Currency_Hedge_Ratio_Statistical_Estimates_Russell_Nov95 (1)

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Unformatted text preview: Research Commentary Russell Research Commentaries provide original research or analysis of specific topics and events. Grant W. Gardner is an analyst in Research in Tacoma. Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? Grant W. Gardner Douglas Stone Douglas Stone is Director of Research at Nicholas Applegate in San Diego, CA. Note: This paper is based partially on research that will appear in a forthcoming article in the Financial Analysts Journal. We wish to thank Amy Barton for programming assistance. This research was begun while Douglas Stone was an analyst at Frank Russell Company. November 1995 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? 1 Introduction An essential element of currency risk management is determining a normal hedge ratio. As explained in two earlier Commentaries [Gardner (1994a,b)], the normal hedge ratio can be thought of as the fraction of a portfolio’s currency exposure that an investor would offset with forward contracts if forced to choose a permanent, constant policy. Because it is a permanent policy, the value of the normal hedge ratio is based on the long-run behavior of currencies and asset returns. If an investor uses passive currency-risk management that ignores trends and short-term fluctuations in currency values, then the normal hedge ratio should be maintained constantly. Alternatively, if the investor attempts to add value by actively managing the portfolio’s currency exposure, then the normal hedge ratio serves as the strategic benchmark used to judge the success of this active currency management. Conceptually, finding the value of the normal hedge ratio is straightforward. As explained in the earlier Commentaries, standard mean-variance optimization can be used to derive a hedge ratio that maximizes the total portfolio’s riskadjusted return. This technique underlies the discussions of currency-risk control found in Froot and Perold (1993), Glen and Jorion (1993), Jorion (1989), Kritzman (1993), Lee (1989), and Nesbitt (1991), among others. To calculate the normal hedge ratio, an investor needs values for expected returns, variances, and covariances that reflect the long-run behavior of currency and asset returns. Of course the “true” values of these parameters are unknown and must be estimated. Even when correct statistical techniques are used and the data sample is large, the parameter estimates contain degrees of error. When these estimates are used to calculate the normal hedge ratio, this error is translated into error in the estimate of the hedge ratio. The potential size of this error is a critical issue. If the estimation error is very large, then the confidence region for the true value of the normal hedge ratio is large and the point estimate is not very useful in constructing a currency hedging strategy.1 Results from the earlier Commentaries suggest that the estimation error may be significant. This Commentary provides a more detailed examination of the estimation error for investors based in the United States, the United Kingdom, and Japan. We find that when the estimate of the normal hedge ratio is based on sample values of the mean returns, standard deviations of returns, and correlations, the error of the estimate is so large as to make it essentially useless as a guide to currency risk management for investors with moderate or high degrees of risk tolerance. This result holds both when the hedge ratio is allowed to take on any value and when it is constrained to be between zero and one. The primary cause of this error is the low precision of the sample mean as an estimate of the expected return on currency. This fact suggests that useful estimates of the normal hedge ratio might be possible if the sample mean was replaced by some more-precise alternative estimate of the expected return on currency. However, we present Monte Carlo results that suggest that any such alternative estimate must be dramatically more precise than the sample mean in order to provide useful estimates of the normal hedge ratio for investors with moderate and high risk tolerance. The overall message of these results is that statistical estimates of the normal hedge ratio probably have little value as a guide to managing currency risk for investors with moderate or high risk tolerance. 