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A stochastic dominance approach to evaluating foreign exchange hedging strategies.

William C. Hunter is Vice-President and Economist of the Federal Reserve Bank of Atlanta, and an Adjunct Professor of Finance at Emory University, Atlanta, Georgia. Stephen G. Timme is an Associate Professor of Finance at Georgia State University, Atlanta, Georgia.

Managers in multinational corporations, and many individuals, find themselves exposed to foreign exchange risk. Importers and exporters, for example, often need to make commitments to buy or sell goods for delivery at some future time, with the payment to be made in a foreign currency. Likewise, multinational corporations operating foreign subsidiaries receive payments from their subsidiaries that are denominated in a foreign currency. The transactions exposure, i.e., the risk associated with exchanging one currency for another, and the translations exposure -- the risk associated with expressing the value of one currency in terms of another, engendered by foreign currency transactions are prime examples of the significant risks that multinational corporations face in their foreign operations. Given these risks, it is clear that a properly designed and executed foreign exchange hedging strategy can play a significant role in determining the financial success of a firm's foreign operations.

The financial implications of decisions based on inaccurate or poorly designed foreign exchange hedging strategies, require that managers frequently evaluate alternative hedging strategies. In response to this need, previous studies have examined the ex-post performance of various foreign exchange hedging strategies. The approach used in these studies is to establish a criterion, i.e., some characteristic of a desirable hedging strategy, and then compare the results obtained using the various competing strategies relative to this criterion. The evaluation criteria are typically stated in terms of rate of return (relative to a never hedge or always hedge strategy), revenues, or profits.

It is clear from the foreign exchange hedging literature that different hedging strategies generally result in different distributions of outcomes (e.g., profits or rate of return). If the decision-maker is risk-averse or if the outcome distributions are normally distributed, then the familiar mean-variance criterion can be used to select the optimal hedging strategy. On the other hand, when the decisionmaker's utility function is not known or the outcome distributions are not normally distributed, the popular mean-variance criterion cannot always discern a unique choice. In these cases, evaluation criteria which are grounded in utility maximization, robust to nonnormal distributions, and capable of yielding optimal hedging strategies for entire classes of risk-averse or risk-neutral decision-makers are clearly desirable.

The evaluation criteria advocated in this paper emanate from standard stochastic dominance rules. It is shown that these criteria provide managers with a practical means for evaluating ex-post foreign exchange hedging strategies since they are applied to real business variables, e.g., profits, revenues, market values, or returns.(1) In addition, the criteria agree with the familiar mean-variance selection rule when the outcome distributions are normally distributed and the decision-maker is risk-averse. Thus, the approach offered in this paper represents an improvement over traditional hedging strategy evaluation methodologies.

In what follows, we quickly review some prominent criteria for ranking uncertain outcomes such as those resulting from employing different hedging strategies. Next is a brief presentation of stochastic dominance rules and how these rules can be used to form forecast evaluation criteria. This is followed by an example illustrating how the stochastic dominance criteria can be used to evaluate hedging strategies based on foreign exchange forecasts. The paper ends with a summary, conclusion, and suggestions for future research.

I. Ranking Uncertain Outcomes: Traditional Criteria

The theory of rational choice under uncertainty deals with the manner in which decision-makers ought to choose among alternative courses of action when the consequences of these actions are uncertain. Among the several ranking or selection criteria advocated in the finance literature, the semi-variance, mean absolute deviation of return, coefficient of variation, and mean-variance are frequently cited.

The semi-variance, which takes into account only below mean returns, while logically attractive, has the disadvantage of being mathematically cumbersome. In addition, if the distribution of outcomes is symmetric, the semi-variance leads to the same ranking of riskiness as the more tractable variance. As a result of these factors, the bulk of the financial literature uses the variance as the measure of risk and the mean-variance rule as the preferred selection criterion.

The mean absolute deviation of returns or outcomes, is a simple weighted average of the absolute values of the deviations of individual outcomes or returns from the expected outcome or return. Like the semi-variance, this measure has not received wide acceptance due to being difficult to manipulate mathematically. The coefficient of variation, on the other hand, has found wider acceptance and is computed by dividing the standard deviation by the mean of the outcome distribution. Although it is often used as a ranking criterion for pair-wise comparisons, it can produce misleading results when outcome distributions are nonsymmetric or decision-makers' preferences depend on moments of the outcome distribution higher than the first two.(2)

II. Stochastic Dominance

Hadar and Russell |12~ and Hanoch and Levy |13~ suggested a method for ordering uncertain prospects (probability distributions) based on stochastic dominance criteria. The strength of this approach is that it requires few restrictions on decision-makers' utility functions, does not require that the specific functional form of the decision-maker's utility function be specified, and requires no assumptions about the probability distribution of outcomes or returns. Thus, the decision rules are much more general than those based on specific types of probability distributions, e.g., mean-variance analysis. Hadar and Russell initially distinguished two types of stochastic dominance -- first degree (FSD) and second degree (SSD). If f(P) and g(P) are the probability density functions of payoffs (returns, profits, etc.) from employing strategies F and G, then the analysis can be illustrated as follows.

