Forex hedge eur/usd usd chf rate

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Related Articles. Partner Links. Related Terms. VaR became very popular due to the fact that Basel Committee assumed VaR as a risk measurement and that the regulatory capital for a loan is correlated to its marginal contribution to VaR. However, the use and acceptance of CVaR have increased because, in contrast to VaR, it meets expected properties. It informs us about how much we could lose if the portfolio return falls beyond VaR.

Moreover, it is a convex risk measurement which makes it easy to use to set optimal strategies in optimization problems. Therefore, using both simultaneously in multiobjective problems is not recommended. In this context, the main aim of this paper is to establish the reduction in the exchange risk borne through the use of natural multi-currency cross-hedging considering VaR and CVaR as measures of market risk.

For this, the mid exchange rates for 10 developed market currencies against the euro from January to December were used. The approach presented in this paper is useful for implementing a multi-currency hedge strategy and it contributes to the literature in several ways. Firstly, it combines the use of VaR and CVaR as measures of risk with the use of multi-currency cross-hedging as instrument of hedging.

The majority of papers in the literature use variance and derivatives, mainly current futures, for these purposes. Secondly, the approach of minimum hedge ratio and the mean-risk hedge are used. Thirdly, a multiobjective genetic algorithm is proposed to determine a mean-VaR hedge ratio. The paper is organized as follows. The second section explains the determination of the hedge ratio and considers different measures of risk. The fourth section describes the multiobjective genetic algorithm used to obtain mean-VaR hedge ratio.

The fifth section presents the data and empirical results. Finally, the last section summarizes the main findings of the research. The hedge decision requires us to establish both the optimal hedge ratio and the risk measure that needs to be reduced. Suppose that there is a set of two currencies with returns r 0 and r 1. Cross-hedging implies that a short long position in a currency is used to hedge a long short position in the other, assuming that both currencies are positively correlated.

Cross-hedging could exploit the correlation with more than one currency in order to reduce the hedge portfolio risk. In the case of n currencies the hedge portfolio return can be expressed as, where r 0 represents a long or short position in a currency, r i represents a long o short position in a currency i in order to hedge and r h represents the hedge portfolio return. One of the most important issues in hedging refers to the determination of the optimal hedge ratios, h i.

The optimal hedge ratio depends critically on the particular objective function to be optimized and the measure of risk considered. The most widely used optimal hedge ratio is the so-called minimum-variance MV hedge ratio. This is a single objective problem where the risk, measured with the variance, is minimized. This MV hedge ratio is derived by minimizing the variance of the hedged portfolio and it is quite simple to understand and estimate.

Nevertheless, the MV hedge ratio ignores the expected return of the hedged portfolio and so, in general, the MV hedge ratio is not consistent with the mean-variance framework. Under return-risk hedge ratios, expected return and risk of the hedged portfolio are considered. Companies determine the expected returns and risk, and as a consequence, the optimal hedging is obtained.

When variance is used to measure risk, this approach is called mean-variance hedge ratio. The different measures of portfolio risk can be characterized in several ways. The most important characteristics refer to the coherence of the proposed measure and to its ability to deal with the asymmetry of the returns function distribution. It is said that a risk measurement is coherent if it satisfies four properties: monotonicity, translation invariance, homogeneity and subadditivity.

Standard deviation and CVaR satisfy the four properties while VaR satisfies three of them but it does not satisfy subadditivity under certain conditions. Risk measures can be also classified in symmetric and asymmetric measures. Symmetric measures are those that do not take into consideration the asymmetry of the return function distribution, such as variance or standard deviation.

Their use is only appropriate when those functions are normally distributed or, at least, symmetric. Some of the advantages of VaR are that it takes into account the asymmetric risk, a temporal period and a confidence level. By definition, VaR is a quantile of the probability distribution of the portfolio value. VaR can be computed by using an analytic method or Delta Normal, a Montecarlo method or a historical simulation method. The analytic method assumes that returns are normally distributed and that VaR is proportional to the variance.

In the Montecarlo method, simulations are carried out to generate returns assuming that the return distribution function is known and not necessarily symmetric. Finally, the historical simulation method does not make any assumption regarding the return distribution function.

It is based on the idea that past behaviour is a good predictor of future behaviour. In this work VaR si computed by using a historical simulation method. The portfolios are built following Eq. Following, Alfaro-Cid et al. CVaR is a risk measure that has many of the advantages of VaR and quite a few less disadvantages. CVaR takes into account asymmetric risk, a temporal period and a confidence level. Rockafellar and Uryasev propose CVaR as a coherent risk measure, it deals with the kurtosis and skewness of the return distribution function and exact solutions may be found for an optimization problem.

CVaR is defined as the average of all losses exceeding the VaR and it is computed as the expected value of r conditional on exceeding the VaR,. CVaR is closely linked to VaR. That is, we obtain CVaR from the entire distribution of historical returns, as the sample mean of r h,j lower than VaR.

