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Currency Hedging for International

Portfolios

Jochen M. Schmittmann

WP/10/151

© 2010 International Monetary Fund WP/10/151

IMF Working Paper

Finance Department

Currency Hedging for International Portfolios

Prepared by Jochen M. Schmittmann1

Authorized for distribution by Ydahlia Metzgen

-XQH 2010

Abstract

This Working Paper should not be reported as representing the views of the IMF.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent

those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are

published to elicit comments and to further debate.

This paper examines the benefits from hedging the currency exposure of international

investments in single- and multi-country equity and bond portfolios from the perspectives of

German, Japanese, British and American investors. Over the period 1975 to 2009, hedging of

currency risk substantially reduced the volatility of foreign investments at a quarterly

investment horizon. Contrary to previous studies, the paper finds that at longer investment

horizons of up to five years the case for hedging for risk reduction purposes remained strong.

In addition to its impact on risk, hedging affected returns in economically meaningful

magnitudes in some cases.

JEL Classification Numbers: G11, G15

Keywords: Currency hedging, international investments, currency risk

Author’s E-Mail Address: schmittmann.j@gmail.com

1 The author would like to thank Ydahlia Metzgen, Gary Steinberg, Patrick Njoroge, Alexander Attie, Helene

Poirson Ward, Robert Price, and seminar participants at the International Monetary Fund for helpful suggestions

and comments. The usual disclaimer applies.

I. Introduction ............................................................................................................................

II. Literature Review ..................................................................................................................

III. Data ......................................................................................................................................

IV. Components of International Investment Returns .............................................................

V. Hedging Currency Risk .......................................................................................................

A. Hedging methodology and notation ........................................................................1

B. Impact of hedging on returns ..................................................................................1

C. Impact of hedging on volatility ...............................................................................1

D. Calculating the forward premium in practice .........................................................

VI. Simple Hedge Ratios .........................................................................................................2

A. Single-country portfolios ........................................................................................2

B. Multi-country portfolios ..........................................................................................2

VII. Optimal Hedge Ratios ......................................................................................................2

A. Single-country portfolios ........................................................................................2

B. Multi-country portfolios ..........................................................................................3

VIII. Hedging and the Investment Horizon .............................................................................3

IX. Conclusion .........................................................................................................................

References ................................................................................................................................4

Tables

1. Summary Statistics...............................................................................................................

2. Quarterly Returns to International Investments ...................................................................1

3a. Variance Decomposition of Quarterly Returns ..................................................................1

3b. Variance Decomposition of Quarterly Returns ..................................................................1

4. Forward Premia versus the U.S. Dollar Derived from Deposit Rates and T-Bills ..............2

5a. Quarterly Returns on Hedged and Unhedged Portfolios ....................................................2

5b. Quarterly Returns on Hedged and Unhedged Portfolios ...................................................2

6a. Quarterly Standard Deviations of Hedged and Unhedged Portfolios ................................2

6b. Quarterly Standard Deviations of Hedged and Unhedged Portfolios ................................2

7. Quarterly Returns on Hedged and Unhedged Equal-Weighted Portfolios ..........................2

8. Quarterly Standard Deviations of Hedged and Unhedged Equal-Weighted Portfolios .......2

9. Estimated Minimum Variance Hedge Ratios ......................................................................

10. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios – Full Sample

..................................................................................................................................................3

11. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios – First Half of

Sample......................................................................................................................................3

12. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios – Second Half

of Sample .................................................................................................................................3

2

13. Variance Ratios of Unhedged and Hedged Returns over Different Horizons ...................3

14a. Estimated Minimum Variance Hedge Ratios over Different Horizons ...........................3

14b. Estimated Minimum Variance Hedge Ratios over Different Horizons ...........................3

3

I. INTRODUCTION

Investors can potentially improve the risk-adjusted performance of their portfolios by

investing internationally. For example, asset pricing models such as the Sharpe-Lintner

CAPM and multi-factor models suggest that investors should hold global portfolios.

Empirically, many authors have documented the gains from international diversification of

investment portfolios (see, for example, Levy and Sarnat (1970) and Ang and Bekaert (2002)

among many others).

For all the apparent benefits of investing internationally, investing abroad confronts investors

with the decision of how to deal with the foreign currency exposure implied by their foreign

investments. Exposure to foreign currencies potentially alters the return and risk profile of

international investments. International investors therefore need to decide whether to retain

or to hedge the implicit currency exposure associated with investing abroad.

In this paper, we consider the impact of hedging currency exposure from the perspectives of

German, Japanese, British and American investors. We analyze the impact of hedging on the

risk and return of bond and equity investments in France, Germany, Japan, the U.K, and the

U.S. Our dataset covers almost the entire period of free-floating exchange rates and includes

the financial crisis of 2007 to 2009. We provide results for simple hedge ratios that are

popular with investors as well as for optimal risk-minimizing hedge ratios that exploit the full

covariance structure.2 Simple hedge ratios include no hedging, half hedging and full

hedging. Many academic studies (for example, Campbell et al. 2010) advocate optimal hedge

ratios but we find that correlation patterns may be time-period specific. While our

methodological approach is in principle not new, our implementation and empirical results

provide new insights.

We obtain our first new result by distinguishing between short and long investment horizons.

Froot (1993) argued that investors with an investment horizon of several years would be

naturally hedged against exchange rate fluctuations by mean-reverting real exchange rates.

Consequently, only investors who are sensitive to short-term volatility over a quarter or a

year should hedge currency risk. Froot’s results are limited to the perspective of a U.K. based

investor investing in the U.S. Most of his data set spans the period prior to the current regime

of free-floating exchange rates. Despite these limitations, Froot’s reasoning has been popular

with investors. To our best knowledge, we are the first study to test whether Froot’s results

carry over to the post-Bretton Woods exchange rate regime and to investors from markets

other than the U.K. Over the last 35 years, we find that the case for hedging is generally not

decreasing with an increasing investment horizon. While in some cases hedging becomes less

effective in reducing risk at investment horizons of up to five years, there are also cases

2 A 2004 survey by Russell/Mellon indicates that only about 13 percent of institutional investors use hedge

ratios other than 0, 50 or 100 percent.

4

where over-hedging, i.e. shorting a currency, is the optimal risk minimizing strategy. In this

context, particularly being short Yen has reduced portfolio risk as the associated carry trade

profits were largely uncorrelated with bond and equity returns thus providing diversification

benefits.

Most studies on currency hedging have taken the position of a U.S. Dollar based investor.

We provide empirical evidence from the perspectives of investors in four major advanced

economies. Our first insight in this regard is that results cannot be generalized from one base

currency to another – an investor’s base currency matters significantly for drawing

conclusions on a currency hedging policy.

Hedging or, more generally, currency exposure affects both return and risk of foreign

investments. From a risk perspective, we find that for bond portfolios full hedging is the

optimal strategy in almost all cases. This is in line with previous studies. The reason is that

exchange rate volatility dominates bond return volatility. For equity investments the risk case

is more complex because covariances of equities and currencies contribute much more to

overall foreign investment risk than in the case of bonds. When we look at investments in

one foreign country at a time we find a particularly strong positive correlation between the

British Pound and equity markets and a negative correlation between the DM/Euro and

equities over our sample period. Consequently, risk-minimizing investors should have

hedged or even over-hedged exposure to the British Pound while maintaining some exposure

to the German currency would have been optimal. In multi-country portfolios, we confirm

that short positions in the British Pound and long positions in the DM/Euro would have been

optimal. For the Yen and the U.S. Dollar we find that over our entire sample full hedging

would have been optimal, with both over- and under-hedging being optimal in sub-periods.

From a return perspective, currency exposure can have important consequences for returns

although the differences in hedged and unhedged returns are not statistically significant for

the most part. This follows directly from the well-documented failure of uncovered interest

rate parity which links interest rate differentials between countries to expected exchange rate

movements. In line with the literature on the forward bias, we find that investors from low

interest rate currencies, particularly the Japanese Yen, would have benefited from keeping

the currency exposure associated with foreign investments.

In addition to differences in country coverage, our results partially differ from previous

studies for two reasons. First, we improve the data quality of hedged return series by using 3-

month bank deposit rates instead of the traditionally used T-bill rates which are not entirely

comparable across countries.3 Second, we cover a longer time period than most studies. This

is important because correlations between currencies and equities/bonds are not stable over

time. In this regard the financial crisis of 2007-2009 stands out. During this time period the

3 Specifically, T-bill maturities for Germany are 12 months and for France a mix of 12 and 3 months. Japanese

T-bills are extremely illiquid before 1999 resulting in stale prices for up to two years. Table 4 in section V

quantifies the impact of using deposit rates instead of T-Bills.

5

Euro, which has tended to move against equity markets up to the crisis, became extremely

pro-cyclical falling along with equity markets. Investors who would have sought exposure to

the Euro based on historical evidence would have incurred substantial currency losses in

addition to losses on their other assets.4

The remainder of the paper is organized as follows. Section II briefly reviews the related

literature. Section III describes our dataset. Section IV decomposes returns and variances of

international investments into exchange rate and asset exposure. Section V describes our

hedging approach. Section VI provides empirical results for half and full hedging. Section

VII discusses risk minimizing hedge ratios. Section VIII analyzes the importance of the

investment horizon for the decision to hedge and section IX concludes.

II. LITERATURE REVIEW

The optimal degree of currency hedging is controversial and depends on the motivation of

investors’ demands for currency. Currency exposure affects portfolio risk but also affects

returns to the extent that returns on foreign currency are not zero.

Based on risk considerations, full hedging of currency risk, i.e. zero demand for currencies, is

optimal assuming that foreign currencies are uncorrelated with other assets (Solnik, 1974).

Perold and Shulman (1988) recommend full hedging of investment related currency risk

based on the assumption that currency returns are zero in the long-run and that correlations of

currencies with other asset classes are close to zero on average. They proclaim currency

hedging as a “free lunch” for investors arguing that it reduces risk without affecting returns.

The additional risk reduction from hedging currency exposure is estimated to be as large as

the gains from diversifying abroad in the first place. Similarly, Eun and Resnick (1988) show

that currency risk is largely undiversifiable and that it reduces the gains from international

diversification. In their study, they highlight the practical problem of estimating the right

amount to hedge. That is, the return on a foreign equity investment is unknown at the time

the hedge arrangement is put into place. Investors can only hedge the expected return not the

actual return. This effect is often neglected, particularly in studies using a log-return

representation which implies continuous hedging.

Campbell et al. (2010) find that the U.S. Dollar, the Euro, and the Swiss France have moved

against world equity markets over the period 1975 to 2005. Therefore they suggest that risk-

minimizing equity investors should seek exposure to these currencies. For bonds full hedging

tends to be optimal in their sample, a finding that we confirm. Similarly, Glen and Jorion

(1993) find that optimal currency hedging substantially reduces risk for equity investors.

Froot (1993) makes the case for not hedging exchange rate risk over long investment

horizons. His argument is based on mean-reversion of real exchange rates to purchasing

4 For example, Campbell et al. (2010) recommend long positions in the U.S. Dollar, the Euro and the Swiss

France for risk-minimizing investors based on data up to 2005.

6

power parity (PPP). He tests the hypothesis that PPP provides an automatic hedge on 200

years of data for a U.K. based investor investing in the U.S. For equities, Froot finds that for

investment horizons beyond two years full hedging does not reduce the variance of returns

compared to no hedge. For bonds, hedging appears to be more useful as full hedging

significantly reduces the variance of returns over holding horizons of up to five years.