First-Degree Stochastic Dominance. FSD assumes that decision-makers prefer more to less, i.e., the marginal utility of the payoff or outcome (returns, profits, etc.) is positive. That is, |delta~U(P)/|delta~P |is greater than or equal to~ 0, where U(|Mathematical Expression Omitted~) is the decision-makers' utility function and P is the payoff associated with a given strategy. FSD holds whenever one cumulative probability distribution lies entirely, or partly, under another. Given that the cumulative probability distribution function can be obtained by integrating the probability density function, the dominance of strategy F over strategy G by FSD can be expressed as:

|Mathematical Expression Omitted~

for all values of P |epsilon~ |a, b~, with strict inequality holding for at least one value of P |epsilon~ |a, b~, where a and b are the minimum and maximum values of the payoffs or outcomes and |P.sub.i~ is the ith payoff. As previously mentioned, the only assumption made regarding the decision-maker's utility function is that it be monotonically increasing in the payoff.

Second-Degree Stochastic Dominance. SSD obtains when the area under one cumulative probability distribution is equal to, or larger than, the area under the other distribution, i.e., the distributions can cross one another. Formally, the criterion states that strategy G dominates strategy F, if:

|Mathematical Expression Omitted~

for all values of P |epsilon~|a, b~, with strict inequality holding for at least one value of P |epsilon~|a, b~. SSD assumes that in addition to preferring more to less, the decision-maker is risk-averse, i.e., the utility function is concave (i.e., |Mathematical Expression Omitted~).

Third-Degree Stochastic Dominance. Third-degree stochastic dominance (TSD) is a little more advanced than FSD and SSD (see Whitmore |31~ for technical details). Essentially, TSD adds the assumption of decreasing absolute risk aversion to the two assumptions underlying SSD. Decreasing absolute risk aversion assumes that the risk premium that a decision-maker or investor is willing to pay in order to eliminate a given risk decreases as the investor's wealth increases (i.e., |Mathematical Expression Omitted~).

Finally, FSD implies SSD, which in turn implies TSD.

III. Stochastic Dominance Hedging Evaluation Criteria

Application of the stochastic dominance methodology to the ex-post evaluation of foreign exchange hedging strategies requires that the decision-maker specify how the argument of his utility function (profits, returns, or revenues, among others) is affected by the use of the different hedging strategies. For example, a decision-maker could calculate the impact of individual or combinations of hedging strategies on profits, wealth, or rate of return. Once the decision-maker specifies the argument to his utility function and how it is affected by use of the different hedging strategies, the observed distributions of outcomes associated with the use of the competing strategies are then ranked by the three stochastic dominance criteria. This procedure will aid the decision-maker in determining which strategies to utilize in the future.

Typically, a decision-maker must consider the use of two or more hedging strategies. In this case, as shown in the example below, repeated pair-wise comparisons are performed to determine those strategies which are not dominated by others. This results in an "efficient" set of strategies. A strategy within a given efficient set does not dominate any other member of that set. Ideally, when comparing numerous strategies, the stochastic dominance criteria would be effective in reducing the number of strategies within each set.

A. Application of Stochastic Dominance to Foreign Exchange Hedging Decisions

In this section, the stochastic dominance criteria are used to evaluate three possible hedging strategies for a hypothetical U.S. exporter selling to foreign importers. This example is for illustrative purposes only. The stochastic dominance criteria is applied to the hedging strategies based on actual data for the British pound, Canadian dollar, French franc, German mark, Japanese yen, and Swiss franc for the period January 1980 through February 1991. The performance of each hedging strategy is evaluated at nonoverlapping one-month and three-month horizons.

B. The Competing Strategies

For expository purposes, the evaluation is couched in terms of an exporter who (each month for the one-month horizon and every three months for the three-month horizon) sells goods in a foreign currency to a foreign importer.(3) The exporter receives the foreign currency at the end of the next month for the one-month horizon and at the end of the third month following the sale for the three-month horizon. For both the one-month and three-month horizons, the exporter considers three hedging strategies.

