Time varying currency risk hedging and the valuation

(1)

0

Dongni Li

11388277

| 15-8-2017

Time varying currency risk

hedging and the valuation

(2)

D. Li Currency risk hedging and the time-varying evaluation

1

Statement of Originality

This document is written by Student Dongni Li who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

(3)

2

Abstract

Based on equally weighted international equity portfolios, I have applied mean-variance optimization on currency risk hedging. This is the most common currency risk hedging strategy which has been used by most of the researchers. However, the currency risk hedging has no improved the portfolio performance in general, as the average portfolio returns are worsened, even though the portfolio risk has been deducted remarkably. As a result, the hedged portfolios underperform the unhedged portfolios, which is usually measured by Sharpe ratio. The time-varying Sharpe ratio difference between hedged and unhedged portfolio illustrate that, the hedged portfolio obtain no better performance than the unhedged ones. So it is better for investors to leave the international equity portfolio always unhedged.

(4)

3

I. Introduction

Once investors hold a portfolio which contains global stocks, their returns have an exposure to currency risks. Some of the investors choose to hedge this risk by adding currency risk hedging assets into their portfolio. However, some others think it is unnecessary to do so, as long as the portfolio is well diversified. Moreover, this hedging activity may become worthless because of the negative returns, but no performance improved.

As currency is also an asset class, which are traded extensively in the market. But currency risk hedging is not the same with the speculation. Speculations are highly dependent on the macro economic situation of a country, such as emerging countries, their economic situation will change substantially when US experiences some adjustment.

Yet, in this paper, I focus my research on using currency as a hedging instrument for international equity portfolios. The most common hedging strategy is mean-variance optimization, which is mainly attempting to decrease the volatility of the portfolio return and maintain the risk premia at the same time. The former researchers have already conducted numerous amount of research to optimizing the currency hedging strategy. They all arrived to the same conclusion that, the currency risk hedging has indeed made the equity portfolio less risky, but the average return of the portfolio was also declined. This lead to a worsened Sharpe ratio for currency hedged portfolios.

Additionally, there are not yet researches implemented to explore the time-varying hedged portfolio performance. If the time-varying currency risk hedging has averagely deducted the performance of the equity portfolios, will this still hold if the portfolio performance is evaluated out-of-sample with the time variance? This is also my main research question. After bringing the time-varying Sharpe ratio of the hedged portfolio in comparison with the one of unhedged portfolio, it shows distinctly the optimal timing for investors to hedge their currency risk accordingly. Moreover, if the currency hedged portfolio performs better in certain period of time, it is not necessary to overperform the unhedged portfolio all the time. The average Sharpe ratio of the whole sample period can be worsened due to the hedging activity. Nevertheless, investors can adjust their currency exposure in different time period— hedging at the best timing and leave the portfolio unhedged the rest of the time.

Eiling et al. (2012B) have constructed their hedged portfolio by add an overlay fashion on top of the equally weighted portfolio. I have referred to their methodology and obtained further the out-of-sample hedged portfolio returns. Further, I continued with rolling window estimation for time-varying Sharpe ratio. The obtained Sharpe ratios are out-of-sample and they are estimated for both hedged and unhedged portfolios. In order to have a clear indication on how much Sharpe ratio has been varied after the minimum variance hedge added, I calculated the difference between hedged and unhedged portfolio returns. At the same time, by applying Eiling et al. (2012B)’s method of moments methodology, I have tested the significance of the real difference between two out-of-sample Sharpe ratios. If the hedged portfolio produces higher Sharpe ratio than the unhedged portfolio at a specific time, and the increment in portfolio performance is statistically significant, then it can lead to a

(6)

5

conclusion that, this is the optimal timing for investors to hedge their currency risk and avoid loss on time. This is the main contribution of my research.

The data I used in my research are based on seven developed countries—US, Euroland as a whole, Australia, Canada, Japan, Switzerland, and UK. These countries’ data are easily accessible and broadly studied by the other researchers. In the paper of Campbell et al. (2010), they have discussed the in-sample and out-of-sample hedging effect on the equity portfolio based on these seven countries. The time period is from February 1975 to April 2017. The most recent financial crisis is also included.

The remainder of the paper is processed as follows. In Section II, the selectively related literatures are discussed. Afterwards, the data description and summary of statistics are found in Section III. Section IV provides the detailed methodology applied for time-varying currency hedged portfolio evaluation. Section V discuss the findings and results. I have also conducted a robustness test in Section VI. The last the conclusions are drawn in Section VII.

(7)

6

II. Literature review

One of the most discussed topic in asset pricing is about the factors which determine the equity returns. In practice, the currency risk could be an influential role to affect the international portfolio performance.

Menkhoff et al. (2012) have added currency momentum in the international asset pricing. Eiling et al. (2012A) have exploited that the global equity portfolio returns are actually mainly changed by the currency and industry risk factors. Under the conditional analysis, their portfolios have been rebalanced both dynamically and statically. When the portfolios were rebalanced with dynamic strategies, the both situation of with and without short-sale constrains have been discussed. The researchers have found that, when investors included currencies in their portfolio, the returns were boosted effectively.

Refer to the currency risk factor, former academical researchers have already investigated its effect in asset pricing. The currency risk premium is defined by De Santis and Gerard (1998). This risk premium has been captured by degrees of risk aversion of a specific country’s investors, and the covariance of the equity excess return and the currency risk. It has significant importance in pricing equity returns while using conditional information.

In addition, De Santis and Gerard (1998) exploited that, in the analysis that allows time and markets variation, the currency risk premium has changed dramatically. Although in the equity market, the magnitude of the risk premium was quite small during the whole sample period, in the subsample period—between 1980 and 1985, the currency risk has dragged the total risk premium to negative values in most of the markets. This finding supports of my time-varying risk evaluation. Because in the same subsample period, the hedged portfolios have overperformed than the unhedged ones, and this is the only period that hedged portfolios obtained higher Sharpe ratio than the unhedged ones.

The literatures have proved that the currency risk cannot be ignored if an international equity investor would like to profit more from the investment. The literatures of currency risk hedging in the international portfolios have provided plenty academical supports on hedging effects.

There are already large amount of researches conducted to determine an optimal hedging strategy for the international equity investors. Roon et al. (2003) have applied the mean-variance hedge in the international equity portfolio, and taking into account on investors’ utility function. Their results revealed that, the dynamic mean-variance currency hedging is essential while the investors are highly risk averse. By adjusting the hedging weight based on the investors’ utility function, they concluded that, the currency risk hedging improves the equity portfolio performance.

Glen and Jorion (1993) have studied the currency risk hedging for both risk minimizing and speculating intentions. The researchers have applied the hedging ratio, which was derived from International Asset Pricing Model (IAPM). This hedging ratio has been used as the fundamental for constructing hedging strategies.

(8)

7

Further, they have constructed portfolios which have maximized the Sharpe ratio. On top of the optimal portfolios, they added currency hedges. Thus, the Sharpe ratio of their hedged portfolios are much higher than the unhedged ones. Nevertheless, the researchers have found little evidence on improvement in equity portfolio performance by hedging the currency risk unconditionally.

Despite unconditional hedging, Glen and Jorion (1993) have also performed conditional optimal hedging. They have also taken into account on the short sale constrains into account. Due to the fact that, shorting position of forward contracts can be argued as a currency hedging with speculative purpose, therefore, the researchers have additionally implemented the research under the situation where short sales are prohibited. Afterwards, they have drawn the conclusion that, dynamic optimal currency hedging with conditional information boosted the efficiency of the portfolio performance, which can be proved by the significantly increased returns without excrescent volatility. The conclusion are also verified both by in-sample and out-of-in-sample analysis.

