Does interest rate exposure explain the low volatility anomaly?
Interest Rate Exposure’s Explanation for Idiosyncratic Volatility with Varying Portfolios Size
This paper investigates the interest rate exposure’s contributions to the portfolios’ idiosyncratic volatility in different portfolio sizes. We show that a large part of unexplained excess return can be explained by the interest rate exposure. However, portfolios in different sizes react differently towards the premium for interest rate exposure.
1. Introduction
The direct relationship between risk and return—in the sense that return should vary positively and proportionately to volatility—is so essential that it is designated as the first law of finance. In his popular guide to asset allocation, Bernstein offered a prescription stemming from an academic insight: “Growth opportunities are usually the source of high betas.” Investors otherwise would never be induced to commit capital to risky assets. However, in reality, high-volatility stocks offer abnormally low returns relative to their less riskier counterparts, which is known as Low-volatility Anomaly. Documented that U.S. stocks with high lagged idiosyncratic volatility earn very low future average returns. As a result to the mispricing by the Fama French factor model (Falkenstein, 1994). The unexplained returns of portfolios sorted on idiosyncratic volatility can go up to 2% a month (Ang et al., 2006) depending on the estimate method.
Most of which related to irrationality and behavioral biases. Many rational explanations have also contributed. For example Ang et al. (2006), where size, book-to-market, leverage, liquidity and several other variables become controlled.
The seminal work of Baker et al. (2012) suggests not only that so-called bond-like stocks commute with government bonds.
However, also that variables known as predictors of bond returns predict return differences between stock portfolios sorted on volatility. New factors have become raised and tested over the years, for example Joost et al. (2019), explained the Low-volatility Anomaly with the construction of the bond premium. Our paper builds on these works and specifically focuses on interest rate exposure within different volatility levels and portfolio sizes. We aim to figure out whether and how the interest rate exposure can support the idiosyncratic volatility, moreover, in portfolios of different market equities.
In this study, following the method introduced by Driessen et al. (2003), we construct the interest factor as the return of an equally-weighted portfolio consisting of U.S. government bonds with various maturities. The measurement of idiosyncratic volatility is based on the standard deviation of the residual returns from the Fama French five-factor model. Then instead of looking for the existing portfolios, we construct twenty-five portfolios over the period from January 1970 to December 2018, by equally dividing all the stocks by the market equity and idiosyncratic volatility. In order to estimate the interest rate exposure, we run standard time series regressions of portfolios returns on interest rate factor within different portfolio sizes.
Combined, Fama-MacBeth regressions become employed to estimate the premium for the interest rate exposure in the equity market.
2. Data and Methodology
2.1 The Data
The sample time period extends from 1970 to 2018. From several sources, we obtain the data used throughout the analysis within the sample time period. For construction of the Interest rate factor, we obtained the monthly return rate of U.S. government bonds with maturity of 1, 2, 5, 10 and 20 years from Wharton Research Data Service (WRDS). From the Federal Reserve Economic Research (FRES), the return rate on a one-month U.S. Treasury Bill.
For the measurement of IVOL, we gathered the five Fama-French factors published on Kenneth French’s website. The Fama-French 5 factors become constructed using the 6 value-weight portfolios formed on size and book-to-market. The 6 value-weight portfolios formed on size and operating profitability. And the 6 value-weight portfolios formed on size and investment.
For the construction of volatility portfolios, we obtained equity market and monthly return for all stocks trading on the NYSE, AMEX and NASDAQ from the Center for Research on Security Prices (CRSP). For delisted companies, the CRSP monthly return file does not include the returns from the delisting month, unless the delisting date is at month-end. We fetched the returns in the delisting month and the market capitalization on the delisting date from the CRSP daily return file and combined those returns with the delisting returns to create the effective delisting month returns. However, If the delisting was for performance related reasons, we set the delisting return equal to –55% if trading on NASDAQ or –30% if trading on the NYSE or AMEX (for an analysis of CRSP delisting bias, see Shumway 1997; Shumway and Warther 1999).
2.2 Constructing the Interest rate factor
There are several approaches to construct interest rate factors to measure interest rate level or yield curve changes, among which a level factor stands out for its strong explanation of the variance of excess bond returns (Driessen et al., 2003). We choose
the most common format of level factor, which is the equal-weighted combination of government bonds with various maturities. With the return of the portfolio representing shifts in the yield curve, the interest rate factor can become written as
(1) =1∑ ,
=1− ,,
where ,is the monthly total return of U.S. government bonds with maturity of 1, 2, 5, 10 and 20 years respectively, and ,is the rate of return on a one-month U.S Treasury Bill. In order to estimate the premiums later, bond returns instead of bond yields are used in the study.
2.3 Measuring idiosyncratic volatility (IVOL)
Earlier literatures, for example Benjamin and Ryan (2017) (USD), concentrate on measuring the idiosyncratic volatility as the monthly standard deviation of the residual returns from the Fama-French three-factor model. In the study, the Fama-French five model is employed instead to filter out more distractors. The standard time-series regression for each stock i can be written as follows:
(2) , − , =
+ ,⋅ + ,⋅ + ,⋅ + ,⋅ + ,⋅ + ,,
where ,refers to the stock return, and , , , , represents the Fama-French market, size, value, factors, respectively. IVOL is the standard deviation of the daily returns ,in each month, which are estimated using rolling six month windows to allow for sufficient observations to preserve accuracy.
