Robust International Portfolio Optimization with Worst-Case Mean-LPM

+is paper proposes a robust international portfolio optimization model with the consideration of worst-case lower partial moment (LPM) and worst-case mean return. In our model, we assume that the distributions and the firstand second-order moments of distributions of returns of assets and exchange rates are all ambiguous. +e proposed model can be reformulated into an equivalent semidefinite programming (SDP) problem, which is computationally tractable. For investigation of the performance of our model, we also give two benchmark models. +e first benchmark model is a scenario-based model which uses historical observations of returns to approximate the future distributions. +e second benchmark model only considers the ambiguity of distributions but does not consider the ambiguity of the firstand second-order moments of distributions. We conduct empirical experiments in a rolling forward way to evaluate the out-of-sample performances of our proposed model, the two benchmark models, and an equally weighted model using the return measures and various risk-adjusted return measures. +e result shows that our model has the best performance. It verifies that investors can obtain benefits when employing the robust model and considering the ambiguity of the firstand second-order moments of distributions.


Introduction
In order to capture the diversification benefits of international financial markets, institutional and individual investors tend to invest part of their money in the financial markets of other countries or regions using different currencies.
e correlations of returns of assets in other countries or regions are often lower than those in just one country, so international asset allocation may reduce risk [1][2][3][4][5]. Generally, the distribution of international portfolio return is asymmetry. It is well known that variance is not an appropriate measure to evaluate the risk of asymmetry distributions, whereas downside risk measures can measure the risk of asymmetry distributions effectively. Among various downside risk measures, lower partial moments (LPM) are comprehensive and sensible [6]. e definition of LPM is introduced by Bawa [7], Bawa and Lindenberg [8], and Fishburn [9]. In order to compute LPM, we need to know the distributions of future security returns beforehand. However, for investors, the distributions of future security returns are usually unknown or cannot be estimated accurately. Even if they acquire the actual distributions of future security returns, the computation of LPM is also a difficult task. To deal with these problems, some researchers employ robust optimization techniques to portfolio selection models using LPM as the risk measure [10][11][12][13][14]. eir researches focus on the portfolio selection problems in one country and do not consider the international portfolio selection problems with the risk of exchange rates. Meanwhile, their researches only consider the worst-case LPM and do not consider the worst-case mean return. Intuitively, the worst-case mean return can also give investors helpful instructions for making investment decisions.
In this paper, we build a robust international portfolio optimization model with worst-case LPM as the risk measure and consider the worst-case mean return. We assume that the distributions and the first-and second-order moments of distributions of future returns of assets and exchange rates are all ambiguous. Using robust optimization techniques, we reformulate our model into an equivalent semidefinite programming (SDP) problem. In order to evaluate the performance of our model, we also give two benchmark models. In the first benchmark model, we use historical return observations to form empirical distributions of future returns and build an international portfolio optimization model based on these empirical distributions.
In the second benchmark model, we assume that the distributions of future returns are ambiguous, but the first-and second-order moments are known. en, we conduct empirical experiments using the return measures and various risk-adjusted return measures to assess the performances of our model, the two benchmark models, and the equally weighted model. is paper is organized as follows. In Section 2, we propose the robust international portfolio optimization model with worst-case LPM and mean return under a distributional ambiguity set where the distributions and the first-and second-order moments are ambiguous. We derive an equivalent SDP reformulation of this model. In Section 3, we present the two benchmark models. In Section 4, we conduct empirical experiments to evaluate the performance of our model in comparison with the two benchmark models and the equally weighted model using the return measures and various risk-adjusted return measures. Section 5 gives the conclusions of this paper.

Notation.
In this paper, vectors are denoted by lowercase boldface letters, and matrices are denoted by uppercase boldface letters. We use R n to denote the space of vectors of real numbers with dimension n and S n to denote the space of symmetric matrices with dimension n. For any two matrices X, Y ∈ S n , we use <X, Y> � trace(XY) to denote the trace scalar product, and the relation X ≽ Y represents that X − Y is positive semidefinite. Random variables are denoted by symbols with tildes, whereas the realizations of them are denoted by symbols without tildes.

