# Low Risk

1. INTRODUCTION

Low-risk investing is a portfolio strategy, where buying low-risk and selling high-risk stocks have historically delivered superior risk-adjusted returns. More precisely, the reported alphas in several publications vs. the FF-3 factor model are impressive**.** Some even argue that the low-risk anomaly is the key challenge of the efficient market hypothesis or to cite it directly from Ang (2014) [1]:

The risk anomaly is that risk - measured by market beta or volatility - is negatively related to returns. Robin Greenwood ... said in 2010, ‘We keep regurgitating the data to find yet one more variation of the size, value, or momentum anomaly, when the Mother of all inefficiencies may be standing right in front of us - the risk anomaly’ .

One of the first studies showing a negative relation between risk and return is Haugen and Heins (1975) [2] and Jensen, Black and Scholes (1972) [3]. In more recent years, Ang et al. (2006) [4] show for US data and Blitz and van Vliet (2007) [5] for global data that the anomaly still exists.

2.OVERVIEW OF DIFFERENT VERSIONS

Generally, several versions of low-risk investing are circulating in the finance literature and are used in investment management practice: low volatility, low beta, low idiosyncratic volatility and minimum variance.

**Low Volatility:**

The volatility is simply obtained by using the standard deviation of the return series of each stock over a defined time window.

**Low Idiosyncratic Volatility:**

In order to obtain the idiosyncratic risk of a stock, we need to correct the return for systematic risks. This can be achieved for stock $i$ by running a time series regression of the following form:

**$r_{i,t}= \beta_{i,Mkt}r_{mkt,t}+ \beta_{i,smb}r_{smb,t}+ \beta_{i,hml}r_{hml,t}+ \beta_{i,umd}r_{umd,t}+ \epsilon_{i,t}\tag{1}$**

and then calculating the standard deviation of the residual from regression above, where both the time window of the regression and the residual needs to be defined. Alternatively, one could use simply a univariate regression with only the market as independent variable.

**Low beta:**

Calculate each stock’s beta at each rebalance date. You could simply use the same regression as above and use the market beta $\beta_{Mkt}$ as a proxy or you follow simply the method used in Frazzini and Pedersen (2014) [6], where the authors make use of the beta decomposition: $\beta_{i,Mkt}=\rho_{i,t}\frac{\sigma_{i,t}}{\sigma_{mkt,t}}$ and measure the correlation($\rho_{i,t}$) over a longer period than the $\sigma$’s as they argue volatilities are in general not as persistent as correlations and hence, obtain a more forward looking measure to reduce errors in the measurement of the $\beta$s.

**Relation of Low vola, low beta and low idiosyncratic volatility:**

The three concepts are related in the following way:

**$\underbrace{\sigma_{i,t}}_{Low Vola} = \sqrt{\underbrace{\beta_i^2}_{(\text{Low Beta})^2}\sigma_{mkt,t}^2 + \underbrace{\sigma_{\epsilon}^2}_{(\text{IVol})^2}}\tag{2}$**

**Minimum variance portfolio:**

Instead of investing in the lowest decile portfolio simply by the previously described criterion, one might directly construct a minimum variance portfolio, by using the full covariance matrix of all available stocks. The covariance could be estimated by a simple $\beta$ - factorization of returns. However, Blitz and van Vliet (2007) [5] find that by simply using the diagonal elements of the covariance (e.g. simple low volatility) one can achieve greater improvements than by using the full covariance structure.

3. PRACTICAL IMPLEMENTATION

The practical implementation of the strategy strongly depends on what type of investor you are. The challenge for a **pure equity, long only investor**, who is benchmarked against the market is to reduce tracking-error or almost equivalently to increase beta, as typically the low-decile portfolio has a $\beta$ around 0.65 and 0.75.

If you are a **pension fund** or **multi-strategy manager** with a sufficiently low equity portion in your benchmark life is a little easier. Let’s assume your benchmark is 60% cash and 40% equity, and furthermore, the $\beta$ of your low volatility portfolio is 0.7, you could simply hold a fraction of 0.4 *1/0.7 of the low volatility portfolio in your portfolio (if your constraints allow you to) to obtain a fairly beta neutral position vs. your benchmark.