The first strategy is referred to as the "hedged" strategy. In following the hedged strategy, the exporter always hedges in the forward market using the appropriate one-month or three-month forward rate prevailing on the day the goods are sold to the foreign importer. As a result, the exporter always hedges the receipt of the foreign currency through a forward contract. The hedged strategy is defined as the base case or the "risk-free" strategy, since the actual nominal value of the final dollar cash flows at the end of a given hedging period is known with certainty. All other strategies are therefore evaluated relative to the dollar cash flows resulting from the hedged strategy.

In the second hedging strategy, it is assumed that the current spot rate is the best forecast of the future spot rate. This results in what is referred to as the "selective" hedging strategy. Under the selective hedging strategy, the exporter sells the foreign currency forward if the current spot rate is less than or equal to the forward rate (Equation (3a)) and does not hedge if the current spot rate exceeds the forward rate (Equation (3b)).

The forward-rate adjusted returns associated with the selective hedging strategy, |Mathematical Expression Omitted~, are stated as

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where |S.sub.t+i~ is the actual spot exchange rate at time t + i; |f.sub.t,t+i~ is the time period t forward contract price of the foreign currency for delivery in i months; and |Mathematical Expression Omitted~ equals the current spot rate, which is the expected future spot rate in time period t + i. The analysis is performed using rates of return to avoid any spurious results associated with usage of the level of rates.

Empirical support for use of the selective hedging strategy is provided in the studies by Alexander and Thomas |1~, Chiang |6~, and Meese and Rogoff |24~, |25~, |26~. These studies provide evidence showing that the current rate tends to be a better predictor of the future spot rate than either the forward rate or forecasts from econometric models. The studies by Eaker and Grant |9~, Eaker and Lenowitz |10~, and Thomas |30~ show that rules based on spot rate forecasting models are effective in making decisions related to currency hedging and speculation, as well as foreign currency borrowings.(4)

The third strategy is one where the exporter never hedges. This strategy is referred to as the "unhedged" strategy. The forward-rate adjusted returns associated with the unhedged strategy, |Mathematical Expression Omitted~, are stated as

|Mathematical Expression Omitted~

Using the unhedged strategy, the exporter sells the foreign currency in the spot market upon its receipt. In the above example, it is assumed that the exporter's exchange risk is managed on a currency-by-currency basis. While this assumption is consistent with the behavior of firms which manage exchange exposure on a currency-by-currency basis, it is not appropriate for those firms which manage their exchange exposure on a diversified portfolio basis. For these firms, the stochastic dominance rules would be used to evaluate different strategies for managing the firm's portfolio of currencies. An example incorporating a portfolio approach is provided after presenting the results of the currency-by-currency analysis.


III. Data and Empirical Results

The data consist of end-of-month spot, one-month, and three-month forward rates. The spot and forward-rate data are the average of the bid and ask quotes for the period January 1980 through February 1991. The data were obtained from the Federal Reserve Board and Data Resources, Inc. (DRI).

Preliminary analyses of the exporter's strategies are shown in Panel A of Exhibit 1 for the one-month horizon and in Panel B of Exhibit 1 for the three-month horizon. Using the selective hedging strategy, the exporter hedges 84% to 95% of the time for the German mark, Swiss franc, and Japanese yen, reflecting the fact that during the sample period these currencies were generally selling at a forward premium. For the French franc, British pound, and Canadian dollar, the exporter is hedged approximately 25% to 35% of the time. The results for both the one-month and three-month horizons show that the selective hedging strategy results in the highest (lowest) mean monthly forward-rate adjusted return (standard deviation) for all currencies. This pattern is made clearer by examination of the coefficient of variation (standard deviation/mean) which suggests that the selective strategy is preferable to the unhedged strategy. However, from examination of the summary statistics, it is not obvious that the distributions are normally distributed or if the selective strategy is efficient relative to the riskless hedged strategy. The dominance of the competing strategies and their risk efficiency can be ascertained using stochastic dominance.

