Advances in Decision Sciences

Volume 2012 (2012), Article ID 341476, 13 pages

doi:10.1155/2012/341476

Research Article

A Simulation Approach to Statistical Estimation of Multiperiod Optimal Portfolios

Hiroshi Shiraishi

The Jikei University School of Medicine, Tokyo 1828570, Japan

Received 24 February 2012; Accepted 9 April 2012

Academic Editor: Kenichiro Tamaki

Copyright © 2012 Hiroshi Shiraishi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 insights into the properties of optimal portfolio policies. Unfortunately, since it is known that the closed-form solutions are obtained only for a few special cases, the recent literature uses a variety of numerical and approximate solution methods to incorporate realistic features into the dynamic portfolio problem such as Ait-Sahalia and Brandet [4] and Brandt et al. [5].

We introduce an procedure to construct the dynamic portfolio weights based on AR bootstrap.

The simulation algorithm is as follows; first, we generate simulation sample paths of the vector random returns by using AR bootstrap. Based on the bootstrapping samples, an optimal portfolio estimator, which is applied from time to the end of trading time , is obtained under a constant relative risk aversion (CRRA) utility function. Note that this optimal portfolio corresponds “myopic” (single period) optimal portfolio. Next we approximate the value function by linear functions of the past observation. This idea is similar to that of [4, 5]. Then, optimal portfolio weight estimators at each trading time are obtained based on the value function. Finally, we construct an optimal investment strategy as a sequence of the optimal portfolio weight estimators.

This paper is organized as follows. We describe the basic idea to solve multiperiod optimal portfolio weights under a CRRA utility function in Section 2. In Section 3, we discuss an algorithm to construct the estimator involving the method of AR bootstrap. The applications of our method are in Section 4.

2. Multiperiod Optimal Portfolio

Suppose the existence of a finite number of risky assets indexed by , . Let denote the random excess returns on assets from time to (suppose that is a value of asset at time . Then, the return is described as ). Suppose too that there exists a risk-free asset with the excess return (Suppose that is a value of risk-free asset at time . Then, the return is described as ). Based on the process and , we consider an investment strategy from time 0 to time where denotes the end of the investment time. Let be vectors of portfolio weight for the risky assets at the beginning of time . Here we assume that the portfolio weights can be rebalanced at the beginning of time and measurable (predictable) with respect to the past information . Here we make the following assumption.

Assumption 2.1. There exists an optimal portfolio weight satisfied with (we assume that the risky assets exclude ultra high-risk and high-return ones, for instance, the asset value may be larger than ), almost surely for each time where .

Then the return of the portfolio from time to is written as (assuming that , the portfolio return is written as + + = ) and the return from time 0 to time (called terminal wealth) is written as

Suppose that a utility function is differentiable, concave, and strictly increasing for each . Consider an investor’s problem

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 :

subject to the terminal condition . 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 is assumed) in order to obtain an optimal solution at each time are

where . These FOCs make up a system of nonlinear equations involving integrals that can in general be solved for 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 denotes the coefficient of relative risk aversion. In this case, the Bellman equation simplifies to

where and .

From this, the value function can be expressed as

and also satisfies a Bellman equation

subject to the terminal condition .

The corresponding FOCs (in terms of ) are

3. Estimation

Suppose that is an -vector AR(1) process defined by

where is a constant -dimensional vector, are independent and identically distributed (i.i.d.) random -dimensional vectors with and ( is a nonsingular by matrix), and is a nonsingular by matrix. We make the following assumption.

Assumption 3.1. on .

Given , the least-squares estimator of is obtained by solving

where ,   and . Then, the error is “recovered” by

Let denote the distribution which puts mass at . Let (for ) be i.i.d. bootstrapped observations from .

Given , define and by

for .

Based on the above for each , we construct an estimator of the optimal portfolio weight as follows.

Step 1. First, we fix the current time which implies that the observed stretch is fixed. Then, we can generate by (3.4).

Step 2. Next, for each , we obtain as the maximizer of

or the solution of

with respect to . Here we introduce a notation “” as an estimator of conditional expectation , which is defined by for any function of . This corresponds to the estimator of myopic (single period) optimal portfolio weight.

Step 3. Next, we construct estimators of . Since it is difficult to express the explicit form of , we parameterize it as linear functions of as follows;

Note that the dimensions of in and are and , respectively. The idea of and is inspired by the parameterization of the conditional expectations in [5].

In order to construct the estimators of , we introduce the conditional least squares estimators of the parameter , that is,

where

Then, by using , we can compute .

Step 4. Based on the above , we obtain as the maximizer of

or the solution of

with respect to . This does not correspond to the estimator of myopic (single period) optimal portfolio weight due to the effect of .

Step 5. In the same manner of Steps 3–4, we can obtain and , recursively, for .

