Robust International Portfolio Management

## Keywords

Risk Management, Robust Optimization, International Portfolio Optimization, Quanto Options, Semidefinite Programming

Unreviewed

# Abstract

We present an international portfolio optimization model where we take into account the two different sources of return of an international asset: the local returns denominated in the local currency, and the returns on the foreign exchange rates. The explicit consideration of the returns on exchange rates introduces non-linearities in the model, both in the objective function (return maximization) and in the triangulation requirement of the foreign exchange rates. The uncertainty associated with both types of returns is incorporated directly in the model by the use of robust optimization techniques. We show that, by using appropriate assumptions regarding the formulation of the uncertainty sets, the proposed model has a semidefinite programming formulation and can be solved efficiently. While robust optimization provides a guaranteed minimum return inside the uncertainty set considered, we also discuss an extension of our formulation with additional guarantees through trading in quanto options for the foreign assets and in equity options for the domestic assets.

# 1. Introduction

The seminal work of Markowitz in 1952 on portfolio optimization initiated great interest and further academic research in the area of risk management. It was only in 1968, however, that this same interest was extended to international portfolios, that is, to portfolios with assets denominated in foreign currency. In his seminal work, Grubel suggests a model that explains how international capital movements are a function not only of the interest rate differential between countries, but also of the growth rate of asset holdings.

International portfolios are attractive from the point of view of risk diversification, as it is expected that assets in the same economy have a higher correlation among themselves than with assets in other countries. Levy and Sarnat (1970) present an estimate of the potential gains on international diversification for the period 1951-67. They conclude that the traditional approach of comparing the returns of the developed countries with those of the developing ones underestimated the impact of international diversification. The low correlation between these two different economies allows the reduction of the portfolio variance.

Eun and Resnick (1988) question previous studies on the benefits of international diversification, as these did not take into account the uncertainty related to the estimation of the returns. Moreover, they alert for the risk associated with fluctuating exchange rates, as unfavorable movements have the potential to override asset gains. With this in mind, they propose a hedging strategy based on the short selling of an expected amount of foreign currency at the forward rate. They show that from the point of view of an US investor, this hedging strategy outperforms unhedged strategies.

More recently, Topaloglou et al. (2007, 2008) present a multi-stage stochastic programming model that jointly determines the asset weights and the corresponding hedge ratios for the international currencies, using the Conditional Value-at-Risk (CVaR) as a risk measure. In their work, the authors include not only forward contracts but also currency and quanto options to hedge against the foreign exchange risk.

We deal with the uncertainty inherent to parameter estimation in international portfolio optimization by applying robust optimization techniques. This idea has initially been developed by Rustem and Howe (2002). We expand on their work by reformulating the problem in a convex tractable framework and by subsequently implementing the model using historical market data.

# 2. Robust International Portfolio Optimization

Our starting point is a US investor who wishes to invest in assets from other countries. In order to calculate his returns, he must not only take into account the asset returns in their domestic currency, but also the returns on the foreign exchange rates. We assume that there are $n$ available assets in the market, denominated in $m$ foreign currencies. The current and the future price of the $i$th asset in its local currency is denoted by $P_i^0$ and $P_i$, respectively. The local return of asset $i$ is then $r^a_i = P_i/P_i^0$. We denote by $E_j$ and $E_j^0$ the future and the current spot exchange rate of the $j$th currency, respectively. Both quantities are expressed in terms of the base currency per unit of the foreign currency $j$. The return on a specific currency $j$ is then described by $r^e_j = E_j/E_j^0$. The total return on any asset $i$ will result from the multiplication of the local returns $r^a_i$ with the respective currency returns $r^e_j$.

We first need to define an auxiliary matrix $\bm{\mathcal O}$ that assigns to each asset exactly one currency. In the Markowitz framework we would want to minimize some risk measure, the portfolio variance, while guaranteeing a minimum expected return, $r_\text{target}$. The formulation of our problem would be:

(1)
\begin{align} \min_{w} \;\;\; \bm{\mathbb{E}\left\lbrace [\text{diag}(r^a)\mathcal{O}r^e]^\prime w - \mathbb{E}([\text{diag}(r^a)\mathcal{O}r^e]^\prime w)\right\rbrace^2} \label{eq:markowitz} \end{align}
(2)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \bm{\mathbb{E}([\text{diag}(r^a)\mathcal{O}r^e]^\prime w)} & \geq & r_\text{target}\\ \bm{w^\prime1} & = & 1\\ \bm{w} & \geq & 0 \end{eqnarray}

where the variable $\bm{w}$ denotes the vector of asset weights in the portfolio.

