Capturing Low Volatility and Market Risk Premium

"Successful investing is about managing risk, not avoiding it."  Benjamin Graham


The capital asset pricing model (“CAPM”) tells us that risk and return are directly related. Generally speaking, an investor must be willing to bear above-average risks in order to earn above-average returns. According to this theory, investors cannot expect to outperform the market, or generate alpha, on a risk-adjusted basis. Per CAPM, the only risk premium comes from the market itself, or beta.

Considerable evidence, however, shows that this relationship does not hold. Other sources of equity risk premia (i.e., alpha factors) do exist. Rosenberg et al. (1985), found a significant positive relationship between average return and book-to-market ratios. Ross (1976) introduced the Arbitrage Pricing Theory, which also demonstrates another anomaly of the Capital Asset Pricing Model, and shows the expected return of a security to be a function of several (macro) factors. Building on these observations, Fama and French (1992) presented their well-known Three Factor model, showing that expected returns are modeled as a linear function of characteristics that are specific to firms: market risk premium, value premium and size premium.

All models have experienced rigorous empirical examination regarding their ability to explain cross-sectional differences in expected returns. Fama and Macbeth (1973) proposed a two-stage regression test that estimates the premium awarded to a particular risk factor exposure by the market. That is, how much return you would expect to receive for a beta exposure to that factor. The first step estimates the beta’s of each asset or portfolios’ return on the factors using time-series regressions:

R_n,t = a_n + b_n,F1F_1,t + b_n,F2F_2,t + … + b_n,FmF_m,t + e_n,t

Where R_t is the excess return (over the risk-free rate) of portfolio n at time t, F is the risk premium associated with the factor, b is the sensitivity of excess returns to factor risk loadings with intercept term a_n, and e_n represents the error terms.

The second stage uses the risk loading estimates as the independent variable to estimate the cross-sectional regressions:

R_i,t = a₀ + y_1,tb_i,F1 + y_2,tb_i,F2 + … + y_m,tb_i,Fm + e_r

Where R_t represents the asset or portfolio returns, b is the risk loading estimates and y is the factor premia with intercept a₀. The asset pricing model is correct if the time-series average of a₀ is not significantly different from zero and the factor loading estimates are not significantly different from the average risk premium associated with the corresponding risk factor. Further, if the model is correct, we should expect that the risk loadings explain the cross-sectional differences in expected returns.

The underlying idea is that if an asset pricing model is perfect, then the intercept should not be significantly different from zero for any asset or portfolio. The primary goal of this paper is to apply the two-step Fama Macbeth procedure to the Fama and French Three-Factor Model using the Main Research Low Volatility (“Proxy”) Index and to assess the results.

The rest of this paper is structured as follows. The first section introduces the methodology. The second part discusses the results obtained by applying the Fama Macbeth procedure. Finally, the last part summarizes the main findings of the paper.

Methodology

Applying Fundamental Indexation® methodology (Arnott et al, 2005), the data were collected from the universe of public companies listed on the New York Stock Exchange from 1983 through 2012. All companies were ranked by factor, and then the top 1000 companies were indexed by fundamental score. A composite index (“proxy”) comprised of each factor index is examined.

The proxy components are available on financial statements, domestically and worldwide, including emerging markets. The sample was selected to cover the past 25 years of market activity with data from the Compustat database. Financial statement and stock price data are from the Center for Research in Security Price (CRSP) database that is linked to corresponding Compustat entries by way of the Compustat/CRSP Merged (CCM) database[1]. Data for the S&P 500 is also gathered from the CRSP database.

For benchmarking purposes, the S&P 500 annual returns[2] are used. Company information was generated using annual data so the indices hoeld until the next year, when more recent financial information is made available for calculating the new index.

Statistical Properties

Exhibit 1 shows the annual returns of the Main Research Index and the S&P 500 between 1988-2012. Over the 25 years covered by Exhibit 1, the proxy index has a compound annual return of 9.41%, with 4.89% annualized volatility. The benchmark S&P 500 shows a compound return of 6.57% and volatility of 17.48%.

Exhibit 1:  Annual Returns of Main Research Index and the S&P 500

      

Source:  S&P 500 January Average data from Compustat/CRSP combined CCM database

Exhibit 2 illustrates the relative performance of the proxy and the S&P 500 including (1) the change in volatility drag, (2) the total risk effect, and (3) the return/risk tradeoff. Between the proxy and the S&P 500, the difference in geometric means is 2.83% annually. The difference in arithmetic means is 1.43% annually, while the difference in volatility drag is 1.40% annually.

The S&P 500 shows a positive return indeed for bearing risk during the 1988-2012 time period. The Main Research Index’s return is reduced by 5.82% annually since it bore less risk than the S&P 500. Adding this to the difference in volatility drag means that the reduced volatility cost the proxy index approximately 4.42% annually.

To offset these risk effects, the proxy index shows improved reward for bearing risk. Main Research produces 1.95 units of return for every unit of risk, whereas the S&P 500’s return to risk ratio is .462 units annually. The impact of this improved return to risk tradeoff results in an incremental performance of 7.25% annually.

