Robust Optimization of a Currency Portfolio

Author

Raquel Fonseca1

Keywords

Robust Optimization, Portfolio Optimization, Currency Hedging, Second-Order Cone Programming

Review Status

Reviewed; revised 27 November 2009.

Abstract

We study a currency investment strategy, where we maximize the return on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques to maximize the return on the portfolio for the worst-case foreign exchange rate scenario.
Currency portfolios differ from stock only portfolios in that a triangular relationship exists among foreign exchange rates to avoid arbitrage. Although the inclusion of such a constraint in the model would lead to a nonconvex problem, we show that by choosing appropriate uncertainty sets for the exchange and the cross exchange rates, we obtain a convex model that can be solved efficiently.
Alongside robust optimization, an additional guarantee is explored by investing in currency options to cover the eventuality that foreign exchange rates materialize outside the specified uncertainty sets.
We present numerical results that show the relationship between the size of the uncertainty sets and the distribution of the investment among currencies and options, and the overall performance of the model in a series of backtesting experiments.

1 Introduction

Since Markowitz's seminal work on portfolio optimization and the benefits of diversification, academic research in portfolio optimization has received great attention and developed to a mature area of operations research. In recent years, researchers have begun to investigate international investment and portfolios that comprise both national and international assets as a further way to increase diversification and reduce risk. It is expected that international assets have a lower correlation with national assets than the latter amongst themselves.

However, international portfolios carry an additional risk related to unfavorable movements of the foreign exchange rates. The issue of hedging the currency risk, and consequently of determining the optimal hedge ratio and of deciding on which financial instrument to use became more and more relevant.

In most of the previous studies, the hedging instrument was always the forward rate, with little attention given to currency options. In 1983, Giddy studied the application of foreign exchange options, the relationship between forward rates and currency options, and their pricing methodology. He concludes that options were a more adequate hedging instrument than forwards when the future revenues were uncertain.

In order to incorporate the uncertainty associated with the estimation of the relevant parameters, we propose to combine robust optimization with currency options to protect against a depreciation of the foreign currencies. Robust optimization differs from other uncertainty reduction techniques by incorporating uncertainty directly in the model, as returns are not assumed deterministic, but as random variables which may be realized within a prespecified uncertainty set.

Although currencies are not commonly seen as investment asset, the added risk of an international portfolio has been thoroughly studied in the literature. The focus of these studies, however, has been on currencies from the perspective of an investor on assets, that is, an investor who manages a portfolio of foreign assets and wishes to account for the currency risk and return. In contrast, this paper focuses on portfolios of currencies and, in particular, on the problem of hedging against a depreciation of the foreign exchange rates.

2 Robust Portfolio Optimization

We consider a portfolio that comprises $n$ different foreign currencies, taking the USD as our base currency. The return on a currency is measured by the ratio between the expected future spot exchange rate and the spot exchange rate today. We denote by $E_i$ and $E_i^0$ the expected future and the current spot exchange rates, respectively. Both quantities are expressed in terms of the base currency per unit of the foreign currency $i$. The expected return on a specific currency $i$ is then described by $e_i = E_i/E_i^0$. In the Markowitz framework we would want to maximize our expected portfolio return given some risk measure, in this case the variance of the portfolio. The formulation of our problem would be:

(1)
\begin{align} \max_{\mathbf{w} \in \mathbb{R}^n} \;\;\; \mathbf{e^\prime w} \end{align}
(2)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathbf{w^\prime \Sigma w} & \leq & \sigma_\textrm{target}\\ \mathbf{1^\prime w} & = & 1\\ \mathbf{w} & \geq & 0 \end{eqnarray}

The variable $\mathbf{w}$ denotes the vector of currency weights in the portfolio, while the parameter $\Sigma$ represents the covariance matrix of the currency returns.