While the Markowitz mean-variance framework has stimulated a significant amount of research and still provides the basis for portfolio management, its assumptions have been subject to criticism. In problem formulation (1), the expected returns have already been estimated and are taken as given. If, however, the materialized returns deviate from the estimates, the determined solution may be far from the optimum or even infeasible. In view of this, we would like to incorporate directly into the model the uncertainty inherent to the estimation of the asset and currency returns. Robust optimization assumes that the returns are random variables, which may materialize in the future within a certain interval. This interval, commonly designated as uncertainty set, reflects the investor's expectations as to how the returns will behave and may be constructed according to some probabilistic measures.

We would like to obtain a solution to our problem that satisfies all the constraints, for all the possible values of the returns within that defined uncertainty set. Hence, we are interested in the worst-case value of the returns for which the solution is still feasible. The robust counterpart of the international portfolio optimization model is:

(3)
\begin{align} \max_{\bm w} \;\;\; \min_{\bm{(r^a,r^e)} \in \Xi} \;\;\; \bm{[\text{diag}(r^a)\mathcal{O}r^e]^\prime w} \end{align}
(4)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \bm{1^\prime w} & = & 1\\ \bm{w} & \geq & 0 \end{eqnarray}

where we defined the uncertainty set $\Xi$ as:

(5)
\begin{eqnarray} \Xi = \left \lbrace \bm{(r^a,r^e)} \geq 0: \bm{Ar^e} \geq 0 \wedge \left( \begin{bmatrix} \bm{r^a}\\ \bm{r^e} \end{bmatrix} - \begin{bmatrix} \bm{\bar{r}^a}\\ \bm{\bar{r}^e} \end{bmatrix} \right)^\prime \Sigma^{-1} \left( \begin{bmatrix} \bm{r^a}\\ \bm{r^e} \end{bmatrix} - \begin{bmatrix} \bm{\bar{r}^a}\\ \bm{\bar{r}^e} \end{bmatrix} \right) \leq \delta^2 \right\rbrace \label{eq:uncertaintyset} \end{eqnarray}

The uncertainty set $\Xi$ defined in (5) results from the intersection of two different sets. The risk associated with the asset and the currency returns is expressed by the uncertainty set:

(6)
\begin{eqnarray} \hat{\Xi} = \left \lbrace \bm{(r^a,r^e)} \geq 0 : \left( \begin{bmatrix} \bm{r^a}\\ \bm{r^e} \end{bmatrix} - \begin{bmatrix} \bm{\bar{r}^a}\\ \bm{\bar{r}^e} \end{bmatrix} \right)^\prime \Sigma^{-1} \left( \begin{bmatrix} \bm{r^a}\\ \bm{r^e} \end{bmatrix} - \begin{bmatrix} \bm{\bar{r}^a}\\ \bm{\bar{r}^e} \end{bmatrix} \right) \leq \delta^2 \right\rbrace, \label{eq:ellipsoid} \end{eqnarray}

where we assume that $\Sigma$ is positive definite. This reflects the idea of a joint confidence interval, where deviations of the returns from their expected values are weighted by the covariance matrix $\Sigma$. The linear system of inequalities $\bm{Ar^e} \geq 0$ reflects the triangular relationship between the foreign exchange rates, which must be respected at all times to prevent arbitrage.

Because we are multiplying two different sources of returns: the local asset and the currency returns, our problem is nonconvex. A common approximation to this problem, initially proposed by Eun and Resnick (1988), is to consider the total return on assets as the sum between the local asset returns and the currency returns. We present an alternative semidefinite programming approach, where a linear function is maximized subject to the constraint that an affine combination of symmetric matrices is positive semidefinite.

