### Value, Size, Momentum And The Average Correlation Of Stock Returns

University of Applied Sciences Darmstadt

Frankfurt School of Finance & Management Gemeinnützige GmbH

November 19, 2015

**Abstract: **

Dynamic average correlations of stock returns are predicted by the volatility of the market excess return and moving average returns of value, size and momentum portfolios. While the influence of market volatility on average correlation is well-known, the role of value, size and momentum appears to be underappreciated. Correlations of stock returns and stock returns share sources of risk like the market volatility, but there are other sources that are distinct. In particular, correlations are increased when value or momentum returns are roughly zero, while strongly negative returns of value or momentum are associated with lower correlations. Using the market volatility and a moving average return of the value portfolio as predictors of average correlation, we obtain a global minimum variance portfolio with a Sharpe ratio that is 1.5% higher relative to the one based on a Dynamic Equicorrelation Garch model, and the difference in portfolio volatility is statistically significant.

### Value, Size, Momentum And The Average Correlation Of Stock Returns – Introduction

Knowledge about the factors that determine stock return correlations is vital for asset allocation and risk management. The fundamental question is: What are the underlying macro-economic sources of correlation risk? In asset pricing, macro-economic and non-diversifiable sources of risk are linked to risk premiums of factor portfolios like value or size. Such factor portfolios do not only describe risk premiums and means of stock returns, they also describe the dynamic behavior and hence the comovement, see Fama and French [2004], Fama and French [1996] and Cochrane [2011]. In this paper we explore whether standard asset pricing factors from Fama and French [1993] and Carhart [1997] are useful to predict stock return correlations.

We develop a dynamic factor model for average correlations of stock returns. Our model is based on the Dynamic Equicorrelation Garch model (DECO) by Engle and Kelly [2012], but instead of an autoregressive process in the spirit of Engle [2002] we use a generalized linear factor model to describe the dynamics of correlations. For factors we consider variances, covariances and average returns (risk premiums) of the market excess return and of the value, size and momentum portfolios from Fama and French [1993] and Carhart [1997]. The most successful predictors of dynamic average correlations are the volatility of the market excess return together with either the moving average return of the value or momentum portfolio; the volatility of value and the moving average return of the size portfolio; the volatility of size and the moving average return of momentum. Thus the conventional argument that correlations are high when market volatility is high or the market trend is negative1 is incomplete, because it ignores the role of time-varying risk premiums2 of value, size, and momentum as additional sources of correlation risk. For example, the period from July 2010 to October 2010 demonstrates that correlations can be high although markets appear stable, suggesting a decoupling of volatility and correlation. We find that these high correlations can be predicted by moving average returns of the value and momentum factor portfolios.

The relation between average correlations and its predictors is strongly nonlinear. Correlations strongly increase when markets become more volatile. Correlations also increase when average returns of value or momentum are zero or slightly negative, and they are lower when average returns are more extreme in either direction, so that the relation appears inversely U-shaped. This inverse U-shape makes it hard to apply results from the asset pricing literature to identify macro-economic sources of correlation risk3. Macro-economic sources of risk are associated with strong losses incurred by factor portfolios, but increased correlations are associated with a neutral performance of factor portfolios or only mild losses. It appears that we cannot completely explain the behavior of correlations by referring to macro-economic sources of risk for stock returns: correlations and stock returns do share some sources of risk, in particular, falling or volatile market prices, but other sources of risk appear to be distinct.

Relative to DECO our model seems to perform better. In out of sample tests we construct a global minimum variance portfolio for the constituents of the S&P 500 based on volatilities and correlations predicted by models. For example, our model with the volatility of the market excess return and the moving average return of the value portfolio as predictors of average correlation yields a global minimum variance portfolio with a 1:2% higher Sharpe ratio on the sample 2007 – 2012 and a 1:5% higher Sharpe ratio in the crisis 2008-09 than a portfolio constructed by DECO. Diebold-Mariano tests from Engle and Colacito [2006] confirm that the difference in portfolio volatility is statistically significant. The relative success of these predictors also holds if we model block-correlated returns.

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