Different from Glen and Jorion (1993), I have applied equally weighted portfolio for my out-of-sample analysis. DeMiguel et al. (2009) have conducted the research in comparison of portfolio performance, which are constructed based on 14 different optimal asset allocation models. The results have confirmed the significantly high efficiency on 1/N portfolio strategy. Here, N is the number of assets.

DeMiguel et al. (2009) have applied also the “rolling window” out-of-sample estimation in their methodology. The results indicated that out-of-sample Sharpe ratios that obtained by the equally weighted portfolios, are more remarkable than the ones obtained from mean-variance optimized portfolios. Because the mean-variance optimization model failed to recognized the estimation risk, and it assigned overweight in the high-return equities, which brought also higher estimation error.

Therefore, for the out-of-sample estimation, the equally weighted portfolio is the best asset allocation strategy.

Moreover, Glen and Jorion (1993) have stated that, currency risk hedging would not be beneficial enough if the returns improvement was sacrificed for seeking a deduction on risk. In the paper of Campbell et al. (2010). They have discovered the reasons to explain the decreased return after currency hedging.

By estimating the currency demand, Campbell et al. (2010) have discovered that, the correlation of the stock return and exchange rate are the essential drivers for portfolios’ currency hedging strategy. When there is a positive correlation, the investors can reduce the return volatility by selling the currencies. It is even better if investors overhedge. Oppositely, if the currency and the equity returns are negatively correlated, the investors should underhedge by holding the currencies.

The currencies which have the hedging effect are the ones moving contrary with the stock market. They investors would like to be compensated less when holding a long position in their portfolio. As a result, investors would long the currencies which offers less excess

(9)

8

returns and short the currencies that provides higher excess returns. This lead to a relatively lower portfolio return compared with the one produced by unhedged portfolio. Thus, even though the risk has been deducted significantly by currency hedged portfolios, the returns are offset decreasingly.

Further, Campbell et al. (2010) have conducted the researches by hedging the currency risk both unconditionally and conditionally, for both whole sample period and subsample periods. They have found that, in the unconditional analysis, the US dollar and Canadian dollar can effectively reduce the volatility of the portfolio as a pair. Investors can basically suffer less risk by holding long position of Canadian dollar and short position of US dollar. Euro produces the similar hedging effect as well. In the tests of subperiods, the results are consistent. The portfolios which have short position in euro and long position in other currencies appeared to be riskier. Moving forward to the conditional analysis, Campbell et al. (2010) have managed the currency risk with incorporating the former period risk-free rate. The results for reserve currencies (euro, US dollar and Swiss Franc) are contrary to the ones obtained from unconditional analysis. A fraction increase in the interest rate of these reserve currencies, will cause an increase in the risk characteristics, but the incremental is relatively small.

The results of Campbell et al. (2010) have encouraged me to raise a following research question. The volatility of the hedged portfolio returns have already achieved a significant deduction, therefore the hedged portfolios’ Sharpe ratios can only be dragged down by the declined portfolio returns. How do the Sharpe ratios of hedged portfolios’ behave over time and out-of-sample?

The research of Eiling et al. (2012B) have provided a proper and simplified methodology for me to understand and conduct my fundamental research. They firstly constructed hedged portfolios by applying currency overlay on the equally weighted equity portfolio. The estimation is out-of-sample, and the optimal weights are determined. Afterwards, the researchers have evaluated the performance of the hedged portfolios, in the perspectives of returns, volatilities, Sharpe ratio and higher moments.

Moreover, they have also provided the details for how to test significance of the out-of-sample estimated the returns, standard deviations, Sharpe ratio and the differences of these elements. The tests are conducted by applying the method of moments methodologies. I have obtained the similar results with them, that Sharpe ratio in hedged portfolios are not improved. By contrast, it is declined.

What more interesting is, that in the research of Eiling et al. (2012B), they have studied the reasons why hedged portfolio returns are declined, which has led to the decrease of the Sharpe ratios. As in the paper of Campbell et al. (2010), they have already pointed out that the correlation of the currency return and the equity returns has deterministic effect on the hedged portfolio returns, thus, Eiling et al. (2012B) have proved it with conditionally varied expected currency returns and the hedge ratio in a mathematical and statistical way. It is exploited that, in the cross-section, the global state variables influence the returns by affecting the expected currency returns and covariance.

(10)

9

So, if the covariances between the conditional expected currency returns and the conditional equity expected returns are high, then the currency carries higher risk, but obtains higher expected returns. However, the minimum variance hedging strategy will underweight this currency. Thus, in a hedged portfolio, the currencies have higher risk premium yet obtain higher expected returns, are mostly shorted. But the less risky currencies are in hold or longed. As a result, the hedged portfolios obtains averagely lower returns than the unhedged ones. This is in line with the results from the unconditional research of Campbell et al. (2010). There are more researches which drew the focus on the currency excess return in cross-section. Lustig and Verdelhan (2007) have conducted the research to explore the relation between the currency risk premia and the US consumption growth. Instead of applying the overlay currency hedging portfolios, they have constructed eight portfolios based on the ranking of interest rates. Further, the researchers have used the consumption-based pricing models to explain the variation of the currency excess returns.

Lustig and Verdelhan (2007) have developed a consumption-based currency risk premia, and then carried out a cross sectional analysis with the Fama-MacBeth approach. They concluded that, aggregate consumption risk indicates the covariance between the excess currency returns and the equity returns. A high interest rate currency moves positively with the US aggregate consumption growth, and it has a high consumption risk beta. When the US consumption falls, the currency returns decreases as well. However, when the currency is considered as the low interest rate currency, such as euro, it moves contradictorily with the consumption growth.

The conclusion of their research is actually consistent with the conclusion of Campbell et al. (2010). A high interest rate currency has higher interest rate as well. Compare with the US, Eurozone and Switzerland, Australia and Canada have relatively higher interest rate, and these countries have resource-based economy, where commodity returns are more influential in equity market returns. These currencies appreciate in the bull market and depreciate in the bear market. So the hedging effect of these currencies are not as effective as the reserve currencies.

Later in their extended discussion on the same paper, they have involved global consumption factors to perceive the relation between time-varying currency return and the consumption risk. They have found that the US consumption growth risk cannot be used as the reference for currency returns. The currency return varies mostly depending on the global factors. But the consumption drop cannot be used as an optimal currency hedging timing indicator.

To sum up, there are still missing literatures on portfolio time-varying performance evaluation. The former researchers have focused mainly on the cross-sectional analysis. The time-varying portfolio performance evaluation is helpful to indicate an optimal hedging timing. Exploring whether it is better for investors to hedge their currencies at a certain period is the main contribution of my research

(11)

10

III. Data and Summary Statistics

The data are downloaded from Datastream. The time period applied in the research is between February 1975 and April 2017. The data are in monthly frequencies. There are seven developed countries are taken into consideration, which are Germany, Switzerland, Japan, US, UK, Australia, and Canada, and Germany is used as a representative of all the Euro Zone countries. These markets are mostly analyzed by former researchers.

The price indices are measured by MSCI total return indices. The exchange rate are also measured by MSCI. The interest rates are one-month deposit interest rates, which are from Thomson Reuters. The indices and interest rates are all in local currency. The US personal consumption expenditure data are from Federal Reserve Bank. The total personal consumption is calculated as the sum of the durable and non-durable goods.

