#### Abstract

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. Second, assuming that is a nonrandom sequence of variables which satisfy Grenander’s conditions, we also derive the asymptotic distribution of . Then an influence of on is evaluated when is a sequence of harmonic functions. Numerical studies will be given, and they show some interesting features.

The paper is organized as follows. Section 2 introduces the optimal portfolio of the form and provides the asymptotic distribution of . Assuming that is a stochastic process, we derive the asymptotics of when has a shot noise in the frequency domain. The influence of on is numerically evaluated in Section 2.2. Assuming that is a nonrandom sequence satisfying Grenander’s conditions, we derive the asymptotic distribution of . Section 3 provides numerical studies for the influence of on when is a sequence of harmonic functions. The appendix gives the proofs of all the theorems.

#### 2. Optimal Portfolio with the Exogenous Variables

Suppose the existence of a finite number of assets indexed by . Let denote the random returns on assets at time , and let denote the exogenous variables influencing on the utility function at time . We write

Since it is empirically observed that is non-Gaussian and dependent, we will assume that it is a non-Gaussian stationary process with the 3^{rd}-order cumulants. Also, suppose that there exists a risk-free asset whose return is denoted by . Let and be the portfolio weights at time , and the portfolio is whose higher-order cumulants are written as

We use Einstein’s summation convention here and throughout the paper. For a utility function , the expected utility can be approximated as by Taylor expansion of order 3. The approximate optimal portfolio may be described as

Solving (2.4), Shiraishi and Taniguchi [1] introduced the optimal portfolio depending on the mean, variance, and the third-order cumulants, and then derived the asymptotic distribution of a sample version estimator. Although our problem is different from that of Shiraishi and Taniguchi [1], we develop the discussion with the methods inspired by them.

Introduce a portfolio estimator function based on observed higher-order cumulants, and assume that the function is -dimensional and measurable, that is,

Let the random process be a -vector linear process generated by where is a -dimensional stationary process such that and , with a nonsingular -matrix, ’s are -matrices, and is the mean vector of . All the components of are real. Assuming that has all order cumulants, let be the joint th-order cumulant of . In what follows we assume that, for each ,

Letting be the joint th-order cumulant of , we define the th-order cumulant spectral density by

which is expressed as where is the th-order cumulant spectral density of , and is the th element of . We introduce the following quantities: where . Write the quantities that appeared in (2.4) by where . Then , where , , , .

First, we derive the asymptotics of the fundamental quantity .

Theorem 2.1. *Under the assumptions,
**
where
**
and the typical element of corresponding to the covariance between and is denoted by , and
*

In what follows we place all the proofs of theorems in the appendix.

Next we discuss the estimation of portfolio . For this we assume that the portfolio function is continuously differentiable. Henceforth, we use a unified estimator for . The -method and Slutsky’s lemma imply the following.

Theorem 2.2. *Under the assumptions
**
where, .*

The quantities ’s are the 3rd-order cumulants of the process, which show the non-Gaussianity. For the returns of five financial stocks IBM, Ford, Merck, HP, and EXXON, we calculated the standardized 3rd-order cumulants where is the sample variance of the stock. Table 1 below shows their values.

From Table 1 we observe that the five returns are non-Gaussian. In view of Theorem 2.1, it is possible to construct the confidence interval for in the following form: where is the upper level- point of and is a consistent estimator of calculated by the method of Keenan [2] and Taniguchi [3].

##### 2.1. Influence of Exogenous Variable

In this subsection we investigate an influence of the exogenous variables on the asymptotics of the portfolio estimator .

Assume that the exogenous variables have “shot noise” in the frequency domain, that is, where is the Dirac delta function with period , and , hence has one peak at .

Theorem 2.3. *For (2.17), denote and in Theorem 2.1 by and , respectively. That is, and represent the asymptotic variance when the exogenous variables are shot noise. Then,
*

##### 2.2. Numerical Studies for Stochastic Exogenous Variables

This subsection provides some numerical examples which show the influence of on .

*Example 2.4. *For a risk-free asset and risky asset , we consider construction of optimal portfolios . Here is the return process of the risky asset, which is generated by
where , . We assume that , and that the exogenous variable in the frequency domain is given by . Write,
then
which are covariances between and , and show an influence on . From Figure 1, it is seen that as tends to 1, and tends to 0, then increases. If tends to −1 and tends to , , then also increases, which entails that the exogenous variables have big influence on the asymptotics of estimators when is close to the unit root of AR(2.2).

*Remark 2.5. * is robust for the shot noise in at .

#### 3. Portfolio Estimation for Nonstochastic Exogenous Variables

So far we assumed that the sequence of exogenous variables is a random stochastic process. In this section, assuming that is a nonrandom sequence, we will propose a portfolio estimator, and elucidate the asymptotics. We introduce the following quantities,

We assume that ’s satisfy Grenander’s conditions (G1)–(G4) with(G1).(G2).(G3) for .(G4) the -matrix is regular.

Under Grenander’s conditions, there exists a Hermitian matrix function with positive semidefinite increments such that

is the regression spectral measure of . Next we discuss the asymptotics for sample versions of and . For this we need the following assumption, There exists constant such that where is the spectral density matrix of .

Theorem 3.1. *Under Grenander’s conditions and the assumption
**
where the -th element of is given by
*

Theorem 3.2. *Under Grenander’s conditions and the assumption
**
where with
*

##### 3.1. Numerical Studies for Nonstochastic Exogenous Variables

Letting and be scalar processes, we investigate an influence of non-stochastic process on . The figures below show influence of harmonic trends on in . In these cases measures the amount of covariance between and .

*Example 3.3. *Let the return process and the exogenous process be generated by
where ’s are i. i. d. *N*(0,1) variables. Next, suppose that consists of harmonic trends with period and the quarter period. We plotted the graph of in Figure 2.*Example 3.4. *Assume that and are generated by
We observe that there exist two peaks in Figure 3. If and , increases rapidly. For further study it may be noted that Cheng et al. [4] discussed statistical estimation of generalized multiparameter likelihood models. Although these models are for independent samples, there is some possibility to apply them to our portfolio problem.

#### Appendix

This section provides the proofs of theorems.

*Proof of Theorem 2.1. *Our setting includes the exogenous variables. Although Shiraishi and Taniguchi’s [1] setting does not include them, we can prove the theorem in line with Shiraishi and Taniguchi [1].

Let
From Fuller [5], it is easy to see that
where
Then we can see that
where is the sum over with and satisfying .

Hence it follows that
In what follows we assume that is an arbitrary permutation of ,
Then we get
By use of Fourier transform, we see that
The other asymptotic covariances are similarly evaluated. Finally, it suffices to prove the asymptotic normality of . For this we prove
where and are the th component of and , respectively. Let
Then, similarly as in Shiraishi and Taniguchi [1] we can see that
which implies the asymptotic normality of .

*Proof of Theorem 3.1. *We write by . Then,
which leads to
Next we evaluate the covariance:
where is the covariance function of and .

The asymptotic normality of can be shown if we prove
Similarly as in Theorem 5.11.1 of Brillinger [6], we can see that

*Proof of Theorem 3.2. *First, it is seen that
We can evaluate the covariance as follows:
Next we derive the asymptotic normality of . For this we prove
Similarly as in Theorem 5.11.1 of Brillinger [6], it is shown that
which proves the asymptotic normality.

#### Acknowledgments

The authors thank Professor Cathy Chen and two referees for their kind comments.