The effect of counterparty risk and systemic risk on the currency carry trade

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The effect of counterparty risk and systemic

risk on the currency carry trade

Author:

Merel van Vendeloo 10548203

Supervisor: prof. dr. C.G.H. (Cees) Diks Second reader: dr. A.C. (Andreas) Rapp

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Statement of Originality

This document is written by Merel van Vendeloo who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is 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.

Acknowledgement

I would like to thank my supervisor Cees Diks for his guidance during the writing of this thesis. Furthermore, I would like to thank all my colleagues at EY for their support. Also, I would like to thank Niels van der Kleij in particular for his time and effort.

Abstract

This study investigates the effect of counterparty risk and systemic risk on the returns of the covered and uncovered carry trade. To investigate the effect of these risk factors on the returns of the currency carry trade, risk proxies are added to the model. The investigated carry trade combinations are made based on their interest rates and the three combinations are the New-Zealand Dollar funded by the Swiss Franc, the Australian Dollar funded by the Japanese Yen and the Norwegian Krone funded by the Singapore Dollar. The returns are estimated with the use of a univariate and a multivariate model. In all cases, several risk proxies have a significant effect on the returns, but the carry trade returns could not be explained solely based on these risk factors. To determine if one model is better at capturing the characteristics of the returns of the carry trade, a forecast comparison is conducted. In this out-of-sample forecast, a univariate model and a multivariate model are compared to each other and to a random walk. The main finding is that the univariate model and the multivariate model are both outperformed by the naive model, which indicates that the random walk is the best predictor of the returns of the carry trade.

Keywords: GARCH, EGARCH, GJR-GARCH, VAR, DCC, forecast, interest rate parity, carry trade, foreign exchange, interest rates

iii

Contents

Abstract ii

1 Introduction 1

2 The FX and the interest rate market 3

2.1 The Interest Rate Parity . . . 3

2.1.1 Covered Interest Rate Parity . . . 3

2.1.2 Uncovered Interest Rate Parity . . . 4

2.2 Forward Premium Puzzle . . . 5

2.3 The currency carry trade . . . 6

2.3.1 The covered currency carry trade . . . 6

2.3.2 The uncovered currency carry trade . . . 7

2.4 The GARCH model . . . 8

2.5 Risk Premium . . . 10

3 Models and methodology 12 3.1 Interest rate parities . . . 12

3.2 Currency carry trade returns . . . 13

3.2.1 Univariate model . . . 13 3.2.2 Multivariate model . . . 15 3.2.3 Forecasting . . . 17 4 Data 19 4.1 Exchange rates . . . 19 4.2 Interest rates . . . 21 4.3 Explanatory variables . . . 23 4.3.1 Systemic risk . . . 23 4.3.2 Counterparty risk . . . 24 5 Empirical analysis 27 5.1 Interest rate parities . . . 27

5.2 Returns . . . 27

5.2.1 Currency returns . . . 29

5.2.2 Currency carry trade . . . 37

5.2.3 Multivariate currency carry trade . . . 46

6 Summary & Conclusions 54

Bibliography 57

A Data 60

1

Chapter 1

Introduction

Since a few years, the interest rates in the Eurozone have reached an all time low. With the recent developments of international financial markets, in particular the integration of the financial markets, the links between the foreign exchange market and the interest rate market have become stronger. Therefore, the amount of research done on the difference between interest rates has increased in recent years (Borio et al., 2016, Du, Tepper, & Verdelhan, 2018, Chang & Schlögl, 2012), which indicates that the interest parity has become a vital consideration for investors.

The profit that can be made by exploiting the difference between interest rates is called a currency carry trade. The existence of a profit indicates a failure of the interest rate parities, since the parity represents a no-arbitrage condition in which investors are indifferent between two interest rates. The interest rate parity has two distinctive forms, namely the uncovered interest rate parity and the covered interest rate parity. The difference lies in the fact that the uncovered interest rate parity does not cover the foreign exchange risk by the use of a forward contract.

The uncovered interest rate parity has failed for over 30 years (Hansen & Hodrick, 1980) and in the literature the failure of this parity is referred to as the "Forward Premium Puzzle" (Fama, 1984). It is found that the forward rate is not only a biased estimator of the future spot rate, but often a negative correlation exist between the two, which leads to a puzzle (Li, Ghoshray, & Morley, 2012). After the Great Financial Crisis in 2007, the deviations of the covered interest rate parity have persisted and have become more significant (Du, Tepper, & Verdelhan, 2018). These deviations can be seen as one of the most significant developments in global financial markets in recent years. Therefore, it is interesting to investigate the returns that can be made by exploiting both the parities and try to understand why the parities do not hold.

This thesis aims at understanding the failure of the uncovered and covered interest rate parity by investigating whether the currency carry trade returns can be described by counterparty risk and systemic risk. The first research question can be stated as: "To what extent can the returns, that can be made from exercising a currency carry trade, be explained in terms of counterparty risk and systemic risk?". To answer this question, a univariate and multivariate model for the returns are estimated and several risk proxies are added to these models to represent the different risk factors. To represent systemic risk, the VIX, MSCI index and the TED spread are added and counterparty risk is represented by the LIBOR-OIS spread for several currencies. The univariate model estimated, is the Generalised

Autoregressive Conditional Heteroskedasticity (GARCH) model proposed by Bollerslev (1986) and the multivariate model consists of the Vector Autoregressive (VAR) model for the mean and the Dynamic Conditional Correlation (DCC) model proposed by Engle (2002) for the variance and conditional correlation.

To compare the univariate model to the multivariate model and to determine which model best describes the currency carry trade returns, a forecast exercise is conducted and this leads to the second research question, namely: "To what extent does the multivariate model outperform the univariate model when forecasting the profitability of currency carry trade?". To answer the second research question, part of the data set is used to estimate the model and the remaining part is used to forecast and to compare the forecast to the actual time series. To evaluate the forecast, the Root Mean Squared Error (RMSE) and the Mean Absolute Error (MAE) are used and the univariate and multivariate models are compared with each other to investigate whether they differ significantly from one another by a modified version of the Diebold-Mariano test proposed by Harvey, Leybourne, & Newbold (1997).

The value-added of this study to the already existing literature lies in the focus of the research. The focus is on the time series characteristics of the profitability of the uncovered and the covered carry trade, instead of directly explaining the forward premium puzzle by investigating the interest rate parities. Another difference is the addition of counterparty risk and systemic risk to the model to explain the returns. Counterparty risk and systemic risk are two risk factors that have increased in importance after the Great Financial Crisis in 2007. Also, in existing research, the covered interest rate parity is often underexposed compared to the uncovered interest rate parity, since its failure is rather recent. Therefore, this study leads to some new insights regarding the failure of the covered interest rate parity.

This study is organized as follows. In Chapter 2, a theoretical background is given on the interest rate parities and their empirical failure. Chapter 3 introduces the models that are used to model the results and in Chapter 4 an overview of the data used to answer the research questions is presented. In Chapter 5 the empirical analysis is described and an answer on the research questions is given, which is followed by a summary and conclusions in Chapter 6.

3

Chapter 2

The FX and the interest rate market

To understand the connection between the foreign exchange (FX) market and the interest rate market and to provide the theoretical background necessary to answer the research questions, a discussion of the extensive literature on both markets is given in this chapter. This chapter is structured as follows. First, an overview is presented of the different forms of the interest rate parity, followed by a summary of the literature on the failure of the uncovered interest rate parity, also known as the "forward premium puzzle". After that, an explanation on how to exploit the forward premium puzzle and how to make a profit of the interest rate differential is given. This is done by performing a so-called "currency carry trade". This chapter concludes by giving the solutions that several researchers have proposed to solve the puzzle, namely the use of a non-linear time series model and the addition of a time-varying risk premium.