The currency return is estimated based on the Covered Interest Rate Parity as

𝑟𝑟€,𝑡𝑡 =

𝑆𝑆𝑡𝑡�$ €� �

𝑆𝑆𝑡𝑡−1�$ €� � ∗ (1 + 𝑟𝑟_{1 + 𝑟𝑟}_{𝑓𝑓,€,𝑡𝑡−1}𝑓𝑓,$,𝑡𝑡−1)

− 1

𝑟𝑟€,𝑡𝑡 is the return of Euro at time t when the investor’s home currency is USD. 𝑆𝑆𝑡𝑡�$ €� � is the

spot exchange rate at time t, which indicates the Dollar price of each Euro. 𝑟𝑟𝑓𝑓,$,𝑡𝑡−1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑓𝑓,€,𝑡𝑡−1 are the risk-free rate at time t-1 in US market and Euro market

respectively. When the home currency becomes the other country’s currency, a cross-exchanging rate is used, which both of the exchange spot rate are all based on US dollar. For example, when using Euro as the home currency, and the exchange rate of sterling respect to Euro is calculated as

𝑆𝑆𝑡𝑡�£ €� � =

𝑆𝑆𝑡𝑡�£ $� �

𝑆𝑆𝑡𝑡�€ $� �

Table 1 reports the full-sample monthly mean and standard deviation from each market, and the returns are all in local currencies. Following, the correlation between the indices returns and currency returns, and the correlation between equity returns and US consumption growth are reported.

The currency returns are estimated based on the interest rates, and not all of these seven markets have their interest rates published from 1975, for example, Japan had its interest rates since August 1978 and Australia started even ten years later, therefore, in order to have the accordant starting point, the missing interest rates are set to zero.

(12)

11

From Table 1 Panel A I can observe that, all of the markets have negative average excess returns. The standard deviation is moving around 6%, except for the Eurozone equities. It has lowest volatility and averagely less negative excess returns. It can be observed from Panel B

Table 1 Summary of statistics

In this table, all of the data are downloaded from Datastream, and from February 1975 till April 2017. The interest rates 𝑟𝑟𝑟𝑟

are the one-month deposit interest rate from Thomson Reuters. The currency exchange rates and the price indices are from MSCI. US consumption growth data are from Federal Reserve Bank and calculated as the sum of total personal expenditures on durable and non-durable goods. The equity index excess returns of currency i are reported based on domestic currency,

and they are estimated by 𝑟𝑟_𝑡𝑡𝑖𝑖= 𝑟𝑟𝑡𝑡𝑖𝑖

𝑟𝑟𝑡𝑡−1𝑖𝑖 − 1 − 𝑟𝑟𝑟𝑟𝑡𝑡

𝑖𝑖_{. Panel A reports the equity index returns and standard deviations (in column)}

in local currency (in row). Panel B reports the currency returns and standard deviation (in column) with USD as the base currency. In Panel C, the first 7 rows present the correlation between the currency excess returns and equity excess returns, the base currencies are given in the most left column and the base markets are given in the heading row. When a country is considered as home country, the currency returns are calculated with this currency as the base, and the equity returns from the other countries are converted to this currency.

Panel A: Equity index excess return in local currency

USD EUR AUD CND JPY CHF GBP

Mean(%) -2,87 -1,85 -5,19 -3,26 -6,08 -1,76 -4,34

Stdv(%) 6,60 5,29 6,52 6,52 6,60 6,15 6,05

Panel B: Currency return (Base:USD)

Mean(%) -1,12 -1,19 0,69 -2,53 -2,21 1,70

Stdv(%) 3,89 6,57 2,49 4,22 4,59 3,96

Panel C: Correlations between equity returns and currency returns

US EUR AUS CAN JAP SWZ UK

USD -0,002 0,304 -0,121 0,088 0,116 0,162 EUR 0,378 0,250 0,211 0,186 0,136 0,362 AUD 0,573 0,323 0,506 0,320 0,343 0,509 CND 0,272 0,071 0,353 0,131 0,129 0,212 JPY 0,238 -0,107 0,281 0,116 0,076 0,256 CHF 0,350 0,004 0,203 0,180 0,169 0,324 GBP 0,102 -0,042 0,293 -0,045 0,089 0,045

that, Canadian dollar returns volatile the least among all currency returns, but offers also higher returns, which is just next to British sterling.

In Panel C, the home countries are indicated at the heading row. When a country is considered as home country, the currency returns are calculated with this currency as the base, and the equity returns from the other countries are converted to this currency.

It can be observed that, most currency returns move accordingly with the base country’s equity market returns. The investors can expect their oversea investments to gain or loss based on the home country index. But this is not true with euro. Euro moves in a contrary way with the US market, Japanese and UK market. This means that investors can invest in

(13)

12

Euro when their investment suffers a loss in these markets, although the hedging correlation with British equity market is quite low, which is indicated by only -0.2% correlation. In the rest of the market, Euro returns have comparatively low correlations with the equity returns, except the Australian ones. Similarly, Japanese Yen returns have rather low correlation on whichever the home country is based, except Australia. Additionally, while considering Australia as the home country, the rest of the currency returns have all relatively higher correlations than Euro and Japanese Yen. Due to the fact that Australian equity market is highly correlated with commodity market, the high correlation can be explained by the comovement of the currency returns and the world equity market.

(14)

13

IV. Methodology

In this section, the methodology for constructing the out-of-sample hedging portfolio is firstly explained. Further, the unhedged and hedged portfolio returns are both evaluated by using Sharpe ratio for each rolling estimation period. Finally, the significance of the difference in time-varying out-of-sample Sharpe ratio has been tested.

A. Constructing hedging portfolio

The equally weighted equity portfolio is the starting point of the methodology, which contains all seven countries’ equity returns and used as the basic unhedged portfolio. Afterwards, the out-of-sample hedged portfolio returns can be obtained and further analyzed. DeMiguel et al. (2009) have concluded that the equally weighted portfolio is referred as the best method to obtain the out-of-sample portfolio returns. This approach has been applied by most of the former currency risk researchers.

Eiling et al. (2012B) have used equally weighted equity portfolio as the base portfolio, afterwards, by applying currency overlay on the unhedged equity portfolio, the hedged portfolio can be obtained. US dollar is used as the base currency, and the equity index returns from the other six markets are converted into US dollar. The currency returns are based on US dollar as well.

To convert the equity index return from the other markets, the exchange rate of US dollar per foreign currency is used. By using euro as an example, I have:

𝑟𝑟𝑡𝑡,€→$ = �1 + 𝑟𝑟𝑡𝑡,€� ⎝ ⎜ ⎜ ⎛ 1 + 𝑆𝑆𝑡𝑡�$ €� � 𝑆𝑆𝑡𝑡−1�$ €� � ⎠ ⎟ ⎟ ⎞ − 1

𝑟𝑟𝑡𝑡,€→$ denote as the equity return from euro converted into US dollar at time t. 𝑟𝑟𝑡𝑡,€ is the

equity return in euro. Afterwards, each equity index are used as an asset, and the weight on each asset is assigned as 1/7. Then unhedged portfolio excess returns are estimated by:

𝑟𝑟𝑡𝑡,𝑢𝑢𝑢𝑢ℎ = 𝑟𝑟𝑡𝑡,$− 𝑟𝑟𝑓𝑓$,𝑡𝑡

Here, 𝑟𝑟_𝑡𝑡,$ is the equally weighted international equity portfolio return at time t and 𝑟𝑟_{𝑓𝑓$,𝑡𝑡} is the US one-month interest rate return at time t.

Further, based on the unhedged portfolio excess returns, the hedged portfolio excess returns are estimated by using every 60 months’ unhedged portfolio excess returns and the currency returns. Thus, the hedged portfolio returns are out-of-sample. Ghysels et al. (2005) have proved that the rolling window is an ideal methodology for unconditional out-of-sample estimations, as it inherent dynamically with the variance process.