We start by rewriting our robust problem (3) in the epigraph form:

(7)
\begin{align} \max_{\bm w,\phi} \;\;\; \phi \end{align}
(8)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \bm{[\text{diag}(r^a)\mathcal{O}r^e]^\prime w} - \phi & \geq & 0, \;\;\; \forall \bm{(r^a,r^e)} \in \Xi \label{eq:innermin} \\ \bm{1^\prime w} & = & 1\\ \bm{w} & \geq & 0, \end{eqnarray}

We show how to replace the semi-infinite inequality constraint by a linear matrix inequality, using the following result:

Approximate S-lemma
Consider $t$ symmetric matrices $\mathcal {W}_l$ with $l = 1,\ldots,t$ and the following propositions:

(i)
$\exists \lambda \in \mathbb{R}^t$ with $\lambda \geq 0$ and $\mathcal{S} - \sum_{l=1}^t \lambda_l \mathcal{W}_l \succeq 0$;
(ii)
$\xi^\prime \mathcal{S} \xi \geq 0$, $\forall \; \xi \in \Xi:= \lbrace \xi \in \mathbb{R}^k: \xi^\prime \mathcal{W}_l \xi \geq 0,l=1,\ldots,t \rbrace$.

For any $t \in \mathbb{N}$, (i) implies (ii).

In order to apply the Approximate S-lemma, we rewrite the constraints that define the support of our uncertain returns in the form:

(9)
\begin{align} \Xi = \lbrace \xi \in \mathbb{R}^k: e_1^\prime \xi = 1,\; \xi^\prime \mathcal{W}_l \xi \geq 0, l = 1,\ldots,t \rbrace, \end{align}

where the first component of the vector $\xi$ is by construction equal to 1.

We then replace the inequality constraint in our original problem with a linear combination of matrices constrained to be positive semidefinite:

(10)
\begin{align} \max_{\bm {w,\lambda},\phi} \;\;\; \phi \end{align}
(11)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathcal{S} - \sum_{l=1}^t \lambda_l \mathcal{W}_l & \succeq & 0\\ \bm{1^\prime w} & = & 1\\ \bm{w,\lambda} & \geq & 0 \end{eqnarray}

where:

(12)
\begin{eqnarray} \mathcal{S} = \begin{bmatrix} -\phi & 0 & 0\\ 0 & 0 & \frac{1}{2}\text{diag}\bm{(w)\mathcal{O}}\\ 0 & \frac{1}{2}\bm{\mathcal{O}}^\prime \text{diag}(\bm{w}) & 0 \end{bmatrix} \end{eqnarray}

The reformulated problem (10) on the decision variables $\bm{w}$ and $\bm{\lambda}$ constitutes a conservative approximation, that is, it provides a lower bound to our original problem (7). Moreover, the semidefinite program is a convex optimization problem, as both its objective function and constraints are convex.

# 3. Downside Risk Protection with Quanto Options

Options are a flexible instrument as they give their buyer the right but not the obligation to buy (call) or sell (put) another asset, called the underlying, at a future date for a specified price, the strike. In an international portfolio, if the investor wishes to be protected against both depreciations of the foreign exchange rate and losses in the value of the assets, he would have to buy both currency and equity options. We propose to use quanto options to overcome these issues. Quanto options or “quantity-adjusting options” are mostly used in foreign exchange markets, where the price of an underlying asset needs to be converted into another underlying asset at a fixed guaranteed rate.

We define the payoff of a quanto put option $Q$ as the difference between the strike price $K$ and the spot price of the underlying asset $P$ at maturity date, translated to the base currency of the investor at a specified exchange rate $\bar{E}$:

(13)
\begin{align} Q = \max \left \lbrace 0,\bar{E} (K - P) \right \rbrace \end{align}

Note that both the strike price $K$ and the price of the asset $P$ are denominated in foreign currency and translated at the fixed foreign exchange rate $\bar{E}$ expressed in units of the base currency per unit of the foreign currency. The exchange rate chosen is usually the forward rate with the same maturity as the option. In 1992, Reiner formally derived a pricing formula for quanto options in the domestic currency, based on the same assumptions as the Black & Scholes model. The key aspect of his formulation lies in the inclusion of the correlation coefficient $\rho$ between the foreign equity and the exchange rate. We define the premium $p^q$ of a quanto put option with expiration date in $T$ periods of time as:

(14)
\begin{align} p^q = \bar{E} \left \lbrace K e^{-rT}N(\sigma_s\sqrt{T} - d_1) - P e^{(r_f - r - \rho \sigma_s \sigma_{fx})T} N(d_1) \right \rbrace \end{align}

where:

(15)
\begin{align} d_1 = \frac{\log \left (P/K \right ) + (r_f - \rho\sigma_s\sigma_{fx} + \sigma_s^2/2)T}{\sigma_s\sqrt{T}}, \end{align}

$\sigma_s$ and $\sigma_{fx}$ denote the standard deviation of the asset price and the foreign exchange rate respectively, $N(\cdot)$ is the standard normal distribution, and $r$ and $r_f$ are the domestic and the foreign risk-free rate respectively. We concentrate solely on the payoff and pricing functions of put options, as our model will only include put options. The inclusion of call options could easily be done following the same approach as for put options. Because we are interested in the potential hedging benefits of options, we choose to include only put options.