Naturally, the question arises what **time window** to use to obtain the criteria described above. In general, we recommend to use rather longer time windows to estimate the FF $\beta$’s - a good starting point is three years of weekly (or even daily) data. The time window for the residuals can then still be chosen rather short, say 1-month (of daily) data, as suggested in Ang et al. (2006) [4], or probably more appropriate, from a practical perspective, is a three year window (daily or weekly data) as used in Blitz and van Vliet (2007) [5] to reduce turnover. Additionally, the significant alphas of the long-short portfolio reported in Ang et al. (2006) [4] are driven more by the short position in high volatility stocks than in the long of the low volatility stocks. Blitz and van Vliet (2007) [5] on the other hand observe a more symmetric effects with using three year estimates. Combing this information, we have another argument for choosing a longer time window measuring the volatility. We note however, that the two studies use different set-ups - Blitz and van Vliet (2007) [5] use simple return volatilities, Ang et al. (2006) [4] – the FF-3 factor corrected idiosyncratic volatilities.

Some critique has been articulated that low-risk investing is partly a result of sector concentration and that the strategy profits from an implied value bias. Asness, Frazzini and Pedersen (2013a) [7] demonstrate that low-risk investing even does the work best when its implemented in a sectorneutral fashion. The authors conclude that neither the sector, nor value is the driving force of the strong performance of low-risk strategies.

If your constraints do not allow to utilize the anomaly, an alternative implementation of the low-risk phenomena could be as proposed in Asness, Frazzini and Pedersen (2013b) [8], where low volatility and low beta contribute to the **safety** **dimension** of **quality** (see the quality section for more details).

4. (POSSIBLE) THEORETICAL EXPLANATIONS

What are possible explanations of the anomaly? Blitz and van Vliet (2007) [5] mention that leverage constraints and/or tracking error aversion of investors are the potential reasons. Alternatively, the authors provide a behavioural argument, where high risk stocks are lottery tickets for retail investors and hence, high in demand and consequently inflate prices (deflate returns) of these stocks. Frazzini and Pedersen (2014) [6] present a theoretical model, where investors are constrained in their investment opportunities, and how this can lead to high $\alpha$s for low $\beta$-stocks.

5. SUMMARY

Low-risk investing can add value to your portfolio, however, it crucially depends on what type of investor you are and to what extent you can actually exploit the strategy. Exposure to the factor can be obtained by building your own low-risk portfolio (as described above) or alternatively, by simply investing into low-risk ETFs (typically called “Index XYZ low volatility ETF” or “ Index XYZ minimum variance ETF”).

References

- Asset Management: A Systematic Approach to Factor Investing,

Ang, Andrew

, (2014) - Risk and the rate of return on financial assets: Some old wine in new bottles,

Haugen, Robert A., and A Heins James

, Journal of Financial and Quantitative Analysis, Volume 10, Number 05, p.775–784, (1975) - The capital asset pricing model: Some empirical tests,

Jensen, Michael C., Black Fischer, and Scholes Myron S.

, (1972) - The cross-section of volatility and expected returns,

Ang, Andrew, Hodrick Robert J., Xing Yuhang, and Zhang Xiaoyan

, The Journal of Finance, Volume 61, Number 1, p.259–299, (2006) - The Volatility Effect,

Blitz, David C., and van Vliet Pim

, The Journal of Portfolio Management, Volume 34(1), p.102-113, (2007) - Betting against beta,

Frazzini, Andrea, and Pedersen Lasse Heje

, Journal of Financial Economics, Volume 111, Number 1, p.1–25, (2014) - Low-Risk Investing Without Industry Bets,

Asness, Cliff, Frazzini Andrea, and Pedersen Lasse H.

, Working Paper, (2013) - Quality Minus Junk,

Asness, Clifford S., Frazzini Andrea, and Pedersen Lasse H.

, Available at SSRN 2312432, (2013)

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