusion of these capital structure measures as risk factors in a Modified Fama French three factor model and Extended Fama–French three factor model enhance the explanatory power relative to the established FF3fm. The study has considered two capital structure ratios such as total liability excluding owners’ equity to total asset (TL/TA) and ratio of long-term debt to total asset (LD/TA) in line with Li et al., (35) to examine the role of capital structure in asset pricing setting in Indian stock market.
The study contributes to the literature in three ways. First, unlike Li et al. (35), who combined momentum with capital structure, this study merges the capital structure with the size factor to check its dynamics. Second, the study goes further by checking if these measures can be used as risk factors. To the best of our knowledge, this is the first study in Indian context to incorporate asset-scaled capital structure measures (LD/TA and TL/TA) rather than the debt-equity ratio in the asset pricing framework both as portfolio sorting variables and as constructed risk factors thereby distinguishing the pricing of the capital structure from that of financial leverage documented by Maiti and Balakrishnan (29).
The paper is structured as follows. Section 2 elaborates the data and methodological procedures applied in the study. Section 3 presents the analysis and interpretation of the findings. Section 4 outlines the conclusions. Section 5 discusses the limitations of the study followed by practical implications which are discussed in Section 6.
Data and methodology
The data consists of month end adjusted closing price for 394 companies from January 2010 to December 2024. The sample companies are part of National Stock Exchange (NSE) 500 index which is a broad-based market index and represents the total market performance of Indian stock market and covers around 90 percentage of market capitalisation and trading volume. If a company does not have data for at least 2 years (24 months) are filtered out from the sample. The financial firms are also not included as the operational definition for assets and liabilities differ from the non-financial firms. The monthly share prices are converted into returns for further estimation and analysis purpose.
Firm characteristics data such as Market Capitalisation (MC) and Price/Book ratio (P/B ratio) which proxies for firm size and value respectively, are used to construct portfolios. MC is used as a measure for market size globally and following that this study has used the logarithmic value of MC to proxy for size factor. However, P/B ratio is generally used in Indian context as a proxy for the value factor instead of Book Equity-to-Market Equity (BE/ME) which is used globally, because of unavailability of data in the original data source. The interpretation of P/B ratio is inverse to the interpretation of BE/ME. Hence, the interpretation of this study with regards to P/B is opposite to that of Fama French (3). Next, following Li et al., (35) the study considered two ratios such as Long-term Debt/Total Assets (LD/TA) and Total Liability/Total Assets (TL/TA) as measure of capital structure. The book value of liability, debt, and asset are considered. BSE – 200 index return is calculated and adjusted for risk free return to arrive at excess market return which is the proxy for market premium. Finally, 91-days T-bill yield is used to measure risk free return which is conventionally considered as risk free short-term financial assets in Indian Money market. All the data are sourced from BLOOMBERG database.
Portfolio construction
The study constructed portfolios by using both single sorting and double sorting technique following Fama and French 1992 and 1993 (2, 3). Single sorted portfolios are constructed for the firm characteristics such as size, value, and capital structure measured by MC, P/B, LD/TA, and TL/TA, respectively. Then the mimicking portfolios such as SMB (Small Minus Big), LMH (Low Minus High), LMXLDTA, and LMXTLTA are sorted using the double sorting technique. Further, this double sorting is also employed to construct 25 stylized portfolios combining size-value and size-capital structure factors. Additionally, the study modifies the FF3fm by replacing the value factor (P/B ratio) with capital structure factors (LD/TA and TL/TA). The motivation for replacing the value factor is that leverage has been identified as one of the potential sources of the value premium (Ferguson and Shockley (37)), suggesting that the value factor and capital structure factors may capture overlapping dimensions of risk. This potential overlap, however, implies that a direct comparison of the Modified Fama French Three-Factor Model (MFF3fm) and FF3fm cannot, by itself, establish whether the capital structure factors contain incremental information beyond that captured by the value factor or merely substitute for it. To address this issue directly, the study additionally estimates a nested Extended Fama French three factor model (EFF3fm), which retains the value factor alongside the two capital structure factors. The reported correlation of 0.677 between the two capital structure factors implies a pairwise variance inflation factor (VIF) of approximately 1.84, calculated as 1/(1 − r2), indicating that the collinearity between TL/TA and LD/TA considered in isolation is well below the conventional threshold of 5. The EFF3fm therefore provides a direct framework for assessing whether the capital structure factors retain explanatory power for portfolio returns after controlling for the conventional value factor.
Single sorted portfolio
Portfolios are constructed using single sorting technique where sample securities of company are sorted based on MC. The portfolios are constructed using the following procedure. Initially the companies are ranked at the end of the June, 2010 (period t) based on MC. Then the sample stocks of the company are categorized into five equal weighted portfolios, where portfolio one (P1) consists of 20 per cent of the sample stocks which are small in size and portfolio five (P5) consist of 20 per cent of the sample stocks which are called big stocks. Then the returns are calculated on these five equal weighted portfolios for 1 year from July 2010 (t) to June 2011 (t + 1). After this the portfolios are rebalanced every June till the end of the study period. The mean excess return on each portfolio is calculated for whole time period from 2010 to 2024. Next, the same procedure is followed to construct quintiles considering the P/B ratio, except that P/B, LD/TA, and TL/TA are accounting-based measures drawn from the financial statements for the fiscal year ending March 2010. Firms are ranked on these variables at the end of June 2010 (period t), the same ranking date used for MC. The lag between fiscal year-end and portfolio formation is introduced intentionally to avoid the look-ahead bias. Because most Indian firms have a March fiscal year-end, portfolios are formed at the end of June on the basis of P/B, LD/TA and TL/TA data from the fiscal year-end of that period. The holding period begins in July, so that the accounting information used to construct the portfolio was known to the market before the return measurement. This timing convention is consistent with standard practice in the asset-pricing literature Fama and French (2, 3) and the study does not use information that was not publicly available. P1 consists of lower 20 percent of the stocks known as value stocks and P5 contains higher 20 percent of the stocks classified as growth stocks. Next, the mean returns on the portfolios are calculated by adopting the procedure followed in the case of MC. Similarly, quintiles are formed using TL/TA and LD/TA ratios which are considered as measures of capital structure. The top 20% stocks are categorised into P5 known as high capital structure stocks whereas the bottom 20% are clubbed into P1 called low capital structure stocks. Then the mean excess return for portfolio is calculated following same procedure as of MC and PB.
Double sorted portfolios
The double sorted portfolios such as size-value, size-LD/TA, and size-TL/TA are constructed to calculate portfolio returns. For construction of these double sorted portfolios, at the end of June of year t, all the sample firms are ranked based on MC to generate five portfolios by following single sorting technique. Similarly, the firms are again ranked based on P/B ratio to construct five quintile portfolios. The intersection of five size-based portfolios and five value-based portfolios results in formation of 25 (5*5) portfolios. Similar procedure is followed to construct 25 size-LD/TA and size-TL/TA portfolios. Next, the excess returns of these portfolios are computed and used as dependent variables to run the risk models as described below.
Risk mimicking portfolios
Next, double sorting technique is used to construct mimicking portfolios such as SMB, LMH, LMXLDTA, and LMXTLTA which mimic the size, value and capital structure factors, respectively. To improve the clarity and ensure the consistent interpretation of the factor portfolios, the study employs the following notation: SMB denotes Small Minus Big, LMH denotes Low Minus High based on the price-to-book (P/B) ratio, LMXLDTA is Low Minus High Long-Term Debt to Total Asset, and LMXTLTA is Low Minus High Total Liability -to Total Asset. The procedure to construct these portfolios are as follows.
First, all the sample securities are ranked based on MC and divided into two groups namely Small (S) and Big (B). Then, in descending order ranking, the bottom 50 percent securities are assigned to small portfolios (S) and the remaining stocks are categorised into big portfolios (B). Simultaneously, the firms are trifurcated into three sub categories based on their P/B ratio such as Low (L), Medium (M), High (H) considering 33.33 percent interval. The bottom 1/3rd (33.33 percent) stocks are called Low value stocks whereas the top 1/3rd stocks are called as high value stocks. The intermediate 33.33 percent stock are placed in medium value stock category. The intersection of two size group and three value groups stocks generate six size-value sorted portfolios (S/L, S/M, S/H, B/L, B/M, and B/H). S/L, S/M, and S/H include the small size stocks whereas B/L, B/M, and B/H consist of big size stock based on MC combined with low, medium, high value stocks, respectively. Finally, using these six portfolios, the mimicking portfolios are constructed which is explained in the subsequent sections.