In order to include quanto options in our robust optimization model, we define as $r^q_{ij}$ the return on the $j$th quanto option on the $i$th foreign asset, given that there are $k$ options available for each asset:

(16)
\begin{align} r^q_{ij} = \max \left \lbrace 0, \frac{\bar{E} \left( K_{ij} - P_i^0 r^a_i \right)}{p^q_{ij}} \right \rbrace \end{align}

As in the previous section, we wish to maximize our portfolio return in view of the worst-case of the asset and the currency returns, assuming that these will materialize in the uncertainty set defined in (5). A new vector of weights $\bm{w_q}$ defines the percentage of the budget allocated to quanto put options. We formulate our hedging model as:

(17)
\begin{align} \max_{\bm w,\bm {w_q},\phi} \;\;\; \phi \end{align}
(18)
\begin{eqnarray} \operatorname{s.t.} \; \bm{[\text{diag}(r^a)\mathcal{O}r^e]^\prime w} + \bm{{r^q}^\prime w_q} - \phi & \geq & 0, \forall \bm{(r^a,r^e)} \in \Xi, \bm{r^q} = f(\bm{r^a}) \label{eq:innermin_options} \\ \bm{1^\prime w + 1^\prime w_q} & = & 1\\ \bm{w,w_q} & \geq & 0 \end{eqnarray}

Note that $\bm{r^q}$, though dependent on two parameters, is interpreted as a vector. Writing the return on the quanto options $\bm{r^q}$ as a function $f(\cdot)$ of the local asset returns $\bm{r^a}$, implies that the constraint on portfolio return must be satisfied for all the random returns in $\Xi$, plus:

(19)
\begin{eqnarray} r^q_{ij} & \geq & \frac{\bar{E} \left( K_{ij} - P_i^0 r^a_i \right)}{p^q_{ij}},\; \forall \;i=1,\ldots,n,\;\forall\; j=1,\ldots,k\\ r^q_{ij} & \geq & 0,\;\mspace{116.25mu} \forall \;i=1,\ldots,n,\;\forall\; j=1,\ldots,k \end{eqnarray}

Again, we will use the Approximate S-lemma to derive an equivalent tractable formulation to the hedging problem (17), and then rewrite the constraints referring to the quanto options in the quadratic form:

(20)
\begin{align} \xi_q^\prime \mathcal{W}_l \xi_q \geq 0,\; \text{with} \; l = 1,\ldots, 2(kn), \end{align}

where the vector $\xi_q$ is augmented by the variables $\bm{r^q}$:

(21)
\begin{align} \xi_q^\prime = \begin{bmatrix} 1 & \bm{r^a} & \bm{r^e} & \bm{r^q} \end{bmatrix}. \end{align}

We are then able to replace the semi-infinite inequality in our hedging model (17) with a linear combination of matrices constrained on their positive semidefiniteness:

(22)
\begin{align} \max_{\bm {w,w_q,\lambda},\phi} \;\;\; \phi \end{align}
(23)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathcal{S} - \sum_{l=1}^t \lambda_l \mathcal{W}_l & \succeq & 0\\ \bm{1^\prime w + 1^\prime w_q} & = & 1\\ \bm{w,w_q,\lambda} & \geq & 0 \end{eqnarray}

where:

(24)
\begin{eqnarray} \mathcal{S} = \begin{bmatrix} -\phi & 0 & 0 & \frac{1}{2} \bm{w_q^\prime}\\ 0 & 0 & \frac{1}{2}\text{diag}(\bm{w})\mathcal{O} & 0\\ 0 & \frac{1}{2}\mathcal{O}^\prime \text{diag}(\bm{w}) & 0 & 0\\ \frac{1}{2}\bm{w_q} & 0 & 0 & 0 \end{bmatrix} \end{eqnarray}

By investing in put options, the investor is able to lock in a certain price of the underlying asset denominated in his domestic currency. In the case of foreign assets, the exchange risk is effectively eliminated by considering a fixed foreign exchange rate in which to translate the respective payoff. Depending on the total investment in options, in particular relative to the amount invested in foreign assets, the investor may benefit from an increased protection even when the asset returns fall outside the uncertainty set considered. In the next section, we perform a series of experiments following the implementation of both the robust and the hedging models.