1 Jobson and Korkie (1980) examine the distribution of estimated portfolio weights in a general mean-variance portfolio problem. This issue is closely related to the concept of “estimation risk” as discussed by Klein and Bawa (1976), and Jorion (1985, 1986). These authors focus on the potential loss in utility of using portfolio weights derived from an optimization that uses sample estimates of parameters. 2 Russell Research Commentary 2 The Normal Hedge Ratio 2 The Normal Hedge Ratio To illustrate the framework used by most institutional investors, currency managers, and consultants for calculating the normal hedge ratio (also called the “strategic” or “policy” hedge ratio), consider a simplified portfolio that contains composite domestic and foreign assets as well as a composite forward currency contract that is used to hedge currency risk.2 The returns on the domestic and foreign assets are represented by the returns on the indexes chosen to represent asset classes. The foreign currency return is likewise represented by the weighted return on the currencies included in the index of foreign assets. The total return on the portfolio, rp , is equal to rp = w d rd + w f r f − w f he , where: (1) wd is the fraction of wealth allocated to the domestic asset, w f is the fraction of wealth allocated to the foreign asset, h is the hedge ratio (the fraction of the value of foreign assets offset with a short position in the forward currency market), e is the currency return (the gain on a long position in foreign currency, normalized by the beginning-of-period spot rate), rd is the return on the domestic asset, and r f is the return on the foreign asset in terms of domestic currency. The normal hedge ratio is the value of the hedge ratio that maximizes the mean-variance utility function U = E rp − where: 12 σp, T (2) E rp is the expected value of the total portfolio return, σ 2 is the variance of the portfolio return, and p T is the investor’s level of risk tolerance. Formally, the optimal value of h should be found simultaneously with the optimal portfolio allocation weights wd and wf . However, in order to simplify the analysis and focus on currency risk, we take the allocation weights as given. We then derive the normal hedge ratio by taking the derivative of equation (2) with respect to h, setting this derivative equal to zero, and then solving for h. Following this * procedure, the normal hedge ratio, h , is h* = 1− where: 1 T E[ e] σ z − ρ ez , 2 wf 2 σe σe (3) E e is the expected value of the currency return, σ e is the standard deviation of the currency return, σ z is the standard deviation of the return of a zero exposure portfolio that is completely hedged, and ρ ez is the correlation coefficient of the return on the zero exposure portfolio and the currency return. The normal hedge ratio given in equation (3) can be negative (indicating that the investor should increase currency exposure beyond what is already in the portfolio) or greater than one (indicating that the portfolio should have a net short position in currency). However, many institutional investors constrain the range 2 This framework is discussed in more detail in Gardner (1994a,b). November 1995 3 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? of the hedge ratio to be between zero (completely unhedged) and one (completely hedged). Because of the quadratic form of the optimization problem, it is easy to show that the optimal value of the constrained hedge ratio is * hconstrained 0 = h * 1 if h * < 0 if 0 ≤ h * ≤ 1 (4) if h > 1 * 3 Estimating the Normal Hedge Ratio To calculate either the unconstrained [equation (3)] or constrained [equation (4)] normal hedge ratio, an investor needs long-run values for the expected currency return, the variances of the returns on currency and the zero-exposure portfolio, and the correlation of these returns. These parameters are unknown, and, consequently, the “true” normal hedge ratio is unknown. However, it can be estimated. The most common way of doing this is to calculate the sample values of the parameters for a long sample and substitute them in equation (3). Table 1 shows the “conventional” estimates of both the unconstrained and constrained normal hedge ratios for investors with various levels of risk tolerance based in the United States, the United Kingdom, and Japan. The portfolios of assets being hedged are typical of Russell clients in each of the three countries. Table 1 Conventional Estimates of the Normal Hedge Ratio Unconstrained