A Maximum Entropy Method for a Robust Portfolio Problem

2.1. Notations and Assumptions

Assume that an investor allocates his initial wealth to n risky assets in a market with transaction costs and dividends, in which transaction costs include capital income tax and basic income tax. Let t_g be the tax rate of marginal capital income. t₀ is the tax rate of marginal basic income. t_f represents the commission rate of transaction. t_s is the stamp tax rate of transaction. The rate of dividends on asset i is specified by d_i. Let ${\tilde{r}}_i$ be the return rate of asset i, which is a random variable. Let r_i denote the mathematical expectation of the random variable of ${\tilde{r}}_i$. σ_ij is the covariance between the random variables ${\tilde{r}}_i$ and ${\tilde{r}}_j$. (σ_ij)_n×n is the matrix of asset volatilities, which can be used to measure the portfolio risk. Let $x_i^0$ and x_i be the proportion invested in asset i of initial portfolio and optimal portfolio respectively. The expected return and risk of a portfolio are respectively given by ${\displaystyle {\sum }_{i=1}^n{r}_i{x}_i}$ and ${\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^n{\sigma }_{ij}{x}_i{x}_j}}$.

Next, we give the following assumptions:

Assumption 1. The covariance matrix (σ_ij)_n×n is strictly positively definite.

Assumption 2. The expected return for asset i is unknown-but-bounded. Instead of defining a box from an uncertain set, we consider that the expected return r_i is located in a known interval. That is, r_i ∈ (a_i, b_i), where a_i > 0, b_i > 0. They can be obtained by simulating from a probability model for future returns.

Assumption 3. The market considered in this paper includes transaction costs and dividends. The transaction cost is the V type function of the volume of transaction. The capital income tax contains the commission c₁(x) and stamp tax c₂(x). They are given by

To make things easier, it is useful to define the following concepts. The net return is the total returns of expected return and dividends after paying the basic income tax and capital income tax. The pre-tax profit represents the total returns of expected return and dividends excluding the commission and stamp tax. The after-tax profit is the value of pre-tax profit after paying the capital income tax.

2.2. Robust Model

On the basis of the above notations and assumptions, we are now ready to formulate the robust portfolio model in the market with transaction costs and dividends.

Generally speaking, the stamp tax can be eliminated as a pre-tax expense in the period. However, the commission can not be ruled out since it is considered as investment cost. Therefore, the total pre-tax profit of a portfolio can be expressed by

Hence, the after-tax profit of a portfolio can be obtained by the following expression:

Substituting expressions c₁(x) and c₂(x) into the above expression, it yields

Define

and

Then the after-tax profit of a portfolio can be expressed as:

It is obvious that R_i represents the rate of net return on risky asset i. Moreover, the parameter k can be explained by the cost of market transaction. Hence, the variance of a portfolio can be formulated as:

Based on the mean-variance framework, in our model we attempt to maximize the minimization over expected returns subject to the constraint that all of the asset expected returns lie within some specified intervals. Therefore, it can be formulated by the following min-max problem:

(1)

where the parameter ω ∈ [0, +∞) characterizes the investor’s risk aversion, ${a^{\prime }}_i=(1-t_g)a_i+(1-t_0)d_i$, ${b^{\prime }}_i=(1-t_g)b_i+(1-t_0)d_i$.

3.1. Maximum Entropy Method

We first consider the following min-max optimal problem:

where x ∈ Ω ⊆ Rⁿ, y ∈ Ω′ = [0, N₁] × [0, N₂] × ⋯ [0, N_m] ⊂ R^m.

We define

Obviously, Problem (2) is equivalent to the following optimization problem:

According to the results by Huang and Shen [18], and its references such as Shannon [19], Skilling and Gull [20], Everett [21], Brooks and Geoffrion [22], Gould [23] and Greenberg and Pierskalla [24], Problem (5) is equivalent to the following problem:

(6)

where λ(y) represents the probability density function of the y. The last constraint is the famous maximum entropy condition. In the following, we will provide the optimal probability density function λ^∗(y) of y according to the maximum entropy constraint.