# 4. Numerical Results

We want to evaluate the performance of both the robust and the hedging models under real market conditions and over a long period of time. To this end, we consider the real index returns and the respective real currency returns in the period from October 1998 until September 2008. Each month we calculate the optimal asset allocation taking the expected asset and currency returns as the mean of the historical returns from the previous twelve months. The upper and lower bounds of the cross-exchange rates were calculated based on their mean returns for the period considered plus the standard deviation for the same period multiplied by a factor of $\pm1.5$. These bounds and the covariance matrix $\Sigma$ are assumed to remain constant throughout this period. At the end of each month, the actual portfolio return is computed based on the materialized returns, and the options (if any) are exercised or left to expiry depending on the spot price of the asset. This procedure is repeated every month, and the accumulated wealth is calculated.

We consider five different options for each asset in the portfolio. In the case of domestic assets, simple put options are included, while for foreign assets, we include quanto options. Because quanto options are mainly traded over-the-counter, there are no records of historical premiums. In order to perform our backtesting experiment, we simulate the options premiums based on the pricing formula developed by Reiner described in the previous section. For the simple put options, we use the Black & Scholes model. We consider five different strike prices in the range of 10% equally distant from the current asset price. The fixed foreign exchange rate $\bar{E}$ is assumed to be the historical forward exchange rate with one month maturity, equal to the option maturity. We consider an annual risk-free rate of 3.32% for the US investor (based on LIBOR annual rates for the same period).

We have solved the robust and the hedging models over the considered period for different sizes of the uncertainty set $\delta$, computed the cumulative gains and compared them with the results obtained from the Markowitz risk minimization model. We have also imposed an expected return constraint of 5% per year. Recall that this expected return must originate only from the asset returns, which prevents the entire budget from being allocated to options.

Figure above depicts the accumulated wealth from October 1998 to September 2008 for the three different models. For this particular data set and parameter choice, the minimum risk model is outperformed by the robust model, while the hedging model dominates both the robust and the minimum risk models. The average annual returns for the robust and the hedging models are 6% and 9% respectively, while the Markowitz model only provides a return of 2.84%. We have also computed the average annual return for different values of the parameter $\delta$ for both the robust and the hedging models, see table below.

$\delta$ Robust Ret. (%) Hedging Ret. (%)
0.5 6.47 8.31
1 6.02 9.03
1.5 5.98 8.89
2 6.06 9.51
2.5 6.01 11.66
3 5.99 12.56
3.5 5.98 12.80

Because we are optimizing for the worst-case scenario, our accumulated portfolio returns, even in times of decreasing asset prices, are never as low as in the Markowitz model. From January 2002 until September 2003, both the Markowitz risk minimization model and the robust model incurred in losses, though not so significant in the latter case. The insurance effect of the put options is clearly seen in that same period. The hedging model is the only model that guarantees an accumulated wealth above 1, that is, without any losses to the investor.

Although the backtesting results seem to point towards a good performance of the hedging model, these results should also be regarded with caution. Because we use simulated option prices, there is a risk of underestimating these prices, which favors the investment in options and could cause an upward-bias of the results. Furthermore, we have not considered the risk of default from the writer of the option, which in the case of over-the-counter traded options might be significant.

# 5. Conclusions

In this paper, we extend the paradigm of robust optimization to the international portfolio allocation problem. We show that, although the naive problem formulation is nonconvex due to the multiplication of asset and currency returns, it has a tractable convex formulation. The model we obtain by employing the approximate S-Lemma is a conservative approximation to our original problem. We further extend the robust optimization approach by complementing it with an investment in quanto options as an additional insurance. Quanto options link a foreign equity option with a forward rate, and they have been shown to be more effective in downside risk protection than the separate consideration of foreign equity and currency options.

The suggested approach can be considered to be more flexible than the standard hedging strategies, as it relies on options and robust optimization, and not exclusively on forward rates. Furthermore, the hedging strategy is implemented from a portfolio perspective and does not depend on the future value of any particular asset or currency. The backtesting results seem to point towards the better performance of the robust model when compared to the classical Markowitz risk minimization model. The hedging model with options outperforms both the robust and the risk minimization models in the considered data set.