Although the Markowitz model stimulated a significant amount of research, the mean-variance framework has also been subject to criticism due its lack of robustness. Model (1) is deterministic: it assumes that the expected returns are given, and it does not account for their random nature. Small changes in the value of the parameters, however, may pull the solution far from the optimum or even render it infeasible. Robust optimization assumes that there is a degree of uncertainty in these estimates: future returns are not certain, but random, and they may take any value within a predetermined uncertainty set. This uncertainty set represents the investor's expectations about the future currency returns and can be constructed according to some probabilistic measures.

Because we would like our solution to be robust to changes in the parameter values, we will maximize our portfolio return in view of the worst-case currency returns within the specified uncertainty set. We formulate the robust counterpart of problem (1) as:

(3)
\begin{align} \max_{\mathbf{w} \in \mathbb{R}^n} \min_{\mathbf{e} \in \Theta_e} \;\;\; \mathbf{e^\prime w} \end{align}
(4)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathbf{1^\prime w} & = & 1\\ \mathbf{w} & \geq & 0 \end{eqnarray}

Parameter e designates a random variable that represents the real currency returns, and which we assume are within the uncertainty set $\Theta$. This uncertainty set can be described in several ways, of which the most widely used ones are range intervals and ellipsoids. In our models, we define $\Theta_e$ as:

(5)
\begin{align} \Theta_\mathbf{e} = \lbrace \mathbf{e} \geq 0 : \mathbf{(e - \bar{e})}^\prime \Sigma^{-1} \mathbf{(e - \bar{e})} \leq \lambda^2 \rbrace, \end{align}

which describes an ellipsoid that is centered at the expected returns $\mathbf{\bar{e}}$ and rotated and scaled by the covariance matrix of the returns.

However, foreign exchange rates have a particular feature that distinguishes them from other investment assets such as stocks or bonds. If we define two exchange rates relative to a base currency, for example, the USD versus the EUR (USD/EUR) and the USD versus the GBP (USD/GBP), then we automatically define an exchange rate between the EUR and the GBP as well. This triangular relationship between exchange rates must be observed at all times, since otherwise arbitrage opportunities would arise and the market mechanisms would drive this relationship back to its equilibrium. Robust optimization, on the other hand, takes into account all possible returns within the uncertainty set. Hence, we need to add a new constraint to the model which enforces this triangular relationship to be respected. With $n$ currencies in the model, the number of cross exchange rates is $n(n-1)/2$. If we define as $X_{ij}$ the cross exchange rate between $E_i$ and $E_j$, that is, $X_{ij}$ is the number of units of currency $i$ that equals one unit of currency $j$, then:

(6)
\begin{align} E_i \cdot \frac{1}{E_j} \cdot X_{ij} = 1 \end{align}

We may modify this equation to express the future exchange rates in terms of the currency returns and the spot exchange rates:

(7)
\begin{eqnarray} E_i^0 e_i \cdot \frac{1}{E_j^0 e_j} \cdot X_{ij}^0 x_{ij} & = 1\\ \Leftrightarrow \quad [E_i^0 \cdot \frac{1}{E_j^0} \cdot X_{ij}^0] \cdot [e_i \cdot \frac{1}{e_j} \cdot x_{ij}] & = 1\\ \Leftrightarrow \mspace{153.25mu} e_i \cdot \frac{1}{e_j} \cdot x_{ij} & = 1 \end{eqnarray}

Including this constraint, however, will make the problem nonconvex. We then define $\Theta_\mathbf{x}$ as the uncertainty set associated with the returns of the cross exchange rates, where:

(8)
\begin{align} \Theta_\mathbf{x} = \lbrace \mathbf{x} \geq 0: \mathbf{l} \leq \mathbf{x} \leq \mathbf{u} \rbrace \end{align}

The returns on the cross exchange rates may be replaced by their corresponding ratio and the nonlinear expression may be simplified to a linear one:

(9)
\begin{eqnarray} l_{ij} \leq & x_{ij} & \leq u_{ij}, \quad \forall i,j = 1,\ldots,n, i \leq j\\ \Leftrightarrow \mspace{32mu} l_{ij} \leq & \frac{e_j}{e_i} & \leq u_{ij}\\ \Leftrightarrow \quad l_{ij}e_i \leq & e_j & \leq u_{ij}e_i \end{eqnarray}

We define $\mathbf{A}$ as the coefficient matrix reflecting all the triangular relationships between the foreign exchange rates.