Normal Hedge Ratio Investor Risk Tolerance T=0 (minimum variance) T=25 T=50 T=100 US Portfolio 1.11 0.18 -0.75 -2.61 UK Portfolio 1.25 1.33 1.41 1.57 Japanese Portfolio -0.58 -0.74 -0.90 -1.22 Constrained Normal Hedge Ratio Investor Risk Tolerance T=0 (minimum variance) T=25 T=50 T=100 US Portfolio 1 0.18 0 0 UK Portfolio 1 1 1 1 Japanese Portfolio 0 0 0 0 US Portfolio: 40% S&P 500 Index 40% Lehman Brothers Aggregate Bond Index 20% Morgan Stanley Capital International Europe, Australia, Far East Index UK Portfolio: 45% FT - A All Share 25% FT - A UK Government All Stocks 30% Capital-Weighted 17-Country Index of Non-UK Equities Japanese Portfolio: 25% TOPIX Index 60% Salomon Brothers BMI Japan Bond Index 15% Capital-Weighted 17-Country Index of Non-Japanese Equities Each of the sample parameters used to calculate the conventional estimates appearing in Table 1 is a random variable which estimates the true value of a 4 Russell Research Commentary 3 Estimating the Normal Hedge Ratio parameter with error. Consequently, the estimated normal hedge ratio is also a random variable that estimates the true normal hedge ratio with error. It is important for the investor to know the potential size of this error in order to know how much confidence to place in the estimate of the hedge ratio. The natural measure of this potential error is the standard deviation of the estimated hedge ratio. With an estimate of this standard deviation and a reasonable idea of other characteristics of the distribution, an investor can construct confidence intervals for the true normal hedge ratio. For some common estimators, such as a sample mean or the coefficients of a least-squares regression, there are well-known formulas for estimating the standard deviation of the estimator when the data satisfies certain assumptions. Unfortunately, even if the underlying sample returns are assumed to have “nice” propertiesindependent draws from identical joint normal distributionsthere is no known formula for estimating the standard deviation of the estimate of either version of the normal hedge ratio.3 In situations where there is no known formula for the standard deviation of an estimator, a bootstrap method can be used.4 Briefly, this method consists of randomly selecting, with replacement, N observations from a data sample of N observations.5 The estimator of interest is then calculated using this randomly selected set of observations. This process of drawing samples and calculating the estimator is repeated many times, creating an empirical distribution of the estimator. The standard deviation of this empirical distribution is then used to estimate the standard deviation of the estimator. Also, the empirical distribution can be used to directly calculate confidence regions. Table 2 shows the means and standard deviations of the bootstrap empirical distributions of the estimates of the constrained and unconstrained normal hedge ratios. The complete distributions for a risk tolerance of 50 are shown in Figures 1 through 6. 3 The estimate of the unconstrained minimum-variance hedge ratio is a regression coefficient whose standard error can be directly calculated. The standard error of the minimum-variance hedge ratio calculated using regression techniques is very close to that of the estimated standard deviation from the bootstrap technique shown in Table 2. 4 The literature on bootstrap methods is very large. Two introductions to this technique are Efron (1982) and Mooney and Duval (1993). 5 The simple methods employed here require that the observations be independent. Conventional tests on the autocorrelations of data series showed no evidence of serial correlation. November 1995 5 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? Table 2 Bootstrap Estimates of the Normal Hedge Ratio Unconstrained Normal Hedge Ratio Investor Risk Tolerance US Portfolio Mean Std Dv T=0 (min. variance) UK Portfolio Mean Std Dv Japanese Portfolio Mean Std Dv 1.11 0.59 1.19 0.45 -0.63 0.88 T=25 -0.16 3.24 1.63 1.79 -0.66 4.60 T=50 -1.44 6.59 2.07 3.71 -0.68 9.09 T=100 -3.99 13.34 2.95 7.60 -0.72 18.16 Constrained Normal Hedge Ratio Investor Risk Tolerance US Portfolio Mean Std Dv UK Portfolio Mean Std Dv Japanese Portfolio Mean Std Dv T=0 (min. variance) 0.83 0.27 0.90 0.21 0.11 0.25 T=25 0.45 0.47 0.75 0.39 0.40 0.48 T=50 0.41 0.48 0.64 0.48 0.44 0.49 T=100 0.39 0.48 0.59 0.48 0.47 0.50 Figure 1 US Portfolio Distribution of Unconstrained Normal Hedge Ratio Cumulative