International portfolio selection model with exchange rate risk

Abstract

This paper discusses an international portfolio selection problem under fuzzy environment. Using the possibilistic theory, we first propose an general international portfolio selection model with exchange rate risk under the assumption that the values of unit investment in risky assets and exchange rates are fuzzy numbers. With this model, the investors can not only consider the foreign investment risk but also the domestic investment risk, which give the investors more selections in facing various risks. Furthermore, we deduce an equivalent model, when investment values of risk assets and exchange rates are triangular fuzzy numbers. Then, an numerical study is carried out with a portfolio of six assets denominated in the local currency. Based on the data, we obtain the portfolio frontier with exchange rate risk, and compare it with the portfolio frontier without domestic asset. The results illustrate the effectiveness of the proposed model.

Keywords

Exchange rate risk fuzzy number possibilistic theory portfolio efficient frontier

1 Introduction

The real financial markets were influenced by many non-probabilistic factors, the return of risky asset is often fuzzy and uncertain. Many researchers worked on the fuzzy portfolio selection problems. Carlsson and Fullér [3] assumed the return of risky asset was fuzzy number, they introduced the possibilistic portfolio model. Zhang et al. [9, 10] deduced the possibilistic mean and possibilistic variance, then they analyzed the possibilistic portfolio selection model when holdings of assets are nonnegative. Zhang et al. [11] proposed a two-period possibilistic portfolio model with bounded constraints on investment proportions. Yiu et al. [13] proposed a dynamic possibilistic portfolio model with cardinality constraint and bankruptcy control.

With the globalization of economic, more and more enterprises participate in the international market operation and competition. The rapid development of economic liberalization and financial innovation have made the fluctuations of exchange rate more frequent and intense. Therefore, these companies not only have to response for fierce international market competition, but also to face the growing exchange rate risk, which will erode their net profits and deeply affect their business operations. Carmichael and Coën [1] built a OLG model with transaction costs to study the international portfolio choice. Xu et al. [12] considered the exchange rate risk into the portfolio selection model, and analyzed a fuzzy multi-objective international portfolio problem. Jiang et al. [4] integrated behavioral portfolio theory into an international portfolio selection model, deduced an analytical solution of the model. However, these papers didn’t consider the domestic investment.

In this paper, we firstly proposed a general international portfolio selection model with exchange rate risk based on possibility theory. This model not only considers multiple foreign markets but also the domestic market. Using the model, investors could better make portfolio selection decisions. Secondly, we derive a equivalent model with exchange rate risk, when the values of unit investment in risk assets and exchange rates are triangular fuzzy numbers. Thirdly, we gave a numerical study and calculate the variances with the different expected returns. Finally, we compare the difference of the efficient frontiers.

The rest of the paper is organized as follows. Section 2 introduces fuzzy number and its properties. Section 3 describes the international portfolio model, and deduces the equivalent form of the model. Section 4 gives a numerical example to illustrate these portfolios. Section 5 presents conclusions.

2 Preliminaries

In this section, we will introduce some concepts about fuzzy number.

Definition. [5] $\tilde{A} = (a, b, c)$ is a triangular fuzzy number, if its membership function $μ_{\tilde{A}}$ is of the following type: $\begin{matrix} μ_{\tilde{A}} (t) = {\begin{matrix} \frac{t - a}{b - a}, & a \leq t \leq b, \\ \frac{t - c}{b - c}, & b \leq t \leq c, \\ 0, & otherwise . \end{matrix} \end{matrix}$ where 0 ≤ a ≤ b ≤ c.

Lemma. [5] If $\tilde{A} = (a_{1}, b_{1}, c_{1})$ and $\tilde{B} = (a_{2}, b_{2}, c_{2})$ are triangular fuzzy numbers, then

Addition: $\tilde{A} + \tilde{B} = (a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2})$ .

Subtraction: $\tilde{A} - \tilde{B} = (a_{1} - c_{2}, b_{1} - b_{2}, c_{1} - a_{2})$ .

Multiplication: $\tilde{A} \times \tilde{B} = (\min (a_{1} a_{2}, a_{1} c_{2}, a_{2} c_{1}, c_{1} c_{2}), b_{1} b_{2}, \max (a_{1} a_{2}, a_{1} c_{2}, a_{2} c_{1}, c_{1} c_{2}))$ .