Robust optimization uses duality theory to reformulate the inner minimization problem of model (3) as a maximization problem for a fixed vector w of weights. The inner minimization problem determines the worst possible outcome of the currency returns and may be formulated as:

(10)
\begin{align} \min_{\mathbf{e} \in \mathbb{R}^n} \;\;\; \mathbf{e^\prime w} \end{align}
(11)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \lVert \Sigma^{-1/2} \mathbf{(e - \bar{e})}\rVert & \leq & \lambda\\ \mathbf{Ae} & \geq & 0\\ \mathbf{e} & \geq & 0, \end{eqnarray}

where the operator $\lVert \cdot \rVert$ denotes the Euclidean two-norm. Problem (10) is a second-order cone program, and in that case strong duality holds, that is, as long as both problems are feasible, the value of the objective function of the dual problem is equal to the value of the objective function in the primal problem. Our problem now becomes:

(12)
\begin{align} \max_\mathbf{w,k,y} \;\;\; \mathbf{\bar{e}^\prime (w - A^\prime k - y)} - \lambda \lVert \Sigma^{1/2} (\mathbf{w - A^\prime k - y})\rVert \end{align}
(13)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathbf{1^\prime w} & = & 1\\ \mathbf{w,k,y} & \geq & 0 \end{eqnarray}

Problems (3) and (12) are equivalent, but (12) constitutes a tractable formulation that can be easily computed with a modern conic optimization software.

Maximizing in view of the worst possible outcome of the future returns ensures the investor with a guarantee that the portfolio value at maturity date will always be at least as high as the objective value of (12). The investor is protected against any depreciation of the foreign exchange rates that materializes within the uncertainty set, and hence robust optimization provides guarantees against the currency risk without the need to enter into any hedging agreement. The main disadvantage of this approach is that it only protects the portfolio value for fluctuations inside the uncertainty set. If the future spot exchange rates fall outside this set, robust optimization does not provide any guarantees. In the next section we present an additional strategy which includes investing in currency options to hedge against the possibility of the foreign exchange rates falling outside the uncertainty set.

3 Hedging and Robust Optimization

Options entitle the investor to a right, and not to an obligation, to buy (call) or sell (put) a particular asset at a specified strike price at a certain point in the future. Currency options are similar to other options, but the strike price considered here is a foreign exchange rate. Buying a put option on EUR versus USD with a strike price of $1.25 gives the right to transform EUR into USD at the rate of $1.25 at the maturity date. Whether the investor chooses to exercise the option will depend on the spot exchange rate at maturity. We consider only European options, therefore options may only be exercised at maturity.

We assume that for each currency the investor has a set of $m$ available put and call options with different premiums and strike prices. We denote by $E_i$ the future spot exchange rate and by $K_{il}$ the strike price of the $l$th option on the $i$th currency. We can compute the payoff $V_{il}$ of the $l$th option on currency $i$ versus the USD as:

(14)
\begin{eqnarray} V_{il}^\textrm{call} & = & \max \lbrace 0, E_i - K_{il} \rbrace \\ V_{il}^\textrm{put} & = & \max \lbrace 0, K_{il} - E_i \rbrace \end{eqnarray}

Assume now that a portfolio comprises one unit of currency $i$ and one put option on currency $i$ with a strike price $K_{il}$. At maturity date, the payoff of the portfolio would be:

(15)
\begin{eqnarray} V_\textrm{port} & = & E_i + \max \lbrace 0, K_{il} - E_i \rbrace \nonumber \\ & = & \max \lbrace E_i, K_{il} \rbrace \end{eqnarray}

Hence, by including a put option corresponding to currency $i$ in the portfolio, we are able to lock the foreign exchange rate at $K_{il}$.