Small Minus Big (SMB) mimics the risk factors in portfolio associated with firm size. It is calculated by subtracting the simple average returns on large stock ‘(B/L, B/M, B/H)’ portfolio from simple average of small stocks (S/L, S/M, S/H) portfolio. SMB is calculated as shown in Equation 1:
SMB = ( S / L + S / M + S / H ) / 3 − ( B / L + B / M + B / H ) / 3 (1)
The risk factors in portfolio associated with firm value is mimicked by LMH portfolio. LMH is calculated by subtracting the simple average returns of high value stocks (S/H and B/H) portfolio from simple average returns of low value stocks (S/L and B/L) portfolio. LMH is calculated as shown in Equation 2:
LMH = ( S / L + B / L ) / 2 − ( S / H + B / H ) / 2 (2)
Similar to SMB and LMH, the study constructs two capital structure factors which mimic the risk related to the capital structure of the firm. The same sorting method used for SMB and LMH is applied here for capital structure variables (LD/TA and TL/TA), and two sets of six portfolios such as (S/LLD/TA, S/MLD/TA, S/HLD/TA, B/LLD/TA, B/MLD/TA, and B/HLD/TA), and (S/LTL/TA, S/MTL/TA, S/HTL/TA, B/LTL/TA, B/MTL/TA, and B/HTL/TA) are obtained by the intersection of the two size groups and three LD/TA and TL/TA groups, respectively. The capital structure mimicking factors (LMXLDTA) and (LMXTLTA) are calculated using Equations 3 and 4, respectively:
LMXLDTA = ( S / L LD / TA + B / L LD / TA ) / 2 − ( S / H LD / TA + B / H LD / TA ) / 2 (3)
LMXLDTA mimics the risk in portfolio returns related to the capital structure of the firms measured by long-term debt to total asset. The same procedure is applied to TL/TA to construct another capital structure factor.
LMXTLTA = ( S / L TL / TA + B / L TL / TA ) / 2 − ( S / H TL / TA + B / H TL / TA ) / 2 (4)
LMXTLTA mimics the risk associated with the total liability to total asset-based measure of capital structure.
After constructing the mimicking portfolios, the study employs three risk models to examine whether the mean excess return of portfolios is captured by these mimicking portfolios. The first risk model is the Fama French three factor model (FF3fm), and the second one is a modified model in which the value factor is replaced with the two capital structure factors, referred to as the Modified Fama–French three Factor Model (MFF3fm) and the third model which retains the value factor alongside the capital structure factors, referred to as Extended Fama French three-Factor Model (EFF3fm). The FF3fm, MFF3fm, and EFF3fm are specified in Equations 5, 6, and 7, respectively.
R Pt − R Ft = a + b ( R Mt − R Ft ) + sSM B t + lLM H t + e t (5)
Where, R Pt − R Ft denotes excess return after excluding the risk-free return on portfolio p in period t, next the R Mt − R Ft represents market premium. Then, SMB and LMH are mimicking the size and value factors of the firms, respectively. a is the intercept used as a measure of abnormal returns where b , s , and lare the sensitivity coefficients. Finally, e t signifies the error terms in the above equation.
R pt − R Ft = a + b ( R Mt − R Ft ) + sSM B t + d 1 LMXLDT A t + d 2 LMXTLT A t + e t (6)
Where, LMXLDTA and LMXTLTA mimic the risk associated with the capital structure of the firms measured by long-term debt to total assets and total liability to total assets respectively, and d1 and d2 are the sensitivity coefficients. The remaining terms in the above equation carry the same meaning as in the FF3fm specification.
The study further estimates the following Extended Fama–French three factor Model (EFF3fm) to test if the capital structure factors still explain portfolio returns in the presence of the value factor.
R pt − R Ft = a + b ( R Mt − R Ft ) + sSM B t + lLM H t + d 1 LMXLDT A t + d 2 LMXTLT A t + e t (7)
Analysis and interpretation
Descriptive statistics of single sorted and mimicking portfolio
The summary statistics for the single-sorted portfolios for size, value and capital structure variables for each quintile portfolio (P1 to P5) are provided in Table 1. The results provide strong evidence about the existence of strong size, value, and capital structure premium in Indian Stock market. The portfolio comprising micro-cap and value stocks outperform portfolios with mega-cap and growth under both size- and value-based sorting. It is evidenced by higher mean excess return (MR) of P1 portfolios, which are 3.3% and 2.0% compared to 0.4% and 1.0% for the corresponding P5 portfolios, respectively. Conversely, for both the capital structure measure, the portfolios including stocks with higher capital structure (P5) outshine the portfolio that consist of stocks with low capital structure (P1). The MR for P5 and P1 portfolios are 1.5%, 1.3%, 1.2%, and 1.2%, respectively. These results are in line with the underlying capital structure theories which says firms with higher debt are exposed to higher financial risk.
Table 1. Summary statistics of size, value, capital structure portfolios.
| Mkt-cap | P1 | P2 | P3 | P4 | P5 |
|---|---|---|---|---|---|
| MR | 0.033 | 0.015 | 0.012 | 0.009 | 0.004 |
| SD | 0.088 | 0.068 | 0.059 | 0.054 | 0.049 |
| T-stat | 4.906 | 3.030 | 2.582 | 2.211 | 1.134 |
| P/B | |||||
| MR | 0.020 | 0.015 | 0.013 | 0.013 | 0.010 |
| SD | 0.086 | 0.070 | 0.058 | 0.054 | 0.048 |
| T-Stat | 3.115 | 2.903 | 3.018 | 3.225 | 2.827 |
| TL/TA | |||||
| MR | 0.012 | 0.015 | 0.016 | 0.015 | 0.015 |
| SD | 0.054 | 0.053 | 0.063 | 0.067 | 0.077 |
| T-Stat | 2.915 | 3.728 | 3.318 | 3.019 | 2.592 |
| LD/TA | |||||
| MR | 0.012 | 0.017 | 0.018 | 0.013 | 0.013 |
| SD | 0.055 | 0.060 | 0.066 | 0.068 | 0.071 |
| T-stat | 3.041 | 3.709 | 3.632 | 2.509 | 2.363 |
*Author’s own calculation.
Further, the mean returns of mimicking portfolios reported in Table 2 corroborate the above findings as all portfolio exhibit positive returns except one portfolio that mimics TL/TA measure of capital structure. The average returns generated by the market, size and value premium are 0.5%, 1.4%, and 0.3% respectively, of which only the size premium is statistically significant with (t = 5.72), consistent with the strong size premium reported by Sehgal and Balakrishnan (11), Balakrishnan (13, 38). In case of the capital structure factors, the LD/TA earns a premium of 0.2% (t = 1.01), whereas the TL/TA premium generates negative returns of −0.5% (t = −2.25), similar to the leverage premium reported by Maiti and Balakrishnan (29).
Table 2. Descriptive statistics of mimicking portfolios.
| Mimicking portfolios | Market return | SMB | LMH | LMXLDTA | LMXTLTA |
|---|---|---|---|---|---|
| Mean | 0.005 | 0.014 | 0.003 | 0.002 | −0.005 |
| SD | 0.049 | 0.032 | 0.042 | 0.026 | 0.030 |
| T-Stat | 1.304 | 5.722 | 0.845 | 1.012 | −2.249 |
*Author’s own calculation.
Results of correlation between risk factors
Table 3 depicts the correlation matrix for the risk factors. The market factor is positively related to the size factor (0.385) and the value (0.513) factors. Size factor of the company is positively related to the value factor (0.546), indicating that small size stocks in the sample tilts towards value. Both the capital structure factors are negatively correlated with market, size and value factors. The negative correlation is strongest with the value factor (−0.646 for LD/TA and −0.671 for TL/TA), which suggest that the firms with higher capital structure behaves like a value firms which is theoretically expected, as financial leverage has been argued to be one of the sources of the value premium Ferguson and Shockley (37). The two capital structure factors are positively correlated with each other (0.677). However, the correlation is well below the unity. Which reveals that LD/TA and TL/TA carry a distinct information and that justifies including both the factors in the MFF3fm.
Table 3. Correlation matrix for risk factors.
| Risk factors | MKT | SMB | LMH | LMXLDTA | LMXTLTA |
|---|---|---|---|---|---|
| MKT | 1 | ||||
| SMB | 0.385 | 1 | |||
| LMH | 0.513 | 0.546 | 1 | ||
| LMXLDTA | −0.424 | −0.282 | −0.646 | 1 | |
| LMXTLTA | −0.480 | −0.411 | −0.671 | 0.677 | 1 |
*Author’s own calculation.
Regression results of Fama–French three factor model
A factor model is considered well specified when alpha (intercepts) obtained from the time-series regression are statistically indistinguishable from the zero. Table 4 presents the results of FF3fm regression for the twenty-five size-value sorted portfolios. The intercept values range from −0.009 and 0.008, and the average absolute alpha (AAA) is estimated to be 0.003. Twenty-three out of twenty-five intercepts are indistinguishable from zero, leaving only three significant alphas at 5 % level (t = 2.12 for P1, t = −3.30 for P7 and t = −1.97 for P8). The market beta coefficients of all the twenty-five portfolios are close to unity and highly statically significant. The coefficients on SMB decline monotonically across the size quintiles within every value column ranging from 1.455 for the small firms to −0.162 for the big firms in low-value column. These patterns confirm that size factor effectively captures the variations in portfolio returns related to the size dimension of the sort. Similarly, the value factor loadings decline systematically across the value columns within each size row, ranging from 0.986 to −0.251 in the small-size row, providing further evidence that the value factor also appropriately captures the variation in portfolio returns associated with the value characteristics. The R2 values of MC_PB portfolios ranges from 0.547 to 0.865. Hence it can be concluded that the FF3fm captures the average returns on the MC_PB sorted portfolios reasonably well. These findings are in line with the earlier evidence of Connor and Sehgal (9), Sehgal and Balakrishnan (11) and Balakrishnan (13).