(15)

14

To apply the minimum variance hedging strategy for each 60-month rolling window, the weights of hedging portfolios can be estimated with using:

𝑟𝑟𝜏𝜏,𝑢𝑢𝑢𝑢ℎ = 𝑎𝑎 + 𝑏𝑏′𝑟𝑟𝜏𝜏,𝑐𝑐+ 𝜖𝜖𝜏𝜏, 𝑟𝑟𝑓𝑓𝑟𝑟 𝜏𝜏 = 𝑡𝑡 − 1, . . 𝑡𝑡 − 60

𝑟𝑟𝜏𝜏,𝑢𝑢𝑢𝑢ℎ is the excess return of unhedged equity portfolio. 𝑟𝑟𝜏𝜏,𝑐𝑐 is the return of the currencies on

USD. The coefficient −𝑏𝑏 is the hedging weight. Therefore, the out-of-sample hedged portfolio returns 𝑟𝑟_𝑡𝑡,ℎ of each 60 months can be estimated by

𝑟𝑟𝑡𝑡,ℎ = 𝑟𝑟𝑡𝑡,𝑢𝑢𝑢𝑢ℎ− 𝑏𝑏′𝑟𝑟𝑡𝑡,𝑐𝑐

In total, I have obtained 446 out-of-sample hedged portfolio returns, and plotted them in an histogram. To compare the results, the figures in Table 2 can be referred.

B. Portfolio evaluation based on Sharpe ratio

The mean and standard deviation of the portfolio returns offer a fundamental comprehension for the portfolio’s performance. Both hedged and unhedged portfolio returns have non-normally distribution. Because of this, the most common tests are no longer applicable. Therefore, the methodology has been applied in testing the significance is based on the method of moments.

To begin with, the non-central moments 𝑚𝑚_𝑘𝑘 need to be determined for both portfolios. The first two non-central moments 𝑚𝑚₁, 𝑚𝑚₂,of the portfolio returns are:

𝑚𝑚1 = 𝜇𝜇;

𝑚𝑚2 = 𝜎𝜎2(𝑟𝑟) + 𝜇𝜇2;

The derivation of non-central moments can be referred in Appendix B.

The most common portfolio evaluation approach is the Sharpe ratio, which is also referred as the reward-to-volatility ratio. In this paper, I use this method as well to assess the hedged portfolio performance, and further to compare the hedged portfolio performance with the unhedged one.

As mentioned before, the normality assumption is released while testing both out-of-sample hedged portfolio returns and unhedged portfolio returns. Thus, the Sharpe ratio is necessary to be applicable for normally distributed returns. By deriving with the first two non-central moments, I can get the Sharpe ratio of rolling window 𝜏𝜏 of a portfolio:

𝑆𝑆𝑆𝑆𝜏𝜏 = 𝐸𝐸(𝑟𝑟𝜏𝜏) [𝐸𝐸(𝑟𝑟𝜏𝜏2) − 𝐸𝐸(𝑟𝑟𝜏𝜏)2] 1 2 = 𝑚𝑚1,𝜏𝜏 �𝑚𝑚2,𝜏𝜏− 𝑚𝑚1,𝜏𝜏2 � 1 2 , 𝑟𝑟𝑓𝑓𝑟𝑟 𝜏𝜏 = 1,2,3 … 60

The Sharpe ratios of both hedged and unhedged portfolio are obtained with the above approach. Further, I can use them to compare the performance of both out-of-sample hedged portfolio returns and the unhedged portfolio returns over time. The rolling window estimation results can be applied to interpret the best hedging timing, and identify the periods when the

(16)

15

currency risk is better to be left unhedged. The mean and the volatility of Sharpe ratio for both portfolios are reported in Table 2. I have plotted the time-varying out-of-sample Sharp ratios for both portfolios and the differences between them in Figure 1. The plots documented the time-varying Sharpe ratios obtained from the portfolios which are constructed based on different home currencies.

In order to test the changing of the Sharpe ratio, the limiting variance of it is needed. Firstly, the matrix of difference between the portfolio return and 𝑖𝑖_𝑡𝑡ℎ non-central moments are needed. It is calculated as:

𝑢𝑢𝑖𝑖,𝑡𝑡 = 𝑟𝑟𝑡𝑡𝑖𝑖 − 𝑚𝑚𝑖𝑖

𝑟𝑟𝑡𝑡𝑖𝑖 is the portfolio return at time t with exponential i, and 𝑚𝑚𝑖𝑖 is the 𝑖𝑖𝑡𝑡ℎ non-central moment of

portfolio return. Since the variance is the second moment of the return distribution. So i=2 and let 𝑈𝑈_{ℎ−𝑢𝑢𝑢𝑢ℎ} denotes as the matrix of this differences. They are obtained by hedged portfolio returns, unhedged portfolio returns, and their non-central moments respectively:

𝑈𝑈ℎ−𝑢𝑢𝑢𝑢ℎ = (𝑟𝑟ℎ− 𝜇𝜇ℎ, 𝑟𝑟ℎ2− 𝑚𝑚2ℎ, 𝑟𝑟𝑢𝑢𝑢𝑢ℎ− 𝜇𝜇𝑢𝑢𝑢𝑢ℎ, 𝑟𝑟𝑢𝑢𝑢𝑢ℎ2 − 𝑚𝑚2𝑢𝑢𝑢𝑢ℎ)

𝜇𝜇ℎ, 𝜇𝜇𝑢𝑢𝑢𝑢ℎ, 𝑚𝑚2ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑚𝑚2𝑢𝑢𝑢𝑢ℎ are the first and second non-central moments of hedged and

unhedged portfolio returns respectively. Then the covariance matrix can be obtained as:

𝛺𝛺 =1_{𝑇𝑇 𝑐𝑐𝑓𝑓𝑐𝑐}(𝑈𝑈ℎ−𝑢𝑢𝑢𝑢ℎ)

Furthermore, to calculate the limiting distribution of the Sharpe ratio difference, the Sharpe ratios’ partial first derivatives with respect to the first two non-central moments 𝑚𝑚₁ and 𝑚𝑚₂ are needed. After taking first derivatives I obtain a 4x1 vector which is denoted as (𝑆𝑆𝑆𝑆� )_{ℎ,𝑢𝑢𝑢𝑢ℎ}. Use 𝛿𝛿 as the true difference between the two estimated Sharpe ratios 𝑆𝑆𝑆𝑆� and 𝑆𝑆𝑆𝑆_ℎ � , which _{𝑢𝑢𝑢𝑢ℎ} is expected to be zero. At last, the limiting distribution of the differences between two portfolios’ Sharpe ratio is:

√𝑇𝑇 ��𝑆𝑆𝑆𝑆� − 𝑆𝑆𝑆𝑆ℎ � � − 𝛿𝛿� → 𝑁𝑁 �0, Ω�𝑆𝑆𝑆𝑆𝑢𝑢𝑢𝑢ℎ � �_{ℎ,𝑢𝑢𝑢𝑢ℎ}� ;

The limiting distribution of the Sharpe ratios’ difference is essential to test the significance in changing Sharpe ratios. While calculating the significance of out-of-sample Sharpe ratios’ difference, I applied the same methodology for each rolling 60-month window. Thus, each out-of-sample Sharpe ratio significance at time t is:

𝑡𝑡 − 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡𝑆𝑆𝑅𝑅ℎ−𝑆𝑆𝑅𝑅𝑢𝑢𝑢𝑢ℎ,𝑡𝑡 =

�𝑆𝑆𝑆𝑆� − 𝑆𝑆𝑆𝑆ℎ,𝑡𝑡 � � − 𝛿𝛿𝑢𝑢𝑢𝑢ℎ,𝑡𝑡 𝑡𝑡

(𝑆𝑆𝑆𝑆)�ℎ,𝑢𝑢𝑢𝑢ℎ,𝑡𝑡′∗ 𝛺𝛺𝑡𝑡∗ (𝑆𝑆𝑆𝑆)�ℎ,𝑢𝑢𝑢𝑢ℎ,𝑡𝑡

After the significance of the difference is determined, it can be used to interpret that, if the hedging portfolio has truly out- or under-performed the unhedged portfolio. I have also highlighted the significant Sharpe ratio differences on the plots of time-varying Sharpe ratios. It can be used to investigate the time-varying portfolio performance variance, as well as to gain the cognization, that in a certain period of time, if the investors are ideally beneficial from the currency risk hedging.