We define as $\mathbf{e^d}$ the vector of returns and as $\mathbf{w^d}$ the vector of weights of the options. If $p_{il}$ is the price of the $l$th put option on currency $i$, then its return can be calculated as:

(16)
\begin{align} e^d_{il} = f(e_i) = \max \left \lbrace 0, \frac{K_{il} - E^0_i e_i}{p_{il}} \right \rbrace, \end{align}

which leads to a simplified expression, that we will be using in the following formulations of our model:

(17)
\begin{align} e^d_{il} = f(e_i) = \max \lbrace 0, a^{il} + b^{il} e_i \rbrace \;\;\;\operatorname{with} \;\;\; a^{il} = \frac{K_{il}}{p_{il}} \;\;\; \operatorname{and} \;\;\; b^{il} = - \frac{E^0_i}{p_{il}} \end{align}

As in the previous section, our investor wishes to maximize the portfolio return in view of the worst-case currency returns, while assuming that these will materialize within the uncertainty set $\Theta_\mathbf{e}$ .

(18)
\begin{align} \max_{\mathbf{w,w^d} \in \mathbb{R}^n} \min_{\stackrel{\mathbf{e} \in \Theta_\mathbf{e}}{\mathbf{e^d}=f(\mathbf{e})}} \;\;\; \mathbf{e^\prime w} + \mathbf{{e^d}^\prime w^d} \label{eq:optionsrobust1} \end{align}
(19)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathbf{1^\prime (w + w^d)} & = & 1\\ \mathbf{w,w^d} & \geq & 0 \end{eqnarray}

Following the same procedure as in the previous section, we will reformulate the inner minimization problem as a maximization problem by using duality theory. The minimization problem is concerned with finding the worst-case currency returns and it is a second-order cone program. Hence, we can replace the inner minimization problem in problem (18) with its dual formulation:

(20)
\begin{align} \max_\mathbf{w,w^d,k,y,u} \;\;\; \mathbf{\bar{e}^\prime (w - A^\prime k - y + b^\prime u)} - \lambda \lVert \Sigma^{1/2} (\mathbf{w - A^\prime k - y + b^\prime u})\rVert + \mathbf{a^\prime u} \end{align}
(21)
\begin{eqnarray} \operatorname{s.t.} \;\;\; \mathbf{1^\prime (w + w^d)} & = & 1\\ \mathbf{u} & \leq & \mathbf{w^d} \\ \mathbf{w,w^d,k,y,u} & \geq & 0 \end{eqnarray}

By using robust optimization, the investor is protected against any depreciation of the foreign exchange rates within the uncertainty set. Adding currency options to the model provides a “cap” on the value of the future foreign exchange rates. Note that neither the currency returns nor the currency option returns enter in the final formulation (20). This is a tractable problem which can be solved efficiently by any second-order cone optimization software.

4 Numerical Results

We want to assess the performance of our model under real market conditions by computing the portfolio returns over a long period of time. To this end, we consider the real currency returns in the period from January 2002 to March 2009 and conduct a backtest with a rolling horizon of twelve months. Every month we compute the estimated average returns $\bar{e}$, based on the historical returns from the previous twelve months, and calculate the optimal portfolio weights. The covariance matrix $\Sigma$ and the triangulation matrix $\mathbf{A}$ are assumed to remain the same throughout the time series. At the end of each month, the portfolio return is computed based on the materialized returns, and the options are exercised or left to expiry depending on the spot rate. This procedure is repeated until March 2009 and the accumulated returns are calculated.