Model Formulation
In the international financial markets, we assume that an investor plans to invest in the stock markets of n foreign countries or overseas regions where people use different currencies to the investor's domestic currency. We denote that the return of the asset in the i-th country or region is s i , and the return of the exchange rate of the i-th country or region is c i , where i � 1, 2, . . . , n. en, the return of the i-th asset in the investor's domestic currency can be obtained as We assume that the weight of money in the domestic currency of the investor invested in the i-th asset is w i , and the sum of w i , i � 1, 2, . . . , n, equals 1. en, the total return of the international portfolio w � (w 1 , w 2 , . . . , w n ) T can be written as For convenience, we denote that ξ � s 1 , s 2 , . . . , s n , c 1 , c 2 , . . . , c n T , which combines the returns of assets and exchange rates in one vector. We also denote that en, (2) can be rewritten as We employ first-order lower partial moment to measure the risk of international portfolios. For a given benchmark return a, the first-order LPM can be written as follows: where P is the distribution of ξ. In practice, investors usually cannot know P accurately beforehand. us, some researchers use historical observations of returns to form an empirical approximation of P. We assume that investors can obtain m historical observations, which are denoted by ξ 1 , ξ 2 , . . ., ξ m . e empirical approximation P of P is typically formed as follows: Under P, the LPM in (6) can be rewritten as If m is relatively large, according to the law of large numbers, the gap between (6) and its scenario-based version (8) can be small. But the number of observations that investors can acquire is usually small and cannot satisfy the requirement of the law of large numbers. Instead, some researchers use historical observations of returns to form a distributional ambiguity set of future returns. A popular ambiguity set that considers the ambiguity of the 2 Mathematical Problems in Engineering distributions and the first-and second-order moments of distributions is proposed by Delage and Ye [15]. is ambiguity set denoted by P 1 can be described as follows: where M is the set of all probability measures on the measurable space (R 2n , B), with B being the Borel σ-algebra on R 2n , μ is the sample-based mean return, Σ is the sample-based covariance matrix, λ 1 reflects the ambiguity size of mean return, and λ 2 reflects the ambiguity size of the covariance matrix. e worst-case LPM with respect to the ambiguity set P 1 denoted by WLPM(w, P 1 ) can be defined as Since the mean return of the international portfolio can also give investors useful instructions for decision-making, we add the worst-case mean return in the objective function of our model. We give the definition of worst-case mean return with respect to P 1 denoted by WReturn(w, P 1 ) as follows: e robust international portfolio optimization model using worst-case LPM as the risk measure and considering worst-case mean return under P 1 can be built as follows: where λ is the risk aversion coefficient of investors and e denotes the vector of 1 s with dimension n. Problem (12) cannot be solved directly; thus, we need to derive its equivalent reformulation, which is computationally tractable. In the following, we first give the equivalent SDP reformulation of WLPM(w, P 1 ) defined by (10); then, we give the equivalent SDP reformulation of WReturn(w, P 1 ) defined by (11).
Theorem 1. WLPM(w, P 1 ) defined by (10) is equal to the optimal objective function value of the following SDP problem: Proof: WLPM(w, P 1 ) defined by (10) is equivalent to the following problem: en, we shall derive the equivalent SDP reformulation of the following: which can be written as

Mathematical Problems in Engineering
We denote the dual variable of constraint (16b) by p 1 , that of constraint (16c) by U, and that of constraint (16d) by where e dual reformulation of problem (16a) can be written as follows: max e Dirac measure δ μ is the measure of mass one at the point μ. Obviously, δ μ lies in the relative interior of the feasible set of problem (16). According to the weak version of Proposition 3.4 in Shapiro [16], we can deduce that there is no dual gap between problems (16) and (19). us, the optimal objective function value of problem (16) is equal to that of problem (19). Constraint (19b) is equivalent to the following two constraints: Equation (20) is equivalent to the matrix inequality (13d), and (21) is equivalent to the matrix inequality (13e).
us, we complete the proof of this theorem. (11) is equal to the optimal objective function value of the following SDP problem: Proof: . WReturn(w, P 1 ) defined by (11) can be written in the following formulation: Similar to the proof of eorem 1, we denote the dual variable of constraint (23b) by q 1 , that of constraint (16c) by Q, and that of constraint (16d) by where e dual reformulation of problem (16) can be written as Similar to the proof of eorem 1, there is no dual gap between problems (23) and (26); thus, the optimal objective function values of the two problems are the same. e equivalent matrix inequality of constraint (26d) is (22d); thus, we complete the proof of this theorem.
With eorems 1 and 2, we can easily obtain the equivalent SDP reformulation of problem RIML (12), and the final formulation is as follows:

Two Benchmark Models
In order to assess the performance of our model RIML, we present two benchmark models in this section. e first benchmark model denoted by SIML is based on empirical distributions approximated by historical samples of returns.
e approximated distribution P is described in (7), and LPM under P is shown in (8). e return of the international portfolio under P can be written as e scenario-based international portfolio optimization model with mean-LPM denoted by SIML is built as where LPM(w, P) is defined by (8) and Return(w, P) is defined by (28). In the second benchmark model denoted by RIML − , we assume that the distributions of returns of assets and exchange rates are ambiguous, but the first-and secondorder moments of distributions are determined beforehand. e corresponding ambiguity set P 2 is described as follows: where the definitions of M, μ, and Σ are the same as those of (9). Under P 2 , RIML − can be written as WReturn w, P 2 � min In the following, we also try to derive the equivalent SDP reformulations of WLPM(w, P 2 ) defined by (32) and WReturn(w, P 2 ) defined by (33). Theorem 3. WLPM(w, P 2 ) defined by (32) is equal to the optimal objective function value of the following SDP problem: Proof. According to (32), WLPM(w, P 2 ) can be rewritten as us, we first study the equivalent SDP reformulation of the following problem: which can be rewritten as We set the dual variable of constraint (37b) as p 1 , that of constraint (37c) as v, and that of constraint (37d) as U. e dual reformulation of problem (37) can be written as follows: max r(w, ξ)) + , ∀ξ ∈ R 2n .