2.1

The Interest Rate Parity

This section introduces the covered interest rate parity and uncovered interest rate parity and discusses their role in the international financial markets. The interest rate parity comprises a no-arbitrage condition between the interest rate market and the foreign exchange market. The interest rate parity establishes an equilibrium in which investors will be indifferent between two different interest rates in two different countries. If the interest rate parities holds, one can state that the interest rates are the main drivers of the foreign exchange market and that therefore the theorem gives an explanation of the value and movements of exchange rates (Suranovic, 2012). Hence, the parity links the foreign exchange market to the interest rate market.

The interest rate parity can be divided into two different forms, namely the covered interest rate parity and the uncovered interest rate parity. The uncovered interest rate parity refers to the parity condition where the investor is subject to foreign exchange risk and the covered interest rate parity refers to the parity condition where investors hedge against foreign exchange risk by using a forward contract. First, the covered interest rate parity is explained in detail, followed by the uncovered interest rate parity.

2.1.1 Covered Interest Rate Parity

In this subsection, the covered interest rate parity is given and an explanation of its interpretation in finance is discussed. The covered interest rate parity (CIP) is a no-arbitrage

condition stating that two interest rates on two otherwise identical assets in two different currencies should be equal once the foreign exchange risk is hedged. The covered interest rate parity as stated by Borio et al. (2016) is given by

Fk,t

St

= 1 + if,k 1 + id,k

, (2.1)

with id,k the domestic interest rate with maturity of k and if,k the foreign interest rate with

maturity k, Stthe spot rate at time t and Fk,tthe forward exchange rate at time t with a maturity

k. The forward and spot rate are taken as the units of foreign currency per one unit of the domestic currency. When the natural logarithms are taken, the CIP can be rewritten as

fk,t− st= ln(1 + if,k) − ln(1 + id,k), (2.2)

with fk,tand stthe logarithms of the forward exchange rate and the spot rate.

Borio et al. (2016) state that the CIP is the closest thing to a physical law in international finance, but since the Global Financial Crisis (GFC) the CIP has failed to hold. The failure of the CIP is made visible by the occurrence of a cross-currency basis spread since 2007. A cross-currency basis spread arises when a cross-currency swap is not traded at par (Hull, 2012). A cross-currency swap is a contract that exchanges floating interest terms of two different currencies at each tenor of the swap term. By the no-arbitrage principle, these two floating rates must be traded at par, which will lead to a basis spread of zero. A violation of the CIP can be indicated by a non-zero spread, since it is more attractive to receive one currency over another. Before the financial crisis, the spread was sufficiently small. Therefore, arbitrage opportunities exploiting this spread are largely canceled out by transaction costs (Chang & Schlögl, 2012). If the CIP fails to hold, one could make a seemingly riskless profit. This riskless profit can be made by exercising a covered carry trade, which will be explained later on in this chapter.

2.1.2 Uncovered Interest Rate Parity

Next to the covered form of the interest parity, it is possible that the interest parity takes on its uncovered form. In the uncovered interest rate parity (UIP), no forwards are used to hedge positions and therefore the investors are subject to foreign exchange risk. The uncovered interest parity can be stated as

St+k

St

= 1 + if,k 1 + id,k

, (2.3)

with St+k the expected spot rate at time t + k, Stthe spot rate at time t and id,k the domestic

interest rate with a maturity of k and if,k the foreign interest rate with the same maturity k. If

the logarithms are taken, the UIP can be rewritten as

Chapter 2. The FX and the interest rate market 5

with st+k and st the natural logarithms of the spot rate. As can be seen from Equation (2.4),

the UIP states that the interest rate differential between two countries should be equal to the exchange rate change. In reality however, low interest rate currencies tend to depreciate relative to high interest rate currencies (Li et al., 2012). If the UIP and the CIP are combined, one finds that the forward exchange rate is an unbiased estimator of the future spot rate

st+k− st= fk,t− st. (2.5)

Extensive literature on the failure of Equation (2.5) can be found and is referred to as the UIP puzzle (Engel, 1996) or the Forward Premium Puzzle (Fama, 1984). In the next subsection, an overview will be given on prior research on this puzzle.

2.2

Forward Premium Puzzle

As already mentioned, the parities do not hold for all currencies in every time period when they are tested on empirical data. The failure of the UIP is also called the "Forward Premium Puzzle" (Fama, 1984). In this subsection a summary of the research done on the puzzle is be given.

The UIP puzzle is one of the most prominent puzzles in international economics and finance (Engel, 1996). The uncovered interest parity states that if the covered interest rate parity holds, both the forward discount and the interest rate differential should be an unbiased predictor of the ex post change in the spot rate if rational expectations are assumed. The puzzle beholds not only that the forward rate is an unbiased estimator for the future spot rate, but even that there is often a negative correlation between the forward rate and the future spot rate (Froot & Thaler, 1990). Multiple reasons have been given for the forward bias puzzle, such as the violation of the rational expectations hypothesis, which is a criterion for the interest parities to hold (Froot & Frankel, 1989; Bachetta & Van Wincoop, 2006), econometric issues (Baillie & Bollerslev, 2000; Maynard & Philips, 2001; Maynard, 2003) and an exchange risk premium interpretation (Engel, 1999; Moore & Roche, 2010). In the remainder of this section, the focus is on the issues of econometric implementation and the existence of an exchange risk premium.

For over 30 years, research has been done in the field of the uncovered interest rate parity. Hansen & Hodrick (1980) were among the first to investigate the failure of the UIP. Their research rejects the FX market efficiency hypothesis, in other words, they reject that speculations in the FX forward market should have zero returns. Meese & Rogoff (1983) conclude that exchange rates follow a "near random walk" and that differences between currencies can be exploited without suffering an exchange rate depreciation. It is called only a "near" random walk, because high-interest-bearing currencies even tend to appreciate.

Nowadays, researchers still investigate the forward premium puzzle. Chinn & Meredith (2004) find that at the short-run horizon, which means using interest rates with a maturity of 12 months or less, the UIP fails to hold for seven different currencies, but that the UIP recovers at long-run horizons. Chinn and Meredith find their solution in the interaction of random FX market shocks with monetary policy reactions that are endogenous. Lustig & Verdelhan (2007)

state that conditional on the foreign interest rates, the aggregate consumption growth risk can be used to explain a large part of the average changes in the exchange rate. They focus on the difference between the returns of high interest rate currencies and low interest rate currencies and conclude that the returns of the high interest rate currencies have a larger impact on the consumption growth risk. Burnside (2007) disagrees with Lustig and Verdelhan and states that in the research of Lustig and Verdelhan the covariance between the excess portfolio return and the stochastic discount factor is insignificant. Therefore, Burnside concludes that it cannot be stated that the consumption growth risk explains the variation in cross-sectional variation in the expected returns. Instead, Burnside et al. (2011) study the UIP puzzle by investigating the characteristics of carry trades. They find that the positive average payoff of a carry trade is uncorrelated with traditional risk factors and reflects a "peso problem", but that the payoff can not be explained by the "peso problem" alone. A peso problem refers to the occurrence of low-probability events that are not present in the sample, such as a potential world disaster. The presence of a peso problem makes it difficult to model the carry trade using only the past. Another explanation of the UIP puzzle, is that it is not caused by a time-varying risk premium, but by monetary policy. Park & Park (2016) state that if monetary policy acts harder on inflation, the relationship between exchange rates and interest rate differentials is more likely to become negative. They found that inflation-targeting economies with interest rates that have a smaller reaction with regarding to inflation after the crisis compared to before the crisis, have a positive coefficient in the UIP regression, while economies that do not have this reduction experience no sign change. A positive coefficient is desirable, since the coefficient is equal to one in the perfect scenario. Monetary policy could therefore play a role in the forward premium puzzle and in its solution.