It can be demonstrated that CVaR is consistent with the minimum variance approximation. See, for example, Baixauli-Soler et al. Given that risk minimization turns the optimization problem in a well-posed problem if CVaR is used, or ill-posed problem if VaR is used, we use a linear programming or a multiobjective genetic algorithm to obtain optimal hedge ratios depending on whether CVaR or VaR is used to measure risk.

The mean-VaR hedge ratio for cross-hedging a long position with long or short positions in other currencies can be obtained by solving the following multiobjective problem:. As it is known, portfolio optimization problems attempt to obtain the smallest risk value for a given return, or the highest return for a certain risk level.

Logically, using VaR as a risk measure, the optimization problem is a typical ill-posed problem, in the sense of Hamard Alexander et al. In contrast to VaR, Rockafellar and Uryasev , demonstrate that, in the case of discrete random variables with T possible outcomes, it is possible to linearize CVaR by introducing a vector of auxiliary variables.

CVaR is replaced by a linear function in the objective function. Then, the mean-CVaR hedge ratio for cross-hedging a long or short position with long or short positions in other currencies can be obtained solving the following linear problem.

Consequently, in our research we have run multiobjective genetic algorithms to minimize VaR problem 4 and linear programmes to minimize CVaR problem 5. In particular, we have considered two scenarios for each currency: i to hedge a loan in a foreign currency with loans or deposit accounts in the rest of the currencies; ii to hedge a deposit in a foreign currency with loans or deposit accounts in the rest of the currencies.

A detailed description of the algorithm used can be found as well in Baixauli-Soler et al. The two objectives to maximize are set to the expected return and to the inverse of the VaR. Multiobjective GA requires to fix some parameters before running the algorithm. The population size used was individuals that evolved along 50 generations. The archive size was set to Every possible solution under evaluation called individual is represented as a vector of n integers h1GA,h2GA,…,hnGA , where n is the number of currencies available to hedge.

The evaluation of individuals follows this flow. First, portfolio returns are calculated as. Once the historical series of portfolio returns is calculated, it is sorted in descending order. The 0. Finally, VaR is calculated as the expected return minus the 0. The data set has been obtained from the Bloomberg Database. Table 1 reports summary statistics for the 10 currency series for the whole period.

A long position and a short position are considered in each currency because the distribution is a non-symmetric distribution. The variance of the returns covers between 0. The excess kurtosis coefficient ranges from 1.

Jarque—Bera test indicates that normality hypothesis cannot be accepted for all currencies. It can be seen that VaR and CVaR values for long and short positions differ since the shape of the upper and lower tail are different. Table 2 contains the hedging effectiveness of the minimum VaR and the minimum CVaR hedging strategy for the two-currency hedge portfolio.

Panel B reports the same information for a short position in each currency. The R-square coefficient has been computed for all two-currency portfolios and select the largest that allows us to obtain the maximum reduction in risk. Panel A shows that two-currency minimum VaR hedging enables us to obtain a considerable reduction in portfolio VaR for nine of the currencies. This reduction goes from On average, the reduction in VaR is In relation to minimum CVaR hedged portfolios, there is reduction for the ten currencies which ranges from 1.

On average, the reduction in CVaR is In both minimum VaR and CVaR hedge portfolios a negative return is obtained for five of the ten hedged portfolios. Panel B shows that, when a short position in the currency is hedged with a long position, the minimum VaR hedge portfolio exhibits a VaR reduction for ten hedged portfolios which ranges from 3. On average the VaR reduction is In both minimum VaR and CVaR hedge portfolios a negative return is obtained for four out of the ten hedged portfolios.

Table 3 shows the risk reduction with minimum VaR and CVaR ten-currency hedging for a given return with regards to two hedge portfolio and total, that is, with regards to VaR and CVaR of each unhedged position in each currency. In particular, the given return is selected to be the minimum VaR and CVaR two-currency efficient portfolio. With regards to reductions in comparison to the two currency hedge portfolios, the most important additional reductions in VaR are for CAD The average reductions are 8.

The number of different currencies added to the hedging portfolio is nine, except in one portfolio CVaR-JPY long where eight currencies are added. Minimum CVaR for a return given ten currecy hedging. When a ten-currency hedging portfolio is considered, the additional reductions in VaR and CVaR are small and can disappear if constraints related to long loans and short deposit accounts positions in some currencies are considered.

Hence, optimal hedge ratio under the objective of minimizing risk does not depend on the number of added currencies. By contrast, optimal hedge ratio consistent on mean-risk framework is very sensitive to the number of added currencies.

This implies that when an expected return is fixed a ten-currency hedging portfolio is significantly more efficient in VaR and CVaR than two-currency hedging portfolio. The AUD long position results are quite similar to the rest of currencies. Movements in exchange rates are a major risk for companies with foreign currency-based activities. Different approaches, such as hedging via forwards, currency swaps, futures options and many other complex financial instruments, have been employed in order to effectively manage risk.

Multi-currency cross hedging is relevant because it greatly expands the opportunity set of risk reducing alternatives. VaR and CVaR have been used for measuring currency risk exposure because they are suitable for asymmetric return distributions.

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