Carlsson and Fullér [2] defined the possibilistic mean of a fuzzy number $\tilde{A}$ with a γ-level set $[\tilde{A}]^{γ} = [l (γ), u (γ)], (γ > 0)$ as $\bar{M} (\tilde{A}) = \int_{0}^{1} γ (l (γ) + u (γ)) d γ$ (1) where $\begin{matrix} l (γ) & = & \inf {x \in R | μ_{\tilde{A}} \geq γ}, \\ u (γ) & = & \sup {x \in R | μ_{\tilde{A}} \geq γ} . \end{matrix}$

Let $\tilde{A}$ and $\tilde{B}$ be fuzzy numbers, and $[\tilde{A}]^{γ} = [l_{1} (γ), u_{1} (γ)]$ , $[\tilde{B}]^{γ} = [l_{2} (γ), u_{2} (γ)]$ , the possibilistic variance and covariance of fuzzy numbers [2, 8] can be defined as

$\begin{matrix} \bar{V} (\tilde{A}) & = & \int_{0}^{1} γ (\bar{M} (\tilde{A}) - l_{1} (γ))^{2} d γ \\ + \int_{0}^{1} γ (\bar{M} (\tilde{A}) - u_{1} (γ))^{2} d γ \end{matrix}$ (2) and

$\begin{matrix} \bar{Cov} (\tilde{A}, \tilde{B}) \\ = \int_{0}^{1} γ [(\bar{M} (\tilde{A}) - l_{1} (γ)) (\bar{M} (\tilde{B}) - l_{2} (γ))] d γ \\ + \int_{0}^{1} γ [(\bar{M} (\tilde{A}) - u_{1} (γ)) (\bar{M} (\tilde{B}) - u_{2} (γ))] d γ \end{matrix}$ (3) respectively.

If $\tilde{A}$ and $\tilde{B}$ are triangular fuzzy numbers, we easily get γ-level sets of $\tilde{A}$ and $\tilde{B}$ , (γ ∈ [0, 1]), that can be computed by $\begin{matrix} [\tilde{A}]^{γ} & = & [a_{1} + (b_{1} - a_{1}) γ, c_{1} + (b_{1} - c_{1}) γ], \\ [\tilde{B}]^{γ} & = & [a_{2} + (b_{2} - a_{2}) γ, c_{2} + (b_{2} - c_{2}) γ] . \end{matrix}$

From Equations (1), (2) and (3), we can easily calculate their possibilistic mean, possibilistic variance and covariance $\bar{M} (\tilde{A}) = \frac{a_{1} + c_{1} + 4 b_{1}}{6}$ (4) $\begin{matrix} \bar{V} (\tilde{A}) \\ = \frac{(b_{1} - a_{1})^{2} + (c_{1} - b_{1})^{2} + (b_{1} - a_{1}) (c_{1} - b_{1})}{18} \end{matrix}$ (5) and

$\begin{matrix} \bar{Cov} (\tilde{A}, \tilde{B}) \\ = \frac{(b_{1} - a_{1}) (b_{2} - a_{2}) + (c_{1} - b_{1}) (c_{2} - b_{2})}{36} \\ + \frac{(c_{1} - a_{1}) (c_{2} - a_{2})}{36} \end{matrix}$ (6)

3 International portfolio selection model with exchange rate risk

3.1 General international portfolio selection model

In this section, we consider the problem with multiple international markets. Suppose an investor wishes to invest his funds to various international portfolio election, he faces not only domestic portfolio risk but also exchange rate risk. We assume that there are n risky assets in foreign markets and each asset’s investment value is denoted by its own currency. Let ${\tilde{ξ}}_{i}$ and ${\tilde{η}}_{i}$ be the fuzzy value of unit investment in asset i and the fuzzy exchange rate of country i, i = 1, 2, ⋯ , n, respectively. Let x_i represent the proportion invested in asset i. The value on domestic asset is denoted by ${\tilde{r}}_{d}$ , which is also a fuzzy number. x_d denotes the proportion invested in domestic asset. Thus the total value of the portfolio return can be expressed as ${\tilde{r}}_{p} = \sum_{i = 1}^{n} x_{i} {\tilde{ξ}}_{i} {\tilde{η}}_{i} + {x_{d} \tilde{r}}_{d}$ where ${\tilde{r}}_{p}$ is a fuzzy number.