A. Test of Normality of Distributions of Outcomes

Determining which hedging strategy is optimal depends on the assumptions concerning the form of the exporter's utility function and, as explained earlier, the distributional properties of the forward-rate adjusted return outcomes associated with each hedging strategy. If the exact functional form of the exporter's utility were known, we could simply choose the hedging strategy which produced the maximum expected utility for the exporter. Unfortunately, exact and logically consistent utility functions are difficult to ascertain for real world decision-makers. However, if the decision-maker is risk-averse and the outcome distributions resulting from the use of the hedging strategies are normal, the simple mean-variance criterion can be employed to select the best strategy. If the latter property does not hold, however, the simple mean-variance criterion may not yield utility-maximizing choices, and more general criteria, which do not depend on the normality assumption, e.g., stochastic dominance, are required. For this reason, we conducted statistical tests on the outcome distributions associated with each risky hedging strategy.

Insights into the distributional properties of the outcomes produced by the selective and unhedged strategies can be obtained from examining the estimated measures of skewness and kurtosis shown in Panels A and B of Exhibit 1. As can be seen in Panels A and B, for all currencies, the measures of kurtosis deviate substantially from the expected value of three for the normal distribution, while the skewness measures, except for the unhedged strategy for the French franc and Canadian dollar, differ from the expected value of zero for the normal distribution. In addition, three tests of normality were conducted. The three tests are: (i) the Kolmogorov-Smirnov test, which measures overall deviations from normality; (ii) the Wald test, which examines the joint hypothesis that the skewness and kurtosis measures reported in Panels A and B of Exhibit 1 are different from their theoretical values of zero and three, respectively, (see Bera and Jarque |3~, |4~); and (iii) the studentized range, which tests the presence of fat tails in the outcome distributions.(5)

The results from the Kolmogorov-Smirnov tests suggest that for the one-month and three-month horizon, the distributions for the selective strategy are nonnormal for all currencies, whereas the outcomes for the unhedged strategy are generally normally distributed. The results for the Wald tests, however, strongly indicate that the distributions of the return outcomes associated with both hedging strategies for the one-month and three-month horizons deviate significantly from normality and are, therefore, characterized by their higher moments. The results for the studentized range are consistent with the Wald test and suggest that the distribution of outcomes is nonnormal for all currencies for the one-month and three-month horizons.

Given these distribution test results and/or the lack of knowledge of the specific form of the exporter's utility function, it is appropriate to extend the analyses in Exhibit 1 with the more general stochastic dominance criteria.

B. Stochastic Dominance Results

The results of applying the FSD, SSD, and TSD to the distributions of forward-rate adjusted returns for the selective and unhedged strategies for the one-month and three-month horizons are reported in Exhibit 2. The stochastic dominance algorithms employed are based on those provided in Levy and Kroll |19~ which modifies the standard stochastic dominance algorithms to include a risk-free asset or investment strategy. Similar to the results of standard portfolio analysis with a riskless asset, the algorithm is able to determine if the risky strategies are members of the efficient set, i.e., no one strategy dominates nor is dominated by another risky strategy or the riskless strategy. Strategies not included in the efficient set clearly should not be chosen, and the manager's choice of which risk efficient strategy to implement will depend on his tastes. Thus, the algorithm allows one to evaluate the dominance of one risky strategy relative to another risky strategy and to evaluate the efficiency of risky strategies relative to a risk-free strategy.(6)

Results for the One-Month Horizon. For FSD, the results in Exhibit 2 show that neither risky strategy dominates for any currency for the one-month horizon. This implies that a dominant strategy can not be ascertained by simply specifying that the decision-maker prefers more to TABULAR DATA OMITTED less. When SSD is invoked, the selective strategy dominates the unhedged strategy for all currencies. This, in turn, results in the selective strategy dominating the unhedged by the TSD criterion. Furthermore, the results also show that the selective strategy is efficient relative to the risk-free strategy.(7)

Results for the Three-Month Horizon. For the three-month horizon, neither the selective nor the unhedged strategy dominates under FSD. When SSD is imposed, the selective strategy dominates the unhedged strategy, for all currencies except the Japanese yen. However, even when imposing TSD, neither the selective nor unhedged strategy dominates for the Japanese yen. In addition, the analysis also shows that the selective strategy is efficient relative to the risk-free hedged strategy for all currencies except the Japanese yen. For the Japanese yen, both the selective and unhedged strategies are inefficient.

In summary, the stochastic dominance comparisons in Exhibit 2 for the one-month and three-month horizons indicate that the selective hedging strategy generally dominates the unhedged strategy and is efficient relative to the risk-free hedged strategy. Only for the three-month horizon for the Japanese yen is the distribution of outcomes from the risky strategies indistinguishable and both inefficient relative to the riskless hedged strategy. Interestingly, the unhedged strategy never dominates the selective strategy nor is it efficient.