Table 4. FF3fm results for size-value sorted portfolios.
| Portfolios | LW | 2 | 3 | 4 | HH | LW | 2 | 3 | 4 | HH |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.008 | 0.005 | 0.005 | 0.001 | 0.005 | 0.858 | 0.943 | 0.969 | 0.869 | 1.005 |
| 2 | −0.002 | −0.009 | 0.001 | −0.002 | −0.001 | 0.988 | 0.884 | 0.993 | 0.846 | 0.891 |
| 3 | −0.004 | −0.005 | 0.004 | 0.000 | −0.001 | 0.933 | 0.915 | 0.838 | 0.958 | 0.895 |
| 4 | 0.001 | 0.001 | 0.000 | 0.001 | −0.002 | 0.967 | 0.923 | 0.884 | 0.900 | 0.883 |
| BG | −0.003 | 0.002 | −0.001 | 0.000 | 0.004 | 0.861 | 0.925 | 0.941 | 0.979 | 0.945 |
| S | L | |||||||||
| SM | 1.455 | 1.375 | 1.664 | 1.518 | 1.411 | 0.986 | 0.609 | 0.074 | −0.080 | −0.251 |
| 2 | 0.765 | 1.196 | 0.925 | 1.038 | 0.964 | 0.731 | 0.123 | −0.078 | −0.310 | −0.326 |
| 3 | 0.496 | 0.741 | 0.450 | 0.642 | 0.715 | 0.584 | 0.197 | −0.024 | −0.354 | −0.554 |
| 4 | 0.060 | 0.234 | 0.292 | 0.391 | 0.495 | 0.699 | 0.267 | −0.028 | −0.288 | −0.451 |
| BG | −0.162 | −0.191 | 0.022 | −0.074 | −0.057 | 0.795 | 0.381 | 0.046 | −0.214 | −0.379 |
| t(α) | t(β) | |||||||||
| SM | 2.123 | 1.477 | 1.428 | 0.402 | 1.613 | 9.982 | 13.059 | 12.688 | 11.590 | 15.443 |
| 2 | −0.602 | −3.297 | 0.376 | −0.908 | −0.264 | 16.253 | 14.100 | 16.359 | 14.997 | 13.332 |
| 3 | −1.555 | −1.978 | 1.210 | 0.186 | −0.335 | 15.410 | 15.395 | 12.998 | 16.334 | 16.535 |
| 4 | 0.540 | 0.565 | −0.008 | 0.406 | −0.762 | 17.118 | 17.294 | 16.301 | 16.187 | 17.401 |
| BG | −1.103 | 0.895 | −0.267 | 0.155 | 1.284 | 16.836 | 21.735 | 20.390 | 20.357 | 14.329 |
| t(S) | t(L) | |||||||||
| SM | 10.971 | 12.340 | 14.120 | 10.723 | 8.045 | 8.896 | 6.539 | 0.754 | −0.831 | −2.994 |
| 2 | 8.162 | 12.363 | 9.872 | 11.927 | 9.347 | 9.337 | 1.521 | −1.001 | −4.264 | −3.781 |
| 3 | 5.308 | 8.079 | 4.521 | 7.100 | 8.554 | 7.487 | 2.577 | −0.283 | −4.687 | −7.945 |
| 4 | 0.691 | 2.837 | 3.490 | 4.564 | 6.317 | 9.593 | 3.889 | −0.402 | −4.016 | −6.898 |
| BG | −2.051 | −2.912 | 0.309 | −0.992 | −0.558 | 12.059 | 6.939 | 0.769 | −3.455 | −4.452 |
| R2 | ||||||||||
| SM | 0.832 | 0.848 | 0.709 | 0.701 | 0.805 | |||||
| 2 | 0.865 | 0.812 | 0.787 | 0.760 | 0.691 | |||||
| 3 | 0.820 | 0.792 | 0.645 | 0.709 | 0.707 | |||||
| 4 | 0.828 | 0.782 | 0.712 | 0.676 | 0.702 | |||||
| BG | 0.834 | 0.834 | 0.776 | 0.727 | 0.547 |
*Author’s own calculation.
Table 5 reports the FF3fm regression results for the twenty-five portfolios formed on size and long-term debt to total assets (LD/TA). The intercept values lie between −0.008 and 0.009 with an AAA value of about 0.003 and the alphas of all the twenty-five portfolios remain close to zero. Only two intercepts differ significantly from zero at the 5 % level (t = 2.06 for P6 and t = −2.64 for P17), and these significant intercepts are scattered rather than being concentrated in any particular LD/TA group. The market beta coefficients are close to one, the SMB loadings, similar to MC_PB sorts, fall monotonically with size. The value factor loadings show no smooth pattern across the LD/TA quintiles, but every size row the highest LD/TA quintile which means the high capital structure portfolios carries a positive and statistically significant factor loadings (t = values between 2.60 and 7.03) and the R2 values of size-LDTA sorts range from 0.631 to 0.865. Hence, it can be concluded that the FF3fm provides an adequate explanation of average returns on size-LD/TA sorts. And long-term debt-based sorting does not generate return patterns beyond what the market beta, size and value factors generally capture.
Table 5. FF3fm results for size-LD/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.005 | 0.008 | 0.004 | 0.003 | 0.002 | 1.034 | 1.000 | 0.907 | 0.812 | 0.766 |
| 2 | 0.001 | −0.004 | −0.003 | −0.008 | −0.002 | 0.866 | 0.913 | 0.933 | 0.977 | 0.934 |
| 3 | −0.002 | 0.000 | −0.001 | −0.002 | −0.006 | 0.836 | 0.916 | 0.934 | 0.935 | 0.935 |
| 4 | −0.002 | 0.009 | 0.005 | −0.003 | 0.001 | 0.815 | 0.818 | 0.930 | 0.973 | 1.029 |
| BG | 0.002 | 0.001 | 0.000 | 0.000 | −0.001 | 0.738 | 0.899 | 0.922 | 1.125 | 0.949 |
| S | L | |||||||||
| SM | 1.493 | 1.426 | 1.665 | 1.656 | 1.488 | 0.267 | 0.320 | 0.087 | 0.277 | 0.464 |
| 2 | 0.804 | 0.971 | 1.186 | 0.921 | 0.945 | 0.070 | 0.116 | 0.010 | 0.012 | 0.249 |
| 3 | 0.643 | 0.578 | 0.647 | 0.604 | 0.506 | −0.076 | −0.375 | −0.063 | 0.101 | 0.560 |
| 4 | 0.477 | 0.309 | 0.304 | 0.394 | 0.060 | −0.191 | −0.018 | −0.095 | 0.132 | 0.465 |
| BG | −0.024 | 0.039 | −0.003 | −0.211 | −0.214 | −0.077 | −0.253 | 0.069 | 0.357 | 0.557 |
| t(α) | t(β) | |||||||||
| SM | 1.469 | 2.058 | 1.334 | 0.875 | 0.682 | 13.898 | 12.526 | 12.843 | 10.562 | 6.499 |
| 2 | 0.380 | −1.191 | −0.899 | −2.636 | −0.633 | 12.689 | 14.015 | 12.116 | 15.143 | 12.578 |
| 3 | −0.731 | −0.145 | −0.208 | −0.480 | −1.774 | 12.997 | 15.566 | 13.484 | 13.565 | 12.997 |
| 4 | −0.682 | 1.682 | 1.631 | −1.242 | 0.378 | 15.853 | 14.369 | 14.878 | 16.304 | 13.578 |
| BG | 0.982 | −0.633 | 0.214 | 0.126 | −0.384 | 15.546 | 20.855 | 21.164 | 22.484 | 15.439 |
| t(S) | t(L) | |||||||||
| SM | 13.001 | 11.574 | 15.270 | 13.959 | 8.180 | 2.787 | 3.111 | 0.959 | 2.796 | 3.055 |
| 2 | 7.639 | 9.659 | 9.981 | 9.247 | 8.245 | 0.800 | 1.376 | 0.100 | 0.142 | 2.603 |
| 3 | 6.477 | 6.365 | 6.055 | 5.681 | 4.563 | −0.913 | −4.946 | −0.708 | 1.134 | 6.043 |
| 4 | 6.013 | 3.523 | 3.155 | 4.278 | 0.514 | −2.874 | −0.246 | −1.173 | 1.712 | 4.756 |
| BG | −0.333 | 0.584 | −0.045 | −2.737 | −2.259 | −1.253 | −4.559 | 1.226 | 5.535 | 7.027 |
| R2 | ||||||||||
| SM | 0.828 | 0.802 | 0.825 | 0.805 | 0.865 | |||||
| 2 | 0.715 | 0.777 | 0.733 | 0.774 | 0.751 | |||||
| 3 | 0.671 | 0.677 | 0.679 | 0.706 | 0.761 | |||||
| 4 | 0.703 | 0.632 | 0.659 | 0.752 | 0.698 | |||||
| BG | 0.631 | 0.739 | 0.79 | 0.831 | 0.749 |
*Author’s own calculation.