Then, the Lagrange function of Problem (6) can be expressed by

where α, μ, ρ > 0 are all the Lagrange multipliers.

Denote

$$T(x,\lambda (y))=\alpha \lambda (y)g(x,y,z)+\mu (\lambda (y)-\frac{1}{M})-\frac{1}{\rho }\lambda (y)\ln \lambda (y),$$

where $M={\displaystyle {\prod }_{i=1}^m{N}_i}$.

According to variational principle, it yields

$$\frac{\partial T(x,\lambda (y))}{\partial \lambda (y)}=0.$$

(7)

Since g(x, y, z) is unrelated to λ(y), it follows from (7) that

$$\alpha g(x,y,z)+\mu -\frac{1}{\rho }[\ln {\lambda }^{*}(y)+1]=0.$$

It is known from (6) that

$${\displaystyle {\int }_0^{N_1}\cdots {\displaystyle {\int }_0^{N_m}{\lambda }^{*}}(y)}d{y}_1\cdots d{y}_m=1.$$

(8)

Thus, it has the following system of equations:

$$\{\begin{array}{c}\alpha g(x,y,z)+\mu -\frac{1}{\rho }[\ln {\lambda }^{*}(y)+1]=0,\\ {\displaystyle {\int }_0^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}{\lambda }^{*}}(y)d{y}_1\cdots d{y}_m=1.\end{array}$$

(9)

From the first equation of (9), we have

$${\lambda }^{*}(y)=\exp \{\alpha \rho g(x,y,z)\}\exp \{\rho \mu +1\}.$$

(10)

Substituting (10) into the second equation of (9), we get

$${\displaystyle {\int }_0^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}\exp \{\alpha \rho g(x,y,z)\}\exp \{\rho \mu +1\}d{y}_1\cdots d{y}_m}=1.$$

Since exp{ρμ + 1} is a constant, it follows that

$$\exp \{\rho \mu +1\}=\frac{1}{{\displaystyle {\int }_0^{N_1}\cdots {\displaystyle {\int }_0^{N_m}\exp }\{\alpha \rho g(x,y,z)\}d{y}_1\cdots d{y}_m}}.$$

(11)

Hence, in view of (11) and (10), we obtain that

$${\lambda }^{*}(y)=\frac{\exp \{\alpha \rho g(x,y,z)\}}{{\displaystyle {\int }_0^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}\exp \{\alpha \rho g(x,y,z)\}d{y}_1}\cdots d{y}_m}.$$

(12)

Substituting (12) into the original Lagrange function, we get

$$L=z+\frac{1}{\rho }\ln {\displaystyle {\int }_{\rho }^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}\exp \{\alpha \rho g(x,y,z)\}d{y}_1\cdots d{y}_m}.$$

It follows from (3) and (4) that

Define

$$F_p(x):=\frac{1}{p}\ln {\displaystyle {\int }_0^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}\exp \{pg(x,y,z)\}d{y}_1\cdots d{y}_m}.$$

Obviously, F_p(x) is a multi-dimensional continuous maximum entropy function. We can derive that

$$L=z(1-\alpha )+\alpha F_p(x),$$

where p = αρ.

Define

$$F(x)=\underset{y\in {\omega }^{\prime }}{max}f(x,y).$$

Our main objective is to prove the convergence of L. If we can prove F_p(x) converges to F (x), then we can get the convergence of the function L. That means L converges to z(1 − α) + αF (x). We find that the value of α does not affect the convergence of L.

Without loss of generality, we set α = 1, then

$$L=F_p(x)=\frac{1}{p}\ln {\displaystyle {\int }_0^{N_1}\cdots }{\displaystyle {\int }_0^{N_m}\exp \{pg(x,y,z)\}d{y}_1\cdots d{y}_m.}$$

Theorem 4. For any m > 1 and p > 1, we have$\left|F_p(x)-F(x)\right|\le \left|\frac{\ln M}{p}\right|$. Moreover, if p → + ∞, then F_p(x) → F(x). Where$F_p\left(x\right)=\frac{1}{p}\ln {\displaystyle {\int }_0^{N_1}\cdots {\displaystyle {\int }_0^{N_m}\exp \left\{pf\left(x,y\right)\right\}d{y}_1\cdots d{y}_m,}}F\left(x\right)={max}_{y\in {\omega }^{\prime }}f\left(x,y\right),M={\displaystyle {\prod }_{i=1}^m{N}_i}$.