Frequency Frequency (%) 100% 8% 90% 7% 80% 6% 70% 5% 60% 50% 4% 40% 3% 30% 2% 20% 1% 10% 0% 0% -35 -30 -25 -20 -15 -10 -5 0 5 10 15 Hedge Ratio 6 Russell Research Commentary 3 Estimating the Normal Hedge Ratio Figure 2 UK Portfolio Distribution of Unconstrained Normal Hedge Ratio Cumulative Frequency Frequency (%) 14% 100% 90% 12% 80% 10% 70% 60% 8% 50% 6% 40% 30% 4% 20% 2% 10% 0% 0% -11 -6 -1 4 9 Hedge Ratio 14 19 24 Figure 3 Japanese Portfolio Distribution of Unconstrained Normal Hedge Ratio Cumulative Frequency Frequency (%) 6% 100% 90% 5% 80% 70% 4% 60% 3% 50% 40% 2% 30% 20% 1% 10% 0% 0% -26 -23 -20 -17 -14 -11 -8 -5 -2 1 4 7 10 13 16 19 22 Hedge Ratio November 1995 7 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? Figure 4 US Portfolio Distribution of Constrained Normal Hedge Ratio Cumulative Frequency 100% Frequency (%) 60% 90% 50% 80% 70% 40% 60% 30% 50% 40% 20% 30% 20% 10% 10% 0% 0% 0 0.010.09 0.100.19 0.200.29 0.300.39 0.400.49 0.500.59 0.600.69 0.700.79 0.800.89 0.900.99 1 Hedge Ratio Figure 5 UK Portfolio Distribution of Constrained Normal Hedge Ratio Cumulative Frequency Frequency (%) 60% 100% 90% 50% 80% 70% 40% 60% 30% 50% 40% 20% 30% 20% 10% 10% 0% 0% 0 0.010.09 0.100.19 0.200.29 0.30 0.39 0.400.49 0.500.59 0.600.69 0.700.79 0.80 0.89 0.900.99 1 Hedge Ratio 8 Russell Research Commentary 4 Improving the Precision of the Estimates Figure 6 Japanese Portfolio Distribution of Constrained Normal Hedge Ratio Cumulative Frequency 100% Frequency (%) 60% 90% 50% 80% 70% 40% 60% 30% 50% 40% 20% 30% 20% 10% 10% 0% 0% 0 0.010.09 0.100.19 0.200.29 0.30 0.39 0.400.49 0.500.59 0.600.69 0.700.79 0.80 0.89 0.900.99 1 Hedge Ratio The information in Table 2 and Figures 1 through 3 clearly shows that the estimation error for the unconstrained normal portfolio is very large when an investor has a moderate or high level of risk tolerance. For example, a US investor with risk tolerance of 50 has a 50% confidence interval (the interval from the 25th to the 75th percentile of the distribution) of (-4.95, 2.94). Because of the magnitude of the estimation error, the conventional point estimate of the unconstrained normal hedge ratio is essentially useless as a practical guide to currency risk management at this level of risk tolerance. The same is true for the constrained normal hedge ratio. Table 2 and Figure 4 show that the 50% confidence interval for a US investor with risk tolerance of 50 spans all the permitted values from zero to one. The results for the United Kingdom and Japan are similar. For investors with extremely low levels of risk tolerance, the conventional estimate of the normal hedge ratio is more useful. By way of comparison, the 50% confidence interval for the unconstrained and constrained minimum-variance hedge ratios of a US investor are (0.72, 1.47) and (0.72, 1), respectively. However, for more typical confidence levels, the confidence intervals are still quite large. For a US investor, the 90% confidence intervals for the unconstrained and constrained minimum-variance hedge ratios are (0.21, 2.12) and (0.21, 1). 4 Improving the Precision of the Estimates The lack of precision of the estimated normal hedge ratio can be traced primarily to the large standard deviation of the sample mean of currency return. The sample standard deviation of the monthly currency return is 2.75% for a US investor. November 1995 9 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? Thus the sample estimate of the standard deviation of the sample mean with 60 observations is  σ (e ) = 2.75 60 = 0.36 . To see the importance of this standard deviation for the precision of the estimated normal hedge ratio, suppose that all of the parameters needed to calculate the normal hedge ratio are known except for the expected value of the currency return. If the expected value of the currency return is estimated by the sample mean, then the standard deviation of the estimated normal hedge ratio for a US investor with risk tolerance of 50 is  Tσ ( e )  σ (h * ) = = 5.87 , (5) 2w f σ 2 e which is only slightly lower that the bootstrap standard deviation of 6.59 given in Table 2.6 The standard deviations of the sample mean monthly currency returns for the United Kingdom and Japan are 0.41 and 0.34, respectively. Thus imprecise estimates of this parameter are a major factor in the imprecise estimates of the normal hedge ratios for these countries as well. Since the primary cause of low precision in the estimate of the normal