A Simulation Approach to Statistical Estimation of Multiperiod Optimal Portfolios

This paper discusses a simulation-based method for solving discrete-time multiperiod portfolio choice problems under AR 1 process. The method is applicable even if the distributions of return processes are unknown. We first generate simulation sample paths of the random returns by using AR bootstrap. Then, for each sample path and each investment time, we obtain an optimal portfolio estimator, which optimizes a constant relative risk aversion CRRA utility function. When an investor considers an optimal investment strategy with portfolio rebalancing, it is convenient to introduce a value function. The most important difference between single-period portfolio choice problems and multiperiod ones is that the value function is time dependent. Our method takes care of the time dependency by using bootstrapped sample paths. Numerical studies are provided to examine the validity of our method. The result shows the necessity to take care of the time dependency of the value function.


Introduction
Portfolio optimization is said to be "myopic" when the investor does not know what will happen beyond the immediate next period.In this framework, basic results about single period portfolio optimization such as mean-variance analysis are justified for short-term investments without portfolio rebalancing.Multiperiod problems are much more realistic than single-period ones.In this framework, we assume that an investor makes a sequence of decisions to maximize a utility function at each time.The fundamental method to solve this problem is the dynamic programming.In this method, a value function which expresses the expected terminal wealth is introduced.The recursive equation with respect to the value function is so-called Bellman equation.The first order conditions FOCs to satisfy the Bellman equation are key tool in order to solve the dynamic problem.
The original literature on dynamic portfolio choice, pioneered by Merton 1 in continuous time and by Samuelson 2 and Fama 3 in discrete time, produced many important

Multiperiod Optimal Portfolio
Suppose the existence of a finite number of risky assets indexed by i, i 1, . . ., m .Let X t X 1 t , . . ., X m t denote the random excess returns on m assets from time t to t 1 suppose that S i t is a value of asset i at time t.Then, the return is described as 1 X i t S i t /S i t−1 .Suppose too that there exists a risk-free asset with the excess return X f Suppose that B t is a value of risk-free asset at time t.Then, the return is described as 1 X f B t /B t − 1 .Based on the process {X t } T t 1 and X f , we consider an investment strategy from time 0 to time T where T ∈ N denotes the end of the investment time.Let w t w 1 t , . . ., w m t be vectors of portfolio weight for the risky assets at the beginning of time t 1.Here we assume that the portfolio weights w t can be rebalanced at the beginning of time t 1 and measurable predictable with respect to the past information F t σ X t , X t−1 , . . . .Here we make the following assumption.

Assumption 2.1. There exists an optimal portfolio weight w
1 we assume that the risky assets exclude ultra high-risk and high-return ones, for instance, the asset value S i t 1 may be larger than 2S i t , almost surely for each time t 0, 1, . . ., T − 1 where e 1, . . ., 1 .
Then the return of the portfolio from time t to t 1 is written as 1 X f w t X t 1 − X f e assuming that S t : S 1 t , . . ., S m t B t e, the portfolio return is written as X f e and the return from time 0 to time T called terminal wealth is written as

2.1
Suppose that a utility function U : x → U x is differentiable, concave, and strictly increasing for each x ∈ R. Consider an investor's problem max Following a formulation by the dynamic programming e.g., Bellman 6 , it is convenient to express the expected terminal wealth in terms of a value function V t :

2.3
subject to the terminal condition V T U W T .The recursive equation 2.3 is the so-called Bellman equation and is the basis for any recursive solution of the dynamic portfolio choice problem.The first-order conditions FOCs here ∂/∂w t E V t 1 |F t E ∂/∂w t V t 1 |F t .isassumed . in order to obtain an optimal solution at each time t are where These FOCs make up a system of nonlinear equations involving integrals that can in general be solved for w t only numerically.
According to the literature e.g., 5 , we can simplify this problem in case of a constant relative risk aversion CRRA utility function, that is, where γ denotes the coefficient of relative risk aversion.In this case, the Bellman equation simplifies to where From this, the value function V t can be expressed as and Ψ t also satisfies a Bellman equation 2.9

Estimation
Suppose that {X t X 1 t , . . ., X m t ; t ∈ Z} is an m-vector AR 1 process defined by where μ μ 1 , . . ., μ m is a constant m-dimensional vector, t 1 t , . . ., m t are independent and identically distributed i.i.d.random m-dimensional vectors with E t 0 and E t t Γ Γ is a nonsingular m by m matrix , and A is a nonsingular m by m matrix.We make the following assumption.
Step 1. First, we fix the current time t which implies that the observed stretch n t is fixed.Then, we can generate Step 2. Next, for each b 0 1, . . ., B, we obtain w b 0 ,t T −1 as the maximizer of or the solution of Advances in Decision Sciences with respect to w.Here we introduce a notation "E * s • " as an estimator of conditional expectation . This w b 0 ,t T −1 corresponds to the estimator of myopic single period optimal portfolio weight.
Step 3. Next, we construct estimators of Ψ T −1 .Since it is difficult to express the explicit form of Ψ T −1 , we parameterize it as linear functions of X T −1 as follows; Note that the dimensions of θ T −1 in Ψ 1 and Ψ 2 are m 1 and m m 1 /2 m 1, respectively.The idea of Ψ 1 and Ψ 2 is inspired by the parameterization of the conditional expectations in 5 .
In order to construct the estimators of Ψ i i 1, 2 , we introduce the conditional least squares estimators of the parameter where

3.10
Then, by using Step 4. Based on the above Ψ i , we obtain w b 0 ,t T −2 as the maximizer of

3.11
Advances in Decision Sciences 7 or the solution of 3.12 with respect to w.This w b 0 ,t T −2 does not correspond to the estimator of myopic single period optimal portfolio weight due to the effect of Ψ i .
Step 6.Then, we define an optimal portfolio weight estimator at time t as w Step 7.For each time t 0, 1, . . ., T − 1, we obtain w t t by Steps 1-6.Finally, we can construct an optimal investment strategy as { w t t } T −1 t 0 .