In contrast to the size-LD/TA sorts, the result considerably changes when the portfolios are formed on size and total liabilities to total asset (TL/TA). Table 6 reports the FF3fm regression results for the twenty-five size-TL/TA sorted portfolios. The AAA values rise to 0.004 and four out of twenty-five intercepts differ significantly from zero at the 5% level, as against only two in each of those size-value and size-LD/TA sorts, with one more intercept at the margin of significance (t = 1.97 for P8). Notably both the significant positive alpha belongs to the smallest size quintile – P6 with 0.010 (t = 2.88) and P16 with 0.011 (t = 2.99). While P22 high TL/TA portfolios records a significant negative alpha of −0.010 (t = −2.65). The unexplained returns are thus concentrated among the small firms, with positive alphas at lower to middles TL/TA levels and one negative alpha at the high TL/TA end. The market beta coefficients are statistically significant. The SMB factor loadings decline monotonically with size, the R2 values range from 0.664 to 0.833. Overall, from the results it can be concluded that the FF3fm leaves a portion of average return pattern on the size-TL/TA unexplained particularly among the small firms, indicating that broad liability-based capital structure measure carries information beyond the size and value. Since FF3fm leaves some intercept of this time series regression unexplained, particularly for size-TL/TA sorts, the study augments model with the capital structure factor and re-estimate all the three set of twenty-five sorts. A model is judged to have improved if the number of significant intercepts falls and AAA values declines or stands stable, and the R2 value the explanatory power of a model increase (Fama and French (39), Maiti and Balakrishnan (29).
Table 6. FF3fm results for size - TL/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.004 | 0.010 | 0.004 | 0.011 | 0.002 | 0.988 | 0.876 | 0.942 | 0.77 | 0.919 |
| 2 | −0.001 | 0.002 | 0.001 | −0.005 | −0.010 | 0.836 | 0.884 | 0.949 | 0.859 | 1.077 |
| 3 | −0.002 | 0.005 | 0.001 | 0.004 | −0.005 | 0.798 | 0.824 | 0.963 | 0.933 | 1.047 |
| 4 | −0.004 | 0.004 | 0.002 | 0.001 | −0.002 | 0.865 | 0.875 | 0.894 | 1.008 | 0.909 |
| BG | 0.001 | −0.002 | 0.001 | 0.004 | 0.000 | 0.798 | 0.856 | 0.932 | 1.063 | 0.965 |
| S | L | |||||||||
| SM | 1.622 | 1.351 | 1.652 | 1.690 | 1.417 | 0.051 | 0.185 | 0.139 | 0.518 | 0.422 |
| 2 | 1.111 | 0.858 | 0.945 | 0.914 | 1.033 | −0.193 | −0.075 | 0.082 | 0.122 | 0.277 |
| 3 | 0.669 | 0.490 | 0.634 | 0.624 | 0.658 | −0.259 | −0.190 | −0.066 | 0.071 | 0.350 |
| 4 | 0.276 | 0.119 | 0.392 | 0.244 | 0.481 | −0.016 | 0.001 | −0.151 | 0.109 | 0.263 |
| BG | −0.089 | 0.054 | −0.072 | −0.221 | −0.112 | 0.022 | −0.109 | 0.111 | 0.307 | 0.285 |
| t(α) | t(β) | |||||||||
| SM | 1.248 | 2.884 | 1.212 | 2.993 | 0.452 | 13.628 | 11.999 | 13.336 | 9.330 | 10.422 |
| 2 | −0.388 | 0.747 | 0.224 | −1.847 | −2.659 | 13.359 | 15.140 | 14.287 | 14.117 | 15.091 |
| 3 | −0.646 | 1.968 | 0.228 | −1.939 | −1.644 | 14.772 | 15.497 | 15.644 | 14.841 | 14.932 |
| 4 | −1.704 | 1.689 | 0.747 | 0.307 | −0.650 | 17.906 | 15.268 | 17.377 | 18.252 | 13.455 |
| BG | 0.365 | −0.996 | 0.369 | 1.665 | 0.099 | 17.770 | 19.747 | 22.537 | 22.204 | 15.703 |
| t(S) | t(L) | |||||||||
| SM | 14.498 | 11.985 | 15.156 | 13.278 | 10.416 | 0.543 | 1.965 | 1.531 | 4.872 | 3.710 |
| 2 | 11.502 | 9.515 | 9.222 | 9.736 | 9.378 | −2.391 | −0.997 | 0.957 | 1.556 | 3.014 |
| 3 | 8.024 | 5.965 | 6.674 | 6.434 | 6.079 | −3.718 | −2.776 | −0.835 | 0.882 | 3.872 |
| 4 | 3.697 | 1.343 | 4.932 | 2.865 | 4.609 | −0.265 | 0.015 | −2.282 | 1.527 | 3.024 |
| BG | −1.276 | 0.812 | −1.125 | −2.989 | −1.180 | 0.379 | −1.945 | 2.073 | 4.969 | 3.602 |
| R2 | ||||||||||
| SM | 0.821 | 0.787 | 0.833 | 0.807 | 0.766 | |||||
| 2 | 0.747 | 0.764 | 0.770 | 0.781 | 0.807 | |||||
| 3 | 0.700 | 0.696 | 0.735 | 0.740 | 0.778 | |||||
| 4 | 0.750 | 0.664 | 0.730 | 0.769 | 0.720 | |||||
| BG | 0.707 | 0.741 | 0.811 | 0.822 | 0.708 |
*Author’s own calculation.
Regression results of modified Fama French three factor model
Next, we run MFF3fm by incorporating the capital structure factors to examine if this model can explain the portfolio returns which remained unexplained by FF3fm. The regression results of twenty-five portfolios based on size-value, size-LD/TA, and size-TL/TA are presented in Tables 7–9 respectively. The results provided in Table 7 clearly indicates the superiority of MFF3fm in explaining the variation in cross section returns of 25 portfolios sorted using size-value. P1 which remained unexplained in FF3fm, is getting explained by MFF3fm indicating the importance of capital structure in driving the portfolio returns of small size and low value stock (see Table 7). Further, the improvement in average R2 by 1% from 75.6% to 76.6% supports the above finding. However, the AAA values remain unchanged at 0.003. Hence, it can be inferred that replacing the value factor with capital structure factors absorbs the additional common variations in returns for size-value sorts even when portfolios are not formed on capital structure.
Table 7. MFF3fm results for size-value sorted portfolios.
| Portfolios | LW | 2 | 3 | 4 | HH | LW | 2 | 3 | 4 | HH |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.008 | 0.004 | 0.004 | 0.002 | 0.006 | 0.899 | 0.994 | 1.023 | 0.848 | 0.947 |
| 2 | −0.003 | −0.009 | 0.001 | −0.001 | 0.001 | 1.042 | 0.855 | 0.997 | 0.807 | 0.814 |
| 3 | −0.005 | −0.004 | 0.005 | 0.001 | 0.001 | 0.975 | 0.914 | 0.783 | 0.926 | 0.819 |
| 4 | 0.000 | 0.001 | 0.000 | 0.002 | 0.000 | 1.028 | 0.929 | 0.894 | 0.858 | 0.792 |
| BG | −0.004 | 0.001 | 0.000 | 0.000 | 0.006 | 0.948 | 0.956 | 0.936 | 0.981 | 0.833 |
| S | D1 | |||||||||
| SM | 1.752 | 1.591 | 1.740 | 1.784 | 1.286 | 0.518 | −0.369 | −0.034 | −0.040 | 0.320 |
| 2 | 1.010 | 1.213 | 0.916 | 0.932 | 0.807 | 0.375 | −0.232 | −0.033 | 0.014 | 0.089 |
| 3 | 0.693 | 0.797 | 0.392 | 0.504 | 0.492 | 0.333 | 0.370 | 0.348 | 0.331 | 0.290 |
| 4 | 0.311 | 0.315 | 0.310 | 0.278 | 0.280 | 0.433 | −0.202 | −0.133 | 0.107 | 0.263 |
| BG | 0.136 | −0.069 | 0.031 | −0.110 | −0.276 | 0.436 | −0.112 | −0.046 | −0.054 | 0.236 |
| D2 | t(α) | |||||||||
| SM | 0.684 | −0.262 | 0.170 | 0.053 | −0.026 | 1.792 | 1.129 | 1.137 | 0.532 | 1.918 |
| 2 | 0.410 | −0.089 | 0.151 | 0.232 | 0.017 | −0.856 | −3.069 | 0.378 | −0.549 | 0.218 |
| 3 | 0.301 | −0.120 | −0.168 | 0.046 | 0.153 | −1.687 | −1.630 | 1.580 | 0.327 | 0.199 |
| 4 | 0.286 | −0.154 | 0.199 | 0.107 | −0.028 | 0.122 | 0.541 | −0.018 | 0.696 | −0.052 |
| BG | 0.301 | 0.276 | −0.043 | 0.345 | −0.193 | −1.423 | 0.518 | −0.210 | 0.216 | 1.849 |
| t(β) | t(S) | |||||||||
| SM | 9.633 | 12.812 | 13.277 | 11.173 | 14.057 | 13.116 | 14.316 | 15.770 | 16.414 | 13.331 |
| 2 | 15.184 | 13.675 | 16.271 | 13.688 | 11.592 | 10.282 | 13.558 | 10.443 | 11.039 | 8.026 |
| 3 | 14.822 | 15.187 | 12.126 | 15.202 | 13.322 | 7.363 | 9.253 | 4.236 | 5.777 | 5.588 |
| 4 | 15.881 | 17.049 | 16.439 | 14.749 | 13.907 | 3.355 | 4.036 | 3.979 | 3.333 | 3.435 |
| BG | 14.838 | 20.780 | 20.074 | 20.381 | 11.926 | 1.484 | −1.054 | 0.458 | −1.594 | −2.756 |
| t(D1) | t(D2) | |||||||||
| SM | 2.583 | −2.210 | −0.207 | −0.244 | 3.830 | 3.557 | −1.641 | 1.069 | 0.340 | −0.188 |
| 2 | 2.538 | −1.725 | −0.253 | 0.113 | 0.592 | 2.900 | 5.964 | 3.200 | 1.912 | 2.121 |
| 3 | 2.353 | 2.315 | 4.343 | 2.526 | 2.189 | 2.224 | −0.969 | −1.266 | 0.363 | 1.212 |
| 4 | 3.111 | −1.722 | −1.133 | 0.856 | 2.147 | 2.144 | −1.373 | 1.773 | 0.893 | −0.235 |
| BG | 3.174 | −1.129 | −0.454 | −0.519 | 1.571 | 2.287 | 2.911 | 0.452 | 3.481 | −1.344 |
| R2 | ||||||||||
| SM | 0.806 | 0.827 | 0.809 | 0.800 | 0.795 | |||||
| 2 | 0.831 | 0.816 | 0.786 | 0.742 | 0.665 | |||||
| 3 | 0.791 | 0.790 | 0.649 | 0.791 | 0.760 | |||||
| 4 | 0.808 | 0.776 | 0.715 | 0.751 | 0.731 | |||||
| BG | 0.746 | 0.809 | 0.775 | 0.731 | 0.647 |
*Author’s own calculation.
