Horizon-based Test of Simple Continuous Time CAPM

Using a capital asset pricing model (CAPM) is understandable for providing a risk-return tradeoff relationship as a basic concept in finance and for offering a coherent story on financial markets. However, taking a continuous time CAPM is less acceptable for giving a coherent story. It's because even the continuous time CAPM tends to be appraised only in theory and not in reality. To confirm that the risk-return tradeoff of a continuous time CAPM is real, financial scholars have tested the CAPM with the data from financial markets. One barrier to testing a continuous time CAPM with the data is that the time interval of the data is discrete, but the theoretical horizon implicitly presumed in a continuous time CAPM is instantaneous. Another barrier is that data horizons affect,risks and returns measured at the data horizons. The risk and return at a data horizon is unstably different from those at another data horizon. Even the risk-return relationship that is sufficiently satisfied at one data horizon often becomes unsatisfied at another horizon. The adversities need be reflected so as to make it known whether a continuous time CAPM has a coherent story. In this paper, we examine how the risk-return tradeoff of a continuous time CAPM depends on a data horizon, over which risks and returns are measured. We also propose a method by which a continuous time CAPM can be tested with the data in discrete time. The version of a simple continuous time CAPM (Cox et al., 1985 a&b) in discrete horizon used in this paper is called a horizon-based continuous time CAPM (HICAPM). The continuous time CAPM refers to a version of CAPM (Sharpe, 1964) in continuous time and nests both of ICAPM (Merton, 1973) and CCAPM (Breeden, 1979). An HICAPM at a given data horizon is sequentially constructed through the following procedure: firstly, we develop the continuous time CAPM governed by the state variable based on certain assumptions made by Cox et al. (1985 a&b). Secondly, we define integrals of the excess instantaneous returns of assets over the instantaneous rate of interest over a given data horizon with respect to stochastic calculus. Thirdly, on the integrals of the excess returns, we define, for our own purposes, some specific moments, including means, variances and covariances among the excess returns of assets, the excess return of the market portfolio, and the instantaneous rate of interest. Next, we insert such moments into the risk return tradeoff implied by the continuous time CAPM to develop an HICAPM of an asset at the given data horizon. The HICAPM consists of the covariance of the excess instantaneous return of the asset with the excess instantaneous return of the market portfolio, the variance of the excess instantaneous return of the asset, and the covariance of the excess instantaneous return of the asset with the instantaneous rate of interest, all of which are defined over the data horizon or the other horizon. Of HICAPMs at data horizons, an HICAPM of multiple horizons is also constructed. We apply the generalized method of moments (GMM) methodology to test HICAPMs and HICAPMs of multiple horizons at data horizons with the data of returns of Fama and French 25 portfolios. The horizons of the data are recorded daily, monthly and quarterly. The empirical results from GMM tests show the followings: first, every HICAPM at a given horizon well explains the expected excess returns at the horizon respectively within a significance level of 1%. Second, a three-factor model at a quarterly horizon has explanatory powers on the expected excess retrns at a monthly horizon within a significance level of 1%. A model of six factors at monthly and quarterly horizons likewise well explains the expected excess returns at a monthly horizon within a significance level of 1%. In addition, we test HICAPMs at data horizons and HICAPMs of multiple horizons with the data of returns of the KOSPI industry index series. Based on the integral form of stochastic calculus, Breeden et al. (1989) used a discrete version of CCAPM to test CCAPM. Longstaff (1989) also developed a three-factor model of the continuous time CAPM used in this paper to fit the data in discrete time. Both the discrete version and the three-factor model are based on raw returns of assets, and they are both weak in explaining the discrete data. In contrast, an HICAPM is based on the excess instantaneous returns over the instantaneous rate of interest, which is equivalent to the state variable in the continuous time CAPM (Cox et al., 1985 b) with respect to the stochastic process. Using an HICAPM sufficiently explains the market data of multiple horizons. We assume that an HICAPM is a cross-sectional three-factor model in market anomalies, and that it is one niche in capital asset pricing models.