(17)

(18)

17

V. Results

In this section, the results of the whole-sample based analysis are presented. Starting with the currency risk hedging effect, I compare the standard deviation and the average return of both portfolios. After that, the time-varying Sharpe ratios are displayed latter in the figure.

A. Currency Hedging effect on international equity portfolios

To begin with, I would like to draw attention on the average return and standard deviation of the hedged portfolios, and compare them with the ones of unhedged equity portfolios. Then, it comes to the Sharpe ratio of both portfolios. All the values obtained are reported in Table 2. The t-stat of each value are reported below it. Each column represents the base currency, which are used to construct equally weighted equity portfolios and currency hedged portfolios.

The standard deviation of hedged and unhedged portfolio are indicated in Panel A. It can be observed that, the minimum variance hedging has achieved lower risk in portfolio returns. All the hedged portfolio returns have lowered and statistically significant standard deviation. Unhedged portfolios returns have the standard deviation around 6,5% to almost 9%, while the returns generate by hedged portfolios have between 5% to 6% standard deviation. The risk deduction by currency risk hedging is between 8,5% to 10,5%. They are all statistically significant at 5% level except for Australian dollar based portfolios.

Moving to Panel B, which reports the portfolios’ average return and the impact of currency hedging on the portfolio mean. The values of mean in both hedged and unhedged portfolios are negative, which holds for all portfolios based on different home currencies. For unhedged portfolios, the average returns vary from -0,6% to -1,2%, and for hedged portfolios, the average returns are around -0,8% to -0,9%. While checking the significance of the portfolio average returns, I exploited that, the hedged portfolio average returns are all statistically significant with higher absolute t-stat value. The unhedged portfolio returns have lower absolute t-stat value, and the one from Japanese yen based equity portfolio is statistically insignificant at 5% level.

Most of the past literature concluded that the average returns of the currency risk hedged portfolios generate no higher average returns than the unhedged portfolios. In my analysis, there is a partially contradict. The mean currency hedged portfolios which constructed based on US dollar, Canadian dollar and British pound have shown a slight increase in the portfolio returns. Although the incremental is rather small—0,02%, this improvement is still statistically significant at the confidence level of 5% for US dollar and pound based hedged portfolios. Canadian dollar based hedged portfolio returns achieve a higher increase as 0,09%, and it is statistically significant at 10% level of confidence.

To acquire a general concept on the effects of currency hedging, it is necessary to combine Panel A and Panel B. Among all of the unhedged portfolios, the ones constructed based on US dollar, Canadian dollar and British sterling are the riskiest. The standard deviations of these 3 unhedged portfolios are all above 8%. The average return are the lowest and around -1%. After the portfolio received a currency overlay with minimum variance hedging strategy, their portfolio returns are lifted, even though with a small fraction, and the volatility is

(19)

18

deducted. The increase of average return and the decrease of standard deviation are also proved to be statistically significant.

Table 2 Currency Hedging effect on international equity portfolios

In this table, the impact of the hedging portfolio in returns’ mean, standard deviation, Sharpe ratio, skewness and kurtosis have been reported. The basic portfolio is the equally weighted international portfolio. Based on that, the minimum variance hedging strategy is applied. The weights to hedge the currency risk are estimated by the overlay strategy. The out-of-sample hedged portfolio returns are estimated by every 60 months’ hedging weights and the unhedged portfolio return. Each column indicates the impacts of hedging based on different home currency. In each panel, the impact on returns’ every category and the t-statistics are presented, as well as the difference in each category between the hedged portfolio and unhedged portfolio. Panel A shows the impact on portfolio returns’ standard deviation, their t-statistics, the difference in standard deviation and the t-statistics for the differences. Panel B reports the impact on portfolio average returns. Panel C reports the impact on Sharpe ratios of each portfolio. The t-stat of each value is reported in the parenthesis under it.

Home currency

Panel A: Impact on portfolio return standard deviation Unhedged portfolio STD 8,15% 7,18% 6,58% 8,12% 7,66% 7,26% 8,63% t-stat (20,60) (21,34) (20,01) (19,06) (28,55) (25,76) (18,95) Hedged portfolio STD 5,86% 5,91% 5,68% 5,90% 5,67% 5,65% 5,93% t-stat (14,89) (15,17) (13,98) (15,06) (14,65) (14,49) (14,77) Difference (Hedged-Unhedged) in STD -2,22% -1,27% -0,90% -2,13% -1,99% -1,61% -2,96% t-stat (-2,64) (-2,37) (-1,77) (-2,60) (-2,54) (-2,28) (-2,60)

Panel B: Impact on portfolio average return Unhedged portfolio MEAN -0,90% -0,77% -0,75% -0,97% -0,61% -0,64% -1,07% t-stat (-2,49) (-2,40) (-2,57) (-2,69) (-1,80) (-1,99) (-2,71) Hedged portfolio MEAN -0,84% -0,84% -0,86% -0,85% -0,83% -0,83% -0,85% t-stat (-3,02) (-2,97) (-3,21) (-2,99) (-3,08) (-3,11) (-3,05) Difference (Hedged-Unhedged) MEAN 0,02% -0,08% -0,06% 0,09% -0,02% -0,02% 0,02% t-stat (4,22) (-1,94) (-1,54) (1,83) (-4,35) (-4,48) (3,37) Panel E:Impact on time-varying Sharpe ratio

Unhedged portfolio SR Mean -1,67 -1,47 -1,67 -1,85 -1,19 -1,14 -1,96 SR STD 0,87 0,75 0,60 0,84 0,93 0,66 0,96 t-stat (8,69) (7,83) (5,85) (7,80) (12,77) (8,31) (8,53) Hedged portfolio SR Mean -2,31 -2,39 -2,29 -2,30 -2,29 -2,31 -2,33 SR STD 0,86 0,96 0,76 0,85 0,82 0,83 0,87 t-stat (5,91) (6,69) (4,83) (5,87) (5,49) (5,44) (6,02)

(20)

19 Difference in SR

(Hedged-Unhedged) -0,756 -1,02 -0,561 -0,551 -1,19 -1,32 -0,511 t-stat (-4,16) (-5,43) (-2,52) (-2,87) (-6,87) (-5,66) (-2,55)

I link this finding with the currency forward demand. The weights and positions of each currency forward can be referred in Table 3. In the research of Campbell et al. (2010), they have summarized that the US dollar, euro and Swiss franc are negatively correlated with the global equity market. Australian dollar, Canadian dollar, yen and pound comove with the world equity market positively. In the hedged portfolio constructed based on US dollar, Canadian dollar and pound, it can be observed that, they all have more than half of their long position invested in US dollar, euro and (or) Swiss franc, and short mostly Australian dollar, Canadian dollar, pounds, and (or) yen. However, this is a general view of the hedging impact on the portfolio returns. Because the portfolio which is constructed based on euro, holds almost the same position with these three “well” hedged portfolio, but its mean is still lower than the unhedged ones. There more statistical analysis to lead to draw a solid conclusion. To evaluate the performance of the portfolios, the Sharpe ratio is used as a reference. I have obtained two vectors of out-of-sample time-varying Sharpe ratio for two portfolios. In Panel C of Table 2, the average Sharpe ratios and standard deviation of Sharpe ratios are reported. The last two rows display the difference between hedged and unhedged portfolios’ Sharpe ratios, the t-stats are reported below.