Table 8. MFF3fm results for size-LD/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.003 | 0.006 | 0.005 | 0.004 | 0.001 | 1.077 | 1.072 | 0.882 | 0.781 | 0.611 |
| 2 | 0.001 | −0.004 | −0.002 | −0.006 | −0.001 | 0.885 | 0.945 | 0.900 | 0.926 | 0.881 |
| 3 | −0.003 | 0.000 | 0.000 | 0.000 | −0.005 | 0.850 | 0.876 | 0.904 | 0.896 | 0.882 |
| 4 | −0.002 | 0.002 | 0.005 | −0.003 | 0.003 | 0.821 | 0.820 | 0.918 | 0.968 | 0.975 |
| BG | 0.001 | 0.001 | 0.001 | 0.000 | −0.001 | 0.783 | 0.879 | 0.915 | 1.132 | 0.929 |
| S | D1 | |||||||||
| SM | 1.541 | 1.533 | 1.666 | 1.729 | 1.558 | 0.532 | 0.356 | −0.101 | −0.500 | 1.410 |
| 2 | 0.829 | 1.021 | 1.169 | 0.915 | 0.991 | 0.595 | 0.080 | −0.162 | −0.471 | 0.518 |
| 3 | 0.582 | 0.423 | 0.612 | 0.631 | 0.623 | 0.610 | 0.359 | 0.107 | 0.474 | 0.579 |
| 4 | 0.415 | 0.289 | 0.264 | 0.453 | 0.155 | 0.295 | 0.195 | 0.081 | 0.379 | 0.574 |
| BG | −0.014 | −0.054 | 0.013 | −0.108 | −0.077 | 0.222 | 0.207 | 0.086 | 0.227 | 0.436 |
| D2 | t(α) | |||||||||
| SM | −0.639 | −0.429 | −0.141 | −0.070 | 0.063 | 1.050 | 1.523 | 1.500 | 1.173 | 1.773 |
| 2 | −0.096 | −0.083 | −0.018 | 0.173 | 0.110 | 0.223 | −1.426 | −0.675 | −2.209 | −0.229 |
| 3 | −0.377 | 0.009 | 0.043 | 0.114 | 0.472 | −1.080 | 0.060 | 0.019 | −0.112 | −1.463 |
| 4 | 0.019 | −0.139 | 0.001 | 0.137 | 0.354 | −0.817 | 0.893 | 1.683 | −1.095 | 0.807 |
| BG | 0.105 | 0.070 | −0.046 | −0.248 | 0.447 | 0.521 | 0.586 | 0.303 | 0.090 | −0.201 |
| t(β) | t(S) | |||||||||
| SM | 14.768 | 13.169 | 12.420 | 10.293 | 5.840 | 14.758 | 13.156 | 16.389 | 15.926 | 10.400 |
| 2 | 12.811 | 14.284 | 11.606 | 14.706 | 12.235 | 8.389 | 10.777 | 10.531 | 10.154 | 9.612 |
| 3 | 13.818 | 14.225 | 12.893 | 13.271 | 12.924 | 6.610 | 4.802 | 6.104 | 6.522 | 6.374 |
| 4 | 15.924 | 14.347 | 14.480 | 16.337 | 13.428 | 5.621 | 3.529 | 2.909 | 5.342 | 1.493 |
| BG | 16.877 | 19.639 | 20.831 | 21.956 | 15.217 | −0.216 | −0.850 | 0.199 | −1.465 | −0.882 |
| t(D1) | t(D2) | |||||||||
| SM | 3.393 | 2.033 | −0.661 | −3.065 | 6.263 | −4.252 | −2.557 | −0.967 | −0.446 | 0.291 |
| 2 | 0.641 | 0.559 | −0.970 | −3.475 | 3.345 | −0.676 | −0.609 | −0.110 | 1.332 | 0.740 |
| 3 | 4.615 | 2.708 | 0.710 | 3.264 | 3.945 | −2.979 | 0.074 | 0.301 | 0.818 | 3.359 |
| 4 | 2.659 | 1.588 | 0.593 | 2.977 | 3.673 | 0.176 | −1.182 | 0.008 | 1.122 | 2.365 |
| BG | 2.230 | 2.149 | 0.909 | 2.046 | 3.316 | 1.105 | 0.757 | −0.507 | −2.332 | 3.550 |
| R2 | ||||||||||
| SM | 0.838 | 0.797 | 0.826 | 0.814 | 0.715 | |||||
| 2 | 0.713 | 0.774 | 0.734 | 0.788 | 0.770 | |||||
| 3 | 0.804 | 0.752 | 0.777 | 0.772 | 0.788 | |||||
| 4 | 0.747 | 0.669 | 0.795 | 0.759 | 0.727 | |||||
| BG | 0.724 | 0.723 | 0.790 | 0.824 | 0.756 |
*Author’s own calculation.
Table 9. MFF3fm results for size-TL/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.003 | 0.008 | 0.003 | 0.012 | 0.004 | 1.066 | 0.885 | 0.966 | 0.705 | 0.792 |
| 2 | −0.002 | 0.002 | 0.001 | −0.004 | 0.004 | 0.878 | 0.893 | 0.935 | 0.817 | 0.995 |
| 3 | −0.002 | 0.003 | 0.001 | −0.004 | −0.004 | 0.800 | 0.824 | 0.968 | 0.874 | 0.961 |
| 4 | −0.004 | 0.003 | 0.001 | 0.002 | −0.001 | 0.900 | 0.886 | 0.914 | 0.949 | 0.839 |
| BG | 0.000 | −0.002 | 0.000 | 0.004 | 0.001 | 0.843 | 0.874 | 0.946 | 1.055 | 0.902 |
| S | D1 | |||||||||
| SM | 1.754 | 1.425 | 1.706 | 1.750 | 1.408 | −0.405 | −0.277 | 0.003 | −0.225 | −0.384 |
| 2 | 1.120 | 0.866 | 0.963 | 0.932 | 1.003 | −0.055 | −0.151 | −0.166 | −0.378 | −0.006 |
| 3 | 0.599 | 0.450 | 0.626 | 0.589 | 0.645 | 0.184 | 0.003 | −0.004 | −0.131 | −0.469 |
| 4 | 0.326 | 0.152 | 0.358 | 0.217 | 0.474 | 0.274 | −0.220 | 0.237 | −0.129 | −0.237 |
| BG | −0.023 | 0.060 | −0.031 | −0.157 | −0.120 | 0.345 | −0.090 | −0.034 | −0.091 | 0.005 |
| D2 | t(α) | |||||||||
| SM | 0.631 | 0.033 | −0.085 | −0.783 | −0.787 | 0.986 | 1.881 | 1.041 | 3.496 | 1.204 |
| 2 | 0.489 | 0.271 | −0.027 | −0.016 | −0.730 | −0.616 | 0.769 | 0.355 | −1.483 | 1.432 |
| 3 | 0.193 | 0.252 | 0.113 | −0.240 | −0.791 | −0.678 | 1.000 | 0.211 | −1.558 | −1.315 |
| 4 | 0.330 | 0.240 | 0.079 | −0.293 | −0.542 | −1.956 | 1.716 | 0.527 | 0.801 | −0.246 |
| BG | 0.295 | 0.306 | −0.057 | −0.367 | −0.663 | 0.039 | −1.110 | 0.217 | 1.704 | 0.492 |
| t(β) | t(S) | |||||||||
| SM | 15.321 | 11.981 | 13.442 | 9.005 | 9.981 | 17.617 | 13.479 | 16.583 | 15.620 | 12.389 |
| 2 | 14.430 | 15.292 | 13.974 | 13.707 | 15.069 | 12.869 | 10.360 | 10.059 | 10.922 | 10.617 |
| 3 | 14.611 | 15.332 | 15.563 | 14.118 | 15.149 | 7.650 | 5.848 | 7.029 | 6.650 | 7.106 |
| 4 | 19.026 | 15.490 | 17.822 | 17.737 | 13.116 | 4.817 | 1.857 | 4.874 | 2.830 | 5.177 |
| BG | 19.110 | 20.505 | 22.422 | 21.972 | 15.855 | −0.362 | 0.984 | −0.513 | −2.289 | −1.471 |
| t(D1) | t(D2) | |||||||||
| SM | −2.705 | −1.747 | 0.016 | −1.337 | −2.247 | 4.408 | 0.219 | −0.576 | −4.856 | −4.816 |
| 2 | −0.420 | −1.200 | −1.151 | −2.946 | −0.041 | 3.907 | 2.256 | −0.197 | −0.129 | −5.373 |
| 3 | 1.564 | 0.025 | −0.031 | −0.980 | −3.504 | 1.715 | 2.277 | 0.886 | −1.884 | −6.054 |
| 4 | 2.712 | −1.785 | 2.148 | −1.125 | −1.993 | 3.387 | 2.040 | 0.747 | −2.659 | −4.110 |
| BG | 3.531 | −0.985 | −0.369 | −0.878 | 0.038 | 3.245 | 3.485 | −0.654 | −3.708 | −5.661 |
| R2 | ||||||||||
| SM | 0.838 | 0.786 | 0.830 | 0.829 | 0.814 | |||||
| 2 | 0.766 | 0.768 | 0.771 | 0.794 | 0.837 | |||||
| 3 | 0.697 | 0.694 | 0.735 | 0.752 | 0.822 | |||||
| 4 | 0.764 | 0.671 | 0.736 | 0.787 | 0.753 | |||||
| BG | 0.722 | 0.753 | 0.807 | 0.824 | 0.754 |
*Author’s own calculation.