Step 6. Then, we define an optimal portfolio weight estimator at time as by Step 4. Note that is obtained as only one solution because is independent of .

Step 7. For each time , we obtain by Steps 1–6. Finally, we can construct an optimal investment strategy as .

4. Examples

In this section we examine our approach numerically. Suppose that there exists a risky asset with the excess return at time and a risk-free asset with the excess return . We assume that is defined by the following univariate AR(1) model:

Let be a portfolio weight for the risky asset at the beginning of time . Suppose that an investor is interested in the investment strategy from time 0 to time . Then the terminal wealth is written as (2.1). Applying our method, the estimator can be obtained by

where is the estimator of optimal portfolio under the CRRA utility function defined by (2.5). In what follows, we examine the effect of for a variety of (initial sample size), (resampling size), (AR parameter), (variance of ), (relative risk aversion parameter), and (defined by (3.7) or (3.8)).

Example 4.1 (myopic (single period) versus dynamic (multiPeriod)). Let , , , , , and . We generate the excess return process by (4.1). First, for each we generate by (3.4) based on (as Step 1). We plot and in Figure 1.

It can be seen that show similar behavior with .

Next, we construct the optimal portfolio estimator along the lines with Steps 2–7. Here we apply the approximated solution for (3.5) or (3.11) following [5], that is,

where

This approximate solution describes a fourth-order expansion of the value function around ( 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 (, Line 1), cumulative portfolio return (, Line 2), and value of utility function (, Line 3) for and 20. The solid line shows the investment only for risk-free asset (i.e., ), the dotted line with shows myopic (single period) portfolio (i.e., ) and the dotted line with + shows dynamic (multiperiod) portfolio by using .

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 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 , we can see that the magnitude of the hedging demands decreases with increased amount of .

Next, we repeat the above algorithm 100 times using the different generated data. Tables 1, 2, and 3 show means, 25 percentiles (), medians (), and 75 percentiles () of terminal wealth () and the values of utility function () for , and .

We can see that for all , the means of terminal wealth are larger than that of risk-free investment (i.e., ). In view of the distribution of , the means are larger than the medians () which shows the asymmetry of the distribution. Among the myopic, dynamic portfolio using and , dynamic portfolio using is the best investment strategy in view of the means of or . There are some cases that the means of for dynamic portfolio using are smaller than those for myopic portfolio. This phenomenon would show the inaccuracy of the approximation of . In addition, in view of the dispersion of , the dynamic portfolio’s one is relatively smaller than the myopic portfolio’s one.

Table 1: Dynamic portfolio returns for .

Myopic
Dynamic ()
Dynamic ()

Mean
()
Mean
()
Mean
()

A: Terminal wealth

1
1.013564
(0.9924, 1.0091, 1.0192)
1.013667
(0.9920, 1.0096, 1.0176)
1.013814
(0.9920, 1.0096, 1.0176)

2
1.024329
(0.9917, 1.0192, 1.0445)
1.024396
(0.9923, 1.0177, 1.0436)
1.024667
(0.9924, 1.0183, 1.0437)

5
1.065896
(1.0021, 1.0504, 1.1125)
1.065988
(1.0000, 1.0509, 1.1115)
1.066355
(0.9999, 1.0505, 1.1106)

10
1.137727
(1.0273, 1.1062, 1.2024)
1.137707
(1.0264, 1.1041, 1.2005)
1.138207
(1.0265, 1.1043, 1.2002)

B: Utility of terminal wealth

1
−0.24158
(−0.257, −0.241, −0.231)
−0.24139
(−0.258, −0.240, −0.233)
−0.24130
(−0.258, −0.240, −0.233)

2
−0.23609
(−0.258, −0.231, −0.210)
−0.23595
(−0.257, −0.233, −0.210)
−0.23578
(−0.257, −0.232, −0.210)

5
−0.21761
(−0.247, −0.205, −0.163)
−0.21761
(−0.249, −0.204, −0.163)
−0.21703
(−0.250, −0.205, −0.164)

10
−0.18349
(−0.224, −0.166, −0.119)
−0.18339
(−0.225, −0.168, −0.120)
−0.18287
(−0.225, −0.168, −0.120)

Table 2: Dynamic portfolio returns for .

