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.
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 [
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
This paper is organized as follows. We describe the basic idea to solve multiperiod optimal portfolio weights under a CRRA utility function in Section
Suppose the existence of a finite number of risky assets indexed by
There exists an optimal portfolio weight
Then the return of the portfolio from time
Suppose that a utility function
Following a formulation by the dynamic programming (e.g., Bellman [
where
According to the literature (e.g., [
From this, the value function
The corresponding FOCs (in terms of
Suppose that
Given
Let
Given
Based on the above
First, we fix the current time Next, for each Next, we construct estimators of
Note that the dimensions of In order to construct the estimators of Then, by using
Based on the above
In the same manner of Steps
Then, we define an optimal portfolio weight estimator at time
For each time
In this section we examine our approach numerically. Suppose that there exists a risky asset with the excess return
Let
It can be seen that
Next, we construct the optimal portfolio estimator
Figure
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
Next, we repeat the above algorithm 100 times using the different generated data. Tables
We can see that for all
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) |
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) |
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) |
Resampled excess return.
Myopic and dynamic portfolio return.
In this example, we examine effect of the initial sample size (
It can be seen that the medians tend to increase with increased amount of
Boxplot1.
In this example, we examine effect of the AR parameter (
Obviously, the medians increase with decreased amount of
Boxplot2.