Similarly in the case of 25 size-LD/TA portfolios, MFF3fm is found to be a better explainer of excess portfolio returns as it is explaining 24 out of 25 portfolios (see Table 8). Furthermore, this finding is corroborated by decline in AAA values from 0.003 to 0.002 and improvement in the R2 values 74.3% to 76.3%. The factor loadings of the LD/TA factor also significant for large number of portfolios confirming the exposure in explaining the portfolio returns. When the portfolios are constructed using size-TL/TA factors, MFF3fm becomes the strongest descriptor of variation in portfolio returns corresponding to FF3fm. The resulting average R2 is 77.2% which is highest among the three models (see Table 9). The number of significant alphas is coming down from four to one and there is also a reduction in AAA from 0.004 to 0.003 (see Table 9). Further, the TL/TA factor loadings decline systematically across the TL/TA quintiles with every size row and are statistically significant for a majority of the portfolios which confirms the capital structure factor works for this TL/TA sorting dimension. Hence, it can be concluded that capital structure plays a crucial role to obtain the explanation about the variation in portfolio return especially when it is applied in modified risk model by replacing the value factor. The MFF3fm outperforms FF3fm for all the combination such as size-value, size-LD/TA, and size-TL/TA confirming the importance of capital structure factor in risk models.
Regression results of extended Fama French three factor model
Next, we test the extended Fama–French three model (EFF3fm), which incorporates the traditional three factors along with capital structure factor to assess its coexistence with established risk factors and incremental relevance of capital structure factor if it replaces the value factor in MFF3fm. The results of EFF3fm are presented in Tables 10, 11, 12 for 25 size-value, size-LD/TA, and size-TL/TA portfolios, respectively. The resulting alpha values, which are indistinguishable from zero in the case of all the combinations might give an illusion that EFF3fm provides stronger explanation of excess portfolio returns. However, a closer look at the results reveal that the number of statistically significant alphas increases from one to two across all the combinations (see Table 13) while the average R2 declines in the case of all the combinations relative to MFF3fm. These findings clearly suggest that the MFF3fm that replaces the value factor with capital structure offers greater explanatory power in capturing the cross-section variation in excess portfolio returns.
Table 10. EFF3fm results for size-value sorted portfolios.
| Portfolios | LW | 2 | 3 | 4 | HH | LW | 2 | 3 | 4 | HH |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.009 | −0.001 | −0.004 | 0.002 | −0.002 | 0.809 | 0.967 | 0.916 | 0.955 | 0.858 |
| 2 | 0.005 | −0.009 | −0.004 | 0.002 | 0.002 | 0.931 | 0.856 | 0.899 | 0.906 | 0.915 |
| 3 | 0.004 | 0.001 | 0.004 | 0.000 | 0.000 | 1.001 | 1.003 | 0.803 | 0.895 | 0.934 |
| 4 | 0.002 | −0.002 | 0.000 | 0.001 | 0.000 | 0.862 | 0.845 | 0.963 | 0.894 | 0.999 |
| BG | 0.005 | 0.000 | −0.001 | −0.001 | 0.005 | 0.985 | 0.864 | 0.887 | 0.855 | 0.897 |
| S | L | |||||||||
| SM | 1.453 | 0.764 | 0.499 | 0.069 | −0.159 | 0.792 | 0.653 | 0.516 | 0.641 | 0.781 |
| 2 | 1.381 | 1.217 | 0.749 | 0.241 | −0.203 | 0.556 | −0.009 | 0.127 | 0.196 | 0.354 |
| 3 | 1.666 | 0.935 | 0.456 | 0.312 | 0.024 | 0.198 | −0.050 | −0.169 | −0.006 | 0.016 |
| 4 | 1.829 | 1.057 | 0.623 | 0.397 | −0.051 | −0.117 | −0.332 | −0.316 | −0.315 | −0.156 |
| BG | 1.411 | 0.973 | 0.714 | 0.489 | −0.065 | −0.331 | −0.440 | −0.588 | −0.554 | −0.559 |
| D1 | D2 | |||||||||
| SM | −0.107 | −0.036 | −0.065 | −0.100 | −0.031 | −0.384 | −0.163 | −0.106 | −0.043 | −0.006 |
| 2 | −0.080 | −0.236 | −0.104 | −0.100 | 0.072 | −0.052 | −0.093 | −0.072 | −0.080 | −0.142 |
| 3 | 0.068 | −0.060 | −0.135 | −0.136 | −0.037 | 0.244 | 0.132 | −0.232 | 0.197 | −0.037 |
| 4 | −0.101 | −0.158 | 0.167 | −0.057 | −0.135 | 0.009 | 0.107 | −0.074 | −0.012 | 0.286 |
| BG | −0.052 | −0.139 | −0.015 | −0.025 | −0.054 | −0.151 | −0.149 | −0.069 | −0.237 | −0.405 |
| t(ɑ) | t(β) | |||||||||
| SM | 2.376 | −0.467 | −1.423 | 0.653 | −1.056 | 9.304 | 15.541 | 14.743 | 16.442 | 16.275 |
| 2 | 1.544 | −3.058 | −1.532 | 0.713 | 0.934 | 12.526 | 13.430 | 14.737 | 16.568 | 21.004 |
| 3 | 1.257 | 0.338 | 1.463 | −0.023 | −0.193 | 12.833 | 16.066 | 12.286 | 16.137 | 19.655 |
| 4 | 0.459 | −0.835 | 0.091 | 0.460 | 0.067 | 11.164 | 14.599 | 15.994 | 15.612 | 20.608 |
| BG | 1.731 | −0.073 | −0.269 | −0.561 | 1.595 | 14.757 | 12.660 | 15.905 | 16.655 | 13.638 |
| t(S) | t(L) | |||||||||
| SM | 10.946 | 8.029 | 5.252 | 0.775 | −1.974 | 5.809 | 6.689 | 5.296 | 7.040 | 9.436 |
| 2 | 12.162 | 12.495 | 8.034 | 2.882 | −3.050 | 4.766 | −0.086 | 1.330 | 2.280 | 5.180 |
| 3 | 13.980 | 9.806 | 4.565 | 3.682 | 0.337 | 1.615 | −0.515 | −1.652 | −0.065 | 0.217 |
| 4 | 15.505 | 11.953 | 6.778 | 4.534 | −0.686 | −0.970 | −3.654 | −3.344 | −3.511 | −2.057 |
| BG | 13.836 | 9.328 | 8.382 | 6.234 | −0.644 | −3.166 | −4.110 | −6.720 | −6.876 | −5.416 |
| t(D1) | t(D2) | |||||||||
| SM | −2.544 | 2.052 | −1.990 | 3.077 | −0.256 | −2.094 | −2.239 | −0.810 | −0.354 | −2.126 |
| 2 | −3.487 | −1.636 | −0.752 | −0.808 | 0.729 | −0.334 | −0.688 | −0.558 | −0.693 | −1.542 |
| 3 | 0.385 | −0.421 | −0.915 | −1.078 | −0.345 | 1.485 | 1.004 | −1.685 | 1.679 | −0.371 |
| 4 | −0.576 | −1.203 | 1.226 | −0.437 | −1.228 | 0.054 | 0.873 | −0.581 | −0.101 | 2.795 |
| BG | −0.342 | −0.898 | −0.122 | −0.211 | −0.363 | −1.075 | −1.034 | −0.583 | −2.186 | −2.914 |
| R2 | ||||||||||
| SM | 0.837 | 0.865 | 0.820 | 0.827 | 0.832 | |||||
| 2 | 0.847 | 0.815 | 0.791 | 0.782 | 0.834 | |||||
| 3 | 0.810 | 0.785 | 0.653 | 0.714 | 0.774 | |||||
| 4 | 0.799 | 0.760 | 0.709 | 0.673 | 0.736 | |||||
| BG | 0.805 | 0.693 | 0.704 | 0.710 | 0.571 |
*Author’s own calculation.