Exhibit 3 charts the 5-year rolling annual return difference between Main Research and the S&P 500.  The rolling annual return difference is shown from 1992-2012, using predictive forecast values for the proxy index from 2013P-2016P. Since 2013, the forecast appears to be significantly consistent with the actual historical performance drift near zero.

Exhibit 2:  Sources of Annual Relative Performance

Reference	Factor	Main Research (%)	S&P 500 (%)	Difference (%)
A	Geometric Mean	9.410	6.578	2.832
B	Arithmetic Mean	9.519	8.087	1.431
C	Volatility Drag (A-B)	-0.109	-1.509	1.400
D	Standard Deviation	4.893	17.484
E	Reward for Risk (B/D)	1.945	.462
F	Impact for Changed Risk Level (Dp-Db)*Eb)			-5.823
G	Impact of Improved Tradeoff ((Ep-Eb)*Dp)			7.255
H	Difference in Arithmetic Means (F+G)			1.432
I	Sum of Risk Effects (C+F)			-4.423
J	Improved Tradeoff, Arguably an Anomaly (G)			7.255
K	Difference in Geometric Means (I+J)			2.832

Exhibit 3:  5-Year Rolling Annual Return Difference (Main Research Index Versus the S&P 500)

Source:  S&P 500 January Average data from Compustat/CRSP combined CCM database

Analysis of Results

To analyze these results, a Fama-Macbeth regression is assessed on the excess returns of the proxy index against the Fama and French factor loadings (market risk premium, value premium, and size premium). Proxy ‘portfolios’ were created from the universe of companies that compose the proxy index, and each portfolio’s excess returns were regressed against the factor loadings. The betas show to what extent each portfolio’s returns can be explained by each factor.

Table 1: Regression One Results

subscript
0	Intercept Coefficient
1	Mkt-Rf Coefficient
2	SMB Coefficient
3	HML Coefficient

P1:P40	β0	β1	β2	β3
Mean	8.69%	12.25%	-2.32%	18.41%
1988:2012	y0	y1	y2	y3
Mean	5.28%	29.13%	-6.26%	6.95%

The results in the cross-sectional regression are inconsistent with what we would expect to find if the model is correct, that the factor premium estimates are equal to the risk premia estimates. The intercept coefficient estimate, which should not be significantly different from zero, is on the contrary highly significant. The t-statistic (3.61) rejects the null hypothesis of equality to zero. In addition, the market risk premium average is significantly different from its beta estimate equivalent. The t-statistic of 2.83 again rejects the null hypothesis that risk equals reward, H₀:  b₁ = y₁.

Table 2: Correlations of Estimates

	Intercept	Mkt-Rf	SMB	HML
Intercept	1.000
Mkt-Rf	.880	1.000
SMB	-.812	-.875	1.000
HML	.836	.923	-.863	1.000

One interesting feature resides in the high correlation of beta estimates between the intercept and factor loadings. In Table 2, this correlation amounts to 0.88 between the intercept and market returns. In addition, the size (-0.812) and value (0.836) factors show a strong negative correlation to each other. This result would imply that the ability of the market premium to explain the cross-sectional differences in the proxy portfolio’s excess returns is captured by both the value premium and market risk premium estimates.

Conclusion

We examined the performance of the Main Research Low Volatility Index and the S&P 500 over a 25-year period. Results show that the proxy strategy achieves incremental performance of 7.25% annually over the S&P with 4.9% annualized volatility. Similar to existing and ample (Lazzara et al, 2016), the persistence of the low volatility anomaly abounds; the proxy strategy’s low market risk exposure is indeed rewarded with equity risk premium.

References

Arnott, R. D., Hsu, J., & Moore, P. (2005). Fundamental Indexation. Financial Analysts Journal , 83-97.

Fama, E. F. & French, K.R. (1992). The cross-section of expected stock returns, The Journal of Finance, 47 (2), 427-465.

Fame, E.F. & Macbeth, J.D. (1973), Risk, return, and equilibrium:  Empirical tests, Journal of Political Economy, 81 (3), 607.

Rosenberg, B, Reid, K., Lanstein, R. (1985), Persuasive evidence of market inefficiency, Journal of Portfolio Management, 11, 9-17.

Ross, Stephen A. (1976), The arbitrage theory of capital asset pricing, Journal of Economic Theory, 13, 341-360.

Disclaimer: Data for the Main Research Low Volatility Index and the S&P 500 are from Compustat/CRSP and CCM database. All data for the indices is for research and informational purposes only. All information for the indices is back-tested. Back-tested performance, which is hypothetical only, is limited because it applies an index methodology and selection of index components in hindsight. No hypothetical approach can account for intangible market factors in general, nor the impact of investor preferences. Hypothetical returns are not indicative of future returns, may differ from, and be lower than, back-tested returns. 

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[1] The historical data were not adjusted for transaction costs, taxes or turnover. Missing observations were adjusted consistently throughout the data set. REITs, ADRs and mutual funds were excluded, as were extreme outliers.

[2] The S&P 500 annual returns are calculated using the January average of each year.