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

Over the long term, portfolios invested in low-volatility equities are expected to pay investors returns that are a function of the market exposure, measured by beta, plus a positive premium or alpha, minus transaction costs (including those due to market impacts):

Beta x equity risk premium + alpha – implementation costs

The premium (or alpha) has been positive over the long term and we expect it to remain positive in the future. As discussed in part 1 of this post, the factors explaining the positive premium have not changed and are not expected to change in the future.

However, with shorter-term horizons, the alpha may not always be positive. The shorter the horizon the more the alpha can vary from its long-term average. This means that the behaviour of portfolios invested in low-volatility stocks becomes less predictable in the case of short-term horizons, because of the volatility of alpha.

Neither is the equity risk premium constant. While in the long term, it can be expected to average 5% annualised, it can change markedly over shorter-term horizons. The recent decades illustrate this well.

The behaviour of portfolios invested in low-volatility stocks is quite different in different market regimes. Since the beta of portfolios invested in low-volatility stocks is lower than 1, in bear markets we should expect an outperformance over the market capitalisation portfolio. The larger the market drop or the lower the beta, the larger the expected outperformance when taking this defensive beta into account.

But the beta is only one component of the performance of portfolios invested in low-volatility stocks. The second component is the premium, or alpha. This is expected to be positive over the longer term. But in the short term, alpha is volatile – it can be positive or negative. By mathematical construction, it is also fully independent from market returns, which means the alpha can be positive or negative, irrespective of which way markets move. The only certainty is that if the alpha is ultimately a positive number on average over the long term, then it must be positive more often than it is negative. Unfortunately, the alpha is also not predictable in the short term. We have no means of saying whether we expect it to be positive or negative over short-term horizons.

Predicting the returns of portfolios invested in low-volatility stocks at short-term horizons is thus not easy. Because of the defensive beta we can say that low-volatility stock portfolios are more likely to outperform the market capitalisation index in bear markets than in bull markets, but it is not certain that they will. It can happen that the alpha is negative in a bear market. Similar, we can say that low-volatility stock portfolios are more likely to underperform the market capitalisation index in bull markets, but again this is not certain because the alpha can be positive in bull markets.

The volatility of the alpha makes it difficult to forecast returns of portfolios invested in low-volatility stocks over short-term horizons. Portfolios will more likely substantially outperform in bear markets and deliver market returns or even underperform the market capitalisation index in bull markets. The more defensive the beta, i.e. how far below 1 it is, the more likely this scenario is.

Investors should thus invest in low-volatility stock funds for the long-term to make sure they have a greater likelihood of capturing the low-volatility positive premium in the long-term and enjoying higher risk-adjusted returns than from investing in the market capitalisation index. Conversely, they should not attempt to use low-volatility funds to implement market timing strategies as the uncertainty of the alpha on short-term horizons is likely to produce unwanted surprises.