Derived from Equations (1), (2) and (3), the possibilistic mean of ${\tilde{r}}_{p}$ can be computed as $\bar{M} ({\tilde{r}}_{p}) = \sum_{i = 1}^{n} x_{i} \bar{M} (\tilde{ξ_{i}} {\tilde{η}}_{i}) + x_{d} \bar{M} ({\tilde{r}}_{d})$

The possibilistic variance of ${\tilde{r}}_{p}$ can be obtained as $\begin{matrix} \bar{V} ({\tilde{r}}_{p}) & = & \bar{V} (\sum_{i = 1}^{n} x_{i} {\tilde{ξ}}_{i} {\tilde{η}}_{i}) + x_{d}^{2} \bar{V} ({\tilde{r}}_{d}) \\ + 2 \sum_{i = 1}^{n} x_{i} x_{d} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{r}}_{d}) \\ = & \sum_{i = 1}^{n} x_{i}^{2} \bar{V} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) + x_{d}^{2} \bar{V} ({\tilde{r}}_{d}) \\ + 2 \sum_{i > j = 1}^{n} x_{i} x_{j} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{ξ}}_{j} {\tilde{η}}_{j}) \\ + 2 \sum_{i = 1}^{n} x_{i} x_{d} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{r}}_{d}) \end{matrix}$ for all x_i ≥ 0, i = 1, 2, …, n, x_d ≥ 0.

Based on mean-variance theory, we consider both the domestic asset and the exchange rate into the portfolio model. The general international portfolio selection model can be formulated as ${\begin{matrix} min & \bar{V} ({\tilde{r}}_{p}) = \sum_{i = 1}^{n} x_{i}^{2} \bar{V} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) \\ + 2 \sum_{i > j = 1}^{n} x_{i} x_{j} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{ξ}}_{j} {\tilde{η}}_{j}) \\ + x_{d}^{2} \bar{V} ({\tilde{r}}_{d}) + 2 \sum_{i = 1}^{n} x_{i} x_{d} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{r}}_{d}) \\ s . t . & \sum_{i = 1}^{n} x_{i} \bar{M} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) + x_{d} \bar{M} ({\tilde{r}}_{d}) \geq \bar{r} \\ \sum_{i = 1}^{n} x_{i} + x_{d} = 1 \\ x_{d} \geq 0, \\ 0 \leq x_{i} \leq u_{i}, i = 1, 2, \dots, n . \end{matrix}$ (7) where $\bar{r}$ is the given unit investment value, u_i represents the upper bound in asset i.

With the model (7), the investors can not only consider the foreign investment risk but also the domestic investment risk, which give the investors more selection in facing various risks. Specially, if the domestic assets aren’t considered in investment period, then x_d = 0, the model (7) is written as the followingform: ${\begin{matrix} min & \bar{V} ({\tilde{r}}_{p}) = \sum_{i = 1}^{n} x_{i}^{2} \bar{V} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) \\ + 2 \sum_{i > j = 1}^{n} x_{i} x_{j} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{ξ}}_{j} {\tilde{η}}_{j}) \\ s . t . & \sum_{i = 1}^{n} x_{i} \bar{M} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) \geq \bar{r} \\ \sum_{i = 1}^{n} x_{i} = 1 \\ 0 \leq x_{i} \leq u_{i}, i = 1, 2, \dots, n . \end{matrix}$ (8)

3.2 International portfolio selection model under triangular distribution

In this section, we assume that the value of unit investment in asset i and the exchange rate are triangular fuzzy numbers denoted by ${\tilde{ξ}}_{i} = (a_{i}, b_{i}, c_{i})$ , and ${\tilde{η}}_{i} = (a_{ei}, b_{ei}, c_{ei})$ , i = 1, 2, ⋯ , n, respectively. We also assume that the value on domestic asset by ${\tilde{r}}_{d} = (a_{d}, b_{d}, c_{d})$ . It is easy to know ${\tilde{ξ}}_{i}$ , ${\tilde{η}}_{i}$ and ${\tilde{r}}_{d}$ are nonnegative fuzzy numbers.

