Statistical Portfolio Estimation under the Utility Function Depending on Exogenous Variables

In the estimation of portfolios, it is natural to assume that the utility function depends on exogenous variable. From this point of view, in this paper, we develop the estimation under the utility function depending on exogenous variable. To estimate the optimal portfolio, we introduce a function of moments of the return process and cumulant between the return processes and exogenous variable, where the function means a generalized version of portfolio weight function. First, assuming that exogenous variable is a random process, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a shot noise in the frequency domain. Second, assuming that exogenous variable is nonstochastic, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a harmonic trend. We also evaluate the influence of exogenous variable on the return process numerically. 1. Introduction In the usual theory of portfolio analysis, optimal portfolios are determined by the mean and the variance of the portfolio return . Several authors proposed estimators of optimal portfolios as functions of the sample mean and the sample variance for independent returns of assets. However, empirical studies show that financial return processes are often dependent and non-Gaussian. Shiraishi and Taniguchi [1] showed that the above estimators are not asymptotically efficient generally if the returns are dependent. Under the non-Gaussianity, if we consider a general utility function , the expected utility should depend on higher-order moments of the return. From this point of view, Shiraishi and Taniguchi [1] proposed the portfolios including higher-order moments of the return. However, empirical studies show that the utility function often depends on exogenous variable . From this point of view, in this paper, we develop the estimation under the utility function depending on exogenous variable. Denote the optimal portfolio estimator by a function where hat means the sample version of . Although Shiraishi and Taniguchi’s [1] setting does not include the exogenous variable in , we can develop the asymptotic theory in the light of their work. First, assuming that is a random process, we derive the asymptotic distribution of . Then, an influence of on the return process is illuminated when has a shot noise in the frequency domain.