Asset Prices
Momentos

Recent empirical work indicates that, in a variety of financial markets, both conditional expectations and conditional variances of asset returns are time-varying. The purpose of this paper is to determine whether these joint fluctuations of conditional first and second moments are consistent with the Sharpe-Lintner- Mossin capital-asset-pricing model. We test the mean-variance model under several different assumptions about the time- variation of conditional second moments of returns, using weekly data from July 1974 to December 1986 on returns to a portfolio composed of dollar, Deutschmark, Sterling, and Swiss franc assets, together with the US equities. The model is estimated constraining risk premia to depend on the time-varying conditional covariance matrix of the residuals of the expected returns equations.

The results indicate that estimated conditional variances cannot explain the observed time-variation of risk premia. Furthermore, the constraints imposed by the static CAPM are always rejected.

The capital asset pricing model (CAPM) explains the price of financial assets in terms of the demand for assets with stochastic returns by risk averse investors. The model predicts that if investors maximize ability function which depends on expected wealth and its variance then in equilibrium the return on an asset depends on the return or a riskless asset, plus a risk premium term, which is the product of the investors' coefficient or risk aversion and the covariances of the asset returns (conditional upon the information available to investors.) Tests of the CAPM have, however, proven disappointing. As in the case in other international financial assets, financial markets display highly volatile and largely unpredictable price movements. These properties make it very difficult to extract statistically reliable estimates of systematic exchange-rate and asset-price movement and are at the root of the generally poor empirical performance of international asset pricing models. Nevertheless, almost all empirical investigations suggest that expected returns on foreign assets vary over time and that the volatility of returns on foreign assets changes over time.

These tests of the CAPM assume that the covariance matrix of asset returns is constant overtime. Movements in risk premia result must therefore be the other factors, such as changes in asset supplies. Tests based on the assumption of a constant covariance matrix have indicated that factors such as asset changes in supplies cannot reproduce the observed movements in risk premia.

The assumption of constant conditional covariance of returns, however, has been proven wrong by the evidence of heteroskedasticity, both in the stock market and in the foreign exchange market. In earlier work we have found that in both the stock market and the foreign exchange market, nominal interest rates help explain much of the variation in the conditional (non- central) second moments of asset returns: this suggests that allowing the conditional second moments to vary our time could improve the empirical performance of the CAPM. Other researchers have found a substantial improvement in the performance of the model once the time variation of conditional second moments is accounted for but then estimates of the coefficient of risk aversion, are still imprecise and they reject in all cases the prameter restrictions associated with the CAPM. If the covariances of asset returns do change overtime, as the evidence suggests, previous empirical tests of the CAPM may be inappropriate and the non constancy may also help explain the behaviour of risk premia. The purpose of this paper is to determine whether the observed fluctuations of the conditional expectations and variances of returns in international financial markets are consistent with the CAPM.

We test the CAPM under several different assumptions about the manner which the conditional covariances of asset returns vary overtime. Since there is no economic model of the fluctuation of variances to rely on, we present a number of alternative specifications of the time-variation of conditional variances and compare their impact on tests of asset pricing. We use weekly data from July 1974 to December 1986 on the rates of return on a portfolio of assets which includes the US dollar, the British pound, the Deutsche mark and the Swiss franc. We estimate the models imposing the theoretical constraint that the the risk premia depend on the time-varying conditional covariance matrix of asset returns (calculated from the residuals of the expected returns equations). Unlike earlier tests of capital asset pricing models, we pool data on the returns in foreign exchange market and the US stock market. This strategy is bound to improve the power of our tests, since returns in the stock market and exchange rates are correlated, and, more importantly, is justified by the sheer size of the stock market in international financial portfolios: in our sample, the average share of the US stock market is .55, versus .31 for dollar-denominated assets, and only .06 for pound sterling and Deutsche mark assets, respectively. We estimate the CAPM using the maximum likelihood techniques under a number of assumptions concerning the variation over time in the variance of returns. Each model was estimated first on the portfolio of the four currencies, and then again with the US stock market as a fifth asset in the portfolio. We also tested the restriction implied by the CAPM, namely that the risk premia are proportional to the conditional covariance matrix of the residuals.

The empirical findings indicate that the specification of the process by which conditional second moments of returns vary over time affects significantly the estimate of the risk aversion parameter, and as a result, affects the estimates of the ex-ante risk premium on various assets. We find that both lagged conditional variances and nominal interest rates have significant predictive ability for second moments of asset returns. We also find that the inclusion of the US stock market in the world portfolio does have an effect on the estimates of the risk aversion parameter. For all the models we estimate, however, the overidentifying restrictions imposed by the CAPM are decisively rejected. We also explored the ability of the CAPM to product the behaviour of risk premia. We obtained "unrestricted" estimates of risk premia for the assets used in our model by regressing asset returns on a constant, the forward premium, lagged squared asset returns and the assets over shown in the portfolio. Although these forecasts are likely to be quite noisy, the size of the fluctuations was remarkable: the excess returns over dollar deposits fluctuate easily between plus and minus 0.4 percent per week, or 20 percent per annum. The ability of the CAPM to reproduce these numbers depends on the volatility of asset supplies, the volatility of conditional second moments, and the size of the risk aversion coefficient. Frankel has argued that, with constant conditional second moments and reasonable risk aversion parameters, the CAPM cannot possibly reproduce the behaviour of unrestricted estimates of the risk premia. Does allowing the conditional second moments to vary over time make the model's predictions closer to the unrestricted estimates? In all cases, the fluctuations of the risk premia predicted by the CAPM are much smaller than, and bear little resemblance to, the unrestricted ones. Finally, since the general pattern of movements in estimated conditional variances (their peaks and troughs) do not differ dramatically across the three models we specify, it appears that the failure of the CAPM can be ascribed to the differences between the fluctuations of conditional variances and those of the unrestricted estimates of risk premia which can not be explained by fluctuations in asset supplies.

Overall, the results of this paper tend to be discouraging to those who believe that the static CAPM is a fair description of the determination of equilibrium returns in world financial markets. However, the evidence also seems to suggest that a thoroughly satisfactory test of the static CAPM would probably require the inclusion of many more assets than those we use, and a much more complete specification of the process followed by conditional second moments. Both of these extensions involve the construction of very large models, that - given the current computational technology - are quite difficult and expensive to estimate.