By the Lemma, the product of ${\tilde{ξ}}_{i}$ and ${\tilde{η}}_{i}$ can be conducted by $\begin{matrix} {\tilde{ξ}}_{i} {\tilde{η}}_{i} & = & (\min (a_{i} a_{ei}, a_{i} c_{ei}, a_{ei} c_{i}, c_{i} c_{ei}), b_{i} b_{ei}, \\ \max (a_{i} a_{ei}, a_{i} c_{ei}, a_{ei} c_{i}, c_{i} c_{ei})) \end{matrix}$

Because ${\tilde{ξ}}_{i}$ and ${\tilde{η}}_{i}$ are nonnegative, we are easy to obtain $\begin{matrix} {\tilde{ξ}}_{i} {\tilde{η}}_{i} = (a_{i} a_{ei}, b_{i} b_{ei}, c_{i} c_{ei}) \end{matrix}$

According to Equations (4) and (5), the possibilistic mean and variance of ${\tilde{ξ}}_{i} {\tilde{η}}_{i}$ are $\bar{M} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) = \frac{a_{i} a_{ei} + c_{i} c_{ei} + 4 b_{i} b_{ei}}{6}$ and $\begin{matrix} \bar{V} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}) & = & \frac{(b_{i} b_{ei} - a_{i} a_{ei})^{2} + (c_{i} c_{ei} - b_{i} b_{ei})^{2}}{18} \\ + \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (c_{i} c_{ei} - b_{i} b_{ei})}{18} \end{matrix}$

By Equation (6), the possibilistic covariance of ${\tilde{ξ}}_{i} {\tilde{η}}_{i}$ is written as $\begin{matrix} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, {\tilde{ξ}}_{j} {\tilde{η}}_{j}) \\ = \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (b_{j} b_{ej} - a_{j} a_{ej})}{36} \\ + \frac{(c_{i} c_{ei} - b_{i} b_{ei}) (c_{j} c_{ej} - b_{j} b_{ej})}{36} \\ + \frac{(c_{i} c_{ei} - a_{i} a_{ei}) (c_{j} c_{ej} - a_{j} a_{ej})}{36} \end{matrix}$

In a similar manner, we can obtain the following formulas: $\begin{matrix} \bar{M} ({\tilde{r}}_{d}) = \frac{a_{d} + c_{d} + 4 b_{d}}{6} \\ \bar{V} ({\tilde{r}}_{d}) \\ = \frac{(b_{d} - a_{d})^{2} + (c_{d} - b_{d})^{2} + (b_{d} - a_{d}) (c_{d} - b_{d})}{18} \end{matrix}$ and $\begin{matrix} \bar{Cov} ({\tilde{ξ}}_{i} {\tilde{η}}_{i}, r_{d}) & = & \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (b_{d} - a_{d})}{36} \\ + \frac{(c_{i} c_{ei} - b_{i} b_{ei}) (c_{d} - b_{d})}{36} \\ + \frac{(c_{i} c_{ei} - a_{i} a_{ei}) (c_{d} - a_{d})}{36} \end{matrix}$

Thus we can get the equivalent form of the model (7), for a triangular fuzzy variable, the model can be converted as ${\begin{matrix} min & \bar{V} ({\tilde{r}}_{p}) = \sum_{i = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei})^{2} + (c_{i} c_{ei} - b_{i} b_{ei})^{2}}{18} x_{i}^{2} \\ + \sum_{i = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (c_{i} c_{ei} - b_{i} b_{ei})}{18} x_{i}^{2} \\ + 2 \sum_{i > j = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (b_{j} b_{ej} - a_{j} a_{ej})}{36} x_{i} x_{j} \\ + 2 \sum_{i > j = 1}^{n} \frac{(c_{i} c_{ei} - b_{i} b_{ei}) (c_{j} c_{ej} - b_{j} b_{ej})}{36} x_{i} x_{j} \\ + 2 \sum_{i > j = 1}^{n} \frac{(c_{i} c_{ei} - a_{i} a_{ei}) (c_{j} c_{ej} - a_{j} a_{ej})}{36} x_{i} x_{j} \\ + \frac{(b_{d} - a_{d})^{2} + (c_{d} - b_{d})^{2} + (b_{d} - a_{d}) (c_{d} - b_{d})}{18} x_{d}^{2} \\ + 2 \sum_{i = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (b_{d} - a_{d}) + (c_{i} c_{ei} - b_{i} b_{ei}) (c_{d} - b_{d})}{36} x_{i} x_{d} \\ + 2 \sum_{i = 1}^{n} \frac{(c_{i} c_{ei} - a_{i} a_{ei}) (c_{d} - a_{d})}{36} x_{i} x_{d} \\ s . t . & \sum_{i = 1}^{n} (\frac{a_{i} a_{ei} + c_{i} c_{ei} + 4 b_{i} b_{ei}}{6}) x_{i} + \frac{a_{d} + c_{d} + 4 b_{d}}{6} x_{d} \geq \bar{r} \\ \sum_{i = 1}^{n} x_{i} + x_{d} = 1 \\ x_{d} \geq 0, \\ 0 \leq x_{i} \leq u_{i}, i = 1, 2, \dots, n . \end{matrix}$ (9)