C. Nonparametric Tests for Differences in Hedging Strategies' Outcome Distributions

Although the results imply that the selective hedging strategy dominates the unhedged strategy for most currencies, the question of whether the outcome distributions produced by this strategy are better in the statistical sense remains. To answer this question, we conducted nonparametric Wilcoxon matched-pair sign-rank tests on the outcome distributions produced by selective and unhedged strategies. The Wilcoxon matched-pair sign-rank test was chosen for the following reasons. First, as indicated earlier, the distributions of outcomes for most currencies are nonnormal, hence, the use of a nonparametric test is warranted. Second, the Wilcoxon test was applied to the ranked distribution of outcomes associated with each strategy and thus, is consistent with stochastic dominance algorithms used to rank the outcome distributions.

The results of Wilcoxon tests for the one-month horizon are reported in Panel A in Exhibit 3. These results show that the distribution of return outcomes for the selective and unhedged strategies is significantly different for all currencies except the Japanese yen. For the yen, however. the number of positive ranks for the selective strategy exceeds, by a wide margin, those for the unhedged strategy. Hence, the results in Panel A of Exhibit 3 are consistent with the stochastic dominance results reported in Exhibit 2.

The Wilcoxon test results for the three-month horizon are reported in Panel B of Exhibit 3. These results suggest that the outcome distributions from the selective and unhedged strategies are statistically different for the French franc, British pound and Canadian dollar, but indistinguishable for the German mark, Swiss franc and Japanese yen. The results for the French franc, British pound, Canadian dollar and Japanese yen are, therefore, consistent with the stochastic dominance results for the three-month horizon reported in Exhibit 2. However, the results suggest that caution should be exhibited in concluding that the selective strategy dominates the unhedged strategy for the German mark and Swiss franc.


D. Application of the Stochastic Dominance Evaluation Criteria Using a Portfolio Approach

Using a portfolio approach, alternative hedging strategies can be evaluated in terms of the distribution of outcomes for the portfolio of currencies in which the firm transacts. The advantage of using a portfolio approach is that it incorporates diversification opportunities across currencies. Compared to the currency-by-currency approach, the portfolio approach may alter the relative risk-return characteristics of the outcome distributions and, in turn, the stochastic dominance ranking of the competing hedging strategies. In using a portfolio approach, the decision-maker would apply the hedging strategies in Equations (3) and (4), or whatever alternative strategies were being examined, on a currency-by-currency basis. Next, the outcomes for the individual currencies would be combined to form a portfolio for each of the alternative hedging strategies. Finally, the distribution of outcomes for each strategy's portfolio would be evaluated using the stochastic dominance criteria. In forming portfolios, a firm would include in the ex-post analysis the mix of currencies they anticipate transacting in the future.

In the example presented here, it is assumed that the firm transacts in all six of the currencies examined in the previous example. Furthermore, it is assumed that the portfolio is comprised of equal proportions of the six currencies. The analysis is again performed using forward rate adjusted returns for the selective and unhedged strategies.

The results of the analysis for the one-month horizon are as follows.(8) The mean (standard deviation) of the forward-rate adjusted distributions for the selective and unhedged strategies is 0.34% (1.42%) and -0.04% (2.98%), respectively. Under FSD, none of the strategies dominate. The selective strategy dominates the unhedged strategy under SSD (and therefore by TSD). Finally, the results also show that the selective strategy is efficient relative to the riskless hedged strategy.

IV. Summary and Conclusion

This paper illustrates how stochastic dominance criteria can be employed to evaluate strategies for hedging foreign exchange risk. By taking account of all points of a hedging strategy's outcome distribution, the approach avoids the problems and paradoxes associated with popular selection procedures based strictly on numerical accuracy or the mean-variance criterion. These problems and paradoxes arise, for example, when the outcome distributions are nonnormal.

We use a simple stylized example from international trade to demonstrate how the stochastic dominance evaluation methodology is easily implemented, why it is theoretically sound, and why it is not restrictive in terms of its underlying assumptions. Using actual foreign exchange rate data for several currencies over the 1980-1991 period, the empirical examples presented in the paper establish the robustness of the methodology in allowing a decisionmaker to select the hedging strategy which maximizes expected utility. Useful extensions of this work would include, for example, evaluation of more complex hedging strategies, alternative decision-making scenarios (e.g., accounting and economic exposure), and explicit recognition of transactions costs.