hedge ratio is the low precision of the sample mean currency return, it follows that if the sample mean could be replaced by an alternative, more-precise estimate of the currency return, then a useful estimate of the normal hedge ratio would be possible. The critical question is whether such an alternative estimate of the expected currency return exists. We address this question in the following way. We first examine how much more precise some alternative estimate must be in order to provide useful confidence intervals for the normal hedge ratio. We then briefly discuss the empirical literature on exchange rate behavior and forecasting in order to evaluate how likely it is that an alternative estimate with this degree of precision exists. To evaluate how precise an estimate of expected currency return must be, we performed a Monte Carlo experiment. We generated 1000 simulated 60-month samples of returns on currency and a zero-exposure US portfolio. These simulated samples were independent draws from a joint normal distribution with parameters equal to the US historical sample described in Table 1. For each of the 1000 simulated samples, the constrained and unconstrained normal hedge ratios were calculated, assuming a risk tolerance of 50. These normal hedge ratios used variances and correlations calculated from the simulated samples. However, rather than using the sample mean of the currency return as the estimate of the expected currency return, we used the sample mean of a “forecast variable” that was constructed to have the same expected value as the currency return but a smaller standard deviation. The normal hedge ratios constructed in this manner represent estimated normal hedge ratios one would obtain if there existed an estimate of the expected currency return that had greater precision than the sample mean. Table 3 shows the results of this Monte Carlo experiment for several versions of the forecast variable that give varying degrees of improvement in precision. For example, the row labeled “50%” gives the mean and standard deviation of the 1000 simulated hedge ratios when the forecast variable used to estimate the expected currency return has a standard deviation that is 50% of the sample mean currency return. Note that the row labeled “0%” are the results when the sample 6 The precision problem caused by using the sample mean currency return as an estimate of expected currency return is an example of a more general problem encountered in using meanvariance optimization in portfolio construction. Sample mean returns of financial assets typically have large standard deviations, and this error causes low precision in the estimates of optimal portfolio weights. This same problem has been discussed in the context of estimation risk. See the references in footnote 1 for alternative estimates to the sample mean that reduce estimation risk. 10 Russell Research Commentary 4 Improving the Precision of the Estimates mean currency return is used as the estimate of the expected currency return. Consequently, these results are very similar to those from the bootstrap estimates for a US investor with T=50 shown in Table 2. Similarly, the row labeled “100%” contains the results for the case where the expected currency return is known. Also shown are the “true” values of the unconstrained and constrained normal hedge ratios that were calculated using the parameter values employed to generate the simulated samples. Table 3 Effects of Improved Estimates of E[e] Reduction in estimate error of E[e] 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Unconstrained Normal Hedge Ratio (True value = -0.75) Mean Std. Dev. -0.69 6.29 -0.68 5.33 -0.86 5.09 -0.65 4.34 -0.81 3.72 -0.94 3.17 -0.75 2.62 -0.86 1.96 -0.89 1.46 -0.83 0.95 -0.81 0.68 Constrained Normal Hedge Ratio (True value = 0.00) Mean Std. Dev. 0.42 0.48 0.43 0.48 0.39 0.47 0.40 0.47 0.36 0.47 0.33 0.45 0.32 0.44 0.24 0.39 0.17 0.33 0.08 0.21 0.04 0.12 Table 4 shows the results for an alternative Monte Carlo experiment. These results were generated in the same manner as those shown in Table 3 except that the values of the variances and covariances were set equal to their true values rather than their sample values. These results represent an extreme case where the investor has complete knowledge of the variances and correlation. Note that for cases where the error reduction is less that 80%, the results of Tables 3 and 4 are quite similar. This fact is indicative of