Next, in order to illustrate that exchange rate risk have an effect on the optimal portfolio strategy, we consider the international portfolio selection model without domestic asset. ${\begin{matrix} min & \bar{V} ({\tilde{r}}_{p}) = \sum_{i = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei})^{2}}{18} x_{i}^{2} \\ + \sum_{i = 1}^{n} \frac{(c_{i} c_{ei} - b_{i} b_{ei})^{2} + (b_{i} b_{ei} - a_{i} a_{ei}) (c_{i} c_{ei} - b_{i} b_{ei})}{18} x_{i}^{2} \\ + 2 \sum_{i > j = 1}^{n} \frac{(b_{i} b_{ei} - a_{i} a_{ei}) (b_{j} b_{ej} - a_{j} a_{ej})}{36} x_{i} x_{j} \\ + 2 \sum_{i > j = 1}^{n} \frac{(c_{i} c_{ei} - b_{i} b_{ei}) (c_{j} c_{ej} - b_{j} b_{ej})}{36} x_{i} x_{j} \\ + 2 \sum_{i > j = 1}^{n} \frac{(c_{i} c_{ei} - a_{i} a_{ei}) (c_{j} c_{ej} - a_{j} a_{ej})}{36} x_{i} x_{j} \\ s . t . & \sum_{i = 1}^{n} (\frac{a_{i} a_{ei} + c_{i} c_{ei} + 4 b_{i} b_{ei}}{6}) x_{i} \geq \bar{r} \\ \sum_{i = 1}^{n} x_{i} = 1 \\ 0 \leq x_{i} \leq u_{i}, i = 1, 2, \dots, n . \end{matrix}$ (10)

4 Numerical example

In this section, we give an example for simulating the real transaction scenario. In the scenario, the investors put their funds in six different countries, each country has their own currency. The index prices are converted to its money prices using their respective exchange rates. We estimate the fuzzy values of unit investment and fuzzy exchange rates with the triangular possibility distributions. Tables 1 and 2 give their parameters of membership functions, respectively.

In order to solve the model (9), we assume that the upper bounds of holdings in risk assets are [0.40,0.40, 0.40,0.50,0.50,0.40]. The optimization software is MATLAB 7.10, with which we conduct our genetic algorithm. The crossover and mutation probabilities were set to 0.95 and 0.05, respectively. The population size was 100, iteration times were 1000. The parameters are set according to a genetic algorithm in [6], and the value of the parameters have been verified in our early work [7]. The experiment results are shown in Table 3.

As shown in Table 3, we can see that when the given unit investment value $\bar{r}$ is 1.05, the investment proportion of asset 2 is the upper bound, assets 4 and 5 are lower bounds. As $\bar{r}$ increases, proportions of asset 1 and asset 6 increase, the proportions of asset 3 decreases. Especially, when $\bar{r} \geq 1.26$ , the proportion of asset 1 is the upper bound, the holding of assets 3 is 0.

Now, we consider the portfolio model without domestic asset. We set the same given unit investment value as Table 3. By running the algorithm, the efficient portfolios of the model (10) can be obtained as shown in Table 4. A comparison of Tables 3 and 4 indicates that when the given unit investment values are identical, the portfolio risks in Table 3 are smaller than those in Table 4. It means that if investors only invest their full funds into foreign markets, they will face more risk. If domestic asset is considered, the portfolio risk will be lowered.

Figure 1 shows the efficient frontier of the international portfolio model in the mean-variance space. Figure 1a illustrates the efficient frontier of model (9). Figure 1b compares the efficient frontiers of the models (9) and (10). In Fig. 1b, the circular parabola depicts the efficient portfolios with domestic asset, the triangle parabola shows the efficient portfolios without domestic asset. Hence the optimal portfolio without domestic asset is more risky for the investor. It means that if investors only consider exchange rate risk, they will bear more portfolio risk and suffer greater losses.

5 Conclusion

In this paper, a general international portfolio model with exchange rate risk is proposed. When the values of unit investment in risk assets and exchange rates are triangular fuzzy numbers, we deduce an equivalent model. Based on an numerical study, we obtain the portfolio frontiers with domestic asset and without domestic asset. We can conclude if investors only consider exchange rate risk and ignore domestic investment risk, they will bear more portfolio risk and suffer greater losses.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant Nos. 71461024 and 61363018.

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