Examples
In this section we examine our approach numerically.Suppose that there exists a risky asset with the excess return X t at time t and a risk-free asset with the excess return X f 0.01.We assume that X t is defined by the following univariate AR 1 model: Let w t be a portfolio weight for the risky asset at the beginning of time t 1. Suppose that an investor is interested in the investment strategy from time 0 to time T .Then the terminal wealth is written as 2.1 .Applying our method, the estimator W T can be obtained by where w t is the estimator of optimal portfolio under the CRRA utility function defined by 2.   show similar behavior with X t .T Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 A: Terminal wealth T Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 A: Terminal wealth T Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 Mean q 0.25 , q 0.5 , q 0.75 A: Terminal wealth Next, we construct the optimal portfolio estimator w t t along the lines with Steps 2-7.Here we apply the approximated solution for 3.5 or 3.11 following 5 , that is,

Advances in Decision Sciences
This approximate solution describes a fourth-order expansion of the value function around 1 X f w s describes a second-order expansion .According to 5 , a second-order expansion of the value function is sometimes not sufficiently accurate, but a fourth-order expansion includes adjustments for the skewness and kurtosis of returns and their effects on the utility of the investor.
Figure 2 shows time series plots for single portfolio return 1 X f w t X t 1 −X f , Line 1 , cumulative portfolio return W T , Line 2 , and value of utility function 1/ 1 − g W 1−g T , Line 3 for γ 5, 10 and 20.The solid line shows the investment only for risk-free asset i.e., w t 0 , the dotted line with shows myopic single period portfolio i.e., Ψ i 1 and the dotted line with shows dynamic multiperiod portfolio by using Ψ 1 .
Regarding the single-portfolio return, we can not argue the best investment strategy among the risk-free, the myopic portfolio and the dynamic portfolio investment.However, to look at the cumulative portfolio return or the value of utility function, it is obviously that the dynamic portfolio investment is the best one.The difference between the myopic and dynamic portfolio is due to Ψ and is called "hedging demands" because by deviating from the single period portfolio choice, the investor tries to hedge against changes in the investment opportunities.In view of the effect of γ, we can see that the magnitude of the hedging demands decreases with increased amount of γ.
We can see that for all T , the means of terminal wealth W T are larger than that of riskfree investment i.e., 1 X f T .In view of the distribution of W T , the means are larger than the medians q 0.5 which shows the asymmetry of the distribution.Among the myopic, dynamic portfolio using Ψ 1 and Ψ 2 , dynamic portfolio using Ψ 2 is the best investment strategy in view of the means of W T or 1/ 1 − g W

1−g
T .There are some cases that the means of W T for dynamic portfolio using Ψ 1 are smaller than those for myopic portfolio.This phenomenon would show the inaccuracy of the approximation of Ψ.In addition, in view of the dispersion of W T , the dynamic portfolio's one is relatively smaller than the myopic portfolio's one.It can be seen that the medians tend to increase with increased amount of n and B. In addition, the wideness of the box plots decreases with increased amount of n and B. This phenomenon shows the accuracy of the approximation of X * t .
Example 4.3 AR Parameter A and variance of ∈ t Γ .In this example, we examine effect of the AR parameter A and the variance of t Γ .Let μ 0.02, n 100, B 100, T 10, and γ 5.In the same manner as Example 4.1, we consider the effect of W T for A 0.01, 0.1, 0.2, and Γ 0.01, 0.05, 0.10.Figure 4 shows the box plots of the terminal wealth W T for each A and Γ.
Obviously, the medians increase with decreased amount of Γ which shows that the investment result is preferred when the amount of t is small.On the other hand, the wideness of the box plots increases with increased amount of A which shows that the difference of the investment result is wide when the amount of A is large.

4 .
Note that w t t is obtained as only one solution because X

Example 4 . 2
sample size n and resampling size B .In this example, we examine effect of the initial sample size n and the resample size B .Let μ 0.02, A 0.1, Γ 0.05, T 10, and γ 5.In the same manner as Example 4.1, we consider the effect of W T for n 10, 100, 1000 and B 5, 20, 100.Figure3shows the box plots of the terminal wealth W T for each n and B.
the error t s : Y t s − A t Y t s−1 , s −n 2, . . ., t. 3.3 Let F t n • denote the distribution which puts mass 1/ n t at t s .Let { b,t * s 5 .In what follows, we examine the effect of W T for a variety of n initial sample size , B resampling size , A AR parameter , Γ variance of t , γ relative risk aversion parameter , and Ψ defined by 3.7 or 3.8 .