Multiscale systematic risk

Abstract

In this paper we propose a new approach to estimating systematic risk (the beta of an asset). The proposed method is based on a wavelet multiscaling approach that decomposes a given time series on a scale-by-scale basis. The empirical results from different economies show that the relationship between the return of a portfolio and its beta becomes stronger as the wavelet scale increases. Therefore, the predictions of the CAPM model should be investigated considering the multiscale nature of risk and return.

Introduction

The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) has received considerable attention in financial studies.¹ In its simplest form, the CAPM predicts that the excess return of a stock (return over the riskless rate of return) should be proportional to the market premium (market return over the riskless rate of return). The proportionality factor is known as the “systematic risk” or the “beta” of an asset.²

Early empirical studies on the CAPM such as Black et al. (1972) and Fama and MacBeth (1973) were supportive of the implications of the model. That is, the average return of high beta stocks was higher than the average return of low beta stocks. Furthermore, the relationship was roughly linear, although the slope was too flat to strongly support the CAPM (Campbell, 2000). Later studies focused on beta estimation issues in detail. Some examples of the concerns on beta estimation are as follows: the stability of beta over time (Harvey, 1989), borrowing constraints (Black, 1972), the impact of structural change and regime switches (Garcia and Ghysels, 1998), the effect of world markets and volatility (Bekaert and Harvey, 1995, Bekaert and Harvey, 1997, Harvey, 1991), non-synchronous data issues (Scholes and Williams, 1977), time horizons of investors (Levhari and Levy, 1977) and the impact of the return interval (Brailsford and Josev, 1997, Brailsford and Faff, 1997, Cohen et al., 1986, Frankfurter et al., 1994, Hawawini, 1983, Handa et al., 1989, Handa et al., 1993).

Studies on the impact of the return interval of beta estimates point out the importance of the time scale issue. An early study by Levhari and Levy (1977) showed that if the analyst used a time horizon shorter than the true one³ , the beta estimates were biased. Fama, 1980, Fama, 1981 provided evidence that the power of macroeconomic variables in explaining the stock prices increased with increasing time length. Handa et al. (1989) reported that different beta estimates were possible for the same stock if different return intervals were considered. Similarly, Handa et al. (1993) rejected the CAPM when monthly returns were used but failed to reject the CAPM if the yearly return interval was employed. Cohen et al. (1986) and references therein provided ample evidence that the beta estimates were sensitive to return intervals. By using Australian equity market data, Brailsford and Faff (1997b) reported that the CAPM (with a GARCH-M specification) was supported for weekly and monthly interval returns while the greatest support was found in the weekly return intervals. The daily return interval in that study did not support the CAPM. Hawawini (1983) proposed a model to overcome the interval effect in beta estimation.

In this paper, we propose a new approach to estimating systematic risk (the beta) in a CAPM. The proposed method is based on wavelet analysis that enables us to decompose a time series, measured at the highest possible frequency, into different time scales. It provides a natural platform on which to investigate the beta behavior (systematic risk) at different time horizons without losing any information. The empirical results from different economies show that the relationship between the return of a portfolio and its beta becomes stronger as the scale increases. Therefore, predictions of the CAPM are more relevant in a multiscale framework as compared to short time horizons.

This paper is structured as follows. The CAPM model is presented in Section 2. Wavelet analysis, the wavelet variance and the wavelet covariance are presented in Section 3. Multiscale beta estimation with S&P 500 stocks in the United States, Financial Times Stock Index (FTSE) stocks in the United Kingdom and DAX stocks in Germany is studied in Section 4. We conclude afterwards.