Factors beyond aggregate market risk are sources of alternative risk premia. Factor timing addresses the question when to receive and when to pay such risk premia. A new method for predicting the performance of cross-sectional equity return factors proposes to focus only on the dominant principal components of a wide array of factors. This dimension reduction seems to be critical for robust estimation. Forecasts of the dominant principal components can serve as the basis of portfolio construction. Empirical evidence suggests that predictability is significant and that market-neutral factor timing is highly valuable for portfolio construction, over and above directional market timing. Factor timing is related to macroeconomic conditions, particularly at business cycle frequency.

Haddad, Valentin and Serhiy Kozak and Shrihari Santosh (2019), “Factor Timing” (December 22, 2019).

The quotes are from the paper. Cursive text and text in brackets have been added for clarity.

The post ties up with this site’s summary on macro trends, particularly the section on “why macro trends matter”.

The importance of factor timing

“We study factor timing, which combines…long-short factor investing and market timing…Stock returns are predictable over time, creating the scope for investors to engage in market timing…Factors beyond the aggregate market are sources of risk premia in the cross-section of assets creating the basis for factor investing.”

“Because the factors completely capture the sources of risks of concern to investors, optimal portfolios can be constructed from only these few factors—the so-called mutual fund theorem. Factor timing strategies are the dynamic counterpart of this observation; as the properties of the factors change, an investor should adjust her portfolio weights accordingly.”

“The optimal portfolio is equivalent to the stochastic discount factor [a pricing kernel based on intertemporal substitution under uncertainty]. Therefore, if factor timing is relevant for the optimal portfolio, we should account for this fact when estimating the stochastic discount factor.”

“The connection between factor timing, the stochastic discount factor, and predictability is best illustrated by the following decomposition…The average maximum conditional Sharpe ratio can be expressed as…an unconditional part reminiscent of a static Sharpe ratio…[and] a second term which encodes predictability [and which] is increasing in the maximum predictive R-squared when forecasting the asset.”

A method for factor timing

“Our main focus is on understanding predictability of cross-sections of returns… our main empirical setting is the cross-section of stock returns…Our empirical analysis focuses on fifty standard stock ‘anomaly’ portfolios that have been put forward in previous work as capturing cross-sectional variation in expected returns.”

“Empirically determining the value of factor timing appears difficult because it requires measuring the predictability of many returns, which opens the door for spurious findings. We propose a new approach to overcome this challenge. Imposing that the implied stochastic discount factor is not too volatile leads us to focus only on the estimation of predictability of the largest principal components of the factors. We find that these statistical restrictions are crucial to construct robust forecasts of factor returns.”

“To summarize, our assumptions lead to the following approach to measure conditional expected returns and engage in factor timing:

  1. Start from a set of pricing factors
  2. Reduce this set of factors to a few dominant components, using principal components analysis.
  3. Produce separate individual forecasts of each of the dominant components
  4. To measure the conditional expected factor returns, apply these forecasts to factors using their loadings on the dominant components.
  5. To engage in factor timing or estimate the stochastic discount factor, use these forecasts to construct the portfolio.”

“The key ingredient for our results is the dimension reduction of the set of portfolios to predict…We focus on the largest sources of variation by restricting our attention to the first five principal components (PCs) of anomaly returns, which explain 60% of the variation in realized returns. This dimension reduction allows for robust estimation of their predictability, and therefore the stochastic discount factor. As such, our approach is a regularization of the left-hand-side of the predictability problem — ‘which factors are predictable?’ — rather than the right-hand side — ‘which variables are useful predictors?’ We take a simple stance on this second issue by using only the book-to-market ratio of each portfolio as a measure to predict its returns.”

“We use our results to construct an optimal factor timing portfolio. This allows us to quantify the investment benefits of factor timing. And, more importantly, we use it to characterize the properties of an stochastic discount factor consistent with the evidence of these factor timing benefits.”

The predictability of anomaly returns

Factor timing is very valuable, above and beyond market timing and factor investing taken separately. The changing conditional properties of the pricing kernel are mostly driven by market-neutral factors.”

“Taking into account the predictability of the factors leads to an estimated stochastic discount factor which exhibits drastically different properties than estimates which assume constant factor premia, the standard approach in previous work.”

Our estimated stochastic discount factor is more volatile: its variance increases from 1.66 to 2.96. Moreover, the benefits to factor timing are strongly time-varying, which results in much more heteroskedasticity of the stochastic discount factor. These fluctuations in stochastic discount factor variance exhibit a very different pattern than estimates which only account for the predictability of market returns. They occur mostly at shorter business-cycle frequencies, and are correlated with different macroeconomic variables.”

“We find that the principal components of anomalies are strongly predictable. For the two most predictable components, their own book-to-market ratios predict future monthly returns with an out-of-sample R2 around 4%, about four times larger than that of predicting the aggregate market return.”

“The substantial anomaly predictability we document in [the table below] also contributes to the recent debate on whether these strategies represent actual investment opportunities or are statistical artifacts, which are largely data-mined.”

The investor value of factor timing

“The predictability of the dominant principal component portfolios captures common variation in risk premia which allows us to form forecasts of individual anomaly returns.”

Timing expected returns provides substantial investment gains; a pure factor timing portfolio achieves a Sharpe ratio of 0.71. This means that the conditional variance of the stochastic discount factor is substantially larger than that inferred from static strategies alone. The benefits from timing market-neutral factors largely outweigh those from timing the aggregate market return and are comparable to those obtained by static factor investing.”

“[The table below] reports measures of the performance for versions of this portfolio under various assumptions. We consider five variations of the optimal timing portfolio. ‘Factor timing’ is the portfolio described above. ‘Factor investing’ sets all return forecasts to their unconditional mean, while ‘market timing’ does the same except for the market return. ‘Anomaly timing’ does the opposite: the market is forecast by its unconditional mean, while anomalies receive dynamic forecasts. Finally, the ‘pure anomaly timing’ portfolio sets the weight on the market to zero and invests in anomalies proportional to the deviation of their forecast to its unconditional average. In other words, this portfolio has zero average loading on all factors, and lets us zoom in on…variation in anomaly expected returns.”

“The first performance metric we consider is the unconditional Sharpe ratio: the ratio of the sample mean and the standard deviation of returns. The factor investing, market timing, anomaly timing, and factor timing portfolio all produce meaningful performance, with Sharpe ratios around 1.2 in sample and between 0.63 and 0.87 out of sample…It is important to remember that the factor timing portfolio is not designed to maximize the unconditional Sharpe ratio. In a world with predictability, this is not an accurate measure of performance improvement…Maximizing the unconditonal Sharpe ratio requires portfolio weights which are highly nonlinear and nonmonotone in conditional expected returns. Still, the sizable Sharpe ratio of the pure anomaly timing portfolio is a first piece of evidence that factor timing is valuable. This portfolio does not engage in any static bets, but obtains Sharpe ratios of 0.71 and 0.77 in and out of sample.”

Empirical relation of factor timing with macroeconomic variables

“The dynamics of the variance of the stochastic discount factor differ from those of the market risk premium. The stochastic discount factor variance evolves mostly at business cycle frequency rather than at longer horizons. However, it is not always related to recessions. “

“[The table below] relates the variance of the stochastic discount factor to a variety of measures of economic conditions. For each measure, we report the coefficient and t-statistics of a univariate regression of the stochastic discount factor variance on the variable.”
N.B.: D/P means dividend-price ratio

“The contribution of various anomalies to the stochastic discount factor exhibit interesting dynamics. For example, the loadings of size and value are procyclical while the loading of momentum is countercyclical.”


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