Practioners tend to be pragmatic in determining hedge ratios. Often they use simple hedge

ratios of 0, 50, and 100 percent. For example, providers of major hedged indices such as

MSCI and S&P hedge each foreign currency in an index fully back into the base currency using

beginning-of-period investment values. A likely reason for practioners not determining optimal

hedge ratios in a portfolio context is the instability of the approach. We are sympathetic to

the notion of ignoring potential correlations of currencies with equities. In our dataset we

find that currency-equity correlations are unstable and fluctuate from plus 40 percent in one

decade to minus 40 percent in the next decade for some currency-equity pairs. Similarly,

Black (1989a) shows that, depending on the input data, hedge ratios over a very wide range

of values can be optimal.

Much of the hedging literature naturally focuses on risk. However, the evidence on the

failure of uncovered interest rate parity (see, for example, Fama (1984) and Engel (1996))

suggests that currency excess returns are not always zero.5 The literature finds that currencies

of countries with low interest rates tend to not appreciate as much as suggested by the parity

condition. The opposite holds for currencies of countries with high interest rates. This effect

is behind the global currency carry trade where investors borrow in a low yielding currency

and lend the proceeds in a high yield currency. Hedging currency risk associated with foreign

investments removes these carry trade profits for investors from low interest rate currencies

while it may enhance the returns to high interest rate currency investors. We find the effect of

currency excess returns to be economically large but statistically insignificant. A second,

speculative impact of currencies and hence hedging on returns results from currency returns

to investors in different countries being quoted in terms of different numeraire currencies.

Black (1989) points out that each party in a currency trade can simultaneously perceive

positive returns. This manifestation of Jensen’s inequality is known as Siegel’s paradox and

can explain symmetric speculative demands for currencies. Campbell et al. (2010) highlight

that the demand for currency generated by this effect is quite small in practice given the high

volatility of currencies.

III. DATA

The sample data covers the period from January 1975 to December 2009. All data series are

available on a monthly basis and we present results for investment horizons of up to five

years. Country stock index returns are provided by Morgan Stanley Capital International

5 The literature has not reached consensus on an explanation for the failure of UIP. Market frictions and

inefficiencies as well as exposure to certain risk types such as crash risk, consumption risk, and inflation risk for

traders trying to exploit the failure have been cited as possible reasons.

7

(MSCI). Each of the indices is value-weighted, formed from all companies in the market that

fulfill minimum requirements for size, liquidity and free-float, and adjusted for dividend

payments on a daily basis. Long-term bond portfolio returns are not available prior to 1986

for all countries. We therefore use the approximation suggested by Campbell, Lo, and

MacKinlay (1997) to obtain holding-period returns from bond yields.6 Government bond

yields as well as spot exchange rates and Consumer Price Indices (CPI) are obtained from the

IMF’s International Financial Statistics (IFS). Three-month deposit rates are obtained from

IFS in the case of Japan and DataStream for Germany, France, U.K. and U.S. With the

introduction of the Euro interest rate differentials between Germany and France have

virtually disappeared and there are of course no more exchange rate movements. We

therefore only present the German perspective in all tables following Table 1.

Table 1 reports arithmetic averages and standard deviations of rolling annual changes/real

returns of the Consumer Price Index (CPI), 3-month deposit rates, stock and bond returns for

the full sample period from 1975 to 2009. Returns are in local currency terms and adjusted

for the local CPI. The table therefore allows for the comparison of returns domestic investors

can expect in their respective markets. Returns to foreigners are addressed in the next section.

Inflation, as measured by the CPI, has been highest in the U.K. with 5.6 percent per year

followed by France (4.4 percent) and the U.S. (4.2 percent). Germany and Japan have

experienced moderate inflation of 2.5 percent and 1.8 percent, respectively. Annualized real

three-month rates on wholesale deposits with banks range from only 0.2 percent in Japan to

3.3 percent in France. Volatility of deposit rates has been low not exceeding 3.1 percent per

year for any country. Real equity market returns to local investors vary substantially across

countries. While a Japanese investor has only earned about 5.4 percent per year, a French

investor has received 9.9 percent over the sample period. The equity premium over long-term

6 Campbell, Lo, MacKinlay (1997) derive a log-linear relation between holding-period returns and yields:

,,

1

,,

where ,, denotes the log return on a coupon bond with coupon rate c and n periods to maturity,

denotes the log yield on the bond at time t, and is its duration, which is approximated as

.

In our calculations, we treat all bonds as having a maturity of 10 years. We assume that bonds are issued at par,

so that the coupon rate equals the yield on the bond.

To ensure the quality of the approximation, we compare results to returns based on the JP Morgan Government

Bond Index (GBI) for the period December 1986 to December 2009. The average absolute annual difference

ranges from 0.15 percent in the case of Germany to 0.48 percent for Japan. Regression analysis confirms that

both return series track each other closely with R-squared around 95 percent. The approximation method

appears to overstate volatility a bit by about 2 percent p.a. Some differences are expected given that bonds

underlying our yield data and bonds in the GBI are not exactly identical in terms of maturity, credit quality and

liquidity. Therefore, we consider the approximation of returns from yields as providing a very good proxy of the

holding period returns an investor would have earned.

8

bonds is just above 1 percent in Japan and substantially below the other markets. Equity

returns are associated with substantial volatility in all countries with volatility being

somewhat lower in the U.K and U.S. than in the other markets. Real returns on long-term

government bonds are between 4 and 5 percent for all countries and volatilities are between 6

and 8 percent.

Table 1. Summary Statistics – Annual 1/

Sources: Author’s estimates, IMF IFS, DataStream.

1/ Arithmetic averages and standard deviations of rolling annual changes/returns in percentages.

Data coverage extends from 1975M1 to 2009M12. Data are on a monthly basis. CPIs and bond

yields are obtained from the IMF’s IFS. Stock market returns are from Morgan Stanly International.

Three-month interbank deposit rates are from IFS for Japan and from DataStream for the other

countries.

IV. COMPONENTS OF INTERNATIONAL INVESTMENT RETURNS

In this section, we examine the effect of currency fluctuations on the return and risk of

foreign investments. After establishing some notation, we present results from the viewpoint

of investors based in Germany, Japan, the U.K., and the U.S.7

Consider an investor who uses a certain base currency and is invested in a foreign currency

investment. Her nominal unhedged return measured from time t - 1 to t is given by:

̃, 11̃1 (1)

7 As mentioned in the previous section, results from the French perspective are identical to results from the

German perspective after the introduction of the Euro.

France Germany Japan

United

Kingdom

United

States

Consumer Price Index (CPI)

Average 4.39 2.51 1.79 5.65 4.19

Standard Deviation 3.99 1.67 2.64 4.74 2.91

Real 3-month Deposit Rate

Average 3.26 2.49 0.19 2.66 2.02

Standard Deviation 2.84 1.57 1.56 3.11 2.24

Real Equity Index (MSCI)

Average 9.89 8.46 5.38 8.45 7.56

Standard Deviation 24.97 24.43 22.57 16.92 17.26

Real Bond Returns (IFS)

Average 4.86 4.90 4.27 5.03 4.17

Standard Deviation 7.82 6.11 6.43 7.23 8.15

9

where is the return in foreign currency on the investment between time t - 1 and t; ̃ is the

percentage change in the base currency per unit of foreign currency over the same period.

The tilde symbol identifies random variables. Equation (1) can be written as

̃, ̃̃ (2)

Since the cross-product in equation (2), ̃, is small in magnitude, ̃, can be approximated

by8

̃, ̃ (3)

Based on equation (3), the variance of foreign investment returns is approximately

̃,̃ 2̃ (4)

As equation (4) shows, exchange rate fluctuations contribute to the variance of unhedged

foreign investment returns through their own variance and their covariance with foreign asset

returns.

The preceding analysis is analog for real returns:

̃, 11̃1 ,

⁄1 (5)

We adjust returns for inflation in an investor’s home market, ,, as opposed to adjusting

returns for the inflation in the market where returns are achieved. The reason is that inflation

in her home market is the relevant measure for an investor that tries to preserve her domestic

purchasing power.

Table 2 presents exchange rate gains/losses, currency excess return, and unhedged equity and

bond returns on a quarterly basis for investors investing in France, Germany, Japan, the U.K.

and the U.S. Currency excess returns are returns from borrowing in domestic currency for 3-

months, lending the proceeds in foreign currency for the same period, and exchanging back

into domestic currency after three months to repay the domestic currency loan. We assume

that investors can borrow and lend at the same rate. In reality, currency excess returns to

investors would be lower because of transaction costs and bid-ask spreads.

8 For quarterly returns ̃ is smaller than 0.07 percent in absolute terms in all base currency-market

combinations considered. However, for returns over longer periods the approximation is less precise.

10

Table 2. Quarterly Returns to International Investments 1/

France Germany Japan

United

Kingdom

United

States

German perspective

Exchange rate gain/loss 2/ -0.41 - 0.62 -0.57 -0.24

Currency excess return 3/ 0.22 - -0.11 0.23 0.04

Equities: unhedged return 4/ 3.10 2.70 2.34 3.29 2.75

Bonds: unhedged return 4/ 1.80 1.84 2.10 2.00 1.72

Japanese perspective

Exchange rate gain/loss 2/ -0.70 -0.27 - -0.89 -0.67

Currency excess return 3/ 0.67 0.46 - 0.64 0.35

Equities: unhedged return 4/ 2.75 2.45 1.70 2.91 2.32

Bonds: unhedged return 4/ 1.51 1.55 1.46 1.66 1.29

U.K. perspective

Exchange rate gain/loss 2/ 0.35 0.79 1.44 - 0.43

Currency excess return 3/ 0.20 0.00 -0.09 - -0.07

Equities: unhedged return 4/ 3.80 3.42 3.21 3.81 3.39

Bonds: unhedged return 4/ 2.60 2.66 2.95 2.59 2.43

U.S. perspective

Exchange rate gain/loss 2/ 0.13 0.57 1.04 -0.11 -

Currency excess return 3/ 0.49 0.29 0.02 0.41 -

Equities: unhedged return 4/ 3.58 3.20 2.73 3.68 3.00

Bonds: unhedged return 4/ 2.36 2.42 2.52 2.47 1.98

Sources: Author’s estimates, IMF IFS, DataStream.

1/ Data coverage extends from 1975M1 to 2009M12. All entries are in percentages.

2/ The exchange rate gain/loss is the change in the investor’s base currency per unit of foreign

currency over one quarter.

3/ The currency excess return is the return to an investor of borrowing in her domestic currency to

invest in foreign currency deposits.

4/ Unhedged stock and bond returns are the sum of local currency returns, exchange rate

gains/losses and interaction between local currency and exchange rate returns.

The following example illustrates the table. A German investor investing in Japan would

have gained 0.62 percent on average per quarter on exchange rate movements. An exchange

rate gain implies a depreciation of the investor’s home currency vis-à-vis the foreign

currency, so in this case the German DM/Euro has on average depreciated against the

Japanese Yen. The currency excess return from borrowing in DMs/Euros and lending in Yen

is, however, a negative 0.11. This implies that the exchange rate gain for the German investor

is more than offset by the lower interest rate earned on the Yen deposit compared to the

11

DM/Euro denominated loan. This is the flipside of the so-called currency carry trade.9 The

gains from favorable currency movements boost the returns to a German investor investing in

the Japanese stock and bond market by about 0.62 percent compared to the domestic returns

of a Japanese investor. The interaction term ̃ adds an additional 0.03 percent. Against

other currencies, German investors generally realized exchange rate losses on foreign

investments as a result of a strong home currency. The losses against investments in France

all predate the introduction of the euro and indicate the depreciation of the Franc against the

DM.

Japanese investors experienced exchange rate losses on investments in all countries

considered in this study on the back of strong Yen appreciation. Currency excess returns

from a Japanese perspective are substantial ranging from 0.35 percent for the U.S. to 0.67

percent for France on a quarterly basis. This is consistent with the fact that the Japanese Yen

has been the funding currency for the global currency carry trade for many years. Positive

currency excess returns imply that the Yen has not appreciated as much as suggested by

uncovered interest rate parity.