Proof. For any x ∈ Ω

Since

$$f\left(x,y\right)-F\left(x\right)=f\left(x,y\right)-\underset{y\in {\omega }^{\prime }}{max}f\left(x,y\right)\le 0,p>1,$$

then we obtain

$$p(f(x,y)-F(x))\le 0.$$

Moreover, we can deduce that

$$\exp \{p(f(x,y)-F(x))\}\le 1.$$

Hence, we arrive at

$$\left|F_p\left(x\right)-F\left(x\right)\right|\le \left|\frac{1}{p}\ln {\displaystyle {\int }_0^{N_1}\cdots {\displaystyle {\int }_0^{N_m}d{y}_1\cdots d{y}_m}}\right|\le \left|\frac{\ln M}{p}\right|.$$

Thus, F_p(x) → F (x), when p → +∞. □

3.2. Optimal Solution

Based on the discussion of subsection 3.1, now we aim to determine the solution of Problem (1). Since

then Problem (1) can be reformulated as follows:

where ${R^{\prime }}_i=R_i-{a^{\prime }}_i$.

It can be seen that min_x∈Ω F_p(x) is an approximation of Problem (13).

Let

According to Theorem 4, Problem (13) can be approximated by the following constrained optimization problem:

For the sake of convenience, denote

Therefore, Problem (14) can be re-written as

Substituting both (15) and (16) into (17), Problem (14) can be reformulated as follows:

Obviously, Problem (18) is a non-smooth optimization problem. So the derivative of the objective function does not exist at some point. Thus, a derivative-free method is needed to solve this problem. In this paper, we adopt the widely used simplex search method (Lagarias et al. [25]), which is available in Matlab by the subroutine ‘fmincon’.

4.1. Comparison with a Classical Mean-Variance Model

The classical mean-variance model under the same market situation of this paper can be formulated as follows:

where the notations of mathematical symbols are the same as this paper.

Now, we provide the market performances for the classical mean-variance model and the robust one. Let’s assign the following parameters: p = 30000, t_g = 0.3, t_s = 0.00002, t_f = 0.00007. For simplification, we set $x_i^0=0$.

By using the subroutine “fmincon”in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of classical mean-variance model in the following Table 3 and Figure 1 respectively.

By using the maximum entropy method and subroutine ‘fmincon’ in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of robust mean-variance model in the following Table 4 and Figure 2 respectively.

Next, we put the efficient frontiers of the robust mean-variance model and classical mean-variance model in the same coordinate plane.

Figure 3 shows the efficient frontiers to the robust model and classical mean-variance one. It indicates that the classical mean-variance efficient frontier lies far below the robust one. An immediate finding is that the investor selecting the mean-variance strategy must take more risk than one selecting the robust strategy under the same conditions of portfolio return. In other words, under the same conditions of portfolio risk, an investor selecting the mean-variance strategy has less return than one selecting the robust strategy. This means that the optimal portfolio in the mean-variance framework is not a good portfolio. More importantly, this robust portfolio strategy will help investors avoid excessive losses when unexpected events happen.

4.2. Comparison without Transaction Costs

The robust mean-variance model without transaction costs can be expressed as follows:

where the notations of mathematical symbols are the same as this paper.

By using the maximum entropy method and subroutine “fmincon” in Matlab, we obtain the optimal strategies, efficient points and efficient frontier of robust mean-variance model without transaction costs in the following Table 5 and Figure 4 respectively.

Comparing Table 4 with Table 5, we find that transaction costs play as a penalty factor for portfolio revision. Furthermore, it tells us that the impact of transaction costs can not be ignored in the real world when portfolio managers choose investment strategy.