Table 11. EFF3fm results for size-LD/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.004 | 0.001 | −0.003 | −0.002 | 0.001 | 1.036 | 0.872 | 0.846 | 0.832 | 0.773 |
| 2 | 0.007 | −0.004 | −0.001 | 0.002 | 0.001 | 1.019 | 0.925 | 0.916 | 0.819 | 0.904 |
| 3 | 0.005 | −0.003 | 0.000 | 0.005 | 0.001 | 0.883 | 0.911 | 0.919 | 0.929 | 0.912 |
| 4 | 0.004 | −0.007 | 0.000 | −0.003 | 0.001 | 0.772 | 0.946 | 0.905 | 0.962 | 1.102 |
| BG | 0.013 | −0.001 | −0.004 | 0.003 | 0.000 | 0.634 | 0.882 | 0.857 | 0.958 | 0.893 |
| S | L | |||||||||
| SM | 1.405 | 0.788 | 0.569 | 0.453 | −0.046 | 0.360 | 0.111 | 0.035 | −0.100 | 0.085 |
| 2 | 1.359 | 0.953 | 0.557 | 0.285 | 0.029 | 0.460 | 0.179 | −0.353 | 0.009 | −0.222 |
| 3 | 1.671 | 1.205 | 0.665 | 0.300 | 0.003 | −0.014 | −0.097 | −0.139 | −0.096 | 0.025 |
| 4 | 1.701 | 0.980 | 0.658 | 0.434 | −0.207 | 0.075 | −0.172 | −0.071 | 0.050 | 0.262 |
| BG | 1.634 | 0.993 | 0.540 | 0.100 | −0.198 | −0.200 | −0.004 | 0.220 | 0.147 | 0.320 |
| D1 | D2 | |||||||||
| SM | 0.719 | 0.153 | 0.629 | 0.243 | 0.266 | −0.503 | −0.054 | −0.364 | −0.019 | 0.137 |
| 2 | 0.595 | 0.173 | 0.175 | 0.200 | 0.092 | −0.255 | −0.015 | −0.124 | −0.136 | −0.014 |
| 3 | −0.108 | −0.212 | −0.179 | 0.031 | −0.073 | −0.147 | −0.054 | −0.009 | −0.035 | −0.036 |
| 4 | −0.461 | −0.560 | −0.511 | −0.353 | −0.091 | −0.041 | 0.108 | 0.087 | 0.156 | −0.148 |
| BG | −1.514 | −0.520 | −0.465 | −0.497 | −0.269 | −0.138 | −0.111 | −0.389 | −0.298 | −0.325 |
| t(ɑ) | t(β) | |||||||||
| SM | 1.308 | 0.296 | −1.049 | −0.905 | 0.604 | 14.340 | 12.423 | 13.496 | 15.908 | 16.413 |
| 2 | 1.843 | −1.305 | −0.211 | 0.896 | 0.367 | 12.762 | 13.825 | 15.182 | 14.055 | 20.404 |
| 3 | 1.483 | −0.730 | −0.073 | 1.609 | 0.327 | 12.206 | 11.545 | 12.928 | 14.411 | 20.377 |
| 4 | 1.214 | −2.341 | −0.160 | −1.053 | 0.332 | 9.997 | 14.858 | 13.153 | 15.943 | 21.621 |
| BG | 2.681 | −0.231 | −1.323 | 0.899 | 0.041 | 5.968 | 12.008 | 12.465 | 13.006 | 14.816 |
| t(S) | t(L) | |||||||||
| SM | 12.730 | 7.344 | 5.940 | 5.663 | −0.643 | 3.178 | 1.004 | 0.356 | −1.222 | 1.145 |
| 2 | 11.139 | 9.329 | 6.036 | 3.203 | 0.432 | 3.674 | 1.707 | −3.730 | 0.104 | −3.190 |
| 3 | 15.111 | 10.000 | 6.119 | 3.048 | 0.044 | −0.121 | −0.781 | −1.246 | −0.948 | 0.359 |
| 4 | 14.414 | 10.079 | 6.259 | 4.708 | −2.660 | 0.621 | −1.718 | −0.662 | 0.533 | 3.280 |
| BG | 10.065 | 8.848 | 5.138 | 0.886 | −2.152 | −1.201 | −0.037 | 2.044 | 1.274 | 3.390 |
| t(D1) | t(D2) | |||||||||
| SM | 4.391 | 0.959 | 4.424 | 2.046 | 2.494 | −3.296 | −0.367 | −2.752 | −0.174 | 1.383 |
| 2 | 3.285 | 1.138 | 1.282 | 1.515 | 0.913 | −1.510 | −0.108 | −0.974 | −1.102 | −0.151 |
| 3 | −0.658 | −1.185 | −1.111 | 0.213 | −0.718 | −0.959 | −0.325 | −0.060 | −0.259 | −0.384 |
| 4 | −2.632 | −3.879 | −3.279 | −2.581 | −0.785 | −0.253 | 0.803 | 0.597 | 1.224 | −1.379 |
| BG | −6.285 | −3.126 | −2.982 | −2.976 | −1.971 | 6.713 | 4.718 | −2.680 | −2.916 | −2.558 |
| R2 | ||||||||||
| SM | 0.846 | 0.713 | 0.703 | 0.708 | 0.654 | |||||
| 2 | 0.811 | 0.776 | 0.677 | 0.668 | 0.737 | |||||
| 3 | 0.825 | 0.734 | 0.678 | 0.655 | 0.789 | |||||
| 4 | 0.813 | 0.791 | 0.722 | 0.758 | 0.833 | |||||
| BG | 0.716 | 0.768 | 0.792 | 0.728 | 0.770 |
*Author’s own calculation.
Table 12. EFF3fm results for size-TL/TA sorted portfolios.
| Portfolios | LC | 2 | 3 | 4 | HC | LC | 2 | 3 | 4 | HC |
|---|---|---|---|---|---|---|---|---|---|---|
| α | β | |||||||||
| SM | 0.004 | −0.002 | −0.002 | −0.004 | 0.000 | 1.042 | 0.882 | 0.819 | 0.892 | 0.829 |
| 2 | 0.010 | 0.002 | 0.005 | 0.004 | −0.002 | 0.869 | 0.898 | 0.840 | 0.885 | 0.878 |
| 3 | 0.004 | 0.001 | 0.001 | 0.001 | 0.001 | 0.947 | 0.934 | 0.972 | 0.918 | 0.933 |
| 4 | 0.013 | −0.004 | −0.005 | 0.002 | 0.004 | 0.683 | 0.824 | 0.888 | 0.959 | 1.032 |
| BG | 0.004 | −0.007 | −0.004 | −0.001 | 0.001 | 0.800 | 0.997 | 0.959 | 0.840 | 0.898 |
| S | L | |||||||||
| SM | 1.675 | 1.134 | 0.663 | 0.301 | −0.069 | 0.210 | −0.036 | −0.169 | 0.066 | 0.123 |
| 2 | 1.374 | 0.883 | 0.503 | 0.148 | 0.075 | 0.134 | −0.046 | −0.141 | 0.011 | −0.039 |
| 3 | 1.643 | 0.960 | 0.640 | 0.372 | −0.075 | 0.167 | 0.007 | −0.036 | −0.039 | 0.116 |
| 4 | 1.679 | 0.954 | 0.634 | 0.251 | −0.231 | 0.187 | −0.057 | −0.117 | −0.091 | 0.197 |
| BG | 1.432 | 1.010 | 0.638 | 0.477 | −0.136 | −0.065 | −0.018 | 0.019 | −0.006 | 0.042 |
| D1 | D2 | |||||||||
| SM | −0.295 | −0.074 | 0.096 | −0.140 | −0.081 | 0.711 | 0.476 | 0.129 | 0.355 | 0.341 |
| 2 | −0.208 | −0.175 | −0.070 | −0.214 | −0.111 | 0.084 | 0.254 | 0.199 | 0.244 | 0.291 |
| 3 | 0.089 | −0.162 | −0.023 | 0.217 | 0.027 | −0.022 | −0.025 | 0.100 | 0.064 | −0.013 |
| 4 | −0.128 | −0.407 | −0.191 | −0.177 | 0.011 | −0.712 | −0.037 | −0.284 | −0.327 | −0.292 |
| BG | −0.417 | −0.015 | −0.059 | −0.140 | 0.026 | −0.812 | −0.737 | −0.784 | −0.544 | −0.648 |
| t(ɑ) | t(β) | |||||||||
| SM | 1.130 | −0.640 | −0.830 | −1.890 | 0.170 | 14.847 | 14.228 | 14.843 | 18.548 | 18.600 |
| 2 | 2.960 | 0.730 | 1.880 | 1.710 | −1.150 | 11.593 | 15.099 | 15.455 | 15.174 | 20.237 |
| 3 | 1.150 | 0.360 | 0.180 | 0.490 | 0.340 | 13.006 | 13.696 | 15.338 | 17.579 | 21.877 |
| 4 | 3.610 | −1.520 | −1.640 | 0.720 | 1.920 | 8.622 | 13.566 | 14.120 | 17.648 | 21.512 |
| BG | 1.160 | −2.430 | −1.290 | −0.250 | 0.520 | 9.889 | 14.812 | 14.827 | 12.876 | 15.480 |
| t(S) | t(L) | |||||||||
| SM | 15.624 | 11.977 | 7.868 | 4.097 | −1.017 | 1.912 | −0.372 | −1.955 | 0.880 | 1.759 |
| 2 | 11.994 | 9.720 | 6.057 | 1.662 | 1.128 | 1.140 | −0.491 | −1.649 | 0.116 | −0.574 |
| 3 | 14.772 | 9.220 | 6.606 | 4.667 | −1.150 | 1.459 | 0.066 | −0.367 | −0.476 | 1.739 |
| 4 | 13.869 | 10.279 | 6.599 | 3.022 | −3.158 | 1.508 | −0.595 | −1.189 | −1.065 | 2.611 |
| BG | 11.593 | 9.825 | 6.459 | 4.782 | −1.532 | −0.509 | −0.168 | 0.188 | −0.061 | 0.463 |
| t(D1) | t(D2) | |||||||||
| SM | −1.998 | −0.525 | 0.770 | −1.281 | −0.806 | 4.801 | 3.636 | 2.110 | 3.496 | 3.627 |
| 2 | −2.237 | −1.294 | −0.569 | −1.620 | −1.124 | 1.931 | 2.023 | 1.733 | 1.985 | 3.177 |
| 3 | 0.539 | −1.047 | −0.160 | 1.829 | 0.278 | −0.145 | −0.170 | 0.745 | 0.581 | −0.143 |
| 4 | −2.711 | −2.958 | −1.343 | −1.433 | 0.105 | −4.254 | −0.290 | −2.144 | −2.852 | −2.886 |
| BG | −2.276 | −0.098 | −0.402 | −0.946 | 3.202 | −4.755 | −5.189 | −5.741 | −3.950 | −5.290 |
| R2 | ||||||||||
| SM | 0.840 | 0.764 | 0.702 | 0.764 | 0.726 | |||||
| 2 | 0.786 | 0.767 | 0.697 | 0.669 | 0.752 | |||||
| 3 | 0.832 | 0.769 | 0.733 | 0.735 | 0.809 | |||||
| 4 | 0.830 | 0.793 | 0.753 | 0.787 | 0.830 | |||||
| BG | 0.813 | 0.836 | 0.821 | 0.752 | 0.752 |
*Author’s own calculation.