the point made earlier that most of the error in the estimate of the normal hedge ratio is due to the lack of precision in estimating the expected value of the currency return. Table 4 Effects of Improved Estimates of E[e] (All parameters except E[e] known) Reduction in estimate error of E[e] 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Unconstrained Normal Hedge Ratio (True value = -0.75) Mean Std. Dev. -0.78 5.81 -0.63 5.10 -0.59 4.92 -0.89 4.20 -0.75 3.50 -0.74 2.99 -0.72 2.40 -0.80 1.75 -0.81 1.18 -0.76 0.59 -0.75 0.00 Constrained Normal Hedge Ratio (True value = 0.00) Mean Std. Dev. 0.41 0.48 0.41 0.48 0.42 0.48 0.36 0.46 0.37 0.46 0.33 0.45 0.31 0.43 0.23 0.39 0.14 0.29 0.03 0.11 0.00 0.00 The results of Tables 3 and 4 show that in order for estimates of the normal hedge ratio to have standard errors that are small enough to make them useful as November 1995 11 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? practical guides to currency risk management, an investor with moderate or high risk tolerance needs some method of estimating the expected return on currency that is dramatically more precise than the sample mean. For example, even if there existed some alternative estimate that eliminated 80% of the estimation error contained in the sample mean, the standard deviation of the estimate of the unconstrained normal hedge ratio exceeds one. The standard deviation of the constrained normal hedge ratio is such that confidence regions for typical confidence levels include most of the interval from zero to one. Empirical research on the behavior of exchange rates offers little encouragement of finding dramatically better estimates of expected currency return. Structural models of currency behavior that condition the expected currency return on “fundamental” financial and macroeconomic variables offer no significant improvement over naive forecasts of no change in the exchange rate. Another strand of exchange rate research examines the time series properties of currencies. An important finding of this research is that the return on currency exhibits serial dependence over certain time intervals. One implication is that technical trading rules for currency may be profitable. However, there is no evidence that this information can be used to construct precise point estimates of the expected value of the return on currency. 7 In sum, useful estimates of the confidence interval of the normal hedge ratio require estimates of the expected return on currency that are much more precise than are currently available. 5 Summary and Final Comments The results of this Commentary strongly suggest that the usual paradigm for estimating a portfolio’s normal hedge ratio is of little practical use to investors with moderate to high risk tolerance. For such investors, the normal hedge ratio cannot be estimated with reasonable precision. The primary cause of this problem is the lack of precise estimates of the return on currency. For investors in the United States and the United Kingdom, these results reinforce the conclusion reached in two earlier Commentaries [Gardner (1994a,b)]. For typical Japanese portfolios, this earlier work concluded that there was support for an unhedged normal hedge ratio. However, the results presented in this Commentary indicate that no such definite conclusion can be justified. The reason for the difference in results is that the earlier research did not fully account for the imprecision involve in estimating the expected return on currency. The obvious implication of these results is that an investor should be very skeptical of normal hedge ratios derived by purely statistical techniques. In terms of being able to estimate the “true” normal hedge ratio that an investor would use if he or she had full knowledge of the probability distribution that generated currency and asset returns, even “best practice” statistical estimates are essentially no better than randomly choosing a number between zero and one. But if this statistical technique does not give a good answer, how should an investor choose the normal hedge ratio? Keep in mind that the procedure described in Section 2 gives a normal hedge ratio that is optimal in the sense that it maximizes the mean-variance utility of an investor. In principle, this is a sound criteria for selecting the hedge ratio. But when it is impossible to know the value of the normal hedge ratio based on this criteria, an investor should feel free to base the choice of the normal hedge ratio on other factors. Sometimes the reasonable choice for the normal hedge ratio is obvious. If an investor has an allocation to foreign assets of less than 10%, currency fluctuations 7 Ramaswami (1993) provides an excellent review of recent empirical evidence on currency behavior and an extensive list of references. 