(38b)
Obviously, there is no dual gap between problems (37) and (38). us, the two problems (37) and (38) have the same optimal objective function value. We note that constraint (38b) is equivalent to the following two inequalities: Mathematical Problems in Engineering e equivalent matrix inequality of (39) is (34b), and that of (40) is (34c). us, we complete the proof of this theorem. □ Theorem 4. WReturn(w, P 2 ) defined by (33) is equal to the optimal objective function value of the following SDP problem: Proof: WReturn(w, P 2 ) defined by (33) can be rewritten as follows: We denote the dual variable of constraint (42b) by q 1 , that of constraint (42c) by h, and that of constraint (42d) by Q. e dual reformulation of problem (42) can be written as Similarly, there is no dual gap between problems (42) and (43). us, the two problems have the same optimal objective function value. e equivalent matrix inequality of (43b) is (41b). Hence, we finish the proof of this theorem.
According to eorems 3 and 4, we can obtain the equivalent SDP reformulation of model RIML − defined by (31), and the final formulation is as follows:  [15], to compute the two parameters λ 1 and λ 2 of the ambiguity set of our model RIML in (9), we need to build uncertainty sets of the returns of assets and exchange rates. We first illustrate the samplebased mean returns, standard deviations, and covariance matrix of returns of assets and exchange rates during the insample period. Table 1 shows sample-based mean returns and standard deviations. From Table 1, we find that the standard deviations of returns of assets are much larger than those of exchange rates. Table 2 shows a sample-based covariance matrix. From Table 2, we find that the covariance of returns of assets is also much larger than those of exchange rates. us, we can conclude that the stock market is more volatile than the currency market. According to this observation, we assume that the size of the uncertainty set of returns of assets is larger than that of exchange rates. Specifically, we set the upper bound of returns of assets as 0.06 and the lower bound of those as − 0.06, whereas we set the upper bound of returns of exchange rates as 0.02 and the lower bound of those as − 0.02.
We conduct rolling forward experiments to assess the out-of-sample performances of models RIML, RIML − , SIML, and EW. We want to set the benchmark return a as the realized mean return of model RIML in the in-sample period dynamically, and then LPM(w, P) in (6) is similar to semivariance. However, if we do not know the benchmark return a beforehand, we cannot compute the portfolio of model RIML; thus, we cannot acquire the realized mean return of model RIML in the in-sample period. Hence, instead, we use the realized mean return of model EW in the in-sample period to approximate that of model RIML in the in-sample period dynamically. Specifically, when the realized mean return of model EW is positive or zero, we set the benchmark return a as three times of it. When the realized mean return of model EW is negative, we set a as a third of it. Now we describe the procedure of our rolling forward experiment as follows. First, we use the 150 return observations in the in-sample period from March 26, 2004, to March 30, 2007, to compute the relevant parameters of models RIML and RIML − , and determine the three optimal portfolios of models RIML, RIML − , and SIML. According to the realized returns of assets and exchange rates in the first week during the out-of-sample period, we can compute the realized returns of the above three optimal portfolios and the equally weighted strategy. en, we move the in-sample period one week forward by adding the new week and delete the first week. Based on the return observations in the new in-sample period, we can also compute the three new optimal portfolios of models RIML, RIML − , and SIML. Using the returns observations in the second week of the out-ofsample period and the optimal portfolios obtained from the in-sample period, we can also derive the realized returns of the four models. We continue this procedure until July 23, 2021. Consequently, we obtain four return series with 733 realized returns of the four models. Based on these four return series, we assess the realized performances of the four models using various performance measures, which are mean return, Sharpe ratio [18], downside Sharpe ratio [19], upside potential and downside risk (UP) ratio [20] [20], the UP ratio can be defined as follows: where r t is the realized return of a portfolio at the t-th period and ρ t is a benchmark return at the same period, t � 1, 2, . . . , K. Obviously, the UP ratio is an appropriate measure to assess the performance of portfolios with asymmetry distributions. Without loss of generality, in our numerical experiments, we set that ρ t � 0, t � 1, 2, . . . , K. For a portfolio w, WLPM(w, P 1 ) in (10) is much larger than WReturn(w, P 1 ) in (11). Hence, in order to balance WLPM(w, P 1 ) and WReturn(w, P 1 ) and acquire a portfolio that has a good performance in terms of risk-adjusted return, we should set the risk aversion coefficient λ small. In our empirical experiments, we consider various cases of λ, which are λ � 0.03, λ � 0.02, λ � 0.01, λ � 0.009, λ � 0.008, λ � 0.007, λ � 0.006, λ � 0.005, λ � 0.004, λ � 0.003, λ � 0.002, and λ � 0.001. Table 3 shows the realized performances of models RIML, RIML − , SIML, and EW in terms of the above various performance measures when λ � 0.03. Table 4 shows the result when λ � 0.02. Table 5 shows the result when λ � 0.01. Table 6 shows the result when λ � 0.009. Table 7 shows the result when λ � 0.008. Table 8 shows the result when λ � 0.007. Table 9 shows the result when λ � 0.006. Table 10 shows the result when λ � 0.005. Table 11 shows the result when λ � 0.004. Table 12 shows the result when λ � 0.003. Table 13 shows the result when λ � 0.002. Table 14 shows the result when λ � 0.001. In order to test whether the Sharpe ratio of our model RIML outperforms those of other models significantly, we employ a significance testing method about Sharpe ratios proposed by Jobson and Korkie [21]. e corresponding one-sided p values are presented in the column of the Sharpe ratio. * * * , * * , and * indicate that the Sharpe ratio of our model RIML outperforms that of the corresponding model significantly at the 1% level, 5% level, and 10% level, respectively. For all cases of the risk aversion coefficient λ, our model RIML consistently performs best in terms of return and risk-adjusted return measures among the four models. e Sharpe ratio of our model RIML is significantly larger than those of the other three models. Interestingly, we find that the realized performances of models RIML − and SIML are very similar. For all performance measures, our model Table 2: Sample-based covariance matrix of returns of assets and exchange rates during the in-sample period.

Conclusions
In this paper, we propose a robust international portfolio optimization model with worst-case LPM and mean return.
In this model, we assume that the distributions and the firstand second-order moments of distributions of future returns of assets and exchange rates are ambiguous. We reformulate the proposed model into an equivalent SDP problem which is computationally tractable. For investigation of the performance of our proposed model, we also give two benchmark models. In the first benchmark model SIML, we use historical returns to form approximations of the distributions of future returns and build a scenario-based international portfolio optimization model under these approximations of distributions. In the second benchmark model RIML − , we assume that the distributions of future     returns are ambiguous, but the first-and second-order moments of distributions are known beforehand. We also reformulate this model into an equivalent SDP problem. We conduct empirical experiments in a rolling forward way using the return measures and various risk-adjusted return measures to compare the out-of-sample performances of the four models RIML, RIML − , SIML, and an equally weighted model EW. e result demonstrates the superiority of our model RIML over other models. It shows that investors can get benefits when accounting for the ambiguity of the firstand second-order moments. It also verifies that robust models outperform scenario-based model and equally weighted model.

Data Availability
e data used to support the findings of this study are available from the database Wind with the web address https://www.wind.com.cn/.

Conflicts of Interest
e authors declare that they have no conflicts of interest.