The failure of the UIP and the existence of the forward premium puzzle have been an area of research for the last 30 years. Recent developments in the financial markets have had an impact on the CIP as well. Since the GFC, even the covered interest rate parity does not hold anymore (Borio et al., 2016). This is evident from the fact that the cross-currency basis spread differs from zero since 2007, as already briefly mentioned before. The failure of both interest rate parities can be exploited by making a profit. This is done by performing a "carry trade", which will be discussed in the next section.

2.3

The currency carry trade

The currency carry trade has an uncovered and a covered form, exploiting the uncovered and covered interest rate parity respectively. Next, a more in depth description of both carry trade strategies is given.

2.3.1 The covered currency carry trade

This subsection focuses on the covered currency carry trade. The covered currency carry trade exists by exploiting the CIP. The CIP hedges for foreign exchange risk and it can be stated that therefore the covered currency carry trade seems like an arbitrage opportunity. The CIP

Chapter 2. The FX and the interest rate market 7

fell apart during the GFC in 2007 and the failure of the CIP can be seen by the existence of a cross-currency basis spread, which started deviating from zero after 2007 (Du et al., 2017). The spread has not been closing since the crisis, which indicates that the opportunity for arbitrage is still present.

The covered currency carry trade return can be written as (Borio et al., 2016)

z_tc= ln(1 + if,k) − ln(1 + id,k) − (fk,t− st), (2.6)

with fk,tthe natural logarithm of the forward spot rate and stthe natural logarithm of the spot

rate both at time t, as the price, in foreign currency, of one unit of the domestic currency. This return can be seen as a long position in a foreign currency and a short position in a domestic currency, for example the USD, while hedging the foreign exchange risk with a forward at time t. Under the CIP, there must hold that

Et(ztc) = 0. (2.7)

As already mentioned, if the CIP holds, the cross-currency basis spread is zero. If the CIP fails, the cross-currency basis b can be represented as (Du, Tepper, & Verdelhan, 2018):

fk,t− st= ln(1 + if,k) − ln(1 + id,k+ b). (2.8)

Equation (2.8) shows that whenever the CIP fails, one party pays the currency basis on top of the cash market rates to borrow the corresponding currency, in this case the domestic currency, while the other party receives an equivalent discount when borrowing the other currency. This difference can be exploited by a covered currency carry trade. In a covered carry trade, the foreign exchange risk is hedged by the use of a forward contract. It is also possible not to hedge this risk by performing an uncovered carry trade. The uncovered carry trade is discussed in the next subsection.

2.3.2 The uncovered currency carry trade

The uncovered carry trade can be seen as selling low interest rate currencies, also known as funding currencies, and investing in high interest rate currencies, also known as investment currencies (Brunnermeier, Nagel, & Pedersen, 2008). If the UIP holds, the returns of this carry trade would on average be zero, since the carry gain due to the interest rate differential is offset by the depreciation of the investment currency. Brunnermeier et al. show that this is empirically not the case and that even the opposite holds, namely that the high interest rate currency appreciates instead of depreciates, which makes a carry trade profitable.

The uncovered carry trade return can be written as (Christiansen, Ranaldo, & Söderlind, 2011)

with st+kand stthe natural logarithm of the spot exchange rate as the price, in foreign currency,

of one unit of the domestic currency. This return can be seen as a long position in a foreign currency and a short position in a domestic currency. Under the UIP, there must hold that

Et(ztu) = 0. (2.10)

Therefore, it can be stated that zc

t can be seen as the abnormal return of a carry trade strategy

with the foreign currency as the investment currency and the domestic currency as the funding currency (Brunnermeier et al., 2008).

The uncovered carry trade is an exploitation of the failure of the UIP. As already mentioned, for over 30 years researchers have been concluding that the UIP does not hold empirically. For that reason, many different ways to model the uncovered carry trade returns have been suggested. Examples of models are the component GARCH-in-mean (Li, Ghoshray, & Morley, 2012), the regular GARCH-in-mean (Santos, Klotzle, & Figueiredo-Pinto, 2016) or a Vector Autoregressive (VAR) model (Jiang et al., 2013). These models are be explained more in depth later in this thesis.

The currency carry trade can be seen as an exploitation of the interest rate parity and is therefore a return on a foreign exchange currency. Theoretically the average return of a currency carry trade should be zero, but empirically positive payoffs are found to be significant. Two reasons for the average positive payoff have been proposed, namely a non-linear relationship between the interest rates and the spot rate differential and the existence of a time-varying risk premium. These two solutions are discussed in the last part of this chapter.

2.4

The GARCH model

In the previous sections, the foreign exchange market has been linked to the interest rate market. Also, it was shown that the failure of the UIP could be due to a non-linear relationship between the interest rate differential and spot rate differential. This section presents a model that is often used to model financial returns, since it takes time-varying volatility into account. Therefore, this model can be used to better understand the returns of the currency carry trade. The models that are taken into consideration are the Autoregressive Conditional Heteroskedasticity (ARCH) model presented by Engle (1982) and the Generalised Autoregressive Conditional Heteroskedasticity (GARCH) model presented by Bollerslev (1986). The ARCH model allows the conditional variance to change as a function of past errors, while leaving the unconditional variance constant. The GARCH model can be seen as an extension of the ARCH model, by allowing the conditional variance to change as a function of past conditional variances as well. Therefore, the generalised version corresponds to an ARMA model for the squared residuals instead of an AR model with error term η = a2

t − σ2t.

Chapter 2. The FX and the interest rate market 9 1986) at= rt− E(rt|Ft−1) = rt− µt, (2.11) at= σtεt, (2.12) σ_t2 = α0+ p X i=1 αia2t−i+ q X j=1 βjσt−j2 , (2.13) εt∼ i.i.d. N (0, 1) and Ft= {σt, σt−1, .., at, at−1, .., εt, εt−1, ..}. (2.14)

With µt the conditional mean of the returns, σt2 the conditional variance of the returns and

therefore of at.

The conditional mean of the returns can be written as an ARMA model, where the number of parameters is determined by tests on autocorrelation. The tests are performed on the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the financial time series. These functions can be used to test for correlation between lagged returns and volatility clustering. Volatility clustering represents the alternation of periods with high volatility with more tranquil periods (Tsay, 2010b). The GARCH model with an ARMA mean specification incorporates both properties, leading to residuals that behave like white noise.

One of the main characteristics of the variance is the fact that it must be positive and this property has to be taken into account when estimating a GARCH model. Several restrictions on the parameters will lead to a positive definite variance matrix. Firstly, some restrictions need to be imposed on the parameters. These restrictions state that all parameters, including the constant, need to be larger than zero. The restrictions are not necessary, but they do alleviate computational difficulty. Secondly, another restriction needs to be imposed on the parameters for the variance process to be stationary, namely that the summation of all parameters except the constant needs to be smaller than one. In other words;Pp

i=1αi+

Pq

j=1βj < 1. Thirdly, the

residuals need to behave like white noise, implying that E(at) = 0and cov(at, as) = 0, t 6= s

(Bollerslev, 1986, p. 310).

