Small caps in international equity portfolios: the effects of variance risk

Abstract

We show that predictable covariances between means and variances of stock returns may have a first order effect on portfolio composition. In an international asset menu that includes both European and North American small capitalization equity indices, we find that a three-state, heteroskedastic regime switching VAR model is required to provide a good fit to weekly return data and to accurately predict the dynamics in the joint density of returns. As a result of the non-linear dynamic features revealed by the data, small cap portfolios become riskier in bear markets, i.e. display negative co-skewness with other stock indices. Because of this property, a power utility investor ought to hold a well-diversified portfolio, despite the high risk premium and Sharpe ratios offered by small capitalization stocks. On the contrary small caps command large optimal weights when the investor ignores variance risk, by incorrectly assuming joint normality of returns. These results provide the missing partial equilibrium rationale for the presence of co-skewness in the empirical asset pricing models that have been proposed to explain the cross-section of stock returns.Keywords: intertemporal portfolio choice; return predictability; co-skewness and co-kurtosis; international portfolio diversification.JEL code: G11, G15, F30, C32. AbstractWe show that predictable covariances between means and variances of stock returns may have a first order effect on portfolio composition. In an international asset menu that includes both European and North American small capitalization equity indices, we find that a three-state,heteroskedastic Markov switching VAR model is required to provide a good fit to weekly return data and to accurately predict the dynamics in the joint density of returns. As a result of the non-linear dynamic features revealed by the data, small cap portfolios become riskier in bear markets, i.e. display negative co-skewness with other stock indices. Because of this property, a power utility investor ought to hold a well-diversified portfolio, despite the high risk premium and Sharpe ratios offered by small capitalization stocks. On the contrary small caps command large optimal weights when the investor ignores variance risk, by incorrectly assuming joint normality of returns. These results provide the missing partial equilibrium rationale for the presence of coskewness in the empirical asset pricing models that have been proposed to explain the cross-section of stock returns.2

The Economic Value of Timing Higher Order (Co-)Moments in Bull and Bear Markets

“…A related literature deals with predicting and timing the volatility of daily data (see Fleming et al, 2001, and references therein), assuming constant expected returns given the short horizon under scrutiny in 2 This confirms previous specification tests performed by Ang and Chen (2009) and Guidolin and Nicodano (2009), who also extend the comparison of Markov switching models to other non-linear models such as multivariate GARCH, GARCH-in mean models, and VAR-EGARCH. 3 However, the predictive density of returns at any future horizon  ≥ 1 is generally not normal and is instead a mixture of normal distributions.…”

“…As far as, the scaled co-skewness coefficients of type   ( 6 =  = 1... ) are concerned, out of 56 possible coefficients, we obtain that only 13 are statistically significant (notice that under multivariate normality, all co-skewness coefficients have to be zero). 22 The empirical evidence of rich co-skewness and co-kurtosis patterns is considerably stronger in panels B and C, for data sets IND and BMINT. However, also in this case a few features are remarkable.…”

“…Lettau and Van Nieuwerburgh (2008) suggest that the presence of changing steady state means in the dividend-price ratio may explain why it proves so difficult to predict stock returns out of sample with such a ratio. Guidolin and Nicodano (2009) find that, both in-sample and out-of-sample, regime switching models with a time-varying covariance matrix fare as well as or better than multivariate GARCH models in an international data set of portfolios sorted on the basis of size. Guidolin and Timmermann (2008b) obtain similar evidence for spread portfolios built by sorting on the basis of both size and the book-to-market ratio.…”

“…21 We limit our analysis to pairs of stocks only for reasons of space, although it is clear that with   2 assets as everywhere in this paper, many more scaled co-skewness and co-kurtosis coefficients have been computed. 22 In the co-skewness matrix, coefficients above the main diagonal refer to the sample covariance between the square of the returns of the row portfolio and the level of returns of the column portfolio; coefficients below the main diagonal refer to the sample covariance between the level of the returns of the column portfolio/index and the square of returns of the row portfolio/index.…”

Optimal portfolio allocation with higher moments

“…This effect becomes important for high volatility. Our findings are in line with those of Das and Uppal (2004) and Guidolin and Nicodano (2005). However, these papers also compute optimal portfolio allocation for multiple risky assets, with cross-higher moments.…”