The British Pound has depreciated on average against all other currencies in this study

resulting in exchange rate gains on foreign investment for British investors. Similarly, U.S.

dollar investors have gained from currency movements, except on their investment in the

U.K.

The preceding discussion considers nominal returns. In real terms domestic inflation needs to

be taken into account when comparing returns. In many cases exchange rate gains/losses

compensate only partially for higher/lower domestic inflation. For example, in the case of the

U.K., a country with high average inflation, domestic stock returns still exceed foreign stock

returns despite substantial exchange rate gains.

Excess currency return pairs are generally above zero because percentage gains/losses are

quoted in different numeraire currencies for investors from different countries, as noted

earlier, an effect known as Siegel’s paradox.

Tables 3a and 3b present the breakdown of the volatility of returns to international investors

into different components. Exchange rate volatility contributes between 16 and 40 percent to

the volatility of investing in foreign stock markets.10 For bond portfolios, exchange rate risk

dominates overall volatility contributing up to 95 percent of total unhedged return volatility.

The larger relative importance of exchange rate risk for bond portfolios compared to equity

portfolios explains why practioners tend to view hedging exchange risk in the case of bonds

as much more important. The covariance of currency returns with bond and equity returns

9 The currency carry trade involves borrowing in a low-yielding currency and lending the proceeds in a high-

yielding currency. The trade is a bet against uncovered interest rate parity (UIP). UIP implies that the interest

differential between a domestic and a foreign market is an estimate of the future exchange rate changes.

10 Excluding German investments in the French stock market as this includes both the pre-and post-Euro time

period.

12

matters generally a lot less for overall investment volatility than currency volatility itself. We

also find covariance structures to be unstable over time in many cases.

Table 3a. Variance Decomposition of Quarterly Returns 1/

(1) (2) (3) (4) (5) (6) (7) (8) (9)

var(

) var

̃

) cov(

,

̃

) cor(

,

̃

) var(

̃

,) (1) / (5) *

100%

(2) / (5)

*100%

[2 * (3) /

(5)]*100%

Additional

terms

German perspective

Stock Market

France 1.19 0.03 0.02 0.10 1.25 94.98 2.50 3.07 -0.54

Germany 1.12 - - - 1.12 100.00 - - -

Japan 0.96 0.37 0.01 0.01 1.38 69.43 26.76 0.86 2.96

U.K. 0.91 0.21 0.07 0.16 1.26 71.61 16.52 11.18 0.69

U.S. 0.63 0.33 0.01 0.02 0.98 64.04 33.61 2.04 0.31

Bond Market

France 0.04 0.03 0.00 -0.10 0.07 60.97 47.78 -10.72 1.98

Germany 0.03 - - - 0.03 100.00 - - -

Japan 0.03 0.37 0.01 0.09 0.43 7.62 85.48 4.70 2.20

U.K. 0.04 0.21 0.00 -0.04 0.25 14.05 83.46 -2.42 4.91

U.S. 0.04 0.33 -0.02 -0.14 0.35 10.93 94.52 -9.06 3.61

Japanese perspective

Stock Market

France 1.19 0.34 -0.03 -0.04 1.43 83.14 23.69 -3.85 -2.98

Germany 1.12 0.34 0.03 0.04 1.49 75.02 22.55 3.44 -1.01

Japan 0.96 - - - 0.96 100.00 - - -

U.K. 0.91 0.43 0.03 0.04 1.38 65.62 30.81 4.01 -0.44

U.S. 0.63 0.35 0.02 0.04 0.99 63.52 34.94 3.65 -2.11

Bond Market

France 0.04 0.34 -0.01 -0.08 0.37 10.70 90.80 -5.05 3.56

Germany 0.03 0.34 -0.01 -0.12 0.35 8.08 95.37 -6.72 3.26

Japan 0.03 - - - 0.03 100.00 - - -

U.K. 0.04 0.43 -0.02 -0.16 0.45 7.89 95.37 -8.71 5.45

U.S. 0.04 0.35 -0.01 -0.07 0.38 9.95 90.14 -3.96 3.87

Sources: Author’s estimates, IMF IFS, DataStream.

1/ Data coverage extends from 1975M1 to 2009M12. All entries are in percentages. Column (1)

contains the variance of local currency returns, column (2) the variance of exchange rate

gains/losses, column (3) the covariance and column (4) the correlation of local currency and

exchange rate returns. Column (5) shows the overall variance of unhedged returns. Columns (6)

through (9) show the percentage contributions of variance components to the overall variance of

unhedged returns. Additional terms in column (9) include the variance of (x * e) the covariance of

(x,x * e) and the covariance of (e,x * e).

13

Table 3b. Variance Decomposition of Quarterly Returns 1/

(1) (2) (3) (4) (5) (6) (7) (8) (9)

var(

) var

̃

) cov(

,

̃

) cor(

,

̃

) var(

̃

,) (1) / (5) *

100%

(2) / (5)

*100%

[2 * (3) /

(5)]*100%

Additional

terms

U.K. perspective

Stock Market

France 1.19 0.21 -0.07 -0.14 1.30 91.52 16.37 -10.88 2.99

Germany 1.12 0.23 -0.10 -0.19 1.18 94.67 19.29 -16.23 2.27

Japan 0.96 0.73 0.05 0.05 1.83 52.22 39.88 4.97 2.93

U.K. 0.91 - - - 0.91 100.00 - - -

U.S. 0.63 0.34 -0.05 -0.10 0.89 70.78 38.36 -10.69 1.55

Bond Market

France 0.04 0.21 0.01 0.08 0.28 14.29 76.17 5.22 4.31

Germany 0.03 0.23 0.01 0.15 0.29 9.77 78.18 8.26 3.79

Japan 0.03 0.73 0.03 0.17 0.84 3.91 87.03 6.38 2.68

U.K. 0.04 - - - 0.04 100.00 - - -

U.S. 0.04 0.34 0.00 0.04 0.40 9.48 84.61 2.18 3.74

U.S. perspective

Stock Market

France 1.19 0.31 -0.06 -0.10 1.42 84.02 21.99 -8.64 2.63

Germany 1.12 0.33 -0.09 -0.15 1.33 84.24 24.85 -13.45 4.35

Japan 0.96 0.37 -0.02 -0.03 1.35 70.70 27.45 -2.92 4.78

U.K. 0.91 0.32 -0.01 -0.02 1.24 73.11 25.50 -1.80 3.19

U.S. 0.63 - - - 0.63 100.00 - - -

Bond Market

France 0.04 0.31 -0.01 -0.07 0.35 11.42 89.06 -4.64 4.16

Germany 0.03 0.33 0.00 -0.02 0.37 7.77 89.96 -1.23 3.49

Japan 0.03 0.37 0.01 0.09 0.44 7.50 84.76 4.69 3.06

U.K. 0.04 0.32 -0.01 -0.11 0.35 10.17 91.36 -6.75 5.21

U.S. 0.04 - - - 0.04 100.00 - - -

Sources: Author’s estimates, IMF IFS, DataStream.

1/ Data coverage extends from 1975M1 to 2009M12. All entries are in percentages. Column (1)

contains the variance of local currency returns, column (2) the variance of exchange rate

gains/losses, column (3) the covariance and column (4) the correlation of local currency and

exchange rate returns. Column (5) shows the overall variance of unhedged returns. Columns (6)

through (9) show the percentage contributions of variance components to the overall variance of

unhedged returns. Additional terms in column (9) include the variance of ( * ̃) the covariance of

(, * ̃) and the covariance of (̃, * ̃).

The covariance between local currency stock market returns and exchange rate movements is

positive in all cases for a German investor. Exchange rate movements are thus found to

reinforce, rather than offset, the stock market movements in this case. From a Japanese

perspective covariances are positive except for the French stock market. For the U.K., with

the exception of the Japanese stock market, and for the U.S. covariances between exchange

rate changes and stock market returns are negative thus offsetting some of the stock market

movement.

14

In the case of bond markets, German and U.S. investors have benefited from negative co-

movement between local currency and exchange rate returns except for Japanese bonds. Yen-

based investors have generally benefited from risk reduction through a negative covariance

between bond and exchange rate returns, while British investors have experienced positive

covariances.

Eun and Resnick (1988) extend the preceding analysis to a portfolio context. As they show,

in the multi-currency case overall portfolio risk of foreign investment depends on (a) the

covariances among stock market returns, (b) the covariances among the exchange rate

changes, and (c) the cross-covariances among the stock market returns and the exchange rate

changes.

V. HEDGING CURRENCY RISK

The previous section has shown the substantial contribution of exchange rate risk to the

overall risk of international investments. It is therefore natural for investors to consider

hedging exchange rate exposure. In this section we develop a framework for calculating

hedged returns. Sections V and VI present empirical evidence.

A. Hedging methodology and notation

One way to implement a currency hedge involves short-term borrowing in foreign currency

and lending the proceeds in the investor’s base currency. A fully hedged investor would

borrow the present value of the expected foreign investment proceeds, i.e. 1 /1

,, where , represents the foreign interest rate, and exchange the proceeds at the spot

exchange rate into domestic currency to invest at the domestic interest rate ,. At maturity

the investor would repay the foreign currency loan valued 1 with the expected

proceeds on the foreign investment. This hedging strategy is imperfect to the extent that the

realization of the return on the foreign investment deviates from its expectation at time t-1.

For example, consider a U.S. Dollar 10 million investment for a Japanese investor. Selling

U.S. Dollar 10 million to buy Yen perfectly hedges the exchange rate exposure for as long as

the value of the investment remains U.S. Dollar 10 million. However, any movement in the

U.S. dollar asset value will reduce the effectiveness of the hedge. For instance, if the value of

the Yen-hedged investment increases to U.S. dollar 12.5 million, the investment remains

hedged only for the original U.S. dollar 10 million. The differential of U.S. dollar 2.5 million

is fully exposed to currency movements. The quality of the hedge depends on the

predictability of the underlying asset’s value which is, inter alia, a function of the

investment’s volatility and the hedge horizon.

The preceding discussion shows that due to estimation risk it is impossible to obtain ex ante

exactly the desired target hedge ratio, i.e. the proportion of an investment’s currency

exposure that is hedged. Eun and Resnick (1988) discuss and test several approaches to

15

estimating in the context of currency hedging. Practioners, however, often simply

hedge the beginning-of-period value of their investments, in effect setting 0.11 We

find that for quarterly returns this approach is sensible given the difficulties associated with

forecasting returns. The data support this view – the average quarterly return due to the

unhedged currency exposure of the difference between beginning- and end-of-period

investment values is below 0.07 percent for all base currency/foreign investment

combinations considered in this paper. In the empirical section, we therefore proceed by only

hedging beginning-of-period investment balances. As we will discuss in section VIII,

estimation risk can, however, have a very large impact on returns over long periods.

The hedge ratio can be varied to arrive at investment portfolios that are over- or under-

hedged to varying degrees. Investors may seek to take active currency risks based on their

views on future currency movements. Many studies have also pointed out that hedging 100

percent of currency exposure is not optimal from a risk minimization standpoint when

currencies and equities/bonds are correlated.

The domestic currency return on the borrowing/lending hedge over the period t-1 to t is given

by

̃,

, 1 (6)

Let Φ be the hedge ratio. The return on a hedged investment is then a combination of the

proportion of the expected investment value the investor chooses to hedge, the proportion of

the expected investment value left unhedged, and the unexpected return on the investment

which is exposed to currency risk:

̃, Φ

11̃1

1Φ

11̃

1̃1

Φ

11

,

1

, 1Φ

11̃

1̃1 (7)

Proceeding by setting 0, equation (7) simplifies to

̃, Φ

,

, 1Φ

1̃1̃1 (8)

11 For an increasing number of bond and equity indices currency hedged versions have become available in

recent years. These hedged indices are usually based on hedging the beginning-of-period balances to 100

percent.