Table 13. Number of significant alphas across FF3fm, MFF3fm, and EFF3fm.
| Model | Sort | Significant α (5%) |
|---|---|---|
| FF3fm | Size-value | 3 |
| Size-LD/TA | 2 | |
| Size-TL/TA | 4 | |
| MFF3fm | Size-value | 1 |
| Size-LD/TA | 1 | |
| Size-TL/TA | 1 | |
| EFF3fm | Size-value | 2 |
| Size-LD/TA | 2 | |
| Size-TL/TA | 3 |
Results of GRS test
The study employs the Gibbons, Ross and Shanken (1989) (40) test to examine whether the twenty-five intercepts are jointly zero for both the models in each sort. Table 14 summarizes the results along with the AAA value and the average R2 values of all three twenty-five sorts for both the model. A lower GRS F-statistics along with the higher average R2 marks the model more efficient Fama–French (39), Dusmanta and Barik (38). The GRS results support the findings of the study. The GRS statistics for MFF3fm are lower in the case of size-value, and size-TL/TA portfolios relative to traditional FF3fm. Similarly, MFF3fm exhibits lower GRS statistics than the EFF3fm for size-LD/TA and size-TL/TA based portfolios. Further, these results are corroborated by higher R2 values. These results provide evidence about the relevance of capital structure factor and suggest that its inclusion enhance the model’s ability to explain the cross-section variation in portfolio returns.
Table 14. GRS test of FF3fm, MFF3fm and EFF3fm for size-value, size-LD/TA and size-TL/TA sorted portfolios.
| FF3fm | F-stat | p-value | AAα | R2 |
|---|---|---|---|---|
| Size - value | 1.921 | 0.009 | 0.003 | 0.756 |
| Size-LD/TA | 1.391 | 0.117 | 0.003 | 0.743 |
| Size-TL/TA | 2.850 | 0.000 | 0.004 | 0.758 |
| MFF3fm | ||||
| Size - value | 1.809 | 0.020 | 0.003 | 0.766 |
| Size-LD/TA | 1.420 | 0.104 | 0.002 | 0.767 |
| Size-TL/TA | 2.687 | 0.000 | 0.003 | 0.772 |
| EFF3fm | ||||
| Size - value | 1.758 | 0.022 | 0.003 | 0.778 |
| Size-LD/TA | 1.530 | 0.064 | 0.003 | 0.754 |
| Size-TL/TA | 2.787 | 0.000 | 0.004 | 0.779 |
*Author’s own calculation.
Conclusion
In emerging economies such as India, the variation in stock returns is shaped by a combination of firm specific factors which extends beyond the conventional market risk factors such as size, value and capital structure. So, examining these factors becomes important for investors to earn higher returns from their investments. The study tries to shed light on whether portfolios constructed using these characteristics generate abnormal returns. Further, the study also examines the importance of capital structure by constructing portfolios considering the capital structure as a factor. Furthermore, the study compares the ability of FF3fm corresponding to MFF3fm which incorporates two capital structure measure such as LD/TA and TL/TA, in explaining the cross-section variation in excess portfolio returns. The study also used well established GRS-Test to check model performance and robustness of the findings. Further, the study estimates a nested Extended Fama–French three factor model (EFF3fm), where the value factor is kept together with the capital structure factors, to test whether the improvement of the MFF3fm is due to genuinely new information from capital structure or is a proxy for value.
The results of the study indicate a strong size, value, and capital structure effect in the Indian equity market. The MFF3fm that comprises capital structure factors in place of value factor outperform both the traditional FF3fm and EFF3fm for all three portfolio combinations on size-value, size-LD/TA, and size-TL/TA as the resulting AAA values are lower and indistinguishable from zero, and the R2 values are higher in the case of MFF3fm model. The GRS results also support the above findings as the resulting GRS statistics are lower for MFF3fm in most of the combination relative to FF3fm and EFF3fm. Undoubtedly, it can be concluded that the impact of capital structure factor is conspicuous in Indian stock market even after the inclusion of value factor (29, 30).
Limitations of the study
The study suffers from certain limitations which can be incorporated in future studies. Firstly, the results of the study are not applicable to the financial firm as the financial firms are excluded from the sample of the study considering the distinct characteristics with regards to the dynamics of capital structure. Secondly, the study has not dealt with survivorship bias during the study period which the future study can incorporate by considering various factors such as delisting, merger of firms, and exclusion of firms from constituents. Lastly, the study has not conducted the sub-period analysis considering the methodological limitation of back testing which the future study can include in their study.
Practical implications
- The study findings suggest that investors can benefit by paying close attention to firm-specific characteristics such as firm size, value, and capital structure while making investment decisions. Investors can consider small-cap and value-oriented firms, and the significant negative capital structure premium indicates that the firms with high total liabilities to total assets earned higher returns by compensating investor for bearing the additional financial risk. Moreover, since the pricing results between the two measures of capital structure LD/TA and TL/TA sorted portfolios differs, investor should be mindful about the choice of capital structure measure and the choice matters while assessing the firms risk level and also return potential.
- The superior performance of MFF3fm indicates that fund managers can use capital structure factors as well as conventional risk models to better assess portfolio performance in the Indian market. The EFF3fm results also show that these factors have additional explanatory power above and beyond value, so that risk models relying exclusively on book-to-market could underestimate return variation related to capital structure, especially for portfolios constructed on total liabilities.
- Since firm characteristics such as size and capital structure holds an impact on stock market returns, the regulating bodies such as SEBI can ensure a better disclosure of financial information. Transparent and accurate reporting of firm’s complete liability structure, including operating liabilities, will protect the investors.
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Footnotes
Edited by: Young Shin Aaron Kim, Stony Brook University, United States
Reviewed by: Mochamad Rizal Yulianto, Universitas Muhammadiyah Sidoarjo, Indonesia
Karkaria Dusmanta, Centurion University, India
Statements
- Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
- Funding
The author(s) declared that financial support was not received for this work and/or its publication.
- Data availability
The data analyzed in this study is subject to the following licenses/restrictions: third party licensing restriction. Requests to access these datasets should be directed to Bloomberg.
- Author contributions
JS: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing – original draft. NB: Conceptualization, Formal analysis, Supervision, Validation, Writing – review & editing.
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Citation
Somu, J. & Barik, N. (2026). Capital structure risk and portfolio returns: evidence from Indian stock market. Frontiers in Applied Mathematics and Statistics, Volume 12. https://doi.org/
- Received
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- 12 - 2026
Copyright
Copyright © 2026 Somu and Barik.
License: https://creativecommons.org/licenses/by/4.0/
*Correspondence: Nirakar Barik
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