2.4 Constructing the Volatility Portfolios
To construct portfolios of different volatility levels, the volatility level of each stock shall be calculated. For each month, all the participating stocks are sorted on the volatility and equally divided into five portfolios, with the first one of the lowest IVOL and the fifth one of the highest IVOL. Then in each portfolio, all the stocks are sorted by the market equity and equally divided into five small portfolios.
From 1970-2018 the least volatile portfolio has an average monthly excess return of 1.199%, which is 2.2 times as that of the most volatile portfolio (Portfolios 5). The difference is most significant in small size portfolios. Comparing the second volatile portfolios (Portfolios 4) with the most volatile portfolios (Portfolio 5), we can find there is a jump in portfolios of every size.
2.5 Measuring Interest rate exposure
To measure the interest rate sensitivity that different volatility portfolios hold, we start a regression of excess portfolio returns on the interest rate factor.
2.6 Measuring contributions from interest rate exposure
In order to assess whether the higher interest rate exposure contributes to the excess return of low-volatility stocks, we estimate the premium of this factor using the two stage Fama-MacBeth estimation. After estimation of the interest factor exposure ,, the cross sectional regression used to estimate the factor premium.
Itis assumed to be constant over time when running the regressions. The cross sectional regression results in monthly estimations for the interest rate premium I,t. We estimate the impact of this exposure on the unexpected excess returns based on the CAPM model.
3. Empirical result
3.1 Interest rate exposure
We find that nineteen out of twenty-five portfolios have positive exposure to the interest factor. The results show a clear downward trend with increasing volatility. If we compare these exposures, we can find there is a similar jump from the second volatile portfolio to the most volatile one. If the historical outperformance of low-volatility portfolios is due to the factor exposure that these portfolios have, assuming a positive premium for the risk, the similarity of the pattern makes sense.
Note that the exposure is assumed constant throughout time so far. However, in reality, the stock-bond return correlation displays very substantial time variation. (Baele et al., 2017). In order to see whether the exposure to interest rates is a function of time, we run the same regressions with a two-year rolling time window. We show the average interest rate exposure of all the portfolios, least volatile portfolio of all size and most volatile portfolios of all size, respectively.
Furthermore, we show the deviation from the average interest rate exposure in both lowest and highest volatility levels.
We find that the interest rate exposure of the most and least 20% volatile portfolios deviate almost symmetrically from the average level. In addition, the direction of deviation varies, which means the correlation between the volatility level and interest rate exposure varies between positive and negative.
In order to reveal the interest rate exposure of portfolios of different sizes. Moreover, for the 20% smallest and biggest portfolios. We calculate the difference in interest rate exposure between the highest and lowest volatile part, respectively.
Furthermore, we find that when the interest rate exposure varies in the same direction with the volatility, the portfolios in big size bear more interest rate exposure. In the contract, when the interest exposure is lowered by volatility level, the portfolios in small size bear more exposure.
3.2 Interest rate premium
In order to test whether the higher interest rate exposure contributes to the excess return of low-volatility stocks, we calculated the average unexplained return for all portfolios based on both CAPM and Fama MacBeth Model (Eq. (5)). We find that when the interest factor is taken into consideration, the unexplained return is more close to zero for portfolios of all volatility levels. Especially the unexplained excess return of 20% most volatile portfolios significantly decreased from -1.12% to -0.27%, showing that interest exposure explains a large part of the unexplained excess return of the portfolios.
In order to find out whether the contributions differ when it comes to a different portfolio size. We calculated the average unexplained return for all 20% small and big size portfolios. Respectively. Moreover, based on both CAPM and Fama MacBeth Model (Eq. (5)). We find that under both portfolio size, the unexplained return is more closed to zero in
most volatility levels. However, the unexplained excess return of 20% most volatile portfolios slightly decreased from -0.01% to -0.101% in small size portfolios, which means in a high volatile level, the interest rate exposure hardly explains the unexplained excess return of small portfolios.
4. Conclusion
After constructing the stock portfolios and interest rate factor over the period from January 1970 to December 2018, by comparing the interest rate risks of the small size portfolios and big size portfolios, we find that when the interest rate exposure varies in the same direction with the volatility, the portfolios in big size bear more interest rate exposure. In the contract, when the interest exposure is lowered by volatility level, the portfolios in small size bear more exposure.
In conclusion, we also estimate whether the interest rate exposure can give an explanation for the idiosyncratic volatility in different size levels. Lastly, we find that although interest exposure explains a large part of the average unexplained excess returns. However, at a 20% most volatile level, interest exposure hardly explains the idiosyncratic volatility of small portfolios.
References
Ang, A., Hodrick, R.J., Xing, Y., Zhang, X. (2009). High idiosyncratic volatility and low returns: International and further U.S. evidence. Journal of Financial Economy. 91(1), 1-23.
Baker, M., Bradley, B., Taliaferro, R. (2014). The low-risk anomaly: a decomposition into micro and macro effects. Finance Anal Journal. 70(2), 43-58.
Coval, J.D., & Moskowitz, T. J. (2001). The geography of investment: Informed trading and asset prices. Journal of Political Economy, 109, 811-841.
Glosten, L. R., Jagannathan, R., N Runkle, D.E. (1993). On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, 48, 1779-1801.
Joost, D., Ivo, K., Korhan, N., Robbert, B. (2019). Does interest rate explain the low-volatility anomaly?. Journal of Banking and Finance, 103, 51-61 Li, Xi, Sullivan, Rodney N., & Garcia-Feijo, L. (2014). The limits to arbitrage and the Low Volatility Anomaly. Financial Analysts Journal, 70, 52-63.
Does interest rate exposure explain the low volatility anomaly?