1This paper applies the stochastic dominance methodology to evaluate foreign exchange hedging decisions. It could also, for example, be applied to the evaluation of foreign exchange forecasts as well as other hedging and forecasting scenarios. The accuracy of both structural and time series forecasting models of exchange rates has been the focus of much empirical research. See, for example, Alexander and Thomas |1~, Boothe and Glassman |5~, Frankel |11~, and Meese and Rogoff |24~, |25~, |26~.

Many evaluations of the accuracy of foreign exchange rate forecast models have been made on the basis of statistical criteria such as the root mean square error, mean absolute error, or the mean squared error, among others. These procedures implicitly assume that statistical criteria are consistent with and optimal for the subsequent use of the forecasts in a managerial decision-making framework. Papers by Jenkins |17~, Makridakis, et al |21~, and Stockman |29~ have questioned the validity of this assumption. The concern is that statistical criteria do not consider the influence of forecasting errors on resulting economic decisions and profits.

The studies by Boothe and Glassman and Stockman and those by Havenner and Modjthahedi |16~, Gerlow and Irwin |14~, Cumby and Modest |7~, Dufey and Mirus |8~, Goodman |15~, Levich |18~, and Shapiro |28~ have, in varying degrees, recognized the need for economic evaluations of exchange rate forecasting models. However, these studies do not establish a generalized utility-based evaluation methodology. A suggestion for future research would be to apply the stochastic dominance approach described in this paper to evaluate exchange rate forecasting models from a utility-based economic perspective.

2Two other criteria -- Roy's |27~ safety first criterion and Baumol's |2~ lower confidence limit criterion -- address investors' aversion to downside risk. However, neither approach enjoys the popularity of Markowitz's mean-variance criterion.

3The example only considers hedging transaction exposure using the forward markets. Since the purpose of this section is to illustrate how the stochastic dominance criteria can be implemented, alternative hedging strategies involving options and futures and/or business variables, such as profits and rate of return, are not examined. Examples of accounting and other types of economic exposure are also not examined. Application of the stochastic dominance criteria could, however, be applied to alternative hedging strategies, business variables, and accounting and economic exposure using the same framework described in this paper.

4Eaker and Grant |9~, in evaluating the performance of two investment hedging strategies using the simple decision rule, noted above, find that while the hedge strategy resulted in less risk than the no-hedge strategy, it also offered a lower rate of return than the no-hedge strategy. Thus, the authors are unable to determine which strategy is the dominant strategy using their mean-variance criterion.

5To conserve space, the statistics from the tests of normality are not presented. These statistics are available from the authors upon request.

6If a risk-free strategy is not available, the risky strategies can be evaluated using the stochastic dominance algorithms in Levy and Samat |20, Ch. 6, App. 6.31~.

7The numerical results establishing the dominance of the selective strategy over the unhedged and the selective's risk efficiency for both the one- and three-month horizons are available upon request.

8The results for the three-month horizon are essentially the same as those for the one-month horizon and are available from the authors upon request.


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21. S. Makridakis, A. Anderson, R. Carbone, R. Fildes, M. Hibon, R. Lewandowski, J. Newton, E. Parzen, and R. Winkler, "The Accuracy of Extrapolation (Time Series) Methods: Results of a Forecasting Competition," Journal of Forecasting (April/June 1982), pp. 111-153.

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23. H. Markowitz, Portfolio Selection, New York, John Wiley and Sons, 1959.

24. R. Meese and K. Rogoff. "Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?" Journal of International Economics (February 1983), pp. 3-24.

25. R. Meese and K. Rogoff, "The Out-of-Sample Failure of Empirical Exchange Rate Models: Sampling Error or Misspecification?" in Exchange Rates and International Macroeconomics, J. Frankel (ed.), Chicago, University of Chicago Press, 1983.

26. R. Meese and K. Rogoff, "Was It Real? The Exchange Rate-Interest Differential Relation over the Modern Floating-Rate Period," Journal of Finance (September 1988), pp. 933-947.

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28. A. Shapiro, Multinational Financial Management, Boston, MA, Allyn and Bacon, Inc., 1989.

29. A.C. Stockman, "Economic Theory and Exchange Rate Forecasts," International Journal of Forecasting (March 1987), pp. 3-15.

30. L. Thomas, "A Winning Strategy for Currency-Futures Speculation," Journal of Portfolio Management (Fall 1985), pp. 65-69.

31. G.H. Whitmore, "Third Degree Stochastic Dominance," American Economic Review (June 1970), pp. 457-459.
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Title Annotation:Special Issue: Corporate Control
Author:Hunter, William C.; Timme, Stephen G.
Publication:Financial Management
Date:Sep 22, 1992
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