16

The same hedged result can be achieved with lower transaction costs by employing currency

forward contracts.12,13 An investor would sell the proportion of expected foreign currency

proceeds that she wishes to hedge in the forward market capturing the forward exchange

premium/discount , equal to Ft-1 / St-1 -1, where Ft-1 and St-1 are, respectively, the forward

and spot exchange rates in domestic currency equivalents. This hedging practice, of course,

also leaves residual foreign exchange exposure through unexpected foreign currency

proceeds. The hedged return based on a forward hedge is therefore given by

̃,

Φ

11

1Φ

11̃

1̃1

(9)

To see that hedging using borrowing/lending and hedging using forwards yields equivalent

results if covered interest rate parity (CIP) holds, note that this arbitrage condition links the

forward premium to interest rates:

,

, 1

(10)

For this relationship to hold, both interest rates must be based on instruments with identical

default risk, maturity, and liquidity. Given equation (10), equation (9) is identical with

equation (7). In the absence of investment barriers, CIP must hold to preclude arbitrage

opportunities. Empirical research generally finds strong evidence of CIP.14

Using equation (10) in equation (8) yields

̃, Φ

1

1Φ

1̃1̃1

̃̃Φ

̃ (11)

A comparison of equation (11) with the unhedged return on an international investment in

equation (2) shows that by hedging exchange rate risk an investor replaces the stochastic gain

or loss on the exchange rate, ̃, with the forward premium/discount,, which is known at

the time of the investment. If the investor hedges 100 percent of the beginning-of-period

exchange rate exposure, equation (11) becomes

̃,

̃ (12)

12 Additional means of implementing a currency hedge include currency options and swaps. However, for

investment management purposes forwards and futures are the instruments of choice for hedging.

13 Note, however, that the two strategies are not completely equivalent except in the continuous time limit.

14 See, for example, Taylor (1987). A recent study by Akram, Rime, and Sarno (2008) using high-frequency

tick-data shows that very short-lived violations of CIP arise. This is, however, of little relevance for our

quarterly analysis.

17

B. Impact of hedging on returns

Currency hedging is sometimes described as a “free lunch” (Perold and Schulman, 1988)

based on the argument that currencies add only volatility but have zero expected returns. In

the preceding notation, currency hedging affects returns if the unconditional expectation of

is different from zero.

From a theoretical perspective, if investors are risk neutral and have rational expectations,

then

0, a relationship known as uncovered interest parity (UIP). UIP implies that

the interest differential between a domestic and a foreign market is an estimate of the future

exchange rate changes. UIP, however, is not a pure arbitrage condition. To see this, suppose

the 3-month U.S. interest rate is 5 percent and the 3-month Euro interest rate is 3 percent.15

Risk neutral, rational investors must expect the U.S. Dollar to depreciate by about 2 percent

over the next 3 months to make both investments equally attractive. If, for example, risk

neutral and rational investors would expect a smaller U.S. Dollar depreciation of 1 percent,

they would borrow in Euros and lend in U.S. Dollars, thus driving up Euro rates and down

U.S. Dollar rates until the interest differential is also equal to 1 percent. This is clearly not a

riskless arbitrage opportunity as exchange rates may not move in line with the parity

condition.

Indeed, a large body of empirical literature finds that UIP does not hold; a failure often

referred to as the forward discount bias.16 Empirically, low interest rate currencies tend to not

appreciate as much as the interest rate differential and high interest rate currencies do not

depreciate as much as the interest rate differential.17 The failure of UIP suggests that in some

cases hedging affects expected mean returns of foreign investments.

C. Impact of hedging on volatility

For most investors, hedging currency exposure is about reducing the volatility of foreign

investments. In subsection A. it was shown that hedging replaces the stochastic exchange rate

gain/loss with the ex ante known forward premium/discount. The volatility of a hedged

return series compared to the equivalent unhedged return series thus depends on the volatility

of ̃ versus the volatility of. In a preview of the findings presented in the empirical

sections, we find that the quarterly volatility of is only about 7 to 16 percent of the

volatility of ̃.18 Hedging, therefore, has the potential to reduce volatility substantially at

15 Rates are quarterly.

16 For example, see the surveys by Engel (1996) and Froot and Thaler (1990).

17 The failure of UIP is the impetus behind the carry-trade in foreign exchange markets.

18 The exception is the bilateral case of Germany and France because exchange rate volatility is not present in

the post-Euro part of the sample.

18

least at short investment horizons. Mean-reverting properties of exchange rate movements

could potentially change this result for longer horizons, an issue addressed in section VIII.

In addition to the volatility of foreign exchange, the correlation of currencies with other

assets matters for the risk properties of investment-related currency exposure. For example, a

foreign currency that tends to depreciate/appreciate relative to the investor’s domestic

currency when the foreign equity market increases/decreases offsets some of the risk of the

underlying investment. Investors should ideally retain some exposure to such a currency. On

the other hand, a currency that is expected to reinforce asset market movements should be

over-hedged, i.e. sold short.

D. Calculating the forward premium in practice

The calculation of hedged returns requires data on interest rates in the investor’s base

currency and in the foreign currency. To be comparable across countries, interest rates should

be based on instruments with the same maturity, credit risk and liquidity.

We consider 3-month deposit rates and 3-month T-bill rates as candidate rates that are

available across countries. A problem with T-Bill rates is that no 3-month paper is issued by

the German government and that France only started issuing 3-month paper in 1989. For

Japan, 3-month government paper was relatively illiquid before 1999 and therefore, the Bank

of Japan deemed the interest rate on these financing bills as not representative of market

conditions in Japan (IMF, 2000). In support of this conclusion, we find that the interest rate

on Japanese T-bills is often stale before 1999, sometimes not changing for up to two years.

For 3-month deposit rates, comparability across France, Germany, Japan, the U.S., and the

U.K. is better than for T-Bills. Rates starting in 1975 are available in DataStream for all

countries except Japan. For Japan the IMF’s International Financial Statistics provide the

relevant deposit time series.

We check the comparability of interest rates across countries by comparing interest-rate-

based forward premia to forward premia derived from forward and spot exchange rates for

the period 1990 to 2009. By covered interest rate parity both calculation approaches should

yield the same result. Any systemic deviation would suggest that the employed interest rates

are not comparable across countries. Table 4 presents the comparison of forward premia

calculated from exchange rates and forward premia based on deposits and T-Bills for the

U.S. dollar. The presented differences are for quarterly premia. Forward premia derived from

deposit rates are generally closer to “true” exchange-rate-based forward premia. The

improvement is particularly large for France, Germany, and Japan. This is consistent with the

French T-Bill rate being partially and the German T-Bill rate being entirely based on 12-

month maturity rates. For Japan, as mentioned above, the problem is likely to be the absence

of a liquid secondary market for T-Bills until 1999. We conclude that deposit rates provide a

more accurate approximation of the forward premium and continue by using deposit rates in

our calculations.

19

Table 4. Forward Premia versus the U.S. Dollar Derived from Deposit Rates and T-Bills

Compared to Exchange Rate based Premia (1990M1-2009M12) 1/

Panel A: Deposit-rate-based versus exchange-rate-based

U.S./U.K. U.S./Japan U.S./France U.S./Germany

Mean absolute deviation 0.08 0.11 0.04 0.04

Standard deviation of deviation 0.12 0.25 0.05 0.05

Panel B: T-Bill-based versus exchange-rate-based

U.S./U.K. U.S./Japan U.S./France U.S./Germany

Mean absolute deviation 0.11 0.19 0.10 0.11

Standard deviation of deviation 0.15 0.19 0.10 0.07

Sources: Author’s estimates.

1/ This table provides a comparison of forward premia against the U.S. dollar derived from forward

and spot exchange rates with forward premia derived from interest rate differentials.

2/ Panel A contains forward premia based on differentials in three-month bank deposit rates

compared to exchange rate based forward rates.

3/ Panel B contains forward premia derived from T-Bill rate differentials compared to exchange rate

based forward rates.

VI. SIMPLE HEDGE RATIOS

In this section, we present empirical evidence for the impact of currency hedging for the full

sample period 1975 to 2009. The academic literature has pointed out that hedge ratios

deviating from 100 percent can be optimal in the presence of correlation between exchange

rate and asset movements. A survey of the hedging policies of institutional investors in major

markets in 2004 by Russell/Mellon suggests however that a majority of investors chooses to

hedge 0, 50, or 100 percent of foreign currency exposure. The reluctance of practioners to

calculate optimal hedge ratios and to treat currencies like other assets in a portfolio

optimization framework may be partially attributable to the instability of hedge ratios (see,

for example, Black 1989a). We proceed by presenting results for unhedged, fully hedged,

and 50 percent hedged portfolios. Optimal hedge ratios are addressed in the next section.

A. Single-country portfolios

Tables 5a and 5b show returns on unhedged and fully hedged single-country portfolios on a

quarterly basis. Returns are additive, therefore, with the results for no hedging and 100

percent hedging, results for any other hedge ratio can be obtained. In almost all cases the null

hypothesis of equal means of hedged and unhedged quarterly returns cannot be rejected at

conventional levels. The substantial sample variance of the return series, especially for

equities, makes it difficult to find statistically significant differences. In economic terms,

many of the return differentials between hedged and unhedged portfolios are, however,

substantial. For example, a Japanese investor in the French stock market would have earned

quarterly returns of 2.75 percent without hedging currency risk and only 2.1 percent on a

20

hedged basis. The difference is not statistically significant but an approximate annual return

differential of 2.6 percent over the last 34 years is very relevant to investors.

Table 5a. Quarterly Returns on Hedged and Unhedged Portfolios 1/

Tests of significance

No

hedge

Full

hedge

No hedge vs. full hedge

T-stat P-value

German perspective

Stock Market

France 3.10 2.88 0.28 78.24

Germany 2.70 - - -

Japan 2.34 2.45 -0.15 87.96

U.K. 3.29 3.07 0.31 75.65

U.S. 2.75 2.72 0.06 95.07

Bond Market

France 1.80 1.59 1.40 16.13

Germany 1.84 - - -

Japan 2.10 2.21 -0.34 73.14

U.K. 2.00 1.78 0.85 39.63

U.S. 1.72 1.68 0.12 90.07

Japanese perspective

Stock Market

France 2.75 2.10 0.84 40.01

Germany 2.45 1.99 0.58 56.19

Japan 1.70 - - -

U.K. 2.91 2.29 0.85 39.63

U.S. 2.32 1.98 0.55 58.39

Bond Market

France 1.51 0.85 2.10 3.61

Germany 1.55 1.09 1.51 13.20

Japan 1.46 - - -

U.K. 1.66 1.03 1.83 6.73

U.S. 1.29 0.95 1.06 29.15

Sources: Author’s estimates.

1/ Quarterly returns on unhedged and fully hedged stock and bond portfolios from the perspectives of

German and Japanese investors. Hedged returns are based on rolling quarterly hedges of beginning-

of-period balances. Reported T-statistics are based on the null hypothesis of equal returns.

For German investors returns are generally higher on an unhedged basis than on a hedged

basis except in the case of investments in Japan. Excluding investments in France, British

investors have yielded higher returns if they chose to hedge currency risk. Japanese and

American investors have experienced generally lower returns if hedged.