Given that currency options are traded mainly over-the-counter, there are no records of historical prices, but only of three different volatilities that may be used to construct the volatility smile and compute the option price. Contrary to the assumptions of the Black & Scholes and the Garman-Kohlhagen models, the volatility is not constant throughout the spectrum of the strike prices, but is higher for “out-of-the-money” and for “in-the-money” options, while it is lower for “at-the-money” options. Moreover, it has been also verified empirically that options with the same exercise price but with different maturities exhibit different implied volatilities, designated as the term structure. The probability distribution of the currency returns, consequently, is not lognormal, but has heavier tails, making it more likely for extreme variations of the returns. The volatility associated to a given strike price may be calculated from the volatility smile, for which there is an approximate expression:

(22)
\begin{align} \sigma(\delta,T) = \sigma_{ATM,T} - 2rr_{T}\left(\delta - \frac{1}{2}\right) + 16str_T\left(\delta - \frac{1}{2}\right)^2 \end{align}

where

(23)
\begin{align} \delta = e^{-r_dT} \Phi \left[\frac{\ln(S/K) + (r_d - r_f + \sigma^2/2) T}{\sigma \sqrt{T}} \right] \end{align}

Expression (23) corresponds to the delta of a call option and is used in the Garman-Kohlhagen model. The quadratic approximation to the volatility smile (22) includes three different volatilities: i) $\sigma$, corresponding to the implied volatility of an at-the-money option (delta = 50); ii) risk reversal (rr), the difference in volatilities between a long out-of-the-money call option and a short out-of-the-money put option (delta = 25); and iii) strangle (str), the average of the volatility of two long out-of-the-money call and put options (delta = 25) minus the volatility of the at-the-money option. The volatility obtained by this expression can then be used in the Garman-Kohlhagen model to calculate the option price. We considered an annual risk free rate of 3.32% for the US investor (based on LIBOR annual rates for the same period).

We have run all of the three models - minimum risk, robust and hedging - over the period considered, rebalancing the portfolio every month and measuring the cumulative gains for different values of the parameters $\omega$ and $\rho$. While the minimum risk model yields an average annual return of 2.8%, the robust model consistently yields a higher return, from 5.7% ($\omega = 80\%$) to 3.9% ($\omega = 30\%$). As the uncertainty set increases the average returns move closer to the values exhibited by the minimum risk model.

backtesting.png

Figure above depicts the accumulated wealth when optimizing the portfolio with the three different models, taking $\omega = 80\%$ and $\rho = 50\%$ (these define the size of the uncertainty set and the insurance level, respectively. Please refer to the full paper for more details). For this particular parameter choice, the minimum risk model is dominated by both the robust and the hedging model, while the hedging model clearly outperforms the robust model, with average annual returns of 14.5% and 5.7% respectively. The hedging model outperforms the robust model for smaller values of the parameter $\rho$. Without any restriction on the minimum return guarantees outside the uncertainty set, we may choose expensive, “deep-in-the-money” options, although only a small number of units. These options will be exercised with high probabillity and yield a high return per unit as well. In contrast, as we impose a higher restriction on the minimum return, that, is, as $\rho$ increases, we also choose options less expensive (i.e.,more “at-the-money”) to be able to buy the necessary number of units to satisfy the constraint. These options will have a 50% chance of being exercised and therefore returns are potentially lower.

5 Conclusions

The suggested approach to the hedging problem has the advantage of being more flexible than the standard hedging strategies, as it relies on options and robust optimization and not on forwards or futures. The backtesting experiment conducted with real market data seems to point towards the overall better performance of the robust and of the hedging model when compared to the Markowitz minimum risk model. Moreover, we observe that when the imposition on the guaranteed portfolio return for the entire support of the currency returns is not too restrictive, the hedging model outperforms the robust model.

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References
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2. Hull, John C. Options, Futures and Other Derivatives. Pearson International Edition, 2006.
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5. Ben-Tal, A.; and Nemirovski, A. Robust Convex Optimization. Mathematics of Operations Research, 23, 769-805, 1998.
6. Boyd, S.; and Vandenberghe, L. Convex optimization. Cambridge University Press, 2004
Weblinks
R. J. Fonseca, W. Wiesemann, S. Zymler and B. Rustem. Robust Optimization of a Currency Portfolio (WPS-012).

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