#### Which returns and volatility assumptions should be used to determine the allocation to low-volatility equities? An in-depth view.

Investors should be paid a premium for taking equity risk. This equity risk premium is often assumed to be 5% annualised in excess of money market rates. This is the annualised return an investor should expect in the long term from investing in the market portfolio, i.e. the portfolio where stocks are weighted according to their market capitalisation. This portfolio represents the aggregate position of all investors in the world. With the historical volatility of equities at about 16% (e.g. since the 1970s), the risk-adjusted return of the market portfolio has been about 0.31.

The Capital Asset Pricing Model (CAPM) introduced in the 1960es tells us that any equity portfolio should generate returns in proportion to its exposure to the market portfolio. The exposure to the market portfolio is measured by beta, which can be estimated from a simple regression of the returns of a portfolio in excess of money market rates against the excess returns of the market in excess of money market rates.

The higher the beta, the higher the expected return to the portfolio; the lower the beta the lower the expected return. A beta of 1 means investors should expected to be paid the equity risk premium.

In theory, investors should only be paid to take market risk, or systematic risk. Any portfolio that deviates from the market portfolio has an additional risk component: idiosyncratic risk. The efficient market hypothesis tells us not to expect a return from taking idiosyncratic risk. Clearly, any additional return earned by an investor who invests in a portfolio that is different from the market portfolio must be paid by the investors who invest in the complementary portfolio. In the end, the sum of all investor portfolios must add up to the market portfolio and the returns earned by all investors must add up to the equity risk premium. The excess returns beyond what is expected from beta must total zero.

A consequence of idiosyncratic risk not paying a return in aggregate is that, in theory at least, we should expect lower risk-adjusted returns as soon as our portfolio deviates from the market portfolio. This is simple arithmetic and arises from the fact that any portfolio different from the market portfolio will have this unpaid idiosyncratic risk component.

However, empirical evidence shows us that this is not always the case. Low-volatility equities are a good example. They have a beta well below 1 and yet have nevertheless been generating higher returns than even the market risk premium. This means that the idiosyncratic risk taken by deviating from the market capitalisation portfolio into low volatility stocks has been paying a positive non-expected premium or alpha. In turn, the idiosyncratic risk taken by deviating from the market capitalisation portfolio into high volatility stocks has been paying a negative premium or alpha. The alpha can be measured from the regression used to estimate beta as the intercept as proposed by Michael Jensen in 1972. This shows that the theory underlying the CAPM does not always hold true in the real world.

Why does CAPM not always hold true? Most likely because it is based on a number of unrealistic assumptions such as that investors are free from constraints and can take unlimited leverage or short-sell any stocks by as much as they want, or that investors incur no taxes or trading costs. Other unrealistic assumptions used in CAPM include the idea that all investors aim at maximising returns for a given level of risk. The example of active fund managers paid to outperform market capitalisation indices shows this is not true. Nevertheless, CAPM remains useful in assessing market exposure and how much alpha a given portfolio or strategy generates.

The reasons for CAPM not always working can be shown to explain why low-volatility stocks, which typically have low beta, return a positive alpha in the long-term and, conversely, why investors will be disappointed from investing in the riskier stocks that return a negative alpha. This is known as the low-volatility anomaly. Bob Haugen and James Heins were the first to find that low-volatility stocks do pay a positive alpha in 1972 from their analysis of US stocks returns between 1926 and 1970. Michael Jensen, Fischer Black, and Myron Scholes also demonstrated in 1972 that any constraint on leverage must result in positive alpha for low-volatility stocks. Since then, many academics papers have shown that low-volatility stocks continue today to pay positive alpha.

We now have all the elements required to answer the question raised in the title: what returns can be expected for funds invested in low-volatil

ity stocks? The first part of the answer comes from the market exposure, i.e. the beta. The second component comes from how much alpha we can expect. The third element of the answer comes from how much the implementation turnover will cost in terms of transaction costs and market impact (the fact that when a stock is bought the price rises due to buying interest and falls back after the transaction, and conversely for selling). The portfolio needs to be rebalanced periodically to remove and replace stocks that become more volatile.

Let’s consider the MSCI Minimum Volatility Index, which is heavily tilted towards low-risk stocks, and compare it to the market capitalisation-weighted MSCI World index as a proxy for the market. The index returns can be downloaded from the MSCI web-site. We can use the T-bill rates from Kenneth French’s web-site as a proxy of the risk-free rate. If we regress the monthly total returns of the MSCI Minimum Volatility Index in excess of the risk-free rate against the excess returns of the MSCI World Index, using USD data between January 1995 and June 2014, we find a beta of 0.68 and an annualised alpha of 2.1% for the MSCI Minimum Volatility index in this period. The equation to estimate long-term expected returns gross of management fees is thus simply:

Beta x Equity Risk Premium + Alpha – Turnover Implementation Costs

If we consider the equity risk premium to be 5% and that the index turnover implementation costs amount to 0.4% [1] then we find:

0.68 x 5.0% + 2.1% - 0.40% = 5.1%

In other words, investors are paid much the same returns as the equity risk premium. But since the volatility of the MSCI Minimum Volatility index was just 11.7% over this period compared with 15.2% for the MSCI World index, the compensation for risk is much higher when investing in the MSCI Minimum Volatility index which had a Sharpe ratio of 0.42 over the period, well above the 0.27 observed for the MSCI World index over the same period.

*This is the long-term expected return. Part 2 of this article will address an obvious further question: what short-term return can be expected?*

[1]Our market impact analysis sees this as a reasonable expected cost, taking into account turnover and market impact for large portfolios circa USD 10 billion invested in the index. This figure is difficult to assess but is almost certain to be larger if a given low volatility strategy gets crowded.