12 Russell Research Commentary 6 References will have only a limited effect on total portfolio return. Unless the return on the foreign segment of the portfolio is important to the investor for reasons other than its contribution to the total portfolio, it seems reasonable to adopt an unhedged normal position. A passive currency policy with an unhedged normal hedge ratio amounts to doing nothing about currency risk. Such a policy avoids any costs involved with monitoring and maintaining a currency position. When a investor has a more substantial commitment to foreign assets, another potential guide to determining the normal hedge ratio is regret.8 Regret occurs when an obvious and very different alternative hedging strategy outperforms the normal strategy over a given time horizon. Regret is always possible because currency returns and asset returns are random, so that an alternative hedging strategy can outperform the optimal normal strategy over a finite horizon. If an investor is sensitive to regret, he or she may want to set the normal hedge ratio in a way that minimizes potential regret. For example, if an investor feels that he or she could be judged against both a fully hedged and a fully unhedged alternative, a 50% hedge ratio minimizes regret. In sum, determining the best choice for a normal hedge ratio requires a thoughtful examination of an investor’s overall goals and constraints. Because of their lack of precision, statistical estimates of the normal hedge ratio provide little guidance in making this choice. 6 References Efron, Bradley. 1982. The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia: Society for Industrial and Applied Mathematics. Froot, Kenneth A., and André F. Perold. 1993. “The Determinants of Optimal Currency Hedging.” Mimeo, Harvard Business School. Gardner, Grant W. 1994a. “Managing Currency Risk in US Pension Plans.” Russell Research Commentary (January). __________ 1994b. “Currency-Risk Management for Non-US Investors.” Russell Research Commentary (April). Gardner, Grant W. and Thierry Wuilloud. 1994. “The Regret Syndrome in Currency Risk Management: A Closer Look.” Russell Research Commentary (August). Glen, Jack and Philippe Jorion. 1993. “Currency Hedging for International Portfolios.” Journal of Finance 48 (December): 1865–1886. Jobson, J. D., and B. Korkie. 1980. “Estimation for Markowitz Efficient Portfolios.” Journal of the American Statistical Association 75 (September): 544–554. Jorion, Philippe. 1985. “International Portfolio Diversification with Estimation Risk.” Journal of Business 58 (July): 259–278. __________ 1986. “Bayes-Stein Estimation for Portfolio Analysis.” Journal of Financial and Quantitative Analysis 21 (September): 279–292. __________ 1989. “Asset Allocation with Hedged and Unhedged Foreign Stocks and Bonds.” The Journal of Portfolio Management (Summer): 49–54. Klein, Roger W., and Vijay S. Bawa. 1976. “The Effect of Estimation Risk on Optimal Portfolio Choice.” Journal of Financial Economics 3 (June): 215– 231. 8 The concept of regret in the context of currency risk management is discussed in detail in Gardner (1994a,b) and Gardner and Wuilloud (1994). November 1995 13 Statistical Estimates of the Normal Currency Hedge Ratio: Best Practice or Best Guess? Kritzman, Mark. 1993. “The Optimal Currency Hedging Policy with Biased Forward Rates.” The Journal of Portfolio Management (Summer): 94–100. Lee, Adrian F. 1989. “Hedging Decisions Within the Overall Asset Allocation Strategy.” In Managing Currency Risk, edited by Mark Kritzman and Katrina F. Sherrerd. Charlottesville, Virginia: Institute of Chartered Financial Analysts. Mooney, Christopher Z., and Robert D. Duval. 1993. Bootstrapping: A Nonparametric Approach to Statistical Inference. Newbury Park, California: Sage Publications. Nesbitt, Stephen L. 1991. “Currency Hedging Rules for Plan Sponsors.” Financial Analysts Journal (March-April): 73–81. Ramaswami, Murali. 1993. Active Currency Management. Charlottesville, Virginia: Institute of Chartered Financial Analysts. 14 Russell Research Commentary ...
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