Myopic
Dynamic ()
Dynamic ()

Mean
()
Mean
()
Mean
()

A: Terminal wealth

1
1.011802
(1.0011, 1.0095, 1.0146)
1.011859
(1.0010, 1.0098, 1.0138)
1.011944
(1.0010, 1.0098, 1.0138)

2
1.022249
(1.0059, 1.0196, 1.0323)
1.022286
(1.0065, 1.0190, 1.0319)
1.022439
(1.0065, 1.0192, 1.0319)

5
1.058344
(1.0276, 1.0512, 1.0825)
1.058373
(1.0254, 1.0509, 1.0823)
1.058584
(1.0253, 1.0507, 1.0818)

10
1.120369
(1.0658, 1.1070, 1.1544)
1.120323
(1.0687, 1.1068, 1.1533)
1.120595
(1.0666, 1.1060, 1.1532)

B: Utility of terminal wealth

1
−0.10224
(−0.109, −0.101, −0.097)
−0.10215
(−0.110, −0.101, −0.098)
−0.10210
(−0.110, −0.101, −0.098)

2
−0.09530
(−0.105, −0.093, −0.083)
−0.09523
(−0.104, −0.093, −0.083)
−0.09515
(−0.104, −0.093, −0.083)

5
−0.07581
(−0.086, −0.070, −0.054)
−0.07582
(−0.088, −0.071, −0.054)
−0.07557
(−0.088, −0.071, −0.054)

10
−0.05007
(−0.062, −0.044, −0.030)
−0.05003
(−0.061, −0.044, −0.030)
−0.04986
(−0.062, −0.044, −0.030)

Table 3: Dynamic portfolio returns for .

Myopic
Dynamic ()
Dynamic ()

Mean
()
Mean
()
Mean
()

A: Terminal wealth

1
1.010905
(1.0055, 1.0097, 1.0123)
1.010934
(1.0055, 1.0099, 1.0119)
1.010979
(1.0054, 1.0099, 1.0119)

2
1.021181
(1.0131, 1.0198, 1.0262)
1.021200
(1.0133, 1.0195, 1.0260)
1.021281
(1.0133, 1.0196, 1.0260)

5
1.054646
(1.0396, 1.0512, 1.0668)
1.054655
(1.0381, 1.0509, 1.0668)
1.054767
(1.0381, 1.0508, 1.0667)

10
1.112289
(1.0853, 1.1062, 1.1297)
1.112256
(1.0876, 1.1060, 1.1291)
1.112396
(1.0865, 1.1056, 1.1290)

B: Utility of terminal wealth

1
−0.04386
(−0.047, −0.043, −0.041)
−0.04382
(−0.047, −0.043, −0.042)
−0.04379
(−0.047, −0.043, −0.042)

2
−0.03705
(−0.041, −0.036, −0.032)
−0.03702
(−0.040, −0.036, −0.032)
−0.03699
(−0.040, −0.036, −0.032)

5
−0.02189
(−0.025, −0.020, −0.015)
−0.02189
(−0.025, −0.020, −0.015)
−0.02181
(−0.025, −0.020, −0.015)

10
−0.00881
(−0.011, −0.007, −0.005)
−0.00880
(−0.010, −0.007, −0.005)
−0.00876
(−0.010, −0.007, −0.005)

Figure 1: Resampled excess return.

Figure 2: Myopic and dynamic portfolio return.

Example 4.2 ((sample size () and resampling size ()). In this example, we examine effect of the initial sample size () and the resample size (). Let , , , , and . In the same manner as Example 4.1, we consider the effect of for and . Figure 3 shows the box plots of the terminal wealth for each and .

It can be seen that the medians tend to increase with increased amount of and . In addition, the wideness of the box plots decreases with increased amount of and . This phenomenon shows the accuracy of the approximation of .

Figure 3: Boxplot1.

Example 4.3 (AR Parameter () and variance of ()). In this example, we examine effect of the AR parameter () and the variance of (). Let , , , , and . In the same manner as Example 4.1, we consider the effect of for , and . Figure 4 shows the box plots of the terminal wealth for each and .

Obviously, the medians increase with decreased amount of which shows that the investment result is preferred when the amount of is small. On the other hand, the wideness of the box plots increases with increased amount of which shows that the difference of the investment result is wide when the amount of is large.

Figure 4: Boxplot2.

References

R. C. Merton, “Lifetime portfolio selection under uncertainty: the continuous-time case,” The Review of Economics and Statistics, vol. 51, no. 3, pp. 247–257, 1969.

P. A. Samuelson, “Lifetime portfolio selection by dynamic stochastic programming,” Review of Economics and Statistics, vol. 51, no. 3, pp. 239–246, 1969.

E. F. Fama, “Multiperiod consumption-investment decisions,” American Economic Review, vol. 60, no. 1, pp. 163–174, 1970.

Y. Aït-Sahalia and M. W. Brandt, “Variable selection for portfolio choice,” Journal of Finance, vol. 56, no. 4, pp. 1297–1351, 2001.

M. W. Brandt, A. Goyal, P. Santa-Clara, and J. R. Stroud, “A simulation approach to dynamic portfolio choice with an application to learning about return predictability,” Review of Financial Studies, vol. 18, no. 3, pp. 831–873, 2005.

R. Bellman, Dynamic Programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, USA, 2010.