برآورد و شبیه سازی اجرت های ریسک در بازارهای سهام و ارز

There is substantial evidence to reject constant-risk-premia financial models. While time-varying risk premia are often mentioned as an alternative, the literature has yet to produce an example that accounts for the important time-series properties of asset returns. We inquire whether mean-variance optimization models can do so. We model asset risk with an absolute-error version of the ARCH-in-mean hypothesis and model hedging motives that derive from variation in future real income and inflation to account for agent heterogeneity. We consider a three-country-and-two-asset world. Our model predicts values for five excess returns relative to the US bill rate. We use a systems approach to estimate the model parameters and then simulate the estimated model to determine if it can account for the important time-series properties of risk premia.

The evidence is substantial to reject models of financial markets that imply that constant risk premia are imbedded in asset prices. While a widely suggested alternative is a risk premium that varies through time, the literature has yet to produce a model with time-varying risk premia that accounts for the important time-series properties of asset returns. The purpose of this paper is to determine whether mean-variance optimization models can do so. We confine our attention to models where agents choose a portfolio that maximizes a function that is increasing in expected wealth and decreasing in the variance of wealth. Kim (1995) discusses the place of these models in financial theory. We focus on the mean-variance optimization (MVO) model because it is simple and because we believe that variation in wealth is a better proxy for risk than variation in consumption flows especially when aggregate data are employed. Much of the recent work with MVO models traces back to Frankel (1982) which assumes that the covariance matrix of asset returns is constant through time. Frankel shows, and others have since confirmed, that constant-covariance MVO models can not explain the variation in excess returns observed in the data. More recent studies have assumed that the return-error covariance matrix is time-varying and generated by an ARCH or GARCH process. Examples include Engel and Rodrigues (1989), Giovannini and Jorion (1989), Ng (1991), and Thomas and Wickens (1993). None of these studies has successfully explained the large variation observed in excess returns. We make several modifications to the standard MVO model. First, we model asset risk with a version of the ARCH-in-mean hypothesis in which return-error variances depend on the absolute value of lagged residuals. This modification implies that the first partial derivatives of the return-error variance with respect to lagged errors are constant rather than increasing in the size of the error and can potentially increase the effect of medium and small errors on risk. Second, we model the dependence of real wealth on non-portfolio real income and inflation. Optimizing agents adjust their portfolios to hedge risk associated with variation in real income and inflation, and we inquire whether allowing for this hedging motive can improve the empirical performance of the model. Third, we consider a three-country-and-two-asset world with bills1 and equity issued in each country so that agents can hold six assets.2 Using the US bill rate as the reference rate, our model explains five risk premia: the expected excess return on equity in each country and the expected excess return on German and Rest-of-the-World bills. The international nature of financial markets suggests that agents of different countries face many common sources of risk. We thus expect more precise estimates of risk premia from a study that simultaneously models risk premia in several countries. While agents of each country are the same ex ante, our model is not a representative-agent model in the standard sense of the term. The time series behavior of real wealth differs across countries because agents value wealth in their own currency and because they face different income and inflation processes. Agents in our model trade assets. Engel (1994) sets out a version of the Solnik (1974) model which also has agents from several countries. Because it ignores real income flows and assumes that price levels are constant, the model implies that agents from different countries hold portfolios with different bond shares but the same equity shares. Engel's test of the Solnik model rejects it relative to an alternative that allows across-country differences in bond and equity portfolio shares. Engel concludes that the Solnik model does not allow for sufficient investor heterogeneity. Our model allows for additional investor heterogeneity by recognizing that investors from different countries face different income and inflation processes and thus have different risks to hedge. The empirical strategy of our study has two parts. First, we derive an expression for the equilibrium value of risk premia, and use a systems approach to obtain joint estimates of the parameters of the risk premium equation, the time-series equations for real income and inflation, and the multivariate ARCH model for the error terms. Second, we treat the estimated model as a data generating process, generate repeated samples, and determine whether these samples embody the stylized facts describing the time series behavior of risk premia. The stylized facts that we hope to explain are documented in 2, 5 and 6. Section 3 sets out the model. Sections 4, 5 and 6 present the data, the estimation results and our simulation findings. Section 7 offers our conclusions.

We investigated whether a class of mean-variance optimization models can generate risk premia that embody the stylized facts reported in the literature. We set out a multi-country model in which real incomes and inflation vary through time and agents want to hedge the risk associated with these variations. Agents from different countries trade assets because their exchange risk and hedging motives differ. We model time-variation in risk by modifying the ARCH process of Bollerslev so that conditional variances depend on the absolute values of lagged residuals. To date, most studies that employ an MVO model and assume a standard ARCH process have not obtained significant estimates of the coefficient of risk aversion. Our modifications to the standard model permit us to obtain an estimate of the coefficient of risk aversion that is reasonable in size and highly statistically significant. Our model can explain the high positive correlation observed across risk premia of different countries and asset types while a typical representative agent model can not do so. Our model can account for observed equity premia. It can also account for the high degree of persistence in equity returns reported in the literature because it produces expected equity returns that are highly serially correlated. Our model likewise produces foreign exchange risk premia that are highly persistent. Our model can account for Fama's findings that foreign exchange risk premia and expected rates of exchange depreciation are negatively correlated. It can not account for the large time-variation in foreign exchange risk premia reported by Fama. In future work, we intend to focus on modifications to the MVO model that can produce higher variability in foreign exchange risk premia, possibly by including information differences among agents.