Bollerslev (1986) proposed to estimate the basic GARCH model with an ARMA mean process by using maximum likelihood. This method leads to consistent estimators of both the ARMA and GARCH parameters, if the right specification of the mean is used. If the normality assumption in Equation (2.14) is violated, robust standard errors can be used to consistently estimate the standard errors of the parameters. The procedure can be represented as

rt= µt(φ) + σt(φ, ψ)εt, (2.15) f (rt|Ft−1; φ, ψ) = 1 √ 2πσt(φ, ψ) exp  −(rt− µt(φ)) 2 2σ2_t(φ, ψ)  , (2.16) `(φ, ψ) = T X t=s+1  −1 2log(2π) − 1 2log(σ 2 t(φ, ψ)) − (rt− µt(φ)2 2σ2 t(φ, ψ)  . (2.17)

The maximum of the likelihood function is found by equating its first derivatives to zero. The GARCH model relies on some assumptions that are not realistic in practice. Two of

these assumptions are the symmetric News Impact Curve (NIC) and a Normal distribution of the error term. A number of extensions are discussed next that can circumvent these assumptions.

Firstly, the focus is on the NIC. The NIC displays the effect of past shocks on the volatility of the returns and therefore measures how new information is incorporated in the model. In the GARCH model, the NIC is assumed to be symmetric, indicating that positive and negative shocks have the same effect on the volatility (Engle & Ng, 1993). Asymmetry implies that positive and negative shocks have a different effect on the return volatility, which is often more realistic, since large negative shocks tend to increase volatility more than large positive shocks (Anatolyev & Petukhov, 2016). Models exist that take this asymmetry into account and two examples are the exponential GARCH (EGARCH) model proposed by Nelson (1991) and the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model proposed by Glosten, Jagannathan, & Runkle (1993). These models can be written as

log(σ_t2) = α0+ p

X

i=1

αi(θεt−i+ γ[|εt−i| − E(|εt−i|)]) + q X j=1 βjlog(σ2t−j), (EGARCH) (2.18) σ_t2 = α0+ p X i=1 (αi+ γi1{at−i<0})a 2 t−i+ q X j=1 βjσ2t−j. (GJR-GARCH) (2.19)

In these models, the terms εt−iand 1{at−i<0}are the cause of the asymmetry in the NIC. If

negative shocks increase the return volatility more than positive shocks, the parameters γ > 0 and θ < 0 need to hold.

Secondly, the distribution does not need to follow the Normal distribution with mean zero and variance σ2t. Klar, Lindner, & Meintanis (2012) state that often the error term of financial

returns displays skewness and has a kurtosis larger than four, indicating that the higher moments are not aligned with the Normal distribution. Therefore, some other distributions should be taken into account when modelling financial returns. Examples of distributions that allow for skewness and excess kurtosis are the Student-t distribution and the Generalised Error Distribution (GED).

In this section, the GARCH model is described as a method to model the returns of the currency carry trade. The use of a more advanced econometric model can help to understand the failure of the interest rate parities. Another solution that has been researched extensively, is the existence of a time-varying risk premium, which will be discussed in the next section.

2.5

Risk Premium

So far, it has been explained that the foreign exchange market and the interest rate market are linked with each other by the interest rate parities, but that empirically these parities do not hold, which led to a puzzle. Furthermore, an explanation has been given on how one can exploit these puzzles and how to make a profit out of them using a currency carry trade and

Chapter 2. The FX and the interest rate market 11

in the previous section a description of the GARCH model has been given, which can lead to a better insight in how the returns of the carry trade behave. In this section another solution of the puzzle is given, namely that the inclusion of a time-varying risk premium could lead to a solution of the puzzle.

Brunnermeier et al. (2008) study the behaviour of carry trades and conclude that the carry trade is subject to crash risk. They find that currency carry trades on average have a positive return and a negative skewness. Stated is that this positive return is a premium for providing liquidity and that the negative skewness is an indication of the crash risk of the currency. These currency crashes are linked to sudden unwindings of carry trades.

Another explanation for the performance of the carry trade, is the existence of the peso problem, which was briefly mentioned before. Fahri & Gabaix (2016) state that high interest rates in a risky country are a compensation for the risk of an exchange rate depreciation in a potential world disaster. However, the peso problem is not widely accepted as a solution to the puzzle, Burnside et al. (2011) state that there remains a substantial payoff in carry trade strategies where there is hedged against potential disaster with the use of options.

Coelho dos Santos et al. (2016) investigate the effect of a country specific risk on the uncovered interest rate parity, while using a CGARCH-M model. They made a distinction between time-varying risk premium and a constant risk premium and there has been concluded that for all investigated currencies, at least one of the two risk premia was present. Chang & Schlögl (2012) examine the effect of foreign exchange market liquidity risk and volatility on the returns of carry trades. In their research, the cross-currency basis spread is used as a proxy for market liquidity risk and was found to be significant in explaining the carry trade return.

Several risk premia have already been taken into account, but there are still some additions that can be investigated. Since the GFC, counterparty risk has been playing a larger role in the global markets. Simaitis et al. (2016) investigate the counterparty risk of FX derivatives, which indicates that this risk also plays a role in the foreign exchange market. Ji & In (2010) use the LIBOR-OIS spread as a proxy for counterparty risk between banks, since the LIBOR-OIS spread can be used as a financial stress measure in interbank markets. Another risk that has been receiving more attention since the GFC is systemic risk. Systemic risk is the risk of collapse of an entire financial system or entire financial market. In case of the foreign exchange market, it could be that there are spillover effects from the stock market or bond market that could lead to the inclusion of a risk premium in the carry trade. Christiansen et al. (2011) have also taken spillover effects from other markets into account by adding MSCI world index as a proxy in their estimation.

This chapter ended with two different solutions that have been given to the puzzle, namely the use of advanced time series models and the inclusion of a time-varying risk premium. In this thesis a combination between the latter is performed. In the next chapter, the methodology is explained.

Chapter 3

Models and methodology

So far, it has been stated that the interest rate parity does not hold in practice and that the difference can be exploited. In this chapter, the methodology and models that are used to describe the interest rate parity and its returns are described. First, a description of the model used to check if the interest rate parity holds, is given, followed by an overview of the models that are used to model the returns of the currency carry trade. Finally, the method used to conduct the forecast comparison is presented.

3.1

Interest rate parities

Before investigating the currency carry trade, it has to be tested whether the interest rate parity holds or not. If the parity does not hold, it is possible to exploit this parity and make a profit. The parities are estimated as follows using Ordinary Least Squares (OLS) in the regressions

st+k− st= αU IP + βU IP ln(1 + if,k) − ln(1 + id,k)
+ εt,U IP, (UIP) (3.1)

fk,t− st= αCIP + βCIP ln(1 + if,k) − ln(1 + id,k)
+ εt,CIP. (CIP) (3.2)

If α = 0 and β = 1, the covered and uncovered interest rate parity hold. Then, the forward price is an unbiased estimator of the future spot rate and the forward premium puzzle does not exist. To jointly test whether α = 0 and β = 1, a Wald test is performed. If α 6= 0, the interest rate parity does not hold, but there is a constant risk premium that has to be taken into account (Santos et al., 2016). So the null hypothesis of the Wald test of α = 0 and β = 1 need to hold for the forward rate to be an unbiased estimator of the future spot rate.

The Wald test tests whether coefficients of a model differ significantly from an imposed restriction. The test statistic follows a chi-squared distribution under the null hypothesis with the number of restrictions as the degrees of freedom (Cameron & Trivedi, 2005). In case of the interest rate parities, there are two restrictions, namely: α = 0 and β = 1, so the Wald statistic follows a chi-squared distribution with two degrees of freedom under the null hypothesis.

It is highly likely that the residuals of Equations (3.1) and (3.2) are autocorrelated. If this is the case, the standard errors of the coefficients are not consistent. For consistent standard errors, the heteroskedasticity and autocorrelation consistent (HAC) standard errors proposed by Newey & West (1987) need to be calculated. The HAC errors also need to be used in the calculation of the Wald statistic.

Chapter 3. Models and methodology 13

As mentioned in the previous chapter, if the interest rate parity does not hold, one has the opportunity to exploit these parities and make a profit. In the remainder of this chapter, an explanation of the method to model these returns is given.

3.2

Currency carry trade returns

If the Wald test rejects the null hypothesis, the interest rate parities do not hold and a return can be made from the interest rate differential. It is possible to view this in a univariate case and in a multivariate case. In the univariate case, only one combination of funding and investment currency is investigated, while in the multivariate case, a portfolio can be made of multiple funding and investment currencies. The GARCH model described in the previous chapter is applicable in the multivariate case as well, but the positive definiteness of the conditional variance matrix is harder to establish when the number of estimated parameters increases (Tsay, 2010c). Therefore, the conditional correlation is modelled in the multivariate analysis. First, the methodology of the univariate case is described, followed by the description of the extension to the multivariate case.

3.2.1 Univariate model

In the univariate case, a carry trade with only one funding currency and one investment currency is investigated. The carry trade returns are a financial time series and are therefore highly likely to exhibit volatility clustering and high correlation between lagged returns. A GARCH model captures these characteristics and is therefore a logical model to consider to model these returns. However, first it needs to be checked if the carry trade returns exhibit volatility clustering and correlation between lagged returns. In the following, it is made clear how to check if the returns have these characteristics.

Firstly, it is described how to check if there is high correlation between lagged returns. High correlation between lagged returns can be modelled by ARMA(p, q) model. The lags p and q can be determined by using the autocorrelation function (ACF) and partial autocorrelation function (PACF). The autocorrelation function and the partial autocorrelation function of order ` can be defined as (Tsay, 2010a, p.39 and p.46):

ρ` = Corr(xt, xt−`) =

Cov(xt, xt−`)

V ar(xt)

, (ACF) (3.3)

xt= φ0,`+ φ1,`xt−1+ ... + φ`,`xt−`+ e`t. (PACF) (3.4)

If the returns follow an AR(p) model, the PACF shows a cutoff point at lag p ( so φp,p = 0 )

and if the returns follow a MA(q) model, the ACF shows a cutoff point at lag q. For an ARMA model, both ACF and PACF show exponential decay. An ARMA model can be written as

zt= φ0+ p X i=1 φizt−i+ at− q X j=1 θjat−j. (3.5)

This model, in the GARCH framework also known as the mean equation, captures the correlation between lagged returns. If the residuals atof the mean equation behave like white

noise, that is, if the conditions Et(at) = 0, V ar(at) = σa2and cov(at, at−`) = 0are met, then the

mean model is correctly specified and all correlations between lagged returns are captured. In this study, external exogenous regressors are added to the ARMA model to investigate which risk premium proxies have a significant effect on the carry trade return. Also, an ARCH-in-mean term is added to investigate if a time-varying risk premium has an effect on the return. The estimated mean equation can be written as

zt= φ0+ p X i=1 φizt−i+ β0x + cσt2+ at− q X j=1 θjat−j. (3.6)

In this model x are external regressors that are added to the model to explain the returns and cσ2_t displays the ARCH-in-mean term.

Secondly, the volatility clustering has to be checked. If volatility clustering is present, the squared residuals from the mean equation, that is a2t, display correlation (Tsay, 2010b). This

correlation can be captured by a GARCH model. As already briefly mentioned, the GARCH model can be seen as an ARMA model for the squared residuals. Therefore, the order of the GARCH model can be determined in the same way as for the ARMA model, by looking at the ACF and PACF of the squared or absolute residuals.

The volatility might be clustered in a way that the effect of at on σ2t+1 is not symmetric

around at = 0, which means that positive shocks do not have the same effect on the future

volatility as negative shocks. In practice, it is more realistic to assume that large negative shocks, like a stock market crash, increase volatility more than a large positive shock. This principle is also known as "the leverage effect". This leverage effect can be captured by the EGARCH model proposed by Nelson (1991) or the GJR-GARCH model proposed by Glosten, Jagannathan, & Runkle (1993), as mentioned earlier. In the case of GARCH modelling, in the right specification, the residuals and squared residuals of the mean equation behave like white noise.

Wang, Chung, & Guo (2013) state that the natural outcome of the distribution of the carry trade is a skewed one, since the interest rate follows a truncated Normal distribution and exchange rates are overall normally distributed. So they investigate the Value-at-Risk properties of several skewed distributions. Therefore, several distributions are considered when modelling the covered and uncovered carry trade, namely the skew-Student-t distribution, the skew-Normal distribution, the Student-t distribution and the Normal distribution. The Student-t and the skew-Student-t distributions are chosen, because, overall, it can be stated that residuals obtained from a GARCH model are non-normal (Jondeau & Rockinger, 2006). It is seen as more realistic to allow for a fat-tail distribution, for instance the Student-t distribution as proposed by Bollerslev & Wooldridge (1992). The GARCH model with Student-t distributed residuals allow for time-varying mean and variance, but does not render higher moments time varying. Hansen (1994) was the first to introduce the skew Student-t distribution. This density generalises the general Student-t distribution, while maintaining the assumption of a zero mean and unit variance. The variant of the skew Student-t distribution

Chapter 3. Models and methodology 15

that is considered is written as (Fernandez & Steel, 1998, p.360) p(ε|γ) = 2 γ +_γ1 n fε γ 

I_[0,inf)() + f (γε)I_{(− inf,0)}(ε)o,

with f () the general Student-t distribution and p() the skew Student-t distribution, γ can be seen as the skew parameter and the distribution loses symmetry when γ 6= 1. The skew-Normal distribution is introduced by O’Hagan & Leonard (1976) and can be written as

p(x) = 2φ(x)Φ(αx).

with φ() the probability density function (PDF) of the Normal distribution, Φ() the cumulative distribution function (CDF) of the Normal distribution and α the skew parameter. If α = 0, the Normal distribution is recovered and no skewness is added to the distribution. If α > 0, the distribution is left skewed and if α < 0, the distribution is right skewed. Based on the Kolmogorov-Smirnov test discussed by Stephens (1974) and the Bayesian Information Criterion (BIC) discussed by Schwarz (1978), the decision is made which distribution is the best fit for the residuals of the carry trade model. The BIC used to chose the distribution of the model can be stated as

BIC = k · log(n) − 2log( ˆL), (3.7) with ˆLthe maximised value of the likelihood, n the number of observations and k the number of parameters estimated.

After the estimation of the models, the models need to be tested for stationarity. The ARMA model has to be checked on weak stationarity, because if the process is not stationary, it can be modelled as a random walk. A test that can be used to check this, is the Augmented Dickey Fuller (ADF) test, which has a null hypothesis of the process being a random walk and an alternative hypothesis of a stationary process (Tsay, 2010a). Not only the mean equation has to be checked, also the variance equation has to be tested. The condition that needs to hold for stationarity in the variance equation is:Pp

i=1αi+Pqj=1βj < 1. This can be tested using a Wald

test with the null hypothesis ofPp

i=1αi+

Pq

j=1βj = 1. If the null hypothesis is not rejected,

the possibility of a unit root cannot be rejected and the variance is persistent and possibly not mean reverting.

3.2.2 Multivariate model

After the univariate analysis of the currency carry trades, a portfolio is made out of three carry trade combinations and a multivariate analysis is conducted. To estimate this multivariate case, a combination of the Vector Autoregressive (VAR) model and the Dynamic Conditional Correlation (DCC) model of Engle (2002) is made. A discussion of these models is given in the remainder of this subsection.

Multivariate time series models can be used to investigate dynamic relationships between multiple variables, without imposing restrictions on the endogeneity or exogeneity of these

variables. VAR models form the leading class of multivariate time series models (Tsay, 2010c) and are often used for investigating the dynamic effect of shocks on future outcomes. The VAR(p) model with the addition of the explanatory variables is defined as

Zt= φ0+ p

X

`=1

Φ`Zt−`+ β0X + at, (3.8)

with ata multivariate white noise series, φ0a vector of intercepts and Φ`a coefficient matrix.

This model is estimated using OLS. The lag selection for the determination of p is based on the Bayesian Information Criterion (BIC) under the assumptions that the model errors are independent and identically distributed according to the Normal distribution. The number of lags p are chosen, such that

BIC(p) = c + log| ˆΣp| +

k2· p · log(T∗)

T∗ (3.9)

is minimised. In this equation, k are the number of elements in Xt, T∗ = T − pand ˆΣp is the

variance matrix of at. If the number of lags are determined, the stationarity conditions need

to be checked. If the model has a unit root, it can be reformulated as a Vector Error Correction model (VECM) and it could lead to cointegration. The Johansen cointegration test determines the cointegrating rank and therefore the degree of stationarity (Tsay, 2010c). This tests follows a Dickey-Fuller-type distribution, which depends on the cointegrating rank.

The VAR model estimates the mean of the multivariate time series. In the multivariate setting, it is possible to estimate the variance of the time series as well. Several properties are desirable when modelling the variance (Tsay, 2010c). For instance, the variance matrix Σtneeds

to be positive definite and the specification needs to be flexible enough to replicate properties such as time variation in correlation, volatility spillover effects, leverage effects and persistence in volatility and covariation. A model that captures a few of these properties is the Dynamic Conditional Correlation (DCC) model by Engle (2002). This model can be written as

Σt= DtρtDt, (3.10) ρij,t= qij,t √ qii,tqjj,t , (3.11) Qt= (1 − α1− β1) ¯Q + α1εt−1ε 0 t−1+ β1Qt−1, (3.12)

with ε = D_t−1at, ¯Qthe unconditional correlation matrix of ε and Qt−1the covariance matrix of

εt−1. Engle (2002) assumes normality to give rise to a likelihood function, but if this assumption

is violated, the Quasi Maximum-Likelihood (QML) properties still hold and the estimation is consistent.

The benefits of the DCC model is that it allows the correlation to vary over time. The model allows for Σtto be positive definite if ¯Qis positive definite and if the coefficients α1and β1are

not negative. Also, it can be stated that the model has mean-reverting correlations if α1+β1 < 1,

which can be tested using a Wald test with the null hypothesis α1+ β1 = 1. The DCC model has

Chapter 3. Models and methodology 17

be a downside if one wants to model portfolios, and another downside is that the DCC model does not allow for volatility spillovers.

When the multivariate currency carry trade is modelled, it can be compared with the univariate model to determine which model captures the properties of the carry trade best. This comparison is done with a forecast, which is explained in next subsection.

3.2.3 Forecasting

To compare the univariate with the multivariate model and to determine which model can be used best to estimate the currency carry trade, a forecast exercise is conducted. This section consists of an explanation on how the forecast is done and how it is determined which forecast outperforms the other.

The forecast method that is used, is the out-of-sample forecast. This beholds that part of the sample is used to estimate the model, namely the in-sample data set, and part of the sample is used to evaluate the forecast, the out-of-sample data set. Roughly 80% of the data set is used as the in-sample set and the remaining serves as a comparison of the forecast. The forecast horizon h is set at one, so at every observation only one period ahead is forecasted. The models "roll" through the out-of-sample data set one observation at a time and forecasting at each time h periods ahead. In other words, the coefficients of the model is re-estimated with one more observation added to the in-sample data set each time the forecast is conducted.

The errors of the out-of-sample forecast can be calculated and these errors determine which model outperforms the other. First, a loss function needs to be described to determine the losses associated with the forecast errors. The loss function has a few properties, namely that it takes the value zero if no error is made, is never negative and is increasing in size as the errors become larger in magnitude. The two loss functions considered are the absolute error g(eit) = |eit| and

the squared error: g(eit) = e2it. With the use of these loss functions, the Diebold-Mariano test

can be performed (Diebold & Mariano, 1995). This test investigates whether the loss differential dt = g(e1t) − g(e2t)is significantly different from zero. If the loss differential does not differ

significantly from zero, the two forecasts have equal accuracy. To test whether this is the case or not, the sample mean loss differential ¯dt = _T1 PT_t=1g(e1t) − g(e2t) is calculated. The null

hypothesis can be stated as H0 : E( ¯dt) = 0and the alternative hypothesis is E( ¯dt) 6= 0, which is

equal to a different level of accuracy for the two forecasts. The Diebold-Mariano test can thus be seen as a special version of the t-test, since the test statistic can be written as

tm,n = ¯ dt q ˆ σ2 m,n/n , (3.13) with ˆσ2

m,n the heteroskedasticity and autocorrelation consistent (HAC) estimator of σm,n2 ,

which is the variance of √n ¯dm,n. The HAC estimator of the variance is used, since often

autocorrelation exist between the differences dt if the forecast horizon h is larger than one

(Harvey, Leybourne, & Newbold, 1997). However, in this study the forecast horizon h will be set to one, leading to a lag selection for the autocovariances of zero, since the HAC variance estimator used by Diebold & Mariano (1995) can be written as V ( ¯d) ≈ n−1γ0 + 2Ph−1k=1γk.

Harvey et al. evaluate the Diebold-Mariano test and suggest some changes regarding the test, namely a slightly different test statistic and a change of the test statistic from the Normal distribution to the Student-t distribution with n−1 degrees of freedom. The test statistic Harvey et al. suggest can be written as

t∗_m,n=hn + 1 − 2h + n

−1_{h(h − 1)}

n

i1₂

tm,n. (3.14)

This modified Diebold-Mariano test is used in the empirical analysis. If the null hypothesis is rejected, one model has a forecast that is more accurate than the other model. Which model is the most accurate one, can be determined by checking the mean of the loss functions. The mean of both loss functions are called the Mean Absolute Error (MAE) and the Root Mean Squared Error (RMSE). These are calculated as

M AE = 1 M − h + 1 T X t=t0+h |et|, (3.15) RM SE = v u u t 1 M − h + 1 T X t=t0+h e2 t, (3.16)

with M the number of observations saved for the out-of-sample dataset, h the forecast horizon and et the forecast error. The model that has the lowest MAE and RMSE, forecasts the most

accurate and therefore outperforms the other model.

In this chapter, models and methodology are explained that are used to conduct the empirical analysis. The models that are described, can be divided in univariate models and multivariate models. The univariate model used, falls in the ARMA-GARCH framework and the multivariate model consists of the VAR model for the mean equation, followed by a DCC model to model the conditional variance and correlation. Lastly, a forecast is conducted to compare the univariate model and the multivariate model and to determine which model captures the characteristics of the uncovered and covered currency carry trade returns best. In the next chapter, the data is introduced on which the analysis is carried out.

19

Chapter 4

Data

This chapter describes the data that are used for the application of the models discussed in the last chapter. The interest rate parity combines the foreign exchange market and the interest rate market. Therefore, data on the exchange spot rates, exchange forward rates and the interest rates are needed to perform the analysis. The data used are of exchange spot rates, exchange forward rates, 3-month interbank interest rates and proxies for the risks associated with the currency carry trade as explanatory variables between January 2008 and March 2018. The data are taken on a weekly basis and are obtained from Bloomberg.

4.1

Exchange rates

In this section a description of the spot rates is given. The foreign currencies are collected against the United States Dollar (USD) and the foreign currencies that are chosen to investigate, are the Euro (EUR/USD), British Pound (GBP/USD), Japanese Yen (JPY/USD), Swiss Franc (CHF/USD), Canadian Dollar (CAD/USD), Australian Dollar (AUD/USD), New Zealand Dollar (NZD/USD), Singapore Dollar (SGD/USD), Danish Krone (DKK/USD), Swedish Krona (SEK/USD), Norwegian Krone (NOK/USD). For all these pairs the spot rate and the forward rate are taken in terms of the foreign currency per one USD. The forward rates have a maturity of 3 months, which matches the maturity of the interest rates that are used.

In column (1) of Tables 4.1 and 4.2, the spot rate and the 3 month forward rate are given in levels. It can be concluded that the mean of the spot rate and 3 month forward rate are close to each other for all currencies. The interested reader is referred to Appendix A, where the graphs of the spot rates and forward rates are given in Figure A.1. In column (2), the difference of the natural logarithm of the spot rate in Table 4.1 and the difference of the natural logarithm of the 3 month forward rate and spot rate in Table 4.2 and their variance, skewness and kurtosis can be found in columns (3), (4) and (5) respectively. The skewness and kurtosis are defined as the standardised third and fourth moment respectively.

It is interesting, to check whether the 3 month forward rate has been an unbiased estimator of the future spot rate in the last ten years, since the forward rate agreed on today, has to be paid in 3 months time. As can be seen in Figure 4.1, there seems to be a lag between the spot rate and forward rate when they are evaluated at the maturity time. The forward rates show more similarities to the current spot rate than to the spot rate 3 months from now. This is affirmed by the relatively small values in column (2) in Table 4.2.

TABLE 4.1: Descriptive statistics of the spot exchange rates and the logarithm of the depreciation of the spot exchange rates

Variance Skewness Kurtosis

St ∆st ∆st ∆st ∆st (1) (2) (3) (4) (5) Euro 0.7861 0.0049 0.0027 0.6924 3.8231 British Pound 0.6524 0.0087 0.0027 1.0020 6.2883 Japanese Yen 0.0102 -0.0007 0.0031 -0.2136 3.4568 Swiss Franc 1.0267 0.0028 0.0023 -0.1784 3.2667 Canadian Dollar 0.8966 -0.0057 0.0022 -0.5846 5.6039 Australian Dollar 1.1662 0.0035 0.0056 1.3554 8.7727 New Zealand Dollar 1.3621 0.0020 0.0047 0.5724 5.1595 Singapore Dollar 0.7497 0.0016 0.0008 -0.5662 3.0897 Danish Krone 0.1726 -0.0049 0.0026 -0.7022 3.8563 Swedish Krona 0.1376 -0.0064 0.0040 -0.7013 5.1106 Norwegian Krone 0.1542 0.0457 0.0215 1.5181 5.7667

This table reports the descriptive statistics of the spot exchange rate. In column (1), the average of the spot exchange rate in levels in the period 2008-2018 can be found. Column (2)

displays the average difference of the logarithm of the spot exchange rate between the exchange rate at time t and the exchange rate at time t + k. This difference is the dependent

variable in the uncovered interest rate parity. Columns (3), (4) and (5) display the second, third and fourth moment respectively.

TABLE4.2: Descriptive statistics of the forward exchange rates and the logarithm of difference between the forward and spot exchange rates

Variance Skewness Kurtosis Fk,t fk,t− st fk,t− st fk,t− st fk,t− st

(1) (2) (3) (4) (5)

Euro 0.7854 -5.71e-04 9.066e-06 0.1749 3.1910 British Pound 0.6526 6.45e-04 7.569e-06 0.8087 4.9351 Japanese Yen 0.0102 1.860e-03 7.118e-06 1.5355 10.1034 Swiss Franc 1.0288 1.974e-03 7.487e-06 0.4316 3.7113 Canadian Dollar 0.8954 -1.324e-03 6.908e-06 -0.1800 8.9367 Australian Dollar 1.1737 6.902e-03 1.833e-05 0.0689 4.0208 New Zealand Dollar 1.3706 6.384e-03 1.442e-05 0.6201 4.6976 Singapore Dollar 0.7496 -1.25e-04 3.101e-06 0.7422 4.9479 Danish Krone 0.1727 0.2281 0.003773 0.6643 3.5330 Swedish Krona 0.1373 -0.2288 0.003678 -0.7906 3.8672 Norwegian Krone 0.1537 -1.2081 0.453521 0.1824 3.5003

This table reports the descriptive statistics of the forward exchange rate. In column (1), the average of the forward exchange rate in levels in the period 2008-2018 can be found. Column (2) displays the average difference of the logarithm of the forward exchange rate and the spot exchange rate. This difference is the dependent variable in the covered interest rate parity.

Columns (3), (4) and (5) display the second, third and fourth moment respectively.

Besides the spot rates and forward rates, the interest rates play a main role in the interest rate parity. In the next section, the interest rates used, are described.

Chapter 4. Data 21

FIGURE4.1: The spot rate and forward rate at maturity time

4.2

Interest rates

As their name already indicates, interest rates play an important role in the interest rate parity. In this section, the interest rates used are explained and some descriptive statistics are given.

The interest rates considered, are the 3 month interbank interest rates of the same eleven currencies as the exchange rates that are used. The interbank interest rates are the rates against which banks lend to each other on a short-term basis. In Table 4.3, the interbank interest rates for all countries and the interest rate differential compared to the USD Libor are given.

TABLE 4.3: Descriptive statistics of the interest rates and the logarithm of the interest rate differential compared to the United States Dollar

Variance Skewness Kurtosis if,t rf,t− rd,t rf,t− rd,t rf,t− rd,t rf,t− rd,t

(1) (2) (3) (4) (5)

EURIBOR 0.8159% 8.21e-04 8.517e-05 4.206e-03 2.8613 LIBOR 1.1454% 3.929e-03 7.064e-05 1.4695 6.2162 JPY LIBOR 0.2288% -5.142e-03 4.166e-05 -1.8842 6.7427 CHF LIBOR 0.0905% -6.425e-03 4.258e-05 -0.9794 2.6443 CAD LIBOR 1.2910% 5.214e-03 2.293e-05 -0.6484 2.6833 AUD LIBOR 3.2009% 0.02431 1.683e-04 -0.0852 2.0330 NZD LIBOR 3.2807% 0.02499 1.185e-04 0.8941 4.8775 SIBOR 0.7281% -3.63e-04 3.449e-05 -2.0823 8.1406 DKK LIBOR 1.0696% 3.354e-03 1.341e-04 0.3990 2.9175 SEK LIBOR 1.1301% 4.013e-03 1.550e-04 -0.2661 2.0585 NIBOR 2.2661% 0.01513 1.013e-04 -0.1644 2.8699

This table reports the descriptive statistics of the interest rates. In column (1), the average of the interest rate in levels in the period 2008-2018 can be found. Column (2) displays the average difference of the logarithm of the foreign interest rate and the USD LIBOR interest rate. This difference is the independent variable in the uncovered and covered interest rate parity. Column (3), (4) and (5) display the second, third and fourth moment respectively.

Column (1) in Table 4.3 displays the average of the interest rate in levels, column (2) gives the average interest rate differential of the foreign currency compared to the USD LIBOR as domestic currency, in this column rf,t = ln(1 + if,t)and rd,t = ln(1 + id,t). In columns (3) and

(4), the variance and skewness of the differential in natural logarithms are given. From this table, it can be concluded that the NZD LIBOR, AUD LIBOR and the NIBOR have the highest interest rates on average and therefore also the highest interest rate differential compared to the USD LIBOR. The JPY LIBOR, CHF LIBOR and the SIBOR have the lowest average interest rates and therefore the lowest mean of the interest rate differential compared to the USD LIBOR. These currencies are most interesting to investigate in a currency carry trade, since the interest rate differential can be seen as the profit if the exchange rate stays the same. For completeness, the graphs of the interest rate differentials can be found in Appendix A in Figure A.2.

The profit made by a currency carry trade could be seen as a compensation for risk. To evaluate which risk factors have a significant effect on the currency carry trade profit, proxies are added to the model. These proxies serve as exogenous explanatory variables and are discussed next.

Chapter 4. Data 23

4.3

Explanatory variables

In this section, the focus is on the risk factors that could explain the positive average profit of the carry trade. As already mentioned in Chapter 2, the risk factors that are focused on in this thesis are systemic risk and counterparty risk. In this section, proxies are suggested that can represent these risks when modelling the carry trade. First, systemic risk is explained, followed by counterparty risk.

4.3.1 Systemic risk

Systemic risk can be described as the risk of a collapse of entire financial market or financial system. In case of the currency carry trade, systemic risk can be displayed as spillovers from other financial markets, like the stock or bond market. Christiansen et al. (2011) propose to use the Morgan Stanley Capital International (MSCI) index as a proxy for the stock market. The MSCI index is a market cap weighted stock index and the MSCI Asia Pacific, MSCI Europe and MSCI USA index are used as proxies in this research. The MSCI Asia Pacific index captures the cap representations of Australia, Hong Kong, Japan, New Zealand, Singapore, India, China, Indonesia, Korea, Malaysia, Pakistan, the Phillipines, Taiwan and Thailand. The MSCI Europe index represents the performance of the large and mid-cap equities in Austria, Belgium, Denmark, Finland, France, Germany, Italy, Ireland, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom. The MSCI USA index captures the performance of the large and mid-cap representations of the US stock market. These three indices together represent the stock markets of the countries for which the currency carry trade is investigated and therefore make a good proxy to see if there is a spillover effect from the stock market on the carry trade return.

Another proxy proposed by Christiansen et al. (2011) is the Chicago Board Options Exchange Volatility Index (CBOE VIX). The CBOE VIX measures the stock market volatility implied by the S&P 500 index option and can be used as an indication of the expected volatility of the market in the near future. It can therefore be used as a proxy to measure the effect of the stock market volatility on the carry trade return.

Another risk factor that could indicate a downfall in the foreign exchange market and could therefore be an explanation of the positive return of the carry trade, is foreign exchange market liquidity risk (Chang & Schlögl, 2012). Chang and Schlögl use the cross-currency basis spread as a proxy for liquidity risk, since the basis spread opened up after 2007, which indicates a shortage of USD in the market. Mancini, Ranaldo, & Wrampelmeyer (2013) argue that even though the large size of the foreign exchange market, there exists significant variation in liquidity across exchange rates. They state that liquidity also contributes to the appreciation of high interest rate currencies, since high interest rates tend to offer exposure to liquidity risk. As a proxy, the cross currency basis spread of the EUR/USD is used, since this basis spread opened up after the crisis.

The last proxy that is added to the mean equation to give an indication of the sensitivity to systemic risk, is the TED-spread. The TED-spread measures the difference between the 3 month USD LIBOR rate and the 3 month Treasury Bill (T-Bill) rate and is overall used to give

an indication of the credit risk in the general economy. Therefore, the TED-spread can be used as an indicator of trust in the general economy and hence as an indicator of systemic risk.

As can be seen in Table 4.4 and Figure 4.2, most of the proxies have a large variance. This can

TABLE4.4: Descriptive statistics of the systemic risk proxies

Average Variance Skewness Kurtosis

(1) (2) (3) (4)

MSCI Index Europe 131.61 393.58 -0.2458 3.0410

MSCI Index Asia Pacific 130.41 317.14 -0.7021 4.1807

MSCI Index USA 46.67 671.75 1.3799 4.1584

CBOE VIX 19.97 102.15 2.3583 10.4155

TED spread(bps) 44.70 2323.01 4.0817 25.4241

Cross-Currency basis spread EUR/USD(bps) -28.56 183.62 -0.0698 2.5502 This table reports the descriptive statistics of the systemic risk proxies. In columns (1), (2), (3)

and (4) the first, second, third and fourth moment are displayed, respectively.

FIGURE4.2: Systemic risk proxies in 2008-2018

be explained by the extreme values during and after the GFC.

Next to systemic risk, the positive return of the currency carry trade can be a compensation for counterparty risk. The proxies that are added to account for counterparty risk, are explained in the next subsection.

4.3.2 Counterparty risk

Since the GFC in 2007, the importance of the role of counterparty risk between banks has become more significant. One phenomenon of the GFC is the increase over the course of the crisis of the spread between the London interbank offer rate (LIBOR) and the overnight indexed swap (OIS) (Ji & In, 2010). The LIBOR-OIS spread is one of the key measures for financial distress and can be seen as a barometer of fear of bank insolvency. A higher spread, indicating a higher LIBOR rate, can be interpreted as a decreasing willingness to lend by banks and can

Chapter 4. Data 25

therefore be seen as an indication of, among other things, creditworthiness of other financial institutions from the banks’ perspective. Hence, the LIBOR-OIS spread with a maturity of 3 months of several currencies are a good proxy to use for modelling counterparty risk. The currencies of which the LIBOR-OIS spread is used, are the US Dollar, Canadian Dollar, Euro, Swiss Franc, Japanese Yen and the British Pound. In Table 4.5, descriptive statistics of the LIBOR-OIS spreads can be found and in Figure 4.3 a graph of the proxies can be seen.

FIGURE 4.3: LIBOR-OIS Spreads for the United States Dollar, the Euro, the Japanese Yen, the Swiss Franc, the British Pound and the Canadian Dollar in

2008-2018

TABLE4.5: Descriptive statistics of the LIBOR-OIS spreads

Average Variance Skewness Kurtosis

(1) (2) (3) (4)

LIBOR-OIS Spread USD 0.755 0.681 2.3129 8.2614 LIBOR-OIS Spread EUR 0.844 1.981 1.9317 5.9185 LIBOR-OIS Spread CHF 0.111 0.874 1.8536 6.2133 LIBOR-OIS Spread GBP 1.160 2.269 2.5998 8.2156 LIBOR-OIS Spread JPY 0.236 0.073 1.6441 4.7856 LIBOR-OIS Spread CAD 1.281 0.550 2.2337 8.0870

This table report the descriptive statistics of the counterparty risk proxies. In columns (1), (2), (3) and (4) the first, second, third and fourth moment are displayed, respectively. As can be seen from the graph and the table, the LIBOR-OIS spread has a peak in the autumn of 2008, around the highlight of the financial crisis. Since 2009, the spread started to close, which indicates that the trust between banks started to increase. Overall, it can be stated that the spread is small around 2018 and the counterparty risk between banks is limited. This chapter introduced the data set that is used to check whether the interest rate parities hold and what can explain the positive payoff of the exploitation of the parities. To model

the parities, data on the spot exchange rates, forward exchange rates and the interest rates of several countries are needed. The returns can be explained as a compensation for a certain risk that the investors are exposed to. The risks that are investigated are systemic risk and counterparty risk. The proxies that are used to represent these risks are introduced in this chapter. In the next chapter, the empirical analysis is conducted with the model from Chapter 3 and the data described in Chapter 4.