Averagely, the Sharpe ratio of unhedged portfolios are less negative than the hedged ones. The unhedged portfolios have Sharpe ratios around -1 to -2, but the hedged portfolios return even lower Shape ratios, which varies around -2,3. The most worsened portfolios are the ones based on Japanese yen and Swiss franc. The average Sharpe ratio of yen based portfolio from -1,14 dropped to -2,31 after adding minimum variance hedge. It suffered a more than 100% fall in portfolio performance. The performance of the portfolio based on Swiss franc has also a dramatical decrease after the currency hedging added, which is around 90%. The portfolios based on US dollar, Canadian dollar and pound have their performance decreased as well.

Table 3 Average currency forward weight in hedged portfolio

In this table, the average weight of each currency forward of each hedged portfolio is reported. The unhedged portfolio contains the equally weighted equities from seven countries. The equity returns are calculated based on the home currency indicated on the heading row. On top of this, investors apply an overlay minimum variance currency hedging strategy. The weight of each currency forward which are used to hedge the global currency risk exposure are reported in each row. The data cover the period from February 1975 to April 2017.

Home currency

USD 0,167 0,395 -0,080 0,250 0,097 0,579 0,249 EUR 0,409 0,079 0,408 0,616 0,398 0,387 0,407 AUD -0,078 -0,079 0,175 -0,079 -0,077 -0,078 -0,079 CND -0,357 -0,295 -0,368 0,101 -0,329 0,160 -0,361 JPY 0,105 0,129 0,074 -0,365 -0,160 0,103 0,161 CHF 0,617 0,604 -0,135 -0,157 0,582 -0,152 0,618 GBP 0,241 -0,122 0,254 0,169 0,220 0,229 -0,162

(21)

20

Although the pound based portfolio and Canadian dollar based portfolio perform not as worse as the yen and Swiss franc based ones, the Sharpe ratio of pound based portfolio has declined from -1,96 to -2,33, resulting in a 20% decrease. The Canadian dollar based portfolio has its Sharpe ratio decreased from -1,85 to -2,30, which worsens 24% in the portfolio performance compare with the equally weighted portfolio which has the currency left unhedged. With time being, the out-of-sample hedged portfolios’ Sharpe ratios are more volatile in general, except for Japanese yen and pound based hedged portfolios. The standard deviation of unhedged portfolios’ Sharpe ratios are between 60% to 96%, and the one of hedged portfolios Share ratios varies from 76% to 96%. Statistically, the minimum variance hedge does not promote the portfolio performance. The decrements of the Sharpe ratios are large, averagely around 50%. It can be summarized that, with the currency risk hedging, the time-varying equity portfolio performance is not bettered, even worsened for at least 50% .

Thus, statistically, the minimum variance added portfolios are indeed less risky. The volatility of the hedged portfolios are lower than the unhedged stock portfolios. The decrements of standard deviations are also statistically significant at 5% confidence level, except for the Australian dollar based portfolio. Whereas, the average return of the most portfolios are worsened, except for US dollar, Canadian dollar and pound based equity portfolios. In these three portfolios the returns are slightly increased and the increments are also statistically significant. But this does not indicate a certain improvement in portfolio performance has been achieved by adding the minimum variance currency hedge. As the Sharpe ratios are overall severely and statistically declined.

B. Portfolio performance evaluated by time-varying out-of-sample Sharpe ratio

In order to investigate more carefully in time-varying Sharpe ratios, the plots of the out-of-sample Sharpe ratios can be used, which are referred to Figure 1.

The out-of-sample Sharpe ratio covers the period from March 1980 to April 2017. Each home- currency based portfolio Sharpe ratios are plotted individually. In each plot, the blue lines are the time-varying Sharpe ratio of unhedged equity portfolios, and red lines are the Sharpe ratios of currency hedged portfolios. The black line indicates the difference between each hedged and unhedged portfolio of Sharpe ratios, and the highlighted scatter are the statistically significant difference with 5% confidence level. I have marked the economic crisis periods in the plots with gray color. They are the early 90s’ financial crisis, which are connected with the unsteady political situation globally, the dot com bubble recession, which refers to the internet booming, and last and the most recent financial credit crisis.

By taking a quick glance, it is obvious that the Sharpe ratios of hedged portfolios are most of the time underperforming the unhedged portfolios, except the first five years— before the early 90s’ crisis started. In this period, the portfolio returns are estimated with the missing interest rate of Australian and Japanese markets. Subsequently, the out-of-sample hedged portfolio returns carry these missing information. As a result, the Japanese and Australian equity market obtain higher excess returns, and during this time, there were not as much market volatility injected as in the upcoming recession. Thus, the hedged portfolio overperforms the unhedged portfolios. The difference is also huge and statistically significant. Following the recession’s approaching, the hedged portfolio’s Sharpe ratio started to drop, while the unhedged portfolio’s Sharpe ratio began to raise. Once the recession has spread in these developed nations, the hedged portfolio performance reached till an extremely

(22)

21

low point. This is can also be influence by the availability of Japanese and Australian one-month interest rate.

During this first recession, the difference of Sharpe ratios are mostly insignificant, except for the portfolios based on Australian dollar and Swiss franc. This is because the huge decrement of the portfolio performance. Compare with the other home-country based portfolios, the distance between hedged and unhedged portfolio Sharpe ratios in these two countries are the largest.

At the end of the recession, the hedged portfolio Sharpe ratio started to increase, and the unhedged portfolio Sharpe ratio fell downwards. Still, the unhedged equity portfolios have higher Sharpe ratio than the one of hedged portfolios. After the 90s’ financial crisis, the hedged portfolios in most of the markets overperformed or equally-performed the unhedged ones, and this appears in most countries, except in Australia and Switzerland. The surpass of Australian dollar based hedged portfolio Sharpe ratio appeared at the mid-90s’, and the Swiss franc based hedged portfolio had never gained a higher Sharpe ratio than the one of unhedged portfolio, even though the difference was decreased. However, none of these declines in Sharpe ratio difference are statistically significant. This indicates that this rehab of hedged portfolio performance was nothing but a random noise.

After the first plotted recession, Sharpe ratios of hedged and unhedged portfolios split again with the time varying. During the whole 90s period, there are no significant out-of-sample Sharpe ratio documented, besides in Euroland, Japan and Switzerland. Compare these three countries’ Sharpe ratio variance and the ones of the rest countries, it can be observed that, in the late 90s, there was a drop in the Australian and US market. This is influenced by the late 90s emerging market crisis, which has made US suffered a huge loss. Canada and UK, as a high integrated market with US and Australia, experienced the hit of the crisis as well. Japan, Switzerland and Euroland have defended their economies in this financial storm with high volume of currency reserves. These three countries had a continuously climbing Sharpe ratios of unhedged portfolios while other countries’ equity performances went downwards. This resulted to a more steady variance of the stock portfolio Sharpe ratios in these three countries. The hedged portfolio also started to have less fluctuations. Thus, from late 90s, the distance between hedged and unhedged portfolio Sharpe ratios were remained large, and the Sharpe ratio differences in these three countries were mainly statistically significant, until the end of the analyze period.

Moreover, Euroland, Japan and Switzerland have lowest short-term interest rate among all of the observed developed nations. The box-and-whisker chart of all countries’ interest rate are illustrated in Figure 2.

The one-month interest rate of these three countries are most clustered around the medium and they are comparatively low than the other countries’ interest rate. Even at some period of time, the interest rate in these countries are negative. It shows a link between the steady economy and the averagely low and less volatile interest rate. In the Sharpe ratio perspective, hedged and unhedged portfolios which are constructed based on euro, yen and Swiss franc, have the most difference in Sharpe ratios statistically significant.

(23)

20

Figure 1 Time-varying out-of-sample Sharpe ratio of hedged and unhedged portfolio and their differences

The figures illustrate the changing of Sharpe ratios over time and their differences, which are estimated based on each home currency. The blue line is the time-varying Sharpe ratio of unhedged portfolio. The red line is the time-varying Sharpe ratio of hedged portfolio. The back line indicates the changing of differences between these two portfolios’ Sharpe ratios and the statistically significant differences at 5% confidence level are also highlighted. The gray area is the economic crisis period. They are the early 90s’ crisis, dot com bubble recession, and the most recent financial credit crisis respectively. The out-of-sample Sharpe ratio is estimated for each 60 month with the excess return of portfolio.

(24)

D. Li Currency risk hedging and the time-varying evaluation 21 Summ ing up the results , the curren cy minim um varian ce hedge has not manag ed the time-varyin g risk. The perfor mance

of the hedged portfolio has been worsened over time, which is measured by out-of-sample Sharpe ratio. In comparison with the unhedged equity portfolio Sharpe ratio, the hedged portfolio has constantly lower Sharpe ratio. The majority of the differences in Sharpe ratios are significant for countries have steady economies, such as Euroland, Japan and Switzerland. The currency risk hedging on equity portfolios are not beneficial for investors from these countries most of time, as well as in the bear market period. In the other countries, there are periods that the investors profit from hedging currency risk, but these gains are mainly obtained by luck, instead of the hedging itself.

Figure 2 Interest rate medium and its spread for each country

In this figure, the medium and the spread of each interest rate is displayed. The data cover the period from February 1975 to April 2017. In each box, the x indicates the average interest rate and the box shows the clustered values around the mean. Two whiskers illustrate the spread of the interest rate to two-sided extreme.

(25)

22

V. Robustness Test

In order to re-investigate the results before drawing the conclusion, I conducted a robustness check. It mainly focus on two different time periods. These two subsample periods are 02/1975-09/1978 and 09/1993-04/2017 respectively. In this robustness test, I have shortened each rolling window from 60 to 30 months. Because of the rolling window estimation, two subsample data have overlapped part. But it will be used for further estimation in portfolio Sharpe ratio, so the overlapping in the data is necessary. Each data set serves to produce the out-of-sample hedged portfolio returns firstly. These returns are estimated for the periods of 08/1977-09/1998 and 02/1996-04/2017 respectively. With both portfolio returns, the Sharpe ratio can be obtained. It shows the estimation of the portfolio performance in the period of 02/1980-09/1998 and 08/1998-04/2017 in each subsamples.

A. Currency Hedging effect on international equity portfolios

First of all, portfolio returns, standard deviation and Sharpe ratio in each subperiods are reported in Table 4. The hedged portfolio returns are obtained by applying the same approach with the whole-sample analysis. The returns are out-of-sample and carried the dynamic variance of each 30-month rolling window period.

Table 4 Currency hedging effect on international equity portfolios in two subsample periods

In this table, the impacts of currency hedging for two separate sample periods are indicated. The out-of-sample estimation is conducted by using every 30-month period as a rolling window. The two subsamples have applied the data range 02/1975-09/1998, and 09/1993-04/2017 respectively. The out-of-sample hedged portfolio returns cover the periods of 08/1977-09/1998 and 02/1996-04/2017 in two subsamples respectively. The out-of-sample Sharpe ratios cover the periods of 02/1980-09/1998 and 08/1998-04/2017 respectively. The differences in standard deviation between equity portfolio return and minimum variance added hedged portfolio return are reported in Panel A. Panel B shows the difference between two portfolios’ average return. Panel C shows the difference in Sharpe ratio between hedged and unhedged portfolios. The difference is always calculated with hedged portfolio items-unhedged portfolio items.

Home currency

Panel A: Difference in portfolio return standard deviation (Hedged-unhedged)

Subperiod 1977-1998 -0,12% 0,07% -0,07% -0,11% -0,10% -0,09% -0,11% t-stat (-2,75) (1,67) (-1,74) (-2,74) (-2,41) (-2,09) (-2,58) Subperiod 1996-2017 -0,07% -0,06% -0,05% -0,06% -0,07% -0,06% -0,07% t-stat (-2,02) (-2,17) (-2,02) (-2,18) (-1,73) (-1,97) (-2,05)

Panel B: Difference in portfolio average return (Hedged-unhedged)

Subperiod 1977-1998 4,56% 0,62% 0,00% 5,23% 1,20% 0,85% 6,50% t-stat (6,21) (1,34) (0,00) (7,33) (1,92) (1,45) (9,66) Subperiod 1996-2017 -2,66% -3,28% -0,59% -2,55% -4,88% -4,16% -1,90% t-stat (-6,00) (-8,31) (-1,75) (-6,76) (-11,13) (-10,34) (-4,55)

Panel C: Difference in portfolio time-varying Sharpe ratio (Hedged-unhedged)

Subperiod 1980-1998 0,298 -0,156 -0,608 0,460 -0,264 -0,658 0,695 t-stat (1,12) (-1,00) (-2,23) (1,48) (-1,06) (-2,33) (1,76) Subperiod 1998-2017 -1,27 -1,50 -0,353 -1,04 -1,79 -1,64 -1,07 t-stat (-4,97) (-5,76) (-1,05) (-3,84) (-7,58) (-6,65) (-3,91)

(26)

23

In Panel A, the standard deviation changes caused by currency hedging is reported for both period. In the first subsample, it can be observed that, all the home-country based portfolios have risk deduction, except for the euro based equity portfolio, which is 0,07% more volatile than the unhedged portfolio. In another words, euro based portfolio became riskier after the minimum variance hedging added. However, the t-statistic shows that this increase in the volatility is not statistically significant. Although Australian dollar based equity portfolio became slightly less risky, this 0,07% risk reduction is insignificant. It is consistent with the whole sample analysis that in Australian dollar based equity portfolio, this decrease in volatility was not initiated by currency risk. In the subsequent period, all portfolios return standard deviation has been reduced with a decrement between 0,07% to 0,05%, and this fraction is mildly smaller than the volatility decrement in the first sample period. Also, all of the decrease in standard deviation are statistically significant, except for Japanese yen based portfolios. The hedged portfolio returns was indeed less volatile, but it was not resulted by minimum variance hedging.

To my surprise, in the first sample period, the average returns of each hedged equity portfolios achieved a higher value than the unhedged ones. Except for euro, Australian dollar and Swiss franc, the rest portfolios have obtained statistically significant better average returns. US dollar, Canadian dollar and pound based equity portfolios have the highest increase in the average portfolio returns, which are 4,56%, 5,23% and 6,50% respectively for these three stock portfolios. But when I examine the average return change in the second sample period, all of the portfolios have negative and statistically significant mean. The stock portfolio based on yen had a significant lift in the average return for 1,20% in the first sample, but the 4,88% decrease in this second period has cancelled out all the gains and even worsened off. Whereas this decrease does not offset the higher portfolio returns achieved in period one, by US dollar, Canadian dollar and pound based hedged portfolios.

Considering the average return and the return volatility in both sample period, the results are consistent with the whole-sample analysis. They indicate that, all of the equity portfolios become indeed less risky after the minimum variance hedge added. The decrement in the volatility is larger in the first sample period. But the deduction is statistically significant for most hedged portfolios in the second sample period. The medium returns for all stock portfolios are increased in the first sample period, yet they are not all statistically significant. Also, the significant drop of the average returns in the second period has offset the gain from the first period. By contrast, the hedged portfolios based on US dollar, Canadian dollar and pound retained the average return improvement, and they are not cancelled out by the decline in the second period. This is in line with the findings of the full-sample analysis. This means that, the currency risk hedging really benefitted investors in these three market, but only in the first sample period.

Moving forward to the Sharpe ratio comparison, I can observe that, US dollar, Canadian dollar and pound based equity portfolios have improved the Sharpe ratio. It is coincident with the results shown in Panel A and B, that these three hedged portfolios had their volatilities lowered and average return increased. But the increase in the portfolio performance is not statistically significant. This is caused by large variance between hedged and unhedged portfolio Sharpe ratio. The increase in the Sharpe ratio is not big enough to make the boosted portfolio performance statistically significant.

(27)

24

The other home-country based portfolios have worsened portfolio performance with the time varying, Australian dollar based equity portfolio and Swiss franc based equity portfolio have the biggest significant decrease in Sharpe ratio compare with the others. Although the portfolio based on Swiss franc had a significant volatility deduction in the first period, the increase in the average returns, which are not gained by currency hedging, has not lead the hedged portfolio a better performance than the unhedged one. In fact, it has been worsened. The decrement of hedged portfolios Sharpe ratio is more severe than the first period. All of the portfolios have worsened Sharpe ratio, and statistically significant, except for Australian dollar based hedged portfolio. The huge fall of hedged portfolio Sharpe ratio in the second subperiod led to the overall decreased portfolio performance in the whole sample. As in the full-sample analysis, the Sharpe ratio of hedged portfolios are much more negative than the one of unhedged portfolios. This degradation of Sharpe ratio is also statistically significant for the whole period.

B. Portfolio performance evaluated by time-varying out-of-sample Sharpe ratio

For both two subsamples, I have plotted the time-varying Sharpe ratios in Figure 3. On the left hand side, there are the time-varying Sharpe ratios estimated in subsample period 1 (02/1980-09/1998). Correspondently, on the right side there are the time-varying Sharpe ratios obtained with data of period 2 (08/1998-04/2017). The two graphs in each row are the plot for the same home-country based hedged and unhedged portfolios.

In the first sample period, the hedged portfolios Sharpe ratio are initially higher than the unhedged portfolio Sharpe ratio. This is true for all the home-country based portfolios. Hedged portfolio has outperformed the unhedged portfolios, and this lasts until 1985 roughly. In this 5-year period, the difference in Sharpe ratio has also reached an extremely high peak compare with the rest of the time. Most of these differences are statistically significant. This can be interpreted as outperformance truly obtained by adding minimum variance currency hedge. Right after the year of 1985, the Sharpe ratio of hedged portfolios fell dramatically, and it was immediately lower than the unhedged portfolio Sharpe ratio. This period’s Sharpe ratio differences are only significant in Australia and Euroland. So in these two countries, the currency risk hedging has causal effect on the decrease of the global equity portfolio performance. While entering the first recession in early 90s, the most of the hedged portfolios have higher Sharpe ratios than unhedged ones. Some hedged portfolios high Sharpe ratio last longer, such as the portfolios constructed based on US dollar, Australian dollar, Canadian dollar and pound. But none of the Sharpe ratio differences in this period are statistically significant. This means that the increase in the portfolio performance in this bearish period is certainly not caused by currency risk hedging. So in the first subsample period, the hedged portfolios have similar Sharpe ratio variance in cross-section. Sharp ratios firstly raised higher than the unhedged portfolio ones with significant difference, and then decreased but again increased in the recession. However, this outperformance of hedged portfolio is not statistically significant.

In the second sample period, the magnitude of all Sharpe ratios have declined with more volatility. In UK, the difference in Sharpe ratio lost significance in the early 20s. Most of the time, the hedged portfolios have lower Sharpe ratio than the unhedged portfolios. By contrast, this is typically false during the two recessions in the 20s. The hedged portfolio based on

(28)

25

Australian dollar and Canadian dollar have higher Sharpe ratios in both crisis. In the recent recession, Sharpe ratios of hedged portfolio in US and UK are higher than unhedged portfolio ones. Unfortunately, the Sharpe ratio differences in these period are all insignificant. So the bettered portfolio performance for hedged portfolios are not resulted by currency risk hedging. But in all markets, the Sharpe ratio of hedged portfolios have less distance with the one of unhedged portfolios. This decrease in distance has been documented in the Japanese and Swiss markets. Moreover, the reduced Sharpe ratio differences are also statistically significant. This demonstrated that the currency risk hedging has at least increased the portfolio performance, although the portfolios perform better with the currency left unhedged in the financial crisis. After integrating the two subsample Sharpe ratio variances, the comparison with the whole-sample Sharpe ratio variances can be discussed. As mentioned before, after separate the sample into two periods, the volatility of the Sharpe ratio and the differences in Sharpe ratios are amplified. The variances with the time being can be illustrated more in detail. The first inconsistent I have discovered is that, the outperformance of the hedged portfolios is mainly concentrated in the period of early 80s, and they are all statistically significant. Although the whole sample estimation has documented the significance, the incremental of the Sharpe ratio caused by adding currency hedging is as twice as much higher in the first subsample period. Moreover, in Australian market, the significance in the decrease in the Sharpe ratio has disappeared after considering only first subsample period. Instead, the Sharpe ratio of hedged portfolio has significantly dropped before the recession. What’s more interesting is, some countries’ hedged portfolios outperformed unhedged portfolios in most recessions, which is contradict in the whole sample analysis. In the full-sample period, the hedged portfolios performed always badly and even worse in the bearish market. Yet, these outperformance is statistically insignificant. Only the shrinkage of the distance between hedged and unhedged portfolio Sharpe ratio are documented in Japanese and Swiss market.

To sum up, in the subsample analysis, the effect of currency risk hedging is more amplified. The currency risk hedging indeed increased the portfolio average return with reducing the risk at the same time in the first sample period for US, Canada and UK market. This is consistent with the findings from the whole sample period studies. By contrary, the discovery from subsample analysis is not completely in line with the whole sample research. The positive and statistically significant difference in hedged and unhedged portfolio Sharpe ratios have much higher values, and during the recession, the hedged portfolios have actually better Sharpe ratio in some countries, although it is statistically insignificant. However, in general, the time-varying portfolio performance is still better while the currency risk left unhedged.

(29)

26

(30)

27

Figure 3 Time-varying Sharpe ratio comparison between two sub sample periods

In this figure, the time-varying out-of-sample Sharpe ratios for both subsample periods are illustrated. Left side plots are Sharpe ratios and the difference in hedged and unhedged portfolio Sharpe ratios for the first sample period (02/1980-09/1998). Right side plots are corresponded home-country based portfolios’ Sharpe ratio for the second sample period (08/1998-04/2017). The out-of-sample Sharpe ratio is estimated with 30-month rolling windows. Blue line and red lines are the time-varying Sharpe ratio of unhedged equity portfolio and currency hedged portfolio respectively. Black line is the Sharpe ratio difference between the hedged and unhedged portfolio. The significant differences in Sharpe ratios are highlighted. The gray area indicates the recession period.

Time varying currency risk hedging and the valuation

Dongni Li

11388277

| 15-8-2017

Time varying currency risk

hedging and the valuation

Statement of Originality

Abstract

Table of Contents

I. Introduction

II. Literature review

III. Data and Summary Statistics

IV. Methodology

V. Results

V. Robustness Test