21

Table 5b. Quarterly Returns on Hedged and Unhedged Portfolios 1/

Tests of significance

No

hedge

Full

hedge

No hedge vs. full hedge

T-stat P-value

U.K. perspective

Stock Market

France 3.80 3.61 0.24 80.70

Germany 3.42 3.43 -0.01 98.89

Japan 3.21 3.31 -0.12 90.52

U.K. 3.81 - - -

U.S. 3.39 3.47 -0.13 89.57

Bond Market

France 2.60 2.41 0.69 49.21

Germany 2.66 2.67 -0.04 97.03

Japan 2.95 3.05 -0.22 82.97

U.K. 2.59 - - -

U.S. 2.43 2.51 -0.24 80.86

U.S. perspective

Stock Market

France 3.58 3.10 0.61 54.30

Germany 3.20 2.92 0.37 71.18

Japan 2.73 2.72 0.02 98.73

U.K. 3.68 3.28 0.56 57.72

U.S. 3.00 - - -

Bond Market

France 2.36 1.88 1.59 11.17

Germany 2.42 2.13 0.93 35.11

Japan 2.52 2.51 0.04 97.15

U.K. 2.47 2.07 1.33 18.51

U.S. 1.98 - - -

Sources: Author’s estimates.

1/ Quarterly returns on unhedged and fully hedged stock and bond portfolios from the perspectives of

British and American investors. Hedged returns are based on rolling quarterly hedges of beginning-of-

period balances. Reported T-statistics are based on the null hypothesis of equal returns.

The, in some cases, economically substantial return differentials between hedged and

unhedged returns point to a failure of UIP. Differences are especially large in the case of

Yen-based investors who would generally have yielded higher returns without hedging.

Going back to Table 2, this finding may surprise given that Japanese investors would have

experienced exchange rate losses against all other currencies. The explanation is that for the

Yen the forward premium, , is generally even more negative than the exchange rate loss,

̃. Interest rate differentials have thus predicted an even larger Yen appreciation than

actually materialized. Japanese investors who chose to remain unhedged on their

international investments in effect engaged in a carry trade speculating that the Yen will not

appreciate as much as suggested by UIP. The Japanese experience also highlights that

22

currency hedging does not allow international investors to access local asset returns as

sometimes stated.

We now turn to the potential of currency hedging to reduce risk. Tables 6a and 6b present

standard deviations for unhedged, half hedged and fully hedged bond and equity portfolios.

Hedging currency risk reduces the risk of international investments in almost all cases

significantly statistically as well as economically. The case for hedging is particularly

apparent for bond portfolios. For bond portfolios hedging 100 percent of currency exposure

is the dominant strategy from a risk reduction standpoint. Hedging is more effective for

bonds because, as table 3 shows, currency risk makes up a large portion of the overall risk of

international bond portfolios. Full hedging reduces risk more than half hedging in all cases

except the French and German stock markets from a U.K. investor perspective and the

German stock market from a U.S. perspective. In these cases it is optimal for risk-minimizing

investors to retain some currency exposure because the foreign currency has on average

moved against the foreign stock market thus providing hedging benefits. Specifically, the

Euro in terms of British Pounds and U.S. Dollars tended to appreciate when the French and

German stock markets have fallen. For the Euro and DM against the British Pound this effect

is present during the entire sample period. It is particularly strong over the last twenty years

and even more during the financial crisis of 2007 to 2009. For investments in the French

stock market, the effect is entirely due to the period after the Euro introduction. Against the

U.S. dollar the Euro and the DM have moved against the German equity market during the

period 1990 to 2007. During the financial crisis this pattern has dramatically changed as the

Euro fell against the U.S. Dollar along with equity markets.19

We conclude that for bonds hedging unequivocally reduces risk at quarterly horizons but,

depending on an investor’s base currency, the risk reduction may come at the price of lower

returns. For equities there is also strong evidence for the effectiveness of hedging to reduce

quarterly return volatility. However, the empirical results to this point also indicate that

hedge ratios other than 100 percent are optimal for equities in some cases where correlations

of currencies with equities are large. We provide more evidence on this in section VII.

19 During the financial crisis of 2007 to 2009 correlation patterns between asset markets and currencies changed

dramatically. Generally the U.S. Dollar and the Japanese Yen appreciated while the Euro and to an even larger

extent the British Pound plummeted along with equity markets. Correlations during this period were extremely

high with up to +/- 0.8.

23

Table 6a. Quarterly Standard Deviations of Hedged and Unhedged Portfolios 1/

Tests of significance

No

hedge

Half

hedge

Full

hedge

No hedge vs. full

hedge

No hedge vs. half

hedge

Half hedge vs. full

hedge

F-stat P-value F-stat P-value F-stat P-value

German perspective

Stock Market

France 11.19 10.94 10.74 1.09 20.06 1.05 32.00 1.04 35.50

Germany 10.58 - - - - - - - -

Japan 11.74 10.50 10.05 1.37 0.08 1.25 1.10 1.09 18.79

U.K. 11.24 10.08 9.36 1.44 0.01 1.24 1.30 1.16 6.62

U.S. 9.92 8.46 7.86 1.59 0.00 1.37 0.06 1.16 6.70

Bond Market

France 2.56 2.09 1.87 1.87 0.00 1.50 0.00 1.24 1.29

Germany 1.69 - - - - - - - -

Japan 6.57 3.74 1.83 12.88 0.00 3.09 0.00 4.16 0.00

U.K. 5.00 2.98 1.89 7.03 0.00 2.82 0.00 2.49 0.00

U.S. 5.91 3.35 2.09 8.01 0.00 3.13 0.00 2.56 0.00

Japanese perspective

Stock Market

France 11.96 10.94 10.63 1.27 0.81 1.19 3.51 1.06 27.66

Germany 12.21 11.04 10.56 1.34 0.16 1.22 1.97 1.09 18.44

Japan 9.79 - - - - - - - -

U.K. 11.75 10.08 9.33 1.58 0.00 1.36 0.09 1.17 5.79

U.S. 9.96 8.37 7.70 1.67 0.00 1.41 0.02 1.18 4.36

Bond Market

France 6.11 3.50 0.85 9.94 0.00 3.04 0.00 3.27 0.00

Germany 5.94 3.28 1.82 10.69 0.00 3.27 0.00 3.27 0.00

Japan 1.81 - - - - - - - -

U.K. 6.68 3.64 1.91 12.25 0.00 3.37 0.00 3.63 0.00

U.S. 6.20 3.49 2.05 9.11 0.00 3.15 0.00 2.89 0.00

Sources: Author’s estimates.

1/ Quarterly standard deviations of returns on unhedged, half hedged and fully hedged stock and

bond portfolios from the perspectives of German and Japanese investors. Hedged returns are based

on rolling quarterly hedges of beginning-of-period balances. Reported F-statistics are based on the

null hypothesis of equal variances.

24

Table 6b. Quarterly Standard Deviations of Hedged and Unhedged Portfolios 1/

Tests of significance

No

hedge

Half

hedge

Full

hedge

No hedge vs. full

hedge

No hedge vs. half

hedge

Half hedge vs. full

hedge

F-stat P-value F-stat P-value F-stat P-value

U.K. perspective

Stock Market

France 11.40 10.94 10.97 1.08 21.65 1.09 19.89 0.99 47.54

Germany 10.87 10.54 10.76 1.02 41.75 1.06 26.37 0.96 33.59

Japan 13.54 11.15 10.11 1.79 0.00 1.48 0.00 1.22 2.34

U.K. 9.51 - - - - - - - -

U.S. 9.43 8.26 8.06 1.37 0.07 1.30 0.34 1.05 30.81

Bond Market

France 5.28 3.21 1.93 7.51 0.00 2.71 0.00 2.77 0.00

Germany 5.40 3.21 1.80 9.02 0.00 2.83 0.00 3.18 0.00

Japan 9.17 5.02 1.87 3.33 0.00 24.02 0.00 7.20 0.00

U.K. 1.88 - - - - - - - -

U.S. 6.35 3.72 2.18 8.48 0.00 2.92 0.00 2.90 0.00

U.S. perspective

Stock Market

France 11.90 11.12 11.03 1.16 6.08 1.14 8.43 1.02 43.18

Germany 11.52 10.84 10.92 1.11 13.46 1.13 10.59 0.99 44.29

Japan 11.64 10.47 10.15 1.31 0.27 1.24 1.51 1.06 26.81

U.K. 11.13 9.98 9.57 1.35 0.10 1.24 1.30 1.09 19.51

U.S. 7.94 - - - - - - - -

Bond Market

France 5.91 3.36 1.85 10.22 0.00 3.10 0.00 3.30 0.00

Germany 6.06 3.36 1.68 13.01 0.00 3.25 0.00 4.00 0.00

Japan 6.62 3.73 1.87 12.51 0.00 3.15 0.00 3.98 0.00

U.K. 5.88 3.30 1.86 9.98 0.00 3.18 0.00 3.14 0.00

U.S. 1.96 - - - - - - - -

Sources: Author’s estimates.

1/ Quarterly standard deviations of returns on unhedged, half hedged and fully hedged stock and

bond portfolios from the perspectives of British and American investors. Hedged returns are based on

rolling quarterly hedges of beginning-of-period balances. Reported F-statistics are based on the null

hypothesis of equal variances.

25

B. Multi-country portfolios

In the previous section, we have presented the effect of hedging currency exposure for

investors invested in a single foreign market. We now turn to the impact of hedging on multi-

country portfolios. Returns on multi-country portfolios are simply weighted averages of

single-country portfolios. For portfolio risk, however, results in the multi-country context

depend on the covariances among the stock/bond market returns, the covariances among the

exchange rate changes, and the cross-covariances among the stock/bond market returns and

the exchange rate changes.

Table 7. Quarterly Returns on Hedged and Unhedged Equal-Weighted Portfolios 1/

Tests of significance

No

hedge

Full

hedge

No hedge vs. full hedge

T-stat P-value

German perspective

Global stock portfolio 2/ 2.84 2.76 0.13 89.95

Global bond portfolio 3/ 1.89 1.82 0.44 66.12

Japanese perspective

Global stock portfolio 2/ 2.43 2.01 0.71 47.57

Global bond portfolio 3/ 1.49 1.08 1.83 6.76

U.K. perspective

Global stock portfolio 2/ 3.53 3.53 0.00 99.98

Global bond portfolio 3/ 2.64 2.64 0.00 99.96

U.S. perspective

Global stock portfolio 2/ 3.24 3.00 0.41 68.34

Global bond portfolio 3/ 2.35 2.11 1.09 27.66

Source: Author’s estimates, DataStream, IMF IFS.

1/ Data coverage extends from 1975M1 to 2009M12.

2/ Global stock portfolios include the MSCI country indices for France, Germany, Japan, the U.K. and

the U.S. in equal proportions.

3/ Global bond portfolios include returns on long-term government bonds for France, Germany,

Japan, the U.K. and the U.S. in equal proportions.

We form portfolios by equally weighting the French, German, Japanese, British, and U.S.

stock and bond markets.20 Table 7 presents quarterly unhedged and hedged returns. Similar to

the single-country analysis in tables 5a and 5b, the differences between unhedged and hedged

returns are statistically not significant in most cases. In economic terms however, Japanese

20 Alternatively portfolios could be value-weighted or tilted towards the investor’s home market.

26

investors and to a lesser extent U.S. Dollar based investors would have yielded substantially

higher returns on unhedged investments as opposed to hedged investments.

We present quarterly standard deviations of unhedged and hedged returns for multi-country

portfolios in Table 8. Hedging currency exposure results in economically and statistically

significant risk reduction in almost all cases. The ratio of variances between unhedged and

hedged returns for diversified portfolios is comparable to the single-country cases presented

in Tables 6a and 6b. We conclude that currency risk is largely undiversifiable.

Table 8. Quarterly Standard Deviations of Hedged and Unhedged Equal-Weighted

Portfolios 1/

Tests of significance

No

hedge

Full

hedge

No hedge vs. full hedge

F-stat P-value

German perspective

Global stock portfolio 2/ 8.72 7.87 1.23 1.87

Global bond portfolio 3/ 3.03 1.52 3.99 0.00

Japanese perspective

Global stock portfolio 2/ 9.03 7.76 1.35 0.10

Global bond portfolio 3/ 4.37 1.56 7.88 0.00

U.K. perspective

Global stock portfolio 2/ 8.47 8.03 1.11 13.55

Global bond portfolio 3/ 4.42 1.58 7.88 0.00

U.S. perspective

Global stock portfolio 2/ 8.65 8.10 1.14 9.24

Global bond portfolio 3/ 4.19 1.47 8.11 0.00

Source: Author’s estimates, DataStream, IMF IFS.

1/ Data coverage extends from 1975M1 to 2009M12.

2/ Global stock portfolios include the MSCI country indices for France, Germany, Japan, the U.K. and

the U.S. in equal proportions.

3/ Global bond portfolios include returns on long-term government bonds for France, Germany,

Japan, the U.K. and the U.S. in equal proportions.

VII. OPTIMAL HEDGE RATIOS

To this point we have only considered no, half, and full hedging of currency risk, which are

by far the most popular hedging strategies with institutional investors. Optimal hedge ratios,

however, are usually defined as the hedge resulting in the greatest risk reduction.

A. Single-country portfolios

We estimate optimal hedge ratios for German, Japanese, British and American investors

investing in foreign equity and bond markets. From equation (11) it follows that minimizing

the variance of a hedged return with respect to the hedge ratio,Φ, is equal to

27

minVar̃̃Φ

̃

(13)

The first three terms in equation (13) are equal to the unhedged return. In order to find the

risk minimizing hedge ratio we perform an OLS estimation of the following equation:

̃,

̃

ε

(14)

where the estimate of is the estimate of the minimum-variance hedge ratio.

We present estimated minimum-variance hedge ratios and associated Newey-West standard

errors to correct for autocorrelation due to overlapping return intervals in Table 9. Optimal

hedge ratios for investments in foreign bond portfolios are essentially one for investors in all

base currencies – this is consistent with results for bonds in section VI. Since correlations

between bond returns and exchange rate movements are in some cases not insignificant, the

reason must be that bond volatility is dominated by exchange rate volatility. For equities the

case is more interesting because the volatility of this asset class is higher so that equity

market – exchange rate correlations matter.

From a German perspective the risk minimizing hedge strategy over the sample period would

have been to hedge about 100 percent of currency exposure in all cases except for

investments in the UK stock market. For the UK stock market German investors should have

hedged 140 percent of currency exposure, i.e. they should have taken a short position in the

British pound. The reason for this is the large positive correlation of 16 percent between the

UK stock market in local currency terms and the DM/euro exchange rate versus the British

pound. The UK stock market has tended to do well/bad when the Pound has

appreciated/depreciated against the German currency. The exchange rate movements have

therefore magnified the stock market movements. A possible explanation is that economic

problems in the U.K as proxied by falling stock prices lead to capital outflows into Germany

and thus a falling pound versus the DM/Euro. Correlations of the German exchange rate

versus the Yen and the U.S. dollar with these countries’ respective stock markets are very

close to zero over the entire period. In the first half of the sample period the correlations are

large and positive, similar to the UK, but this is offset by large negative correlations in the

latter part of the sample.

For Japanese investors risk minimizing hedge ratios are statistically indistinguishable from

one in all cases. In sub-periods there are strong positive and negative correlations between

the Yen exchange rate and foreign stock markets but overall there is no consistent effect so

that correlations for the entire sample period are close to zero.

As a mirror image to German investors, U.K. investors should have retained some exposure

to the German currency. As mentioned before, the German stock market has tended to do

well/bad when the British Pound has appreciated/depreciated against the German currency. A

similar effect exists for investments in the French stock market but this is entirely due to the

second half of the sample period after the Euro introduction. The DM/euro thus has been a

“safe haven” currency for British investors – it has done well during falling stock markets.

28

Similar to British investors, but to a lesser extent, risk-minimizing U.S. investors should have

slightly under-hedged their stock market investments in Germany and France. For

investments in Japan and the U.K. the optimal hedge ratio is indistinguishable from one.

Table 9. Estimated Minimum Variance Hedge Ratios

Stocks Bonds

Minimum

variance

hedge ratio 1/

Standard

errors of MV

hedge ratio 2/

Minimum

variance

hedge ratio 1/

Standard

errors of MV

hedge ratio 2/

German perspective

France - - - -

Germany - - - -

Japan 1.00 0.23 1.03 0.02

U.K. 1.39 0.12 0.99 0.03

U.S. 1.04 0.13 0.95 0.03

Japanese perspective

France 0.95 0.13 1.01 0.02

Germany 1.05 0.14 0.97 0.02

Japan - - - -

U.K. 1.08 0.09 0.97 0.02

U.S. 1.05 0.10 0.97 0.03

U.K. perspective

France 0.72 0.18 1.05 0.03

Germany 0.55 0.14 1.05 0.03

Japan 1.05 0.12 1.04 0.02

U.K. - - - -

U.S. 0.84 0.15 1.01 0.03

U.S. perspective

France 0.82 0.18 1.00 0.02

Germany 0.70 0.16 1.00 0.02

Japan 0.92 0.14 1.02 0.02

U.K. 1.00 0.14 0.98 0.02

U.S. - - - -

Source: Author’s estimates, DataStream, IMF IFS.

1/ Minimum-variance hedge ratios for quarterly returns are obtained by regressing the unhedged

return on the row stock and bond markets on the associated exchange rate gain minus the forward

premium. All regressions include an intercept. We run monthly regressions on overlapping quarterly

returns.

2/ Standard errors are corrected for autocorrelation due to overlapping intervals using the Newey-

West procedure.

29

B. Multi-country portfolios

The simultaneous estimation of minimum variance hedge ratios for portfolios containing

investments in several countries allows investors to achieve optimal results by exploiting the

full covariance structure. A potential danger is over-fitting to the sample. This would be a

problem if covariances are time period specific. For this reason we present results for the first

and second halves of our sample period in addition to full sample results.

Our estimation approach for multi-country portfolios is an extension of the single-country

case. The hedged return on a portfolio of N different countries with weight , for country i

at time t can be written as

̃,

∑,̃,,

∑,Φ,̃,

,

N

(15)

We perform an OLS estimation of the following equation to find the hedge ratios that

minimize portfolio variance

∑,̃,,

αβ

̃,

,β

̃,

,β

Ñ,

,

(16)

Estimates of the optimal hedge ratio for currency exposure associated with investing in

country i are then obtained from Φβ

/ω.

Table 10 contains estimated hedge ratios and associated standard errors for stock and bond

portfolios formed by equally weighting the markets of Germany, Japan, the U.K. and the U.S

over the entire sample period. Table 11 presents results for the first half of the sample

ranging from 1975M1 to 1992M7, and Table 12 presents the second sub-period from

1992M8 to 2009M12. As in the single country case, hedge ratios below one imply that a risk-

minimizing investor would retain some exposure to the foreign currency whereas hedge

ratios above one would imply that the investor should over-hedge, i.e. short, the foreign

currency. Negative hedge ratios indicate that it is optimal for risk minimizing investors to

seek active exposure to these currencies beyond the exposure associated with the unhedged

foreign bond/equity investment.

Over the full sample, risk-minimizing equity investors should have over-hedged exposure to

the British Pound. This result is in line with our finding that the British Pound tends to be

pro-cyclical. This effect is, however, driven entirely by the second half of our sample. In the

first half, full hedging of exposure to the Pound would have been optimal. There is strong

evidence that exposure to the DM and its successor, the Euro, is an optimal strategy for risk

minimizing investors. Negative coefficients on DM/Euro exposure indicate that investors

should not only not have hedged but sought additional exposure to the DM/Euro. This result

is particularly strong for the second half of our sample. For the Yen and the U.S. Dollar,

optimal hedge ratios are not statistically different from one at conventional levels of

significance. There is some indication that over-hedging both currencies was optimal in the

first half of our sample, while under-hedging was optimal in the second half. In unreported

results for the financial crisis of 2007 to 2009, we find that the Yen and the U.S. Dollar have

moved against equity markets with correlations jumping to the range of negative 50 to 70

30

percent. The Euro did not provide the hedging benefits it has exhibited over our full sample

during the crisis. On the contrary, the Euro was extremely pro-cyclical falling against Yen

and U.S. dollar along with stock markets. This experience provides a caveat that currency

correlations as well as asset market – currency cross-correlations are unstable and may break

down when investors need diversification most.

For investors in global bond portfolios, estimates of optimal hedge ratios are more precise as

standard errors are much smaller. Similar to the single-country case, hedging currency

exposure fully is optimal with the exception of the British Pound to which risk-minimizing

bond investors should have retained some exposure.

Table 10. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios –

Full Sample 1975M1-2009M12 1/

Currencies

Germany Japan

United

Kingdom United States

German perspective

Global stock portfolio 2/ - 0.62 (0.60) 2.37 (0.57) 1.39 (0.53)

Global bond portfolio 3/ - 1.14 (0.09) 0.73 (0.10) 1.02 (0.09)

Japanese perspective

Global stock portfolio 2/ -0.49 (0.46) - 2.29 (0.52) 1.42 (0.52)

Global bond portfolio 3/ 1.13 (0.11) - 0.78 (0.11) 0.96 (0.10)

U.K. perspective

Global stock portfolio 2/ -0.57 (0.46) 1.25 (0.29) - 0.94 (0.61)

Global bond portfolio 3/ 1.20 (0.10) 0.94 (0.07) - 1.12 (0.11)

U.S. perspective

Global stock portfolio 2/ -0.50 (0.50) 0.71 (0.54) 2.34 (0.57) -

Global bond portfolio 3/ 1.09 (0.10) 1.11 (0.08) 0.80 (0.11) -

Source: Author’s estimates.

1/ For investors from each base currency perspective, minimum-variance hedge ratios for quarterly

returns are obtained by estimating equation 16. All regressions include an intercept. We run monthly

regressions on overlapping returns. Standard errors are corrected for autocorrelation due to

overlapping intervals using the Newey-West procedure.

2/ Global stock portfolios include the MSCI country indices for Germany, Japan, the U.K. and the U.S.

in equal proportions.

3/ Global bond portfolios include returns on long-term government bonds for Germany, Japan, the

U.K. and the U.S. in equal proportions.

31

Table 11. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios –

First Half 1975M1 – 1992M7 1/

Currencies 2/

Germany Japan

United

Kingdom United States

German perspective

Global stock portfolio 3/ - 1.55 (0.45) 1.01 (0.49) 1.55 (0.64)

Global bond portfolio 4/ - 1.38 (0.13) 0.67 (0.14) 0.90 (0.12)

Japanese perspective

Global stock portfolio 3/ -0.12 (0.51) - 1.01 (0.49) 1.48 (0.64)

Global bond portfolio 4/ 1.04 (0.16) - 0.65 (0.16) 0.83 (0.14)

U.K. perspective

Global stock portfolio 3/ 0.12 (0.55) 1.64 (0.36) - 1.12 (0.58)

Global bond portfolio 4/ 1.27 (0.14) 0.92 (0.11) - 1.01 (0.13)

U.S. perspective

Global stock portfolio 3/ -0.07 (0.51) 1.55 (0.48) 0.94 (0.53) -

Global bond portfolio 4/ 1.06 (0.13) 1.37 (0.13) 0.67 (0.14) -

Source: Author’s estimates.

For footnotes see table 10.

Table 12. Estimated Minimum Variance Hedge Ratios for Multi-Country Portfolios –

Second Half 1992M8 – 2009M12 1/

Currencies 2/

Germany Japan

United

Kingdom United States

German perspective

Global stock portfolio 3/ - 0.36 (0.68) 4.02 (1.06) 0.74 (0.92)

Global bond portfolio 4/ - 0.94 (0.12) 0.64 (0.14) 1.22 (0.14)

Japanese perspective

Global stock portfolio 3/ -1.31 (0.70) - 3.99 (0.98) 0.86 (0.87)

Global bond portfolio 4/ 1.24 (0.13) - 0.75 (0.12) 1.17 (0.13)

U.K. perspective

Global stock portfolio 3/ -1.51 (0.66) 1.19 (0.34) - 0.21 (1.24)

Global bond portfolio 4/ 1.18 (0.11) 0.91 (0.09) - 1.31 (0.13)

U.S. perspective

Global stock portfolio 3/ -1.46 (0.74) 0.46 (0.61) 4.21 (0.95) -

Global bond portfolio 4/ 1.13 (0.12) 0.88 (0.11) 0.80 (0.1) -

Source: Author’s estimates.

For footnotes see table 10.

32

VIII. HEDGING AND THE INVESTMENT HORIZON

Thus far our analysis has been based on quarterly returns and their associated variances. We

demonstrated empirically that hedging in almost all cases reduces risk at a quarterly return

horizon. In this section, we turn to the question of whether the preceding results apply at

longer investment horizons, an issue of relevance for long-term investors such as

endowments. In doing so, we consider investment horizons of up to 5 years while continuing

to hedge returns using three-month interest rates.

At investment horizons longer than one quarter, results on the efficacy of currency hedging

for reducing the risk of a foreign investment are potentially different depending on the

properties of exchange rates over longer horizons as compared to short horizons. At

relatively short horizons exchange rate fluctuations are dominated by changes in real

exchange rates. However, Purchasing Power Parity (PPP) suggests that real exchange rates

are mean-reverting over long horizons.

There is a vast literature on whether PPP holds but some consensus appears to have emerged

that real exchange rates mean revert over long horizons.21 A problem of traditional empirical

tests is lack of power to reject the random walk hypothesis for exchange rates. One approach

to circumvent this is by using very long sample periods (100 to 200 years) – these studies

find support for PPP.22 Recently, studies that incorporate nominal price rigidities, transaction

costs, and non-linear adjustments are able to detect evidence in favor of PPP over shorter

sample periods.

Froot (1993) applies the insights from research on real exchange rate mean reversion using

long-term data sets to currency hedging. Based on empirical evidence over 200 years from

the perspective of a British investor investing in the United States, Froot argues that for long-

term investors mean reversion towards PPP provides a “natural hedge”. Specifically, he finds

that for horizons of more than two years, the volatility of a hedged portfolio of stocks

exceeds the volatility of the equivalent unhedged portfolio. For bond portfolios the cross-

over point is about seven years.23 In Froot’s data set the risk reduction potential of currency

hedging decreases almost monotonically with an investor’s time horizon, leading him to

conclude that “no hedging at all is likely to be best for those who care primarily about long-

horizon moments.”

Although, to our best knowledge, there are no further studies substantiating Froot’s findings,

his analysis has been influential with practioners. Froot’s empirical analysis is limited to the

21 Survey articles on this literature are Froot and Rogoff (1995), and Taylor and Taylor (2004).

22 For example, Frankel and Rose (1996), and Lothian and Taylor (1996).

23 Froot uses returns adjusted for the investor’s home country inflation. In unreported results, we do not find that

adjusting nominal returns for domestic inflation changes the results by much or in a systemic way. We therefore

present results for nominal returns.

33

a

case of a U.K. based investor investing only in the U.S. An additional caveat pertains to the

200 year dataset which includes periods with very different exchange rate regimes.

We proceed by testing the proposition that hedging is less effective at long investment

horizons on our free-floating exchange rate data set covering the perspectives of German,

Japanese, British and American investors. A potential problem with our 35 year data set is

that we do not have sufficient independent observations. For instance, in the case of a 5 year

investment horizon we have only seven independent return intervals. This could limit the

statistical significance of our results at long horizons. We acknowledge that the use of rolling

returns implies overweighting of the observations in the middle of the sample. We maintain

our quarterly hedging strategy and calculate hedged returns over k-periods as the product of

quarterly returns: ̃,

∏1 ̃,

1 .

Table 13 presents the ratio of the variance of unhedged returns to the variance of hedged

returns at investment horizons ranging from one quarter to 5 years. We do not provide p-

values for the F-Stats because of the autocorrelation due to overlapping returns.

For investments in foreign stock markets, the evolution of the relative variance of unhedged

returns to hedged returns varies across base currencies and stock markets. However, there is

clearly no general pattern of a decrease of the variance ratio with the investment horizon. In

many cases, the variance ratio even increases with longer investment horizons, particularly

for investments in the U.S. stock market. A large and monotonous fall in the variance of

hedged to unhedged stock investments is only present for U.K. investors investing in the

Japanese stock market.

For investments in bond portfolios variance ratios decrease strongly, albeit not

monotonously, between quarterly and five year horizons. It is noteworthy that the variance

ratio decreases particularly strongly going from one quarter to one year. The decrease in the

relative variance of unhedged portfolios comes however from very high levels in favor of

currency hedging. Even at a five year investment horizon the case for hedging bond

portfolios is very strong, with the unhedged variance being larger than the hedged variance

by a factor of three and higher in many cases.

34

Table 13. Variance Ratios of Unhedged and Hedged Returns over Different Horizons 1/

Horizon

1 Quarter 1 Year 2 Years 3 Years 4 Years 5 Years

German perspective

Stock Market

France 1.09 1.10 1.16 1.22 1.23 1.25

Germany - - - - - -

Japan 1.37 1.60 1.42 1.28 1.35 1.41

U.K. 1.44 1.61 1.62 1.60 1.63 1.49

U.S. 1.59 1.80 2.06 2.44 2.51 2.53

Bond Market

France 1.87 1.42 1.48 1.48 1.48 1.50

Germany - - - - - -

Japan 12.88 6.01 4.11 3.67 3.41 2.94

U.K. 7.03 3.29 2.91 2.73 3.51 3.86

U.S. 8.01 3.27 3.06 3.74 3.58 3.45

Japanese perspective

Stock Market

France 1.27 1.32 1.47 1.32 1.11 1.15

Germany 1.34 1.27 1.34 1.22 1.12 1.16

Japan - - - - - -

U.K. 1.58 1.78 2.05 1.98 1.74 1.30

U.S. 1.67 1.84 2.37 2.77 2.70 2.34

Bond Market

France 9.94 5.17 4.22 3.65 3.14 3.29

Germany 10.69 4.15 2.55 2.21 2.55 3.00

Japan - - - - - -

U.K. 12.25 5.60 5.11 4.94 4.99 4.69

U.S. 9.11 3.82 3.22 3.28 2.58 1.89

U.K. perspective

Stock Market

France 1.08 1.06 1.04 1.05 1.18 1.30

Germany 1.02 0.97 0.99 0.93 0.99 1.11

Japan 1.79 1.96 1.41 1.30 1.27 1.07

U.K. - - - - - -

U.S. 1.37 1.35 1.31 1.45 1.45 1.45

Bond Market

France 7.51 3.77 3.50 3.15 3.26 3.26

Germany 9.02 4.61 4.00 3.32 2.97 2.77

Japan 24.02 10.33 5.69 4.34 3.97 2.72

U.K. - - - - - -

U.S. 8.48 4.00 3.54 3.76 3.32 2.69

U.S. perspective

Stock Market

France 1.16 1.38 1.61 1.46 1.44 1.51

Germany 1.11 1.26 1.25 1.00 0.94 1.04

Japan 1.31 1.58 1.99 2.02 1.95 1.74

U.K. 1.35 1.45 1.55 1.40 1.31 1.01

U.S. - - - - - -

Bond Market

France 10.22 6.02 7.22 6.62 5.92 6.04

Germany 13.01 6.26 6.22 5.29 4.25 3.65

Japan 12.51 6.55 5.98 5.26 4.00 2.85

U.K. 9.98 4.23 3.43 2.95 2.43 1.62

U.S. - - - - - -

Source: Author’s estimates.

1/ The ratio of the variance of unhedged and fully hedged returns. Variances are calculated over

rolling return intervals ranging from one quarter to five years. Hedged returns are based on rolling

quarterly hedges of beginning-of-period balances.

35

To determine optimal hedge ratios at horizons beyond one quarter, we switch to using log-

returns.24 The advantage of continuously compounded returns is that return components scale

up additively over time which allows us to estimate the minimum-variance hedge through a

regression of the k-period unhedged return on the contemporaneous currency excess return:25

̃,

(15)

where the log return on the hedge,

, is given by the sum of the quarterly log hedge

returns:

∑

,

̃

,

, (16)

In regression (15), is the minimum variance hedge ratio for the k-period return. Froot

shows that if 1/2 the variance of hedged and unhedged returns is equal.26 For 1/2,

the variance of hedged returns is smaller than that of unhedged returns and the reverse if

1/2.

Tables 14a and 14b present the results of the OLS estimation of equation (15) along with

heteroskedasticity and autocorrelation robust standard errors. Standard errors generally

increase with the investment horizon as there are less data points and autocorrelation

becomes more of a problem. As expected from table 13, there is no general decrease in the

minimum variance hedge ratio as the investment horizon increases.

For German investors, the hypothesis of a hedge ratio of one cannot be rejected for any of the

foreign stock markets at any horizons. For investments in Japanese stocks and bonds there

24 We have not used log-returns in our quarterly analysis because it entails the unrealistic assumption of

continuous hedging. Continuous hedging means that hedges are “perfect” and there is no estimation problem. In

that case, we have for the hedged return ̃,

instead of ̃,

̃. There is

therefore a difference between the hedge result assumed in academic studies that use continuously compounded

returns and the actual experience of investors whose hedges are necessarily imperfect due to ex ante unknown

returns. The difference between continuous hedging and quarterly hedging can be substantial over longer

periods – for some five year periods we find differences in returns between quarterly hedging and continuous

hedging of up to 120 percent. This shows that estimation risk is not a triviality that should lightly be assumed

away in favor of mathematical simplicity. For our regression purposes we can assess the importance of the

continuous hedge assumption by comparing quarterly returns in tables 9 and 14a/b. Differences are small and

do not affect the interpretation of the results.

25 The non-log regression framework we have used for quarterly returns in equation (14) does not easily scale

up to multiple periods.

26 With log returns the hedged return can be written as ̃,̃,

. Equality of the variance of unhedged

and hedged returns, ̃,̃,

), then implies

2̃,,

).

Because the OLS estimator of in equation (15) is ̃,,

/

equality of

variances implies 1/2.

36

seems to be even a case for shorting the Yen versus the German currency. Only for

investments in U.S. bonds is the 95 percent interval of the hedge ratio consistently below

one. For investments in all markets we can reject that the variance of hedged and unhedged

returns is equal, i.e. 1/2, at the 5 percent level.

Table 14a. Estimated Minimum Variance Hedge Ratios over Different Horizons

Horizon

1 Quarter 1 Year 2 Years 3 Years 4 Years 5 Years

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

German perspective

Stock Market

France - - - - - -

Germany - - - - - -

Japan 1.02 (0.20) 1.34 (0.23) 1.43 (0.29) 1.66 (0.36) 1.75 (0.43) 1.93 (0.56)

U.K. 1.34 (0.12) 1.32 (0.19) 1.19 (0.27) 1.21 (0.27) 1.14 (0.26) 1.10 (0.30)

U.S. 1.04 (0.13) 1.01 (0.16) 1.07 (0.18) 1.20 (0.18) 1.17 (0.18) 1.17 (0.18)

Bond Market

France - - - - - -

Germany - - - - - -

Japan 1.02 (0.02) 1.11 (0.07) 1.24 (0.12) 1.37 (0.16) 1.44 (0.17) 1.55 (0.14)

U.K. 0.96 (0.03) 0.88 (0.05) 0.82 (0.10) 0.82 (0.14) 0.89 (0.13) 0.96 (0.15)

U.S. 0.93 (0.03) 0.83 (0.07) 0.79 (0.09) 0.83 (0.09) 0.82 (0.08) 0.80 (0.08)

Japanese perspective

Stock Market

France 0.99 (0.16) 0.95 (0.26) 0.91 (0.39) 0.60 (0.40) 0.33 (0.35) 0.30 (0.31)

Germany 1.08 (0.17) 0.85 (0.27) 0.74 (0.43) 0.42 (0.49) 0.26 (0.41) 0.21 (0.35)

Japan - - - - - -

U.K. 1.08 (0.11) 1.01 (0.18) 0.91 (0.22) 0.78 (0.22) 0.67 (0.27) 0.48 (0.32)

U.S. 1.05 (0.11) 0.96 (0.17) 1.01 (0.24) 0.99 (0.24) 0.94 (0.25) 0.93 (0.26)

Bond Market

France 0.99 (0.02) 0.97 (0.06) 1.00 (0.08) 0.97 (0.08) 0.96 (0.09) 1.06 (0.14)

Germany 0.96 (0.02) 0.88 (0.07) 0.82 (0.10) 0.75 (0.12) 0.76 (0.11) 0.77 (0.11)

Japan - - - - - -

U.K. 0.94 (0.02) 0.83 (0.05) 0.78 (0.07) 0.75 (0.08) 0.73 (0.08) 0.74 (0.08)

U.S. 0.95 (0.03) 0.83 (0.07) 0.75 (0.08) 0.71 (0.08) 0.67 (0.06) 0.62 (0.06)

Source: Author’s estimates.

1/ Minimum variance hedge ratios are estimated by regressing unhedged returns on the inverse of

the return on a currency hedge (the domestic currency return of borrowing in foreign currency to hold

domestic deposits). All regressions include an intercept. Standard errors are corrected for

autocorrelation due to overlapping intervals using the Newey-West procedure.

From a Japanese perspective, currency hedging indeed appears to be less effective at

increasing investment horizons except for the French bond market. Particularly for the

French, German, and U.K. stock markets minimum variance hedge ratios fall to only 30, 21,

and 48 percent, respectively. Large standard errors allow however for only very imprecise

estimates in the case of stocks. For foreign bond investments, standard errors are smaller, so

that in the case of the U.S. and the U.K. less than full hedging is the risk minimizing hedge

37

strategy at the 5 percent confidence level for horizons greater than one quarter. However,

even in these cases we can reject the hypothesis of equal variances of hedged and unhedged

returns in favor of hedged returns having a smaller variance.

For U.K. based stock market investors, the case for fully hedging investments in France and

Germany appears relatively weak although very large standard errors make any interpretation

difficult and even over-hedging is a statistical possibility. For investments in the Japanese

and U.S. stock markets as well as for investments in foreign bond markets, if anything,

hedging seems to become more effective at longer horizons. From a U.S. perspective, there is

statistically significant evidence that less than full hedging is optimal for the German and

U.K stock and bond markets.

Table 14b. Estimated Minimum Variance Hedge Ratios over Different Horizons

Horizon

1 Quarter 1 Year 2 Years 3 Years 4 Years 5 Years

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

Minimum

Variance

Hedge Ratio

1/

U.K. perspective

Stock Market

France 0.66 (0.16) 0.43 (0.37) 0.17 (0.45) 0.12 (0.52) 0.34 (0.56) 0.56 (0.63)

Germany 0.53 (0.13) 0.34 (0.30) 0.34 (0.40) 0.27 (0.45) 0.38 (0.47) 0.50 (0.57)

Japan 1.06 (0.11) 1.21 (0.21) 1.21 (0.27) 1.42 (0.28) 1.49 (0.36) 1.67 (0.66)

U.K. - - - - - -

U.S. 0.85 (0.15) 0.75 (0.23) 0.73 (0.22) 0.88 (0.18) 0.94 (0.17) 1.07 (0.20)

Bond Market

France 1.02 (0.03) 1.10 (0.10) 1.13 (0.16) 1.17 (0.21) 1.23 (0.21) 1.30 (0.23)

Germany 1.03 (0.03) 1.11 (0.07) 1.15 (0.13) 1.12 (0.16) 1.06 (0.13) 1.03 (0.16)

Japan 1.03 (0.02) 1.10 (0.04) 1.15 (0.06) 1.26 (0.08) 1.37 (0.10) 1.58 (0.16)

U.K. - - - - - -

U.S. 0.99 (0.03) 0.99 (0.09) 0.99 (0.12) 0.99 (0.12) 0.98 (0.11) 0.96 (0.14)

U.S. perspective

Stock Market

France 0.80 (0.18) 0.86 (0.31) 0.77 (0.34) 0.63 (0.32) 0.61 (0.32) 0.57 (0.32)

Germany 0.68 (0.17) 0.65 (0.29) 0.50 (0.28) 0.38 (0.26) 0.39 (0.23) 0.40 (0.22)

Japan 0.89 (0.14) 1.05 (0.22) 1.14 (0.34) 1.21 (0.42) 1.31 (0.56) 1.35 (0.71)

U.K. 0.96 (0.15) 0.98 (0.23) 0.88 (0.26) 0.70 (0.26) 0.62 (0.25) 0.39 (0.22)

U.S. - - - - - -

Bond Market

France 0.98 (0.02) 0.98 (0.07) 1.04 (0.10) 1.03 (0.10) 1.03 (0.09) 1.03 (0.10)

Germany 0.98 (0.02) 0.96 (0.07) 0.96 (0.09) 0.91 (0.10) 0.87 (0.08) 0.82 (0.06)

Japan 1.00 (0.02) 1.03 (0.07) 1.08 (0.10) 1.09 (0.12) 1.07 (0.15) 1.04 (0.17)

U.K. 0.95 (0.02) 0.86 (0.07) 0.79 (0.10) 0.71 (0.12) 0.67 (0.09) 0.58 (0.07)

U.S. - - - - - -

Source: Author’s estimates.

1/ Minimum variance hedge ratios are estimated by regressing unhedged returns on the inverse of

the return on a currency hedge (the domestic currency return of borrowing in foreign currency to hold

domestic deposits). All regressions include an intercept. Standard errors are corrected for

autocorrelation due to overlapping intervals using the Newey-West procedure.

38

In conclusion, currency hedging appears to effectively reduce the variance of foreign

investment returns not only at short investment horizons but also at horizons of up to 5 years

in most cases. At the same time, at long investment horizons less than full hedging is in some

cases optimal. There are, however, also a few cases where over-hedging is potentially an

effective risk minimizing strategy at long horizons. Foreign investments in the Japanese

stock market stand out in this respect. The explanation appears to lie in the profitability of

using the Yen as a funding currency for the carry trade. These carry trade profits were almost

uncorrelated with Japanese stock and bond market returns over the last 35 years. Thus going

short in Yen has on average generated uncorrelated positive returns for stock market

investors and provided some diversification. This effect also explains why the efficacy of

currency hedging decreases more from a Japanese perspective than it does from any other

perspective.

Our results are in stark contrast to Froot’s findings who concludes that “there is no evidence

at relatively long horizons that currency hedging provides a reduction in return variation”.

We show that results depend on investors’ base currency and investment targets. In general,

going completely unhedged does not appear to be the appropriate risk-minimizing strategy

even at investment horizons of up to 5 years.

IX. CONCLUSION

In this paper we study the impact of hedging the currency risk of foreign stock and bond

market investments from the perspectives of German, Japanese, British, and American

investors over the period from 1975 to 2009.

We demonstrate the importance of currency risk for international investors. Currency risk

contributes up to 40 percent to the overall risk of single-country foreign equity investments

and up to 95 percent of the overall risk of single-country foreign bond investments. Hedging

currency exposure is equivalent to replacing the very volatile and stochastic exchange rate

component of international investment returns with the ex ante known and much less volatile

forward premium or discount. For single-country portfolios, in addition to the exchange rate

volatility itself, the correlations of currencies with bonds and equities are a second channel

through which currency exposure affects foreign investment risk. For multi-country

portfolios, results depend on the covariances among asset market returns, the covariances

among the exchange rate changes, and the cross-covariances among the asset market returns

and the exchange rate changes.

At quarterly horizons, the case for hedging currency risk associated with investments in one

foreign country at a time is very strong, particularly for bonds. No exposure to currencies is

generally the variance minimizing strategy for international bond investors because exchange

rate volatility dominates covariances. For equities, minimum variance hedge ratios are

statistically indistinguishable from full hedging at the 5 percent level with the exception of a

German investor investing in the U.K. stock market and a U.K. investor investing in

Germany. These exceptions to full hedging are a result of the relatively large positive

39

correlation between the British Pound in terms of the DM/Euro and the British and German

stock markets.

Currency excess returns are not zero in most cases. The decision to hedge currency risk

versus maintaining active positions in currencies thus impacts returns on foreign investments.

While currency excess returns are small compared to returns on equities or bonds and return

differences between hedged and unhedged portfolios are usually not statistically significant,

some cases stand out. We show that Japanese investors would generally have benefited from

keeping the currency exposure associated with their investments. By doing so, they would

have boosted their return by engaging in a carry trade. This finding is in line with the status

of the Yen as the funding currency of the global currency carry trade over the last decades.

Hedged returns were generally lower than unhedged returns for Japanese investors because

forward rates anticipated a stronger Yen appreciation than actually materialized.

In a multi-country portfolio context, we confirm that full hedging is the dominant strategy for

bond investors. A slight exposure to the British Pound is the only exception. For multi-

country equity portfolios we find that risk-minimizing global investors not only should have

retained DM/Euro exposure but should have sought additional exposure to these currencies to

benefit from the DM/Euro moving against equity markets on average. Conversely, short

positions in the British Pound would have been optimal as this currency tended to move in

line with equity markets for the first half of the sample period. For the Yen and the U.S.

Dollar hedge ratios are close to one over the entire period from 1975 to 2009, with slight

over-hedging being optimal in the first half of the sample and slight under-hedging being

optimal in the second.

We point out that correlations of currencies with other assets are unstable over time.

Correlations during the financial crisis of 2007 to 2009 highlight this point. During this

period the Euro fell along with global equity markets. This is in stark contrast with the

currency’s strong counter-cyclical properties in the previous decades. We are therefore

cautious in recommending optimal hedge ratios calibrated on historical data for practical

portfolio purposes.

Contrary to evidence by Froot (1993) we find that the investment horizon is of limited

importance for the decision to hedge currency risk. Froot argued that mean reversion in real

exchange rates would provide a “natural hedge” over long return intervals. We do not find a

general pattern for horizons ranging from one quarter to five years that would justify the

recommendation to investors with long investment horizons to hedge significantly less. For

bonds, hedged returns are less volatile than unhedged returns at all horizons. In some cases,

less than full hedging becomes optimal at longer horizons, most notably for U.S. investors

investing in U.K. bonds and Japanese investors in German, British and U.S. bond portfolios.

However, there are also cases where over-hedging, i.e. shorting the foreign currency,

becomes optimal. The cases that stand out most in this regard are German and British

investors in Japanese bonds. For equities there are a few, statistically not significant, cases

where the variance of hedged portfolios exceeds the variance of unhedged portfolios at long

horizons. These are U.S. investors invested in German stocks and Japanese investors with

positions in the German and British stock markets. In some instances, particularly for

40

investments in Japan, over hedging becomes increasingly attractive at longer horizons for

risk minimization purposes. The reason for this appears to lie in the diversification benefits

of carry trade profits for investors with short positions in the Japanese currency. We conclude

that there is no clear general relation between the investment horizon and the effectiveness of

currency hedging.

41

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