Can traders benefit from the currency carry trade during crisis periods?
An empirical study of dynamic carry trades developed using VIX and VXY
Sofie Skovsgaard Mogensen (101394) Nina Chaikhot Nielsen (91672) Cand. Merc. Finance & Investments Cand. Merc. Finance & Investments
September 2020
Supervisor: Johan Stax Jakobsen Number of pages: 88 Number of characters: 160,849
Carry investors exploit interest rate differentials by going long on the currencies with a high-interest rate and short the currencies with a low-interest rate. The profitability of the strategy is well-documented, but the real picture is more complex. The findings of this thesis showed that the carry trade generated a positive return up to 2.98% per annum over the period from June 1999 to May 2020. Over the period the performance has alternated between periods of profitability as well as periods of large losses. Crisis periods appear to be the main reason for sudden and large negative returns. During the financial crisis carry investors lost up to 20% on their invested capital. The focal question of this thesis is, therefore, whether it is possible to improve upon the carry trade during volatile periods. This is investigated using the implied volatility indices VIX and VXY to time the currency carry trade strategies. Applying different dynamic carry trade strategies, which take alternative positions during volatile periods, it turns out that the performance of strategies rebalanced monthly generally improved over the full sample period and across different sub-periods.
In fact, one of the alternative strategies generated a positive return up to 7.23%. As most of the impact can be contributed to the improvement observed over the financial crisis, excluding this period from the analysis would have led to worse performance for some of the strategies. Hence, the findings of this cast doubt on the actual improvement obtained from the implementation of the implied volatility as a timing signal.
1 Introduction 4
1.1 Motivation . . . 4
1.2 Problem Statement . . . 5
1.3 Contribution . . . 6
1.4 Delimitation . . . 7
1.5 Chapter outline . . . 8
2 Theoretical Background 9 2.1 The interest parity . . . 9
2.1.1 Covered interest rate parity . . . 10
2.1.2 Uncovered interest rate parity . . . 11
2.2 Carry Trade . . . 13
2.3 Carry trade during stressed periods . . . 15
3 Data 17 3.1 Exchange rates . . . 17
3.2 Forward discounts and spot rate changes . . . 18
3.3 Volatility indices . . . 20
4 Methodology 22 4.1 Portfolio construction . . . 22
4.2 Carry trade returns . . . 24
4.3 Trading signals . . . 27
4.3.1 Threshold . . . 27
4.4 Crisis robust strategy . . . 28
4.5 Performance Measures . . . 30
4.6 Hypothesis testing . . . 31
4.6.1 Nominal individual test . . . 32
4.6.2 Bootstrapping . . . 32
5.1.1 Characteristics of returns . . . 36
5.1.2 Cumulative returns . . . 38
5.1.3 Carry trade performance . . . 41
5.2 Crisis Indicator . . . 47
5.2.1 VIX and VXY . . . 47
5.2.2 Crisis periods . . . 48
5.3 Crisis Robust Carry Trade . . . 50
5.3.1 VIX signal . . . 51
5.3.2 VXY signal . . . 56
5.3.3 Impact of crisis periods . . . 60
5.3.4 Bootstrap hypothesis testing . . . 63
6 Robustness test 68 6.1 Threshold analysis . . . 68
6.1.1 VIX signal . . . 69
6.1.2 VXY signal . . . 70
6.2 Timing signal implied volatility . . . 72
6.2.1 VIX signal . . . 72
6.2.2 VXY signal . . . 77
6.3 Change in implied volatility . . . 80
6.3.1 VIX signal . . . 81
6.3.2 VXY signal . . . 83
7 Conclusion 85 8 Further Research 87 A Abbreviations 93 A.1 Strategy abbreviations . . . 93
B Tables 94 B.1 Descriptive statistics . . . 94
B.2 Carry trade Performance Metric . . . 97
B.3 Kurtosis and skewness . . . 97
B.4 Fraction of negative returns . . . 99
B.5 Bootstrap confidence interval of sub samples using VIX . . . 103
B.6 Bootstrap confidence interval of sub samples using VXY . . . . 105
B.7 Timing signal implied volatility . . . 107
B.9 Sub-sample test of the dynamic strategies based on the change in VXY . . . 113
C Figures 115
C.1 Autocorrelation function plots . . . 115 C.2 Cumulative returns . . . 118 C.3 Plot of the performance measures using different timing signals
of VIX for different standard deviations . . . 119 C.4 Plot of the performance measures using different timing signals
of VXY for different standard deviations . . . 125 C.5 Histogram of bootstrapped 3-, 6- and 12 month rebalance carry
strategies . . . 130
D Python codes 131
D.1 Portfolio construction and carry trade return calculations based on 1-month rebalance . . . 131 D.2 Exit- and Reverse Strategy 1-month rebalance based on VIX . . 148 D.3 Bootstrap resample and confidence interval of static carry trade 162
Introduction
1.1 Motivation
The foreign exchange (FX) market is considered to be the largest financial market, with an average daily turnover close to 5.1 trillion US dollar (BIS, 2019). Its popularity has entailed a large number of well-founded investment strategies being unique for the FX Market, including the FX carry trade.
Carry investors exploit interest rate differentials by going long on the cur- rencies with a high-interest rate and short the currencies with a low-interest rate. The essence of the carry trade is to exploit the anomaly emerging from the deviation of the uncovered interest parity. Empirical studies have docu- mented the profitability of carry trade primarily because high-yield currencies tend to appreciate rather than depreciate against the low-yield currency. This phenomenon is also known as the forward-premium puzzle as it violates the simple risk-neutral efficient market hypothesis (Fama, 1984). However, it ap- pears that the real picture of carry trade performance is much more complex.
The performance of carry trade has alternated between periods of profitability and losses. Brunnermeier et al. (2008) found that periods of good performance are correlated with subsequent periods of low volatility and vice versa. In fact, during the financial crisis in 2008 carry investors suffered losses of up to 20%
of invested capital (Egbers and Swinkels, 2015). Although carry trades are profitable over long periods, in reality, investors presumably do not tolerate a trading strategy which underperforms over several years. Traders are often subject to performance measures of their trading decisions and rarely stick to a strategy that has demonstrated unsatisfactory results over longer periods.
Consequently, a crucial element to the carry trade in practice is to consider the possibility of adjusting the carry trade during volatile periods. Standing on the edge of a new crisis caused by the global pandemic, COVID-19, carry
investors are facing new obstacles and challenges. The pandemic has led to moves in the FX market, which compare to those observed during the financial crisis. The uncertainty in the global economy, which directly impacts the FX market, only stresses the importance of adjusting the carry trade in periods of high volatility.
1.2 Problem Statement
In light of the motivation, the question of whether traders reliably can time the carry strategy to prevent times of high volatility appear highly relevant. With the possibility of predicting crisis periods, investors may be able to avoid the unsatisfying results that follow. The focal question of this thesis, is based on the associated problem being whether it is possible to develop a crisis robust carry trade strategy using the implied volatility indices VIX and VXY1. Based on the motivation and problem statement, the following research questions arise:
• What is the profitability of FX carry trade during the period from June 1999 to May 2020 and are the results driven by certain periods?
• How does the implementation of the timing signals VIX and VXY impact the performance of the FX carry trade?
• Are the findings driven by specific model- and parameter choices?
1VIX is the implied volatility on the US equity market, and VXY is the option implied volatility index from G7 currencies
1.3 Contribution
Despite the acknowledgement of the poor performance during crisis periods, few studies explore the possibility of constructing carry trade strategies that perform well during volatile periods. Among these, all have failed to find a convincing crisis robust carry trade strategy. This thesis attempts to solve this, based on the problem statement presented in the previous section, by ap- plying well-known theory and key theoretical findings from previous academic papers within the same line of literature. To develop a crisis robust carry trade strategy, one of the critical elements, is to have an indicator which can antici- pate negative carry returns. Swinkels and Egbers (2015) and Briere and Drut (2009) used the implied volatility index, VIX. According to Brunnermeier et al.
(2008), the index is empirically related to future carry trade losses. Another critical element is to have an alternative strategy that works complementary to the carry trade strategy during times of carry trade losses. Briere and Drut (2009) used the purchasing power parity-based strategy, while Swinkels and Egbers (2015) investigated the impact of unwinding the carry trade positions during crisis periods. The main contributions of this thesis are described in summary below:
• Briere and Drut (2009) and Swinkels and Egbers (2015) investigate the performance up until 2008 and 2014. Hence, previous studies exclude the recent economic crisis, COVID-19, which led to the historically highest level of VIX. The strategies developed in this thesis are tested during a sample period, which includes the recent economic crisis.
• The crisis periods in Briere and Drut (2009) are determined using the entire sample period, leading to backward-looking bias. Swinkels and Egbers (2015) avoid this issue, by using an expanding rolling window, up until the closing value one day-lagged. Considering that spikes might be both level and context-dependent, the crisis periods in this thesis will be estimated using a moving rolling window.
• Empirical studies indicate that asset returns share some statistical prop- erties common to a wide variety of markets and instruments (Cont, 2000).
As these can lead to an inadequate statistical inference of the perfor- mance using a simple nominal test, the stationary bootstrap method of Politis and Romano (1994) is applied on the FX carry return series.
1.4 Delimitation
The analysis of the carry trade strategy is based on a sample of the 11 most de- veloped currencies2. These are selected to avoid using currencies in the trading strategies, which in practice would have been subject to capacity constraints.
It is possible to modify the carry trade in various ways. However, the intention of this thesis is not to explore the carry trade to its full potential. Instead, the focus is to evaluate a selected number of strategies based on a limited set of models and parameters. These parameters are set to identify crisis periods and avoid losses that follow from the carry trade during volatile periods. To accommodate the results are not depended on the specific modelling choices, a sensitivity analysis is carried out.
The parameters are selected to obtain larger comparability to previous litera- ture, while at the same time contributing to the literature with new aspects.
This includes using the implied volatility indices, VIX and VXY to identify cri- sis periods similar to previous literature, but with different parameter choices.
On end-of-month days, where it was not possible to obtain a price for either of the indices, the previous day was used. This means a notable amount of end-of-month values has been replaced by the most recent available data.
The conclusions are exclusively drawn upon the analysis prepared and the available empiri from the selected time frame between 30-06-1999 and 29-05- 2020. Although this gives an indication of whether the indices can be used to avoid losses during volatile periods, this can not ensure that historical patterns will repeat itself in the future.
2These are known as the G11 currencies, which is the G10 currencies with the addition of the Danish krone
1.5 Chapter outline
The remaining chapters of the thesis are structured as follows:
In Chapter 2, the theoretical background is described to create the founda- tion of the carry trade. This includes a description of the interest rate parity conditions, and in particular the inconsistency of the uncovered interest rate parity, which is the premise of the carry trade. Throughout the theoretical section, relevant papers within the same line of literature, including Burnside et al. (2008), Brunnermeier et al. (2008) and Egbers and Swinkels (2015) are used to understand the opportunities of modifying the carry trade, to obtain a crisis robust carry trade.
In Chapter 3 and 4, the data and methodology used throughout the thesis are described. This gives a detailed understanding and reasoning behind the selected model choices. In addition, this part creates the foundation of the results, presented in the following chapters.
Chapter 5 and 6 consist of a three-part analysis. In the first and second part of the analysis, in Chapter 5, the performance of the different strategies are evaluated with different rebalancing periods and number of currencies. The results are assessed both statistically and economically, covering all the relevant aspects of the reported figures. While the first part presents the findings of the static carry trade, the second part evaluate the performance of two types of dynamic strategies and the possibility of improving upon the static carry trade with these. The performance of the static carry trade and dynamic strategies are further elucidated, by decomposing the sample period into three periods.
All the statistical tests are confirmed by applying a stationary bootstrap on the returns.
The focal point of the last part, Chapter 6, is the robustness analysis, in which a sensitivity analysis of the results is carried out. The Findings of this section is applied to assess whether the results are robust to the selected model choices and in this sense, reliable.
Theoretical Background
In this chapter, the theoretical background is described to create the foun- dation for the methodology and the analysis. First, the interest rate parity condition and in particular, the violation of the uncovered interest parity is described and examined. This is followed by an overview of selected literature centred around carry trades. Lastly, an overview of literature focusing on carry trade during stressed periods is introduced.
2.1 The interest parity
The concept of the Interest Rate Parity (IRP) received prominent exposition by the theories of Keynes (1923), mainly because of the rapid expansion of organised trading in forward exchange following World War I. IRP is used to explain the no-arbitrage condition representing the equilibrium in which investors are indifferent between interest rates available on bank deposits in two different countries. Hence, the pay-off of holding assets denominated in domestic currency at the domestic rate should compare to the value of holding assets denominated in foreign currency at the foreign rate in each future state of the world. If the pay-off of the two strategies is not the same, investors are able to make arbitrage profits by taking a long position in the strategy with a higher pay-off and short the position with a lower pay-off. The condition of IRP can take two different forms: Covered Interest Parity (CIP) and Uncovered Interest Parity (UIP) which is described in further detail in the remainder of Section 2.1 (Isard, 2006).
2.1.1 Covered interest rate parity
Although the understanding of the forward exchange market existed within many banking circles during the second half of the nineteenth century, the literature during this time mainly dealt with spot exchange rates. The intro- duction of forward exchange trading gave investors the opportunity to elim- inate the exposure to the future rate, by entering into a forward contract at a rate Ft,k, which gave rise to the notion of CIP. That is, the investors leave their foreign currency position uncovered at time t and then re-convert into domestic currency at the forward rate at time t+ 1. The market equilibrium requires the following condition of CIP:
1 +idt = Ft,k
St (1 +ift), (2.1)
where idt is the domestic interest rate, ift is the foreign interest rate, St is the spot exchange rate and Ft,k is the forward rate at time t maturing at time k (Isard, 2006). Figure 2.1 provides with a visual presentation of the UIP.
Figure (2.1) Visual representation of the CIP
This figure presents the required market equilibrium of the covered interest rate parity, meaning the returns from investing domestic currency are equal to the
returns from investing in foreign currency.
Numerous papers have described the apparent deviations of the CIP. Akram et al. (2008) used high-frequency data from 2004, between February and Septem- ber, that showed the arbitrage opportunities could be obtained and thereby
the violation of the CIP. Akram et al. (2008) similarly found a short-lived violation of the CIP and the size of the CIP violation to be economically significant. Frenkel and Levich (1977) found that the deviations of CIP be- tween the Deutsche mark, US dollar and Pound sterling can be explained by the transaction costs. The findings suggested that when including transaction cost, the CIP did not appear to exploit opportunities for profit. Summarising the academic papers, the findings indicate that the deviations do appear, but that the arbitrage opportunities are short-lived. Additionally, strategies that try to exploit the CIP violation are generally not expected to earn substantial return after including transaction cost.
2.1.2 Uncovered interest rate parity
If the condition is satisfied without the use of forward contracts, the interest rate parity is said to be uncovered. UIP states that interest rate differentials in the two countries are expected to be compensated by adjustments in the spot exchange rate, thereby leading to an expectation of zero excess returns.
Since it is clear that the perception of the future spot rates also reflects the forward rates, UIP postulates that the forward exchange rate is forced into equality with the expected future spot exchange rate by market forces. The difference is, essentially, that investors re-convert into domestic currency at the spot exchange rate St+k instead of the forward rate at time t+k.
1 +idt = E(St+k)
St (1 +ift). (2.2)
As for CIP, the investors are indifferent between the domestic and foreign in- terest rate if UIP holds (Isard, 2006). However, the UIP is not as empirically verified as the CIP, rather a number of empirical studies suggest that exchange rates do not compensate for the interest differentials as much as predicted by the UIP. Instead, the high-interest rate currencies tend to appreciate, whereas low-interest rate currencies tend to depreciate. In academic literature, this is typically referred to as the forward premium puzzle (Fama, 1984). While it is not possible to enforce an arbitrage strategy, as the position is not hedged against exchange rate uncertainty, this is the premise of the carry trade in the currency market.
Applying the data described in the later data Section 3, the validity of UIP
is investigated using an OLS regression analysis of the change in spot rates against the forward rates minus spot rates, as a proxy for the interest rate differential
.
∆st+1 =α+β(ft−st) +t+1, (2.3)
where st and ft denote the natural logarithm of the spot rate and forward rate at time t. The reasoning behind using forward discounts as a proxy for interest rate differentials is described in Section 2.2. If UIP holds, the coefficientβ should be equal to unity,α to zero and the error term white noise (Olmo and Pilbeam, 2009). As can be seen in Table 2.1, α is not significantly different from zero at a 5% significance level. This applies for all currency pairs over the period 28-05-1999 to 29-05-2020. However, the null hypothesis of the β coefficient is rejected for all currency pairs at a significance level of 5%. In addition, β is negative for all currencies, except for JPY and CAD with 0.045 and 0.015, respectively. The negative β coefficients imply that high-interest rate currencies tend to appreciate rather than depreciate. Hence, the hypothesis tests indicate a violation of the UIP condition.
Table (2.1) Hypothesis test of UIP
This table reports OLS estimates of the regression of 1-day spot exchange rate changes on 1-day forward premium with USD as base currency between 28-05-1999
and 29-05-2020. The z-test is the test statistics of the null hypothesis thatβ=1 and α=0. Positiveβ-coefficients are marked with a grey coloured cell.
DKK GBP AUD CAD EUR JPY NZD NOK SEK CHF
alpha 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 std. err 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 z-test -0.376 0.637 0.647 -0.221 -0.329 0.043 0.087 0.459 0.003 -0.647 p-value 70.69% 52.41% 51.76% 82.49% 74.19% 96.57% 93.07% 64.62% 99.76% 51.76%
beta -0.053 -0.018 -0.053 0.045 -0.040 0.015 -0.021 -0.011 -0.055 -0.098 std. err 0.063 0.073 0.072 0.108 0.069 0.054 0.077 0.064 0.064 0.079 z-test -16.742 -14.022 -14.584 -8.847 -15.075 -18.413 -13.344 -15.817 -16.401 -13.921 p-value 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
As in Fama (1984), it is common to investigate the validity of UIP using statistical tests based on regression models. However, the statistical tests
have been questioned by several papers. One example is Baillie and Bollerslev (2000), who simulated the regression models to investigate the validity of the UIP. The findings of prior literature, such as those in Fama (1984), were found to be overstating the statistical significance to reject the hypothesis of the forward rate being an unbiased predictor of the future spot rate. Among other papers in favour of the UIP, is Flood and Rose (2002) who examined the effect of the financial crisis on the UIP. Their results showed higher validity of the UIP during crisis periods, with higher interest rates depreciating as in line with the UIP. Lothian and Wu (2011) provided weak empirical results for the violations of the UIP using data between 1800-1999 for the USDGPB and FFGBP. Although the critiques of the deviations found in the UIP might be justified, the literature in favour of the deviation of the UIP is predominant (NBIM, 2014).
2.2 Carry Trade
The carry trade is a cross-currency position exploiting interest rate differen- tials. Investors borrow funds at a low-interest rate currency (funding currency) and buy a high-interest rate currency (target currency). Heath et al. (2007) stated that the profitability of carry trade emerged from the failure of the UIP. Carry trade is only profitable if the gain from the interest differential is not overcome by the movements in the exchange rate in the short or medium term. Heath et al. (2007) also investigated the increasing growth in the FX market since 2004 and found that carry trade contributed to increased activity in currencies, including Japanese Yen and Swiss Franc. Darvas (2009) studied all the possible pairs of 11 major currencies over the period 1976 to 2008 and found that holding carry trade positions consistently throughout the period, resulted in significant excess return. Additionally, Darvas (2009) conclude that the failure of the UIP is exploitable for almost all of the currency pairs after including transaction costs.
The return of the carry trade highly depends on the appreciation and depre- ciation of currencies. If the target currency depreciates relative to the funding currency, the carry trade return decreases. If instead, the target currency appreciates, the return on the carry trade increases (Menkhoff et al., 2012).
As was seen in seen Table 2.1, high-interest rate currencies tend to appreci- ate rather than depreciate relative to low-interest rate currencies, which has a positive impact on the performance of carry trades. Carry trades are usually
leveraged, which makes them particularly subject to changes in the interest rate, as well as the exchange rate.
Burnside et al. (2008) emphasised an important feature of the carry trade, which was the diversification opportunity among different currency pairs. Burn- side et al. (2008) developed an equally-weighted portfolio of 23 currency pairs which resulted in a decrease in volatility and an increase in the portfolio Sharpe Ratio. In addition to equally-weighted portfolios, Briere and Drut (2009) cre- ated two portfolios, one consisting of high-interest rate currency pairs and the other of low-interest rate currency pairs. Briere and Drut (2009) found that the portfolio with highest-interest rate differential displayed the largest devia- tion from the UIP.
The currency excess return obtained from going long in foreign currency for- ward at time t and selling the currency in the spot market at time t+ 1, is defined as
Rt+1 = Ft−St+1
St = Ft−St
St −St+1−St
St , (2.4)
whereFt denoted the forward exchange rate, andSt denote the spot exchange rate. The excess return on a foreign currency can be decomposed into two parts: The forward discount rate, which is the difference between the forward and spot rate scaled by the spot rate and the change in the spot rate. Further- more, under the CIP condition shown in Equation 2.2, the forward discount rate must equal the interest rate differential such that
F Dt= Ft−St St
=ift −idt, (2.5)
where ift is the interest rate of the foreign country and idt is the interest rate of the domestic country. The forward discount rate is commonly used as a proxy for the interest rate differential within this line literature (Filippou and Taylor, 2016).
2.3 Carry trade during stressed periods
As described in the previous section, is there a broad agreement among aca- demic literature that the carry trade produces positive excess returns as de- scribed in the previous section. However, the results are often based on longer periods in which the returns vary considerably. Briere and Drut (2009) found that Carry trade strategies, based on 28 different currency pairs, alternated between periods of profitability and losses, in which crisis periods appeared to have a significant impact on the performance. Swinkels and Egbers (2015) report that, during the financial crisis, the carry strategy suffered losses up to 20% of their invested capital. Brunnermeier et al. (2008) found evidence of carry trade being subject to crash risk, as the exchange rate movements between high-interest rate currencies and low interest-rate currencies are neg- atively skewed. Taking note of this, the downside risk should be considered when creating a carry trade strategy.
Roche and Rockinger (2003) proposed a crisis-robust strategy which involves taking the opposite positions of the carry trade during periods of carry trade losses. That is, buying the low-interest rate currency and selling the high- interest rate currency. The strategy is often referred to as an event-driven strategy, that under the right market conditions can be highly profitable or used as a hedge to carry trade positions. Under financial distress, investors historically have avoided high-yielding risky currencies on a massive level by switching their funds to safe and defensive investments. Brunnermeier et al.
(2008) found a positive correlation between currency crashes and stock market volatility. This may be explained by the higher capital requirements, leading traders to cut back on their carry trading. In addition, the findings indicated that the higher the level of VIX, the higher the expected carry returns. Con- trolling for this, lead to a reduction in the FX return predictability of the interest rates, which alleviates the violation of the UIP.
To apply the crisis robust strategy, a critical element is therefore to have an indicator which is able to anticipate periods of currency crashes. Gyntelberg and Remolona (2007) considered two commonly used measures, Value-at-risk (VaR) and expected shortfall (ES). While both VaR and ES focus on the downside risk, the volatility of returns is the most common measure of risk in financial markets. In particular, many studies indicate that implied volatility, measured by the Chicago Board Options Exchange’s (CBOE) Volatility In- dex (VIX), is an accurate measure of perception of future risk: the higher the
concern of increased volatility in the future, the greater the VIX (Briere and Drut, 2009). The index was initially constructed from the implied volatility of near-the-money options of the S&P100, but in 2003, CBOE changed this to the S&P500, which is considered to be the benchmark of the US stock market return (Siriopoulos and Fassas, 2012).
Among 13 different indices, Siriopoulos and Fassas (2012) consider VIX among the leading source of uncertainty as the changes in implied volatility of the US stock market are passed on to other markets. Various studies within this line of literature have used VIX as a crash indicator. Briere and Drut (2009) used VIX as an indicator for when to switch between the carry trade strategy and the purchasing power parity based strategy. The findings showed that applying this strategy instead of the pure carry trade would have lead to Substantially better performance, as it was possible to avoid most of the negative carry-trade returns. Egbers and Swinkels (2015) similarly used VIX as an indicator, but instead of using the power parity based strategy, the positions were simply closed or reversed.
Data
In this chapter, the data used to construct the carry trade strategies as well as the timing signals used for crisis determination is introduced. Additionally, the descriptive statistics of the spot rate changes and forward discounts used in the analysis are summarised and described.
3.1 Exchange rates
All data on forward and exchange rates are sourced from Thomson Reuters DataStream, which is the standard data source in this line of literature. It consists of the World Market Reuters series of the bid-, mid- and ask quotes of spot exchange rates and 1-, 3-, 6- and 12 month forward rates. In line with most literature on UIP, the forward rates and spot rates are expressed relative to the USD, in which the USD is used as the base currency in the quotation of spot and forward rates. This means that an increase in the exchange rate indicates an appreciation of the USD and vice versa.
The sample consist of the ten most developed currencies: Euro (EUR), Pound sterling (GBP), Japanese yen (JPY), Australian dollar (AUD), New Zealand dollar (NZD), Canadian dollar (CAD), Swiss franc (CHF), Norwegian krone (NOK), Swedish krona (SEK) and Danish krone (DKK). Although the most developed currencies, according to G11, included different currencies before 2013, the listed ones will be used throughout the entire analysis. The rea- soning behind the inclusion of developed currencies lies in the high liquidity, which may alleviate potential issues of capacity constraints (Brunnermeier et al., 2008). Although investors in carry trade did not necessarily invest in currencies relative to USD, this sheds light on the profitability of this particu- lar strategy. While most studies start in the early 1980s, this sample runs from
the period between 28-05-1999 and 29-05-2020 to reflect the characteristics of data in more recent periods. The relatively long period with 253 end-of-month observations allows for periods of fairly high-interest rate differentials, as well as low-interest rate differentials as seen in Figure 3.1. The mean, minimum, maximum, standard deviation, kurtosis and skewness of the spot- and forward exchange rates for all ten currency pairs are reported in Figure B.1 and B.2, Appendix B.1.
3.2 Forward discounts and spot rate changes
As explained in Section 2.2, the profit of carry trade strategy is based on the forward discount rates (i.e. interest rate differentials) and changes in exchange rates. To better understand the profit, Table 3.1 presents an overview of the end-of-month 1-month forward discounts and spot rate changes over the whole sample period. Among all ten currencies, the highest average spot rate changes occurred in the NOK, GBP and SEK with 0.136%, 0.136% and 0.092%
respectively, meaning these three currencies depreciated the most against the USD dollar. On the other hand, CHF and JPY on average appreciated against the USD, with average spot rate changes of -0.138% and -0.0088% respectively.
It is important to note the discrepancies in the returns and volatility. While the second-highest average spot rate change is found in the GBP, this currency also had the lowest volatility measured by its standard deviation of 0.025. On the other hand, the lowest average spot rate change is found in the CAD, and this had the second-lowest volatility of 0.026. Except for CHF, all currencies are right-skewed, with the most profound skew of 0.777 found in the spot changes of the CAD. The kurtosis is computed as the Fisher’s kurtosis, by subtracting three. Hence, the kurtosis defined for a normal distribution is zero. For all currencies, the kurtosis is well above zero, indicating that the tails of the distributions are much fatter compared to those of normal distributions.
The 1-month forward discounts in Table 3.1 range greatly between the different currencies. The highest mean forward discount per month occurred in the NZD, AUD and NOK with 0.197%, 0.168% and 0.072% respectively.
The lowest occurred in JPY, CHF and EUR with -0.182%, -0.145% and - 0.056% respectively. If the forward discount rate is negative, this indicates that the foreign interest rate on average was lower than the US interest rate, whereas a positive forward discount rate indicates that the foreign interest rate on average was higher than the US interest rate. The descriptive statistics of the 3-, 6-, and 12-month forward discount rates reported in Figure B.3 in
Table (3.1) Descriptive statistics of spot exchange rate change and forward discounts
This table reports the mean, minimum, maximum, standard deviations as well as the kurtosis and skewness for the spot rate change and currency forwards discounts. The figures are based on the end-of-month mid prices available in the sample period between 28-05-1999 and 29-05-2020. The three highest and lowest
mean returns are marked with coloured cells.
1-month spot rate change
Mean Min Max Std. dev. Kurtosis Skewness AUD 0.057% -8.393% 19.561% 0.036 3.132 0.753 CAD 0.006% -7.769% 14.358% 0.026 3.890 0.777 DKK 0.018% -8.790% 10.560% 0.028 1.276 0.289 EUR 0.017% -8.715% 10.772% 0.028 1.294 0.300 JPY -0.009% -7.497% 8.947% 0.027 0.741 0.291 NZD 0.013% -12.247% 15.266% 0.038 1.717 0.537 NOK 0.136% -6.951% 13.868% 0.033 1.270 0.434 SEK 0.092% -8.680% 11.784% 0.032 0.695 0.166 CHF -0.138% -12.374% 12.629% 0.029 2.200 - 0.020 GBP 0.136% -8.107% 10.317% 0.025 1.487 0.498
1-month forward discount
AUD 0.168% -0.117% 0.445% 0.001 -1.081 -0.100 CAD 0.009% -0.171% 0.201% 0.001 -0.616 0.130 DKK -0.052% -0.369% 0.253% 0.001 -0.842 -0.005 EUR -0.056% -0.334% 0.163% 0.001 -1.000 -0.131 JPY -0.182% 0.613% -0.009% 0.002 -0.353 -0.876 NZD 0.197% -0.124% 0.552% 0.001 -0.448 -0.236
NOK 0.072% -0.236% 0.491% 0.002 0.004 0.480
SEK -0.037% -0.339% 0.232% 0.002 -1.316 0.036 CHF -0.145% -0.395% 0.002% 0.001 -1.220 -0.472 GBP 0.031% -0.267% 0.291% 0.001 -0.277 0.386
Appendix B.1 display similar patterns. From visual inspection of Figure 3.1, it seems that interest rate forward discounts varied a lot over time across all ten currencies as well. The figure presents the evolution of the forward discounts for all ten currencies against the USD from 28-05-1999 to 29-05-2020.
The difference between the two periods can be explained by the substantial decrease in the interest rate environment after the financial crisis. A central bank undertakes monetary policy, such as lowering and raising interest rates, which is central in controlling the economic growth of a country. In response to the financial crisis, a number of central banks lowered the interest rate, to reduce the cost of borrowing in order to stimulate borrowing, investments and growth (Bauer and Neely, 2014).
Figure (3.1) Interest rate differentials
This figure presents the forward discount rate as a proxy for the interest rate differential of the ten different currencies against the USD in the period from 30-06-1999 to 29-05-2020. The horizontal line indicates where the interest rate
differentials are equal to zero.
As seen in Figure 3.1 there is still a clear spread between the high- and low- interest rate currencies in the years after the crisis. However, in recent years, the interest rates differentials rather fluctuate together towards zero. Conse- quently, this decreased the interest buffer to protect against adverse currency movements and a, therefore, higher risk of incurring losses when following the carry trade strategy.
3.3 Volatility indices
As described in Section 2.3, VIX is among the leading sources of uncertainty in this line of literature, despite it being based on the S&P500 index. Therefore, it is not necessarily an accurate estimator of risk in currency markets. For this reason, the JP Morgan G7 Volatility Index (VXY) is used as an additional crisis indicator. Among several other papers, Koch (2014) and Egbers and Swinkels (2015) use VXY as a proxy for foreign exchange risk. Contrary to VIX, this index expands the geographical coverage of risk proxy beyond the US, as it is based on seven different currencies (Koch, 2014). The index is measured by the aggregated volatility through a turnover-weighted index of the currencies of three-month at-the-money options. Daily data of both indices in the period
two indices, the definition of crisis periods solely rely on market data.
Figure (3.2) VIX and VXY implied volatility levels
This figure presents the available daily implied volatility levels of the indices, VIX CBOE volatility index and JPMorgan’s VXY index between the 02-01-1997 and
29-05-2020.
Figure 3.2 presents the evolution of the two volatility indices. From visual inspection, it seems that the implied volatility of the indices tends to cluster together. Volatility clustering is a well-known phenomenon in which prices in financial markets tend to exhibit periods of substantial volatility followed by periods of relative tranquillity as first noted by Mandelbrot (1963). In November 2008, VIX shot up to a closing level of 80.86 as a consequence of the Lehman collapse during the financial crisis. The level is lower on all other days during the sample periods, except for the recent COVID-19 crisis where VIX reached its highest closing level of 82.69 in March 2020. VXY reached its highest level of 24.46 in October 2008 but did not get any higher than 15.01 during the COVID-19 crisis. It seems both indices peaked during the known crisis periods that occurred in the past. Although the indices seem to capture some of the same crisis periods, the VIX index is not necessarily representative of the risk currency markets. In fact, the correlation between the two indices is 70.85%1, indicating some differences in the perception of risk between the two markets. The construction of the signals is described in Section 4.3.
1The value is based on the Pearson correlation coefficient of daily data between 02-01- 1997 and 29-05-2020, on dates where the price was available for both VIX and VXY indices
Methodology
In Chapter 4 all the methods and models are described. The first sections describe the methodologies used to construct the carry trade portfolios and the intuition behind the return calculations. The following sections focus on the methodologies used to construct the crisis robust carry trade, including the trading signal and the alternative strategies used in crisis periods. The final sections describe the performance measures used in this thesis as well as the statistical tests.
4.1 Portfolio construction
To get a broader picture of the carry trade performance, different model se- lections will be considered in this thesis. To start with, four groups of carry trade strategies are constructed according to the rebalancing horizon: 1-,3-,6- and 12 months. These tenors are frequently used when quoting FX forwards.
At the end of each rebalancing periodt, the currencies are reallocated into two portfolios based on their interest rate differential. In this thesis, the analysis is conducted using the natural logarithm of the forward and spot prices. The forward discount (i.e. the interest rate differentials) in Equation 2.5 is then expressed as
F Dt=ln(Ft)−ln(St). (4.1)
Based on the methodologies used by Menkhoff et al. (2012) and Taylor and Wang (2019), the ten currencies described in Section 3.1 are ranked from high- est to the lowest forward discount rate and sorted into portfolios with equal weight. The ”high” portfolio consists of currencies with the highest forward
discount rate (positive interest rate differential), and the ”low” portfolio con- sists of currencies with the lowest forward discount rate (negative interest rate differential). Carry trade is often referred to as the excess return of a portfolio with extremely high- and low-interest rates. This approach has its disadvan- tage, as the carry trade strategy only includes information on the highest and lowest interest rate currencies, consequently excluding relevant information from the remaining currencies with moderate interest rate differentials. Figure 3.1 showed a change in the interest rate environment after the financial crisis, where interest rate differentials fluctuated towards zero. The gap between ex- tremely positive and extreme negative interest rate differential has diminished over the recent years, making it more difficult to distinguish between the two categories. In addition, Burnside et al. (2008) found that diversified portfolios of multiple carry trades lead to a significant risk reduction. To consider this, five different strategies are constructed for each of the four rebalancing hori- zons, such that the total number of carry trade strategies amounts to 20. The first strategy for each rebalancing horizon can hold up to 1 currency in the high and low portfolio, whereas the remaining can hold up to 2-, 3-, 4- and 5 currencies respectively.
In Figure 3.1 it can be seen that the majority of the forward discounts rates are positive in some periods, whereas in other periods all forward discount rates are negative. Consequently, the number of currencies allocated to each of the portfolios will be lower (or even zero) than the number of currencies the portfolio can hold. The actual number of currencies allocated to the high (low) portfolio, is therefore highly depended on the number of positive (negative) in- terest rate differentials at each rebalancing period. In Table 3.1, shown in the previous data section, NZD, AUD, and NOK displayed the highest monthly forward discount rates. Hence, these currencies should, for the most time, be allocated to the high portfolio, whereas currencies such as JPY, CHF and EUR for the most time, should be allocated to the low portfolio.
In Table 4.1 below, the correlation for the return of the ten currency pairs is reported. The carry returns are calculated based on the methodology de- scribed in the following Section 4.2. Evaluating Table 4.1, a larger proportion of the currency pairs exhibit low correlation and a few exhibit notably high correlation. Among those with high correlation are AUD with NZD and DKK with EUR and SEK. The high correlation between AUD and NZD can be ex- plained by the closely linked economies, as Australia is New Zealand’s biggest trading partner. The high correlation between SEK, DKK and EUR might similarly be due to the large volume of mutual trading, which increases the de-
pendencies across the countries. In addition, is the DKK pegged to the EUR, which might explain why the two currencies exhibit the highest correlation of 92%. The actual impact of the correlation is expected to be low, for the portfolio including all or most currencies, as the correlation of returns is low for the majority of the currencies. Portfolios containing fewer currencies may be more exposed to currency-specific risk if the currencies included have a high correlation.
Table (4.1) Correlation matrix for the ten currency pairs This table reports the Pearson correlation coefficients of the ten different currencies
used in this thesis. The results are based on the monthly prices available in the sample period between 30-06-1999 and 29-05-2020. Correlation coefficients higher
than the absolute value of 60% are marked by grey coloured cells.
AUD CAD CHF DKK EUR GBP JPY NOK NZD SEK AUD 100%
CAD 39% 100%
CHF -43% -12% 100%
DKK 21% 14% -6% 100%
EUR 20% 18% -5% 92% 100%
GBP 36% 33% -17% 27% 30% 100%
JPY -13% 4% 35% -4% -2% -10% 100%
NOK 48% 38% -33% 25% 28% 35% 6% 100%
NZD 78% 28% -46% 17% 17% 31% -13% 35% 100%
SEK 24% 37% -11% 63% 69% 41% 9% 44% 27% 100%
Even though diversification reduces some currency-specific risk, carry trades are known to being subject to crash risk. Brunnermeier et al. (2008) found no evidence that the negative skewness or excess kurtosis gets diversified away, as more currencies are included in the portfolio using an equal-weighted portfolio strategy. In Figure 3.1 it can also be seen that all interest rate differential dropped sharply between 2008 and 2009, which exemplify the crash risk of carry trade. In the latter sections alternative strategies are introduced, as an attempt to accommodate the crash-risk observed for carry trade.
4.2 Carry trade returns
Currency excess returns
As stated earlier, this thesis uses natural log values and thus the portfolio excess returns from buying foreign currency in the forward market at time t and selling it in the spot market at time t+ 1, is calculated as
whereft denotes the natural log of the forward exchange rate, and st denotes the natural log of the spot exchange rate both in units of foreign currency per US dollar. This corresponds to the natural log forward discount minus the change in the spot rate: Rt+1 =ft−st−∆st+1. An increase in ∆st+1 therefore reflects an appreciation of the domestic currency. As described in Section 2.2, the forward discount is equal to the interest rate differential: ft−st=ift −idt under normal conditions. Hence, the natural log currency excess return of Equation 2.4, approximately equals the interest rate differential minus the rate of depreciation:
Rt+1 ≈ift −idt −∆st+1. (4.3) Transaction cost
In order to estimate the actual realised excess return of the portfolio, transac- tion costs are taken into account by incurring bid and ask quotes on the spot the same way as in Menkhoff et al. (2012). The monthly net currency excess return for an investor that enters a long position in the foreign currency is computed as
Rlongt+1 =ftbid−saskt+1. (4.4) The first part of the equation reflects the investor buying the foreign currency, which is equivalent to selling a dollar forward at the bid price ftbid at time t.
The second part reflects the investor selling the foreign currency in the spot market in month t+ 1, which is equivalent to buying dollars at the ask price saskt+1. The return thereby corresponds to borrowing the domestic currency, then exchange for and hold the foreign currency and then exchange it back to the domestic currency period att+ 1. Recall that the currencies are allocated into a ”high” and ”low” portfolio at each time step t. If a currency stays in the ”high” portfolio” in both timet and t+ 1, the return is instead calculated using the mid price
Rlongt+1 =ftbid−smidt+1. (4.5) For each currency in the ”low” portfolio, the net natural log excess return in periodt+ 1 is similarly calculated as
Rshortt+1 =−ftask +sbidt+1. (4.6)
In this case, traders borrow the foreign currency, then exchange for and hold
the domestic currency and then exchange back to the foreign currency. As for the currencies in the ”high” portfolio, the return for currencies which stays in the ”low” portfolio, in the subsequent time step, is calculated as
Rshortt+1 =−ftmid+sbidt+1. (4.7)
After calculating the return of the currencies in the high and low portfolio, the carry trade portfolio is found by taking the equally-weighted average of all the high and low currencies in period t+ 1. By taking the equally-weighted average, the construction of the carry trade portfolio follows the methodology used in the majority of existing literature.
Return series
Using the formulas above result in sample observations of 252, 84, 42, 21 for the 1-, 3-, 6- and 12- month rebalancing periods respectively. The small number of observations for 3-, 6- and 12- month rebalance periods may lead to insufficient test statistics. To accommodate this concern, the length of the return series is extended by calculating the monthly returns between each rebalance period.
The total return at each rebalance period can be described as
k
X
i=1
Rlongt+i =
k
X
i=1
(ft+i−1,k+1−i−ft+i,k−i) =ft,k−ft+k,0 =ft,k−st+k, (4.8)
where ft,k corresponds to ftbid, and ft+k,0 or st+k corresponds to saskt+1 or smidt+1 depending on whether the currency remains in the ”high” portfolio.
Similarly, the total return at each rebalance period in the ”low” portfolios can be described as
k
X
i=1
Rshortt+i =
k
X
i=1
(−ft+i−1,k+1−i+ft+i,k−i) =−ft,k+ft+k,0 =−ft,k+st+k, (4.9)
where ft,k corresponds to ftbid, and ft+k,0 or st+k corresponds to smidt+1 or sbidt+1 depending on whether the currency remains in ”low” portfolio.
To calculate the monthly return, the monthly forward rates are therefore lin- early interpolated1 to obtain all monthly data points for 1- to 12- month for- ward rates between 05-28-1999 and 05-29-2020, in accordance with Equation
4.8 and 4.9.
4.3 Trading signals
The poor performance of the carry trade during crisis periods described in Section 2.3, suggests an improvement of the carry trade during stressed periods.
Hence, this section presents methods using implied volatility of the equity and currency market to implement dynamic strategies, from which the investment decisions rely on. To identify the crisis periods, the two indices VIX and VXY will be used respectively as timing indicators. This thesis uses the level of VIX and VXY in order to calculate the signals shown in the formula
signalt=
(1 if indext−1 ≤thresholdt−1
0 if indext−1 > thresholdt−1, (4.10)
where thet-subscript indicates that the signal depends on the closing value in the previous periodt−1. The signals are based solely on available information up until the day the signal is triggered, to avoid the possibility of look-ahead bias in the models. Moreover, the signals are based on information from a rolling window over time, in order to include the latest information in each calculation of the signal. For instance, what felt like a blip during the financial crisis, may have felt like a jolt during calmer periods. During the crisis period, the VIX went up to a closing level of 82.69, which had never happened before 2008, as seen in Figure 3.2. Therefore, what carry traders experience as a spike is presumable both level- and context-dependent.
4.3.1 Threshold
The threshold follows the methodology used by Briere and Drut (2009), Dunis and Miao (2007) and Egbers and Swinkels (2015), where the crisis periods are identified as periods where the index is more than a certain standard devia- tion greater than its historical mean. The larger the difference between the closing index level and the mean of the closing levels, the higher the standard deviation. The moving standard deviation at time k can mathematically be expressed as
ˆ
σindexk = v u u u t
k
P
t=k−m+1
(indext−index)2
m−1 ∀k=1,...,n, (4.11)
where indext denote the implied volatility level, ¯R denote the average of all the implied volatility levels in the window andm denote the window size.
The standard deviation used in previous literature varies between 1 and 2.
Under the assumption of normally distributed returns, a standard deviation of 2 corresponds to approximately 97.5 percentile, so the threshold represents the highly extreme volatile periods, which occur only 2.5% of the time. Hence, with a stand deviation of 2, it is possible to identify deviations vast beyond its normal. On the other hand, with a standard deviation of 1, the volatile periods are defined such that they occur approximately 16% of the time. In order to avoid exclusions of material crises, but at the same time avoiding noise, the crisis periods are defined as periods in which the index is more than 1.5 standard deviations greater than its moving average.
The window size varies greatly in previous literature. Egbers and Swinkels (2015) use an expanding rolling window, whereas Briere and Drut (2009) use the entire sample period. Both papers apply long periods, but using the whole period would lead to look-ahead bias, as the index level would not be known to the traders. In addition, it is important to take into account that spikes might be both level- and context-dependent. The rolling window is therefore based on observations over a shorter period of 1.5 years. As the average number of trading days per year over the sample period of VIX is 2532, 1.5 years correspond to 378 days.
4.4 Crisis robust strategy
The volatility signals are used to generate two dynamic strategies, namely the exit strategy, where all carry trade positions are closed in volatile periods and the ”reverse” carry trade strategy, where the static carry trade is reversed if the current volatility is higher than the thresholds (Dunis and Miao, 2007).
The profitability of the carry trade depends on the corresponding interest rate differential. A passive trading strategy like the carry trade has its downsides by ignoring relevant market information. Market players observe market move- ments closely and trade actively upon the market information. These dynamic
2253 is the conventional number of trading days
strategies enable investigation of the ability to improve upon the static carry trade using timing signals.
Exit strategy
Since carry trade tends to perform poorly under high volatility market condi- tions, Dunis and Miao (2007) use a filter to exit the market when the volatility is higher than the threshold. There are both advantages and disadvantages of such a strategy, as investors can miss out on the market recovery by go- ing cash in times of high volatility. This type of strategy profits from the potential strong performance of carry trade in calm periods and cut losses in volatile market periods. Essentially, this is beneficial for traders being under performance pressure. The trading strategy can be expressed by
Rt+1exit=
Rcarryt+1 if signalt = 1 0 if signalt = 0,
(4.12)
This means that if the volatility index is lower than the threshold, the trader should stick to the carry trade, and if the volatility index is higher than the threshold, the trader should sell her funds. Every time the dynamic strategy switch between no trade and carry trade, the transaction costs are incorpo- rated using Equation 4.4 and Equation 4.6 for the high- and low-interest rate portfolio.
Reverse carry trade
The main advantage of the no-trading strategy is its simplicity. However, many other alternative strategies are possible. Roche and Rockinger (2003) proposed to reverse trading signals under high volatility periods and was later employed on carry trade by Dunis and Miao (2007). As opposed to the carry trade, the investors short the high-interest rate currency and buys the low-interest rate currency in reserve carry trade strategy. Formally this strategy can be written as
Rreverset+1 =
Rt+1carry if signalt = 1
−Rcarryt+1 if signalt = 0,
(4.13)
where Rcarryt+1 denote the carry trade return and −Rcarryt+1 denote the reversed carry trade return. The reverse carry trade is a more aggressive timing strategy compared to the exit strategy. Instead of just avoiding losses by closing the
positions, this strategy is designed to benefit from the negative return of the carry trade, by taking the opposite positions. Therefore, this strategy requires being especially proficient in predicting periods of negative returns.
4.5 Performance Measures
Profitability measures can be obtained by various approaches and are used to evaluate the carry trade performance. The performance measures are com- puted using historic data and are assumed to have some predictive ability.
This section introduces the two well-known performance measures, mean re- turn and Sharpe Ratio. Given a sample of historical returns, the mean excess return of a strategy is calculated using the formula
R¯= 1 T
T
X
t=1
Rt, (4.14)
where T is the total number of observations and Rt = R1, R2, ..., RT is the historical excess return of the strategy. Both the mean return and the Sharpe ratio are specified in annualised measurements. A common practice is to mul- tiply the mean return by the number of periods that makes up one year. In this thesis, the returns are based on monthly observations. Thus, the annualised mean return is obtained by multiplying by 12. The annualised performance measures enable a comparison of findings within this thesis, as well as with previous studies.
The Sharpe Ratio was first introduced by William F. Sharpe in 1966 and is widely accepted across academia. The performance measure estimates the performance of an investment compared to a risk-free investment after adjust- ing for risk (Sharpe, 1994). Formally the Sharpe ratio is the average excess return per unit of risk and is calculated as
SR= R¯
ˆ
σ, (4.15)
where ¯R denote the mean excess return of the strategy and ˆσ denote standard deviation returns defined as
ˆ σ =
s PT
t=1(Rt−R)¯ 2
N −1 . (4.16)
In order to get an annualised Sharpe ratio, the standard deviation is mul- tiplied by √
12, and mean return found in Equation 4.14 is multiplied by 12.
The Sharpe Ratio is defined as scale-independent, since changes in the notional value simply multiply the mean and standard deviation by a common factor, leaving the Sharpe Ratio unaffected. The greater the Sharpe ratio is, the bet- ter the risk-adjusted performance of the investment is. On the other hand, a negative Sharpe Ratio can mean one of two things: The risk-free rate is greater than the investment return or, the return is expected to be negative. Thus one cannot definitely deduce any meaningful information in such case. Modern portfolio theory claims that adding assets to a portfolio reduces portfolio risk without lowering the return. Diversified portfolios have, in that sense lower Sharpe ratios compared to less diversified portfolios. In general, high Sharpe ratios were documented for currency carry trades by Nelly and Weller (2013).
Additionally, studies have found that carry trade yields a Sharpe ratio that is as high as those obtained through the equity market (NBIM, 2014). The calculation of the Sharpe ratio relies on the assumption that the risk is equal to the standard deviation. However, financial returns are, in most cases, skewed away from the average. Moreover, the standard deviation assumes that price movements in either direction have equal weights, even though most financial data are skewed. Despite the potential implication of the returns being nor- mally distributed, the Share ratio is used in this thesis, in order to compare the profitability of carry trade with previous findings.
4.6 Hypothesis testing
The performance measures are evaluated statistically using the z-score from a simple individual test, in order to compare to prior studies. However, time- series of returns are often characterised by volatility clustering and excess kur- tosis. Consequently, statistical tests based on the assumption of independence and normally distributed properties may be inadequate. As we shall later see, the carry trade returns are no exception. To overcome this and enable more adequate inferences, the stationary bootstrapping of Politis and Romano (1994) is applied. The methodologies used to evaluate the performance of the returns are described in more details in the remainder of this section.
4.6.1 Nominal individual test
For a given strategy, the performance measures are tested individually for which ¯R = 0, by considering the following two-sided hypothesis
H0 : ¯R= 0 vs. H1 : ¯R6= 0,
where ¯R denote the mean return. At a given significance level, α, the null hypothesis is either rejected or not. To evaluate the hypothesis, the Z-score that follows a standard normal distribution is used. Using the notation from the null hypothesis, the Z-score is given as
Z =
R¯−0
pσˆ2/n ∼aN(0,1), (4.17) where n is the sample size and ˆσ is an estimator of the standard deviation, which corresponds to Equation 4.16. Applying the central limit theorem, the distribution will approximately be normally distributed as n → ∞.3 The p- value is the probability that a realisation from the standard normal distribution will be more extreme than the realised Z-score, given that the null hypothesis is true. The p-value is compared to a significance level of 5% throughout the thesis and based on the following formula
p-value =P(Z <− |z| |H0) +P(Z >|z| |H0) = 2(1−Φ(|z|)), (4.18) wherez is the realisation of theZ-score and Φ(·) is the cumulative distribution function (cdf) for the standard normal distribution.
4.6.2 Bootstrapping
The characteristics of the returns are described in the later Section 5.1.1. In the section, the returns indicate fat-tailed and left-skewed distributions, sug- gesting lack of normality. The bootstrapping method is therefore applied, to enable more adequate inferences when analysing the carry trade returns. Knsch (1989) proposed the block bootstrap method, which maintains the dependency structure that is prevalent in the carry trade returns. This is specifically done
3In contrast to the Z-score, the t-test assumes that the mean return of the sample is normally distributed. To circumvent this assumption, theZ-score is the preferred statistical measure.
by dividing the data into overlapping blocks of equal length and resampling the respective blocks of the original data. However, this method does not ensure stationary resamples, even if the original series is stationary. In Section 5.1.1, the obtained autocorrelation in the returns implies that the carry returns are in fact stationary, meaning that the block bootstrap is not applicable to the data.
Politis and Romano (1994) suggested an alteration of the block bootstrap in order to tackle the problem of non-stationarity, known as the stationary bootstrap. Instead of using blocks with a constant length, the stationary bootstrap combines blocks of random length, where the length of each block is approximated by the geometrical distribution with parameter q ∈ [0,1]. The attempt of this modification is to retain the stationary property of the original series in the resampled pseudo-time series. As the block size have an average length of 1/q, it is closely related to the stationary bootstrap parameterq. The block size, therefore, plays an important role, in the way that if it is too small (andq is too large), the dependency of the returns series may not be captured sufficiently. On the other hand, if its too large (andqis too small), the number of resamples may not be large enough in regards to the obtaining a finite sample distribution (Shao and Politis, 2018). Below, the actual construction of the stationary bootstrap is presented. Using the notation of Hansen (2005), the pseudo-time series are denotedd∗b,t ≡dτb,t, where each time series, b= 1, ..., B are resamples ofdt.
To implement the stationary bootstrap4,B resamples from two random B×n matrices, Uand V are constructed. The two matrices consist respectively of the elements, ub,t and vb,t, that are independently and uniformly distributed on [0,1] such that
ub,t, vb,t∼ U[0,1]. (4.19) The first element of each resample is defined by the first element ofU, such that τ{b,1}=dnub,1e. Here d·e denotes the ”ceiling” operator to the nearest integer.
The remaining elements of each resample can then be generated based on a recursive procedure, which depends on the probability parameter q
τb,t=
dnub,te if vb,t< q 1{τb,t−1<n}τb,t−1 + 1 if vb,t≥q.
(4.20)
This procedure continues untilnobservations are reached within the resample.
4Thearch package by Kevin Sheppard is used to apply the stationary bootstrap
