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return versus beta

The low-volatility anomaly[Note 1] is the observation that portfolios of low-volatility stocks have higher risk-adjusted returns than portfolios with high-volatility stocks in most markets studied. The capital asset pricing model made some predictions of return versus beta. First, return should be a linear function of beta, and nothing else. Also, the return of a stock with average beta should be the average return of stocks (this is easy to show given the first assumption). Second, the intercept should be equal to the risk-free rate. Then the slope can be computed from these two points. Almost immediately these predictions were challenged on the grounds that they are empirically not true. Studies find that the correct slope is either less than predicted, not significantly different from zero, or even negative. Also, additional factors are predictive of return independent of beta.[1]

Black proposed a theory where there is a zero-beta return which is different from the risk-free return. This fits the data better since the zero-beta return is different from the risk-free return. It still presumes, on principle, that there is higher return for higher beta.

The low-volatility anomaly has now been found in the United States over an 85-year period and in global markets for at least the past 20 years.[when?][2][3][4]

Research challenging CAPM's underlying assumptions about risk has been mounting for decades.[5] One challenge was in 1972, when Jensen, Black and Scholes published a study showing what CAPM would look like if one could not borrow at a risk-free rate. Their results indicated that the relationship between beta and realized return was flatter than predicted by CAPM.[6][Note 2]

Shortly after, Robert Haugen and A. James Heins produced a working paper titled “On the Evidence Supporting the Existence of Risk Premiums in the Capital Market”.[Note 3] Studying the period from 1926 to 1971, they concluded that "over the long run stock portfolios with lesser variance in monthly returns have experienced greater average returns than their ‘riskier’ counterparts".[7]

The evidence of the anomaly has been mounting due to numerous studies by both academics and practitioners which confirm the presence of the anomaly throughout the forty years since its initial discovery.[Note 4] Examples include Baker and Haugen (1991),[8][Note 5] Chan, Karceski and Lakonishok (1999),[9] Jangannathan and Ma (2003),[10] Clarke De Silva and Thorley, (2006)[11] and Baker, Bradley and Wurgler (2011).[12]

For global equity markets, Blitz and van Vliet (2007),[13][Note 6]Nielsen and Aylursubramanian (2008),[14] Carvalho, Xiao, Moulin (2011),[15] Blitz, Pang, van Vliet (2012),[16] Baker and Haugen (2012),[17][Note 7] all find similar results.


  1. ^ Also known as the minimum variance anomaly or minimum volatility anomaly.
  2. ^ There is currently an empirical debate going between researchers of the anomaly. Some argue that the risk-return relation is positive, others flat and negative. See
  3. ^ See Available at SSRN:
  4. ^ There are various explanations for the effect: agency effects (Haugen), leverage constraints (Black), benchmarks (Blitz, van Vliet, Baker, Wurgler)
  5. ^ Haugen Baker (1991) covers the period from 1972 until 1989. Baker and Haugen (2012) covers 1990 to 2011.
  6. ^ See
  7. ^ See


  1. ^ "Betting Against Beta" (PDF).
  2. ^ "Low Risk, High Return? - May 2014 - SagePoint Financial" (PDF). SagePoint Financial.
  3. ^ "Why Low Beta Stocks Are Worth a Look | Portfolio Investing Blog: Portfolioist".
  4. ^ "The Greatest Anomaly in Finance: Low-Beta Stocks Outperform". Investing Daily.
  5. ^ Arnott, Robert, (1983) “What Hath MPT Wrought: Which Risks Reap Rewards?,” The Journal of Portfolio Management, Fall 1983, pp. 5–11; Fama, Eugene, Kenneth French (1992), “The Cross-Section of Expected Stock Returns”, Journal of Finance, Vol. 47, No. 2, June 1992, pp. 427- 465; see Roll, Richard, S.A. Ross, (1994), “On the Cross-Sectional Relation Between Expected Returns and Betas”, Journal of Finance, March 1994, pp. 101–121; see Ang, Andrew, Robert J. Hodrick, Yuhang Xing & Xiaoyan Zhang (2006), “The cross section of volatility and expected returns”, Journal of Finance, Vol. LXI, No. 1, February 2006, pp. 259–299; see also Best, Michael J., Robert R. Grauer (1992), “Positively Weighted Minimum-Variance Portfolios and the Structure of Asset Expected Returns”, The Journal of Financial and Quantitative Analysis, Vol. 27, No. 4 (Dec., 1992), pp. 513–537; see Frazzini, Andrea and Lasse H. Pedersen (2010) “Betting Against Beta” NBER working paper series.
  6. ^ Jensen, Michael C., Black, Fischer and Scholes, Myron S.(1972), “The Capital Asset Pricing Model: Some Empirical Tests”, Studies in the theory of Capital Markets, Praeger Publishers Inc., 1972; see also Fama, Eugene F., James D. MacBeth, “Risk, Return, and Equilibrium: Empirical Tests”, The Journal of Political Economy, Vol. 81, No. 3. (May – Jun., 1973), pp. 607–636.
  7. ^ Haugen, Robert A., and A. James Heins (1975), “Risk and the Rate of Return on Financial Assets: Some Old Wine in New Bottles.” Journal of Financial and Quantitative Analysis, Vol. 10, No. 5 (December): pp.775–784, see also Haugen, Robert A., and A. James Heins, (1972) “On the Evidence Supporting the Existence of Risk Premiums in the Capital Markets”, Wisconsin Working Paper, December 1972.
  8. ^ R. Haugen, and Nardin Baker (1991), “The Efficient Market Inefficiency of Capitalization-Weighted Stock Portfolios”, Journal of Portfolio Management, vol. 17, No.1, pp. 35–40, see also Baker, N. and R. Haugen (2012) “Low Risk Stocks Outperform within All Observable Markets of the World”.
  9. ^ Chan, L., J. Karceski, and J. Lakonishok (1999), “On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model”, Review of Financial Studies, 12, pp. 937–974.
  10. ^ Jagannathan R. and T. Ma (2003). “Risk reduction in large portfolios: Why imposing the wrong constrains helps”, The Journal of Finance, 58(4), pp. 1651–1684.
  11. ^ Clarke, Roger, Harindra de Silva & Steven Thorley (2006), “Minimum-variance portfolios in the US equity market”, Journal of Portfolio Management, Fall 2006, Vol. 33, No. 1, pp.10–24.
  12. ^ Baker, Malcolm, Brendan Bradley, and Jeffrey Wurgler (2011), “Benchmarks as Limits to Arbitrage: Understanding the Low-Volatility Anomaly”, Financial Analyst Journal, Vol. 67, No. 1, pp. 40–54.
  13. ^ Blitz, David C., and Pim van Vliet. (2007), “The Volatility Effect: Lower Risk without Lower Return.” Journal of Portfolio Management, vol. 34, No. 1, Fall 2007, pp. 102–113.
  14. ^ Nielsen, F and R. Aylur Subramanian, (2008), “Far From the Madding Crowd – Volatility Efficient Indexes”, MSCI Research Insight.
  15. ^ Carvalho, Raul Leote de, Lu Xiao, and Pierre Moulin,(2011) “Demystifying Equity Risk-Based Strategies: A Simple Alpha Plus Beta Description”, The Journal of Portfolio Management”, September 13, 2011.
  16. ^ Blitz, David, Pang, Juan and Van Vliet, Pim, “The Volatility Effect in Emerging Markets” (April 10, 2012). Available at SSRN:
  17. ^ Baker, Nardin and Haugen, Robert A., “Low Risk Stocks Outperform within All Observable Markets of the World” (April 27, 2012). Available at SSRN: