Determinants of Carrying Trade with Indian Rupee: Variables and Methods

Abstract

We have taken six variables to ascertain their contribution in finding the direction and values of carry trade returns in Indian rupee. These variables are as follows: foreign exchange volatility; interest rate difference; volatility in equity market; interest rate difference volatility; liquidity in foreign exchange and commodity index. We have employed three methods—ordinary least square, multivariate adaptive regression splines (MARS) and autoregressive integrated moving average (ARIMA-X) to find the predictive capacity of these above-mentioned variables. We find that interest rate differential and volatility in equity market significantly affect carry trade returns with spot exchange rates. Thus, traders can take the indication from volatility in equity market and interest rate difference for the return of carry trade returns.

Keywords

Carry trade determinants of carry trade MARS Diebold Mariano Test emerging market

Introduction

Carry trade is an arbitrage strategy in financial markets in which traders borrow currency with low interest rate and sell it to buy currency with high interest rate. The currency with high interest rate is then lent over the same time period, leading to profits from the difference in interest rates. The borrowed currency with low interest rate is called funding currency, while currency with high interest rate is called target currency or investment currency. Normally, as per no arbitrage argument of covered/uncovered interest rate parity theory, this trade should give no return due to appreciation of low-yield currency and/or proportional depreciation of the high-yield currency (Brunnermeier et al., 2008). This theory is one of the main building blocks of international finance. However, in many empirical studies (Menkhoff et al., 2012) as well as practitioners’ experience, this theory is found to be violated. High-yield currency appreciates relative to depreciation of low-yield currency, eventually resulting in gain in trade (Spronk et al., 2013). This trade has been profitably conducted in spite of having no proper economic explanation, and hence it is termed as the ‘forward premium puzzle’.

This study undertakes the challenge to find the variable, which can be a predictor for carry trade return, using Indian rupees. In this direction, first, we have selected the said variables, which have been frequently quoted as predictors of carry trade in studies carried out for carry trade, involving foreign currencies. This study helps traders to keep an eye on the volatility of equity market for the guide on returns from carry trade. As we know that the returns from carry trade are subject to foreign exchange risk, it would be advisable to predict the foreign exchange rate in order to reduce the risk. But doing so is itself difficult and risk ridden, so we try to take another route, that is, to detect the variables that can foretell the return itself. Recently, profits from carry trade have dipped, as interest rates of all major currencies are near zero, and are not high enough to cover the risks involved. Simultaneously, however, opportunities for carry trade have risen in emerging markets. In view of various macroeconomic upheavals around the globe, it would be of interest to traders and academicians alike, as to which factors significantly contribute to carry trade returns. But, any search for significant predictors should also encompass the search for an appropriate method that identifies such predictors. Accordingly, this study aims to identify both a suitable method that can correctly detect significant predictors and ascertain such predictors therefrom.

Many studies have proposed explanatory variables such as exchange rate volatility (Baillie & Chang, 2011), volatility in equity market (Alam et al., 2017; Brunnermeier et al., 2008), inflation (Kim, 2015), index of industrial production—IIP (Laborda et al., 2014), commodity index (Ready et al., 2017), difference in libor rate and T- bill interest rate (TED) spread (Brunnermeier et al., 2008) and liquidity of foreign exchange market (Menkhoff et al., 2012). However, these studies typically use ordinary least squares (OLS) regression to identify factors that significantly affect carry trade returns. Such regressions require two underlying assumptions: (a) the dependent–independent variable relation is linear and (b) errors follow normal distribution. Yet, these are unsupported assumptions and may lead to spurious results. In this study, therefore, two methods are explored to identify the one that is more suited to the search for predictors of carry trade returns. First, multivariate adaptive regression spline (MARS) method, introduced by Friedman (1991), is employed. This method can reveal high-dimensional data dependencies between the independent and dependent variables. By joining piecewise curves, the method allows bends, interactions and various other non-linear flexibilities in the functional relation. Further, being a non-parametric method, it also does not necessitate assumption of any underlying distribution.

In the OLS method, the results suggest that the relationship is not linear and so the relationship is non-linear, and MARS is used to take this into consideration. One key aspect of carry trade returns is overlooked in the investigation for predictors of its returns—its time-series nature (Bakshi & Panayotov, 2013). Both regression and MARS assume cross-sectional data. Multivariate autoregressive integrated moving average (ARIMA), or ARIMA-X, on the other hand, takes into account the time-series nature of carry trade returns while allowing the inclusion of independent variables in the functional relationship. Hence, this study also employs ARIMA-X method to explore predictors of carry trade returns.

In this study, therefore, OLS regression, MARS and ARIMA-X have been employed to examine the role of explanatory variables such as liquidity of foreign exchange market, interest rate difference and its volatility, volatility in equity market returns, innovation in foreign exchange (proposed by Menkoff et al., 2012) and commodity futures index. Performance of various models is compared using three metrics—adjusted R², root mean square error and Diebold–Mariano (DM) test. While no method is found to have better fit than others, we observe that irrespective of the methods, higher interest rate difference and volatility in equity market affect carry trade return over 3 months. Thus, this result is robust to methods applied. It is also robust to carry trades undertaken with spot exchange rates as well as exchange rate futures. But no factor is found to affect return over 1 month, where carry trade is undertaken using 1-month exchange rate futures.

This article makes two contributions. First, it shows that the search for explanatory variable for carry trades should not be left to one method alone, as different methods throw up different significant variables. So, only those variables, which emerge as significant across methods, should be considered as convincing. Second, it demonstrates that, regardless of the method used, carry trades over 3 months offer higher returns when interest rate differentials are higher, and volatility in equity market is higher. Thus, it is in the interest of traders that volatility in equity market is used as the indicator for the returns of carry trade. But carry trades with 1-month exchange rate futures have no robust explanatory variables.

Literature Review

For studying predictability of carry trade returns, various studies have considered the effect of macroeconomic variables (especially exchange rates). These variables try to capture the risk and returns exhibited by the underlying component markets, viz forex markets and interest rate markets. Equity markets and commodity markets have also remained under consideration; these are alternative markets where domestic currency and foreign currencies, respectively, could have been invested and, thus, present an opportunity set. The predictors of carry trade returns that have consistently emerged from these studies are volatility in exchange rates, interest rate difference, commodity index returns, and volatility and liquidity in equity markets. This study considers these variables and adds volatility in interest rate differences as a potential predictor. The motivations and expected relations are discussed in the following paragraphs.

Carry traders book profitable returns when the currency with high interest rate appreciates, implying that such returns are dependent on the exchange rate differential. Hence, volatility in exchange rates should impact carry trade returns. This has been repeatedly borne out by many studies. Menkhoff et al. (2012) and Clarida et al. (2009) show that carry trade returns are high when foreign exchange volatility are low and vice versa. Clarida et al. (2009) have used Fama regression to establish the aforementioned fact, and a study has been carried on Group of Ten (G10) currencies. Ichiue and Koyama (2011) suggest that devaluation of low-interest-rate currencies and a low-volatility environment are reciprocally dependent on each other, mainly for shorter horizons. Kim (2015), using the Markov regime-shifting model, find that carry trade returns are low when there is high volatility in foreign exchange market. Spronk et al. (2013) report a negative relationship between carry trade returns and exchange rate volatility. Thus, volatility in exchange rate is expected to affect carry trade returns negatively.

Like exchange rate differential, carry trade return is also a direct result of interest rate differential (Brunnermeier et al., 2008). They report that carry trade returns are high when lagged interest rate differential is high. Lustig et al. (2011) find that from the perspective of a US investor, currencies with high interest rates earn significantly higher average returns than currencies with low interest rates. Hattori and Shin (2007) show that the interest rate difference was the main force behind the yen carry trade. So, it is expected that interest rate difference positively affects the carry trade.

Extant literature has always considered interest rate differential, but not its volatility, although volatility in exchange rates has emerged as a consistent predictor. In this study, volatility in interest differential is also taken as a potential explanatory variable.

Global investors shift funds to more profitable assets. So, equity markets are direct competitors of such funds against carry trades (Liu & Yang, 2017). Hence, it can be expected that high volatility in equity returns would have spillovers in carry trade markets (Fung et al., 2013). Brunnermeier et al. (2008) show that higher levels of volatility index (VIX) foretells higher returns for high-interest currencies and lower returns for low-interest-rate currencies. Luo et al. (2009) use VIX as the volatility indicator in their global quantile risk indicator (GQRI). The GQRI has been found to be inversely related to carry trade returns. GQRI performed better than VIX in prediction of foreign exchange volatility. Nirei and Sushko (2011) show that the likelihood of volatility in foreign exchange market rises with higher speculative incentive and uncertainty as proxied by interest rate difference and VIX. Christiansen et al. (2011) use VIX as a regime indicator and report that carry trade returns during high volatility periods are significantly higher than those during low volatility periods. Hoffman (2012) regresses carry trade return on change in VIX and find that carry trade returns are high when VIX is high. Thus, it is expected that volatility in equity market has a positive relationship with carry trade return.

Liquidity in foreign exchange markets would indicate the level of transaction costs to be incurred in undertaking a carry trade strategy. Low levels of liquidity lead to higher transaction costs, eroding realized returns. Accordingly, Mancini-Griffoli and Ronaldo (2011) suggest that high bid–ask spread implies lower return to carry trade. Using bid–ask spread of foreign exchange as a liquidity measure, Menkoff et al. (2012) find that foreign exchange liquidity has a negative relationship with carry trade return when regressed alone, whereas it becomes insignificant when regressed along with volatility proxies. Bakshi and Panayotov (2013) report mixed results for liquidity–carry trade return relationship. Mancini et al. (2013) show that funding currencies offer an insurance against liquidity risk. Hence, it is expected that carry trade returns are negatively impacted by foreign exchange liquidity.

Ready et al. (2017) show that countries exporting raw commodities have a higher interest rate than countries exporting finished goods. These interest rate differences translate into carry trade returns. Commodity indices have been used by Bakshi and Panayatov (2013) as a factor that affects carry trade return positively. They justify this inclusion by stating that their currencies under investigation (Australian Dollar and NZD) are commodity currencies. The volatilities of the commodity index and the carry trade index were found to be significantly correlated in the sample of Gochoco-Bautista et al. (2014), with a correlation coefficient of 0.458 (t = 11.7). Passari (2015) explores the relationship between exchange rate and commodity price and builds a carry trade strategy on the previous day commodity price movement. Byrne et al. (2019) show that carry trade returns can be explained by agricultural commodities and metal commodities. Based on these studies, we expect that commodity index return has a positive relationship with carry trade return.

In this study, interest is to find the factors, which affect the carry trade returns, using Indian rupees. The aforementioned factors are valid for the countries whose economic conditions and monetary policies are different from that of India. In such conditions, it is imperative to ascertain whether the factors that affect the returns in a developed country are also effective in the Indian context.

Research Variables and Research Methods

In this article, the effect of various explanatory variables on carry trade return between Indian rupees and other foreign currencies has been explored with the help of three methods, namely OLS, MARS and ARIMAX. We, first, derive the formulae used to calculate carry trade returns, then present the methods employed to find the relationship between the returns and various explanatory variables and, finally, discuss the criteria on which we choose the best method.

Estimation of Carry Trade Returns Using Only Spot Exchange Rates

In extant literature (Burnside et al., 2008; Menkhoff et al., 2012), carry trade return is calculated as:

Z_{t} = (i_{t - 1}^{d} - i_{t - 1}^{f}) - Δ s_{t}

(1)

Here, Zt is carry trade return at time t;

$i_{t - 1}^{d}$ is the logarithm of the domestic interest rate at time t − 1;

$i_{t - 1}^{f}$ is the logarithm of foreign interest rate at time t − 1;

Δst = st − st − 1; and

st = log(nominal exchange rate).

The exchange rate is taken as the number of units of foreign currency for every Indian rupee.

However, in this study, we take a different approach to calculate the gains from carry trade. This has been discussed in Ranjana and Barai (2021). We assume that at time t − 1, if a trader observes that domestic interest rates are more than foreign interest rates (i.e., it−1d > it−1f), a trader will borrow x units of foreign currency on a continuously compounded interest rate of it−1f, sell it at spot exchange rates of St−1bid to receive ( $\frac{x}{S_{t - 1}^{bid}}$ ) units of domestic currency and invest it at a continuously compounded interest rate of it−1d. On time t, he will need to return ( $x e^{i_{t - 1}^{f} Δ t}$ ) units of foreign currency, where Δt is the difference in periods (t − 1) to t. This will require ( $\frac{x e^{i_{t - 1}^{f} Δ t}}{S_{t}^{ask}}$ ) units of domestic currency. From his investments, he should receive ( $\frac{x}{S_{t - 1}^{bid}} e^{i_{t - 1}^{d} Δ t}$ ) units of domestic currency. As per the uncovered interest rate parity theory, the amounts received in this transaction should equal the amounts to be paid. That is:

(\frac{x}{S_{t - 1}^{b i d}} e^{i_{t - 1}^{d} Δ t}) = (\frac{x e^{i_{t - 1}^{f} Δ t}}{S_{t}^{a s k}}) .

Rearranging, and taking natural logarithms on either side, we obtain:

(i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t = \ln (S_{t - 1}^{bid}) - \ln (S_{t}^{ask}) .

If, however, the uncovered interest rate parity theory does not hold, the aforementioned equality is violated, and we get the gains from carry trade as follows:

Z_{t} = (i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t - [\ln (S_{t - 1}^{bid}) - \ln (S_{t}^{ask})] when i_{t - 1}^{d} > i_{t - 1}^{f}

On the other hand, if foreign interest rates are more than domestic interest rates (i.e., $i_{t - 1}^{d}$ , < $i_{t - 1}^{f}$ ), a trader will undertake the opposite transactions, and then, the gains from carry trade are as follows:

Z_{t} = (i_{t - 1}^{f} - i_{t - 1}^{d}) Δ t - [\ln (S_{t}^{bid}) - \ln (S_{t - 1}^{ask})] when i_{t - 1}^{d} < i_{t - 1}^{f}

Combining the above-mentioned two equations, we get the carry trade gains as follows:

\begin{matrix} Z_{t} = (i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t - [\ln (S_{t - 1}^{b i d}) - \ln (S_{t}^{a s k})] w h e n i_{t - 1}^{d} > i_{t - 1}^{f} \\ = (i_{t - 1}^{f} - i_{t - 1}^{d}) Δ t - [\ln (S_{t}^{b i d}) - \ln (S_{t - 1}^{a s k})] w h e n i_{t - 1}^{f} < i_{t - 1}^{d} \end{matrix}

(2)

Here, we use Equation (2) to calculate carry trade gains, instead of the more commonly used Equation (1).

Carry Trade Returns Using Spot and Forward Exchange Rates

Forward currency markets offer two distinct advantages in carry trade transactions. First, exchange rate risk in carry trade is virtually removed, since all prices and interest rates are known at the time of undertaking the trade, and these forward contracts are subject to minimal default and risk; second, carry trade is easy to implement in these markets. In India, forwards on currencies exist for only four currencies, and those traded are generally with maturities less than 1 year.

In extant literature, using futures contracts, carry trade return is calculated as follows (Barroso & Santa-Clara, 2015):

Z_{t} = \frac{F_{t, t + 1}}{S_{t + 1}} - 1

(3)

Here, Zt is carry trade return at time t,

$F_{t, t + 1}$ is the future exchange rate at time t and

$S_{t + 1}$ is the spot exchange rate time t + 1.

Exchange rate is taken as the number of units of foreign currency for every Indian rupee.

In this study, a different approach is taken for the calculation of gains from carry trade.

We assume that at time t − 1, if a trader observes that domestic interest rates are more than foreign interest rates (i.e. $i_{t - 1}^{d}$ ,> $i_{t - 1}^{f}$ ), a trader will borrow x units of foreign currency on a continuously compounded interest rate of $i_{t - 1}^{f},$ sell it at spot exchange rates of St−1bid to receive ( $\frac{x}{S_{t - 1}^{bid}}$ ) units of domestic currency, invest it at a continuously compounded interest rate of $i_{t - 1}^{d}$ and enter into a forward contract to sell foreign currency at time t, at contracted exchange rate of $f_{t}^{ask}$ on time t, he will need to return ( $x e^{i_{t - 1}^{f} Δ t}$ ) units of foreign currency, where Dt is the difference in time periods (t − 1) to t. He will execute the forward contract for time t at time t − 1. This will require ( $\frac{x e^{i_{t - 1}^{f} Δ t}}{f_{t}^{ask}}$ ) units of domestic currency. From his investments, he should receive ( $\frac{x}{S_{t - 1}^{bid}} e^{i_{t - 1}^{d} Δ t}$ ) units of domestic currency. As per the covered interest rate parity theory, the amounts received in this transaction should equal the amounts to be paid. That is,

(\frac{x}{S_{t - 1}^{b i d}} e^{i_{t - 1}^{d} Δ t}) = (\frac{x e^{i_{t_{t}}^{f} Δ t}}{f_{t}^{a s k}}) .

Rearranging, and taking natural logarithms on either side, we obtain:

(i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t = ln (S_{t - 1}^{bid}) - ln (f_{t - 1}^{ask})

If, however, the uncovered interest rate parity theory does not hold, the above-mentioned equality is violated, and we get the gains from carry trade as follows:

Z_{t} = (i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t - [\ln (S_{t - 1}^{bid}) - \ln (f_{t - 1}^{ask})] when i_{t - 1}^{d} > i_{t - 1}^{f}

If, however, foreign interest rates are more than domestic interest rates (i.e., it−1d < it−1f), a trader will undertake the opposite transactions, and then, the gains from carry trade are as follows:

Z_{t} = (i_{t - 1}^{f} - i_{t - 1}^{d}) Δ t - [\ln (f_{t 1}^{bid}) - \ln (S_{t - 1}^{ask})] when i_{t - 1}^{f} > i_{t - 1}^{d}

Combining the aforementioned two equations, we get the carry trade gains as follows:

\begin{matrix} Z_{t} = (i_{t - 1}^{d} - i_{t - 1}^{f}) Δ t - [\ln (S_{t - 1}^{b i d}) - \ln (f_{t - 1}^{a s k})] w h e n i_{t - 1}^{d} > i_{t - 1}^{f} \\ = (i_{t - 1}^{f} - i_{t - 1}^{d}) Δ t - [\ln (f_{t - 1}^{b i d}) - \ln (S_{t -}^{a s k})] w h e n i_{t - 1}^{d} < i_{t - 1}^{f} \end{matrix}

(4)

Ordinary Least Square Method

Under OLS, the multivariate regression function is written as follows:

{CTR}_{i} = β_{0} + β_{i}^{T} x^{+} u_{i}

(5)

CTRi is the carry trade return of the ith currency with Indian rupees, x is the vector of m number of explanatory variables and bι is the vector of parameters to be estimated. The parameters are estimated by minimizing the sum of squared errors (SSE).

Multivariate Adaptive Regression Spline Model

The MARS method was proposed by Friedman (1991) and has since been used to model a highly non-linear relationship between dependent and independent variables. In this method, a set of piecewise linear functions, called splines, are defined using basis functions between adjacent knots. The knots are assumed to lie within the domain of the training data set. The knot positions, and unknown parameters of the splines, are ascertained by smoothening the splines at the knot points. Essentially, forward and backward iterations are conducted and nodes and linear spline functions identified such that SSE is minimized. In the forward step, overestimation of function is carried out using too many nodes; then in the backward step, the nodes with fewer contributions to the overall fitting, that is, whose SSE is maximum, are removed. So, this method is called adaptive spline method.

MARS is formed of segment functions, which are defined as follows:

g (x_{i}) = \{\begin{matrix} u - x_{i}, i f u > x_{i} \\ 0, o t h e r w i s e \end{matrix}

(6)

Here, u is a knot whose location is to be determined, ${x}_{i}$ is a vector of m explanatory variables and subscript i denotes one of n number of observations. The MARS model can be represented as a linear combination of basis functions as follows:

\hat{CTR (x)} = β_{0} + \sum_{i = 1}^{m} \sum_{p = 1}^{P} β_{p} B_{p} (x_{i}) + \sum_{i, j = 1}^{m} \sum_{p = 1}^{P} β_{p} B_{p} (x_{i}, x_{j}) + \sum_{i, j, k = 1}^{m} \sum_{p = 1}^{P} β_{p} B_{p} (x_{i}, x_{j}, x_{k}) + \dots

(7)

Here, B denotes basis function in one variable, two variables, till m explanatory variables. Its functional form involves interaction of linear functions and segment functions defined in Equation (5). A forward and backward algorithm has been proposed by Friedman (1991) to compute the parameters, and also the potential basis functions. A metric, termed as generalized cross validation (GCV), is used to compare the best basis functions to be retained in the final model. The GCV is expressed as:

G C V (M) = \frac{1}{N} \sum_{i = 1}^{N} {[C T R_{i} - C T R (x_{i})]}^{2} / {[1 \frac{C (M)}{N}]}^{2}

(8)

C(M) = trace (B (BTB)−1BT) + 1 is a complexity penalty, where B is the M x N data matrix of the basic functions. The model with least GCV will be the final model.

Autoregressive Integrated Moving Average-X Model

Stationary time-series are commonly modelled using ARIMA. These models are augmented by including exogenous variables as follows:

\begin{array}{l} Δ^{d} C T R_{t} = \sum_{i = 1}^{p} \emptyset_{i} {\ddot{A}}^{d} C T R_{t i} + \sum_{j = 1}^{q} θ_{j} \in_{- j} + \sum_{m = 1}^{M} β_{m} X_{m, t} + \in_{t} \\ \in_{t - j} \sim N (0, σ^{2}) \end{array}

(9)

Here p is the number of autoregressive lags, d is the degree of differencing and q is the number of moving average lags. The parameters of this model are estimated by the maximum likelihood method.

Model Accuracy Estimation

In this study, three measures have been used to evaluate the accuracy of the forecasts—adjusted R², root mean squared error (RMSE) and the DM test. The methods are outlined as follows:

Adjusted R2: First, errors are calculated as difference between fitted values and actual values. These errors are squared and added. These are designated as SSres. Next, the difference between actual values and series average are calculated, squared and added. These are designated as SStot. Finally, adjusted R² is estimated as follows:

A d j R^{2} = 1 - \frac{S S_{r e s} / (n - p - 1)}{S S_{t o t} / (n - 1)}

(10)

Here, n is the number of data points in the series, and p is the number of independent variables without including the intercept.

RMSE: First, errors are calculated as the difference between fitted values and actual values. These errors are squared, averaged, and, finally, the square root of the average gives the RMSE.

DM test: This test helps us to compare forecasts from two methods. The basic idea is as follows:

Let Xt be the series we forecast. There are two h-period ahead forecast series, say $X_{t + h | t}^{1}$ from ARIMA-X and $X_{t + h | t}^{2}$ from MARS. Then, the forecast errors can be written as $ϵ_{t + h | t}^{1} = X_{t + h} - X_{t + h | t}^{1}$ and $ϵ_{t + h | t}^{2} = X_{t + h} - X_{t + h | t}^{2}$ . Assuming a square loss function on the forecast errors, L( $ϵ_{t + h | t}^{i}$ ), i = 1,2, the difference between the losses is given as:

d_{t} = L (ϵ_{t + h | t}^{1}) - L (ϵ_{t + h | t}^{2}), t = 1, \dots, T .

DM test makes the null hypothesis that the forecasting models perform equivalently, and hence, the expected losses from the two models are the same, that is, E(dt) = 0. The test statistic used is given as DM $\frac{\bar{d}}{\sqrt{\frac{2 μ {\hat{f}}_{d} (0)}{T}}} ‼ N (0, 1)$ =, where $\bar{d}$ is the average of dt, and

{\hat{f}}_{d} (0) = \frac{1}{2 π} ({\hat{γ}}_{d} (0) + 2 \sum_{k = 1}^{h - 1} {\hat{γ}}_{d} (k)) {\hat{γ}}_{d} (k) = \frac{1}{T} \sum_{t = k + 1}^{T} (d_{t} - d) (d_{t - k} - d)

(11)

Data

We have considered carry trade as the dependent variable and exchange rate volatility, interest rate difference, volatility in equity market, liquidity and commodity index as independent variables. The proxies used to estimate these variables are discussed in this section.

Dependent variable: carry trade

In this study, we have selected currencies from both developed and emerging economies. Accordingly, currencies from the USA (USD), the UK (GBP), Europe (Euro) and Japan (JPY), Australia (AUD) and New Zealand (NZD) have been selected from developed economies and currencies from China (CNY), Hong Kong (HKD), Russia (RBL), Brazil (BRL) and Singapore (SGD) from developing economies. We have downloaded the interest rate from Bloomberg for these currencies. We calculate the carry trade gains from each of these currencies vis-à-vis Indian rupees (INR) as per Equation (18). We obtain spot exchange rates of each of these currencies against INR and 3-month interest rates of all countries, including India from Bloomberg Database.

Only four currencies are studied for carry trade using exchange rate future: USD, EURO, GBP and JPY. This is because futures contract exist in Indian exchanges only against these four currencies. Carry trade with exchange rate futures are studied for 1-month return and 3-month return. The period of study of futures extends from January 2010 to March 2018.The interest rate and future exchange rate are downloaded from Bloomberg.

Independent variable 1.

Volatility in Exchange rates (σFX): Following Menkhoff et al. (2012), foreign exchange volatility in week t is estimated as follows:

σ_{t}^{F X} = \frac{1}{T_{t}} \sum_{τ \in T_{t}} \sum_{k \in K_{τ}} \frac{| r_{τ}^{k} |}{K_{τ}}

(12)

Here, |rtk| is the exchange rate, $K_{τ}$ is the number of currencies available on day τ and Tt denotes the number of trading days in week t. Exchange rates have been taken from Bloomberg. For volatility innovation, AR(1) is estimated for the volatility level $σ_{t}^{FX}$ calculated from Equation (12), and residuals are taken as proxy for innovation. This innovation is similar to the innovation used by Menkhoff et al. (2012).

Interest rate difference (∆IR): The 3-month and 1-month interest rates of various countries have been downloaded from Bloomberg.

Volatility in interest rate difference (σ∆IR): Volatility in interest rates in any week t is calculated as follows:

σ_{t}^{\bar{A} I R} = \frac{1}{T_{t}} \sum_{τ \in T_{t}} \sum_{k \in K_{τ}} \frac{| Δ i_{τ}^{k} |}{K_{τ}}

(13)

where, $Δ i_{τ}^{k}$ is the difference in interest rate between Indian rupees and kth currency at time τ, is the number of currencies available in day τ and Tt denotes the number of trading day in week t. The interest rates have been downloaded from Bloomberg. For volatility innovation, AR(1) is estimated for the volatility level, and residuals are the main proxy for innovation. It has been adopted from Menkhoff et al. (2012).

Volatility in equity market (σEq): India VIX (n.d.) is published by the National Stock Exchange (NSE) of India since 2008. It is a volatility index computed from the volatility in option contracts on NIFTY (Benchmark index of NSE), using the method propounded by Chicago Board of Options Exchange (CBOE). Data on India VIX have been downloaded from the NSE website on a daily basis to be used as proxy for volatility in equity market.

Liquidity in foreign exchange (LIQFX): Bid–ask spread of exchange rates has been taken as a measure of liquidity in foreign exchange. This data were downloaded from Bloomberg.

Commodity index (ComInd): Commodity futures MCXCOMDEX (MCX INDIA, n.d.) from Multi-Commodity Exchange (MCX) of India has been used as a proxy for commodity index. This index is composed of the most liquid commodities traded on the index. These data were downloaded from the MCX of India.

The study is based on the time period from 1 April 2006 to 31 March 2018. Over this period, daily data on all the above-mentioned variables and proxies were collected, except for exchange rate futures as noted.

For equity, daily levels of Nifty50, a free-float, market capitalization–based stock index have been downloaded from NSE of India.

Results and Discussions

Table 1 provides the descriptive statistics of the various variables employed in this study. We observe that the returns of carry trades using spot exchange rates are largely positive. However, carry trade returns from developed economies are higher, on average, and have lower standard deviations, lower positive skewness and lower kurtosis than those from Brazil, Russia, India, China and South Africa (BRICS) and South Asian economies. In fact, carry trade returns from currencies of China, Russia and Singapore emerge as negative, with large negative skewness and very high excess kurtosis, making them very poor investment choices. When we compare carry trade returns using spot exchange rates vis-à-vis carry trade returns using exchange rate futures, we find that the former offer higher returns at the cost of higher risk. Furthermore, the excess kurtosis of the former are much lower than the latter, implying lower tail risks.

Table 1.

Descriptive Statistics of Carry Trade Returns.

Carry Trade	With Spot Exchange Rates				With 3-month Exchange Rate Futures				With 1-month Exchange Rate Futures
Carry Trade	Avg.	σ	Skew.	Exc.Kurt.	Avg.	σ	Skew.	Kurt.	Avg.	σ	Skew.	Kurt.
USD	0.04	0.17	1.03	1.58	0.005	0.004	0.83	2.28	0.003	0.003	0.24	3.53
GBP	0.01	0.20	0.33	0.49	0.003	0.008	−1.3	3.99	−0.002	0.005	0.14	0.96
EUR	0.04	0.20	−0.06	0.76	0.004	0	−1.2	5.06	0.003	0.005	0.14	3.44
JPY	0.03	0.28	0.63	1.26	0.001	0.009	−1.2	2.52	0.003	0.005	0.31	2.90
AUD	0.07	0.54	0.34	974
NZD	0.08	0.22	−0.38	0.31
CNY	−0.69	5.46	−3.9	18.5
HKD	0.02	−0.18	1.02	1.66
RUB	−3.3	7.93	−0.9	0.73
BRL	0.08	0.73	−14.0	282
SGD	−0.11	2.23	−13.0	185

Source: The authors.

Table 2 presents the explanatory variables that emerge as significant from three methods, namely OLS, MARS and ARIMA, with the dependent variable as carry trade, using spot exchange rate. We, first, compare model accuracy from the three measures, namely adjusted R², RMSE and DM test. If we compare the results of OLS, we do not find that consistency. ARIMA-X also does not clearly give out the variables. These performance metrics provide conflicting indications. For example, for dependent variable INRUSD carry trades, adjusted R² indicates that ARIMAX is the best model, while OLS is the worst. RMSE, on the other hand, indicates that MARS is the best model, while OLS is the worst. However, DM statistics implies that OLS is the most accurate method, while MARS provides least accuracy. Similar conflicts are observable with each of the currencies. To summarize, adjusted R² indicates that ARIMAX performs the best in all the 11 regressions we conduct. It offers the highest values of adjusted R², implying that this method explains most of the variance in the dependent variables. In terms of RMSE, 10 of the 11 regressions have the lowest RMSE from the OLS model. Thus, RMSE suggests that OLS fitted values are the worst accurate. Finally, DM test throws up a mixed result. It shows that in 5 of the 11 regressions, ARIMAX has the highest accuracy; in 4 of the 11 regressions, OLS has the highest accuracy; in one regression, MARS has the highest accuracy; and in one case, three methods have comparable levels of accuracy.

Table 2.

Significant Predictors for Carry Trade Using Spot Exchange Rate.

Sl. No.	Carry Trade	OLS	MARS	ARIMAX
1.	INRUSD	∆IR, −LIQ^FX	∆IR, σ^Eq, LIQ^FX	σ ^∆IR
		Adj R² = 0.137;RMSE = 0.043	Adj R² = 0.282;RMSE = 0.688	Adj R² = 0.899;RMSE = 0.258
		DM_OLS-MARS = −11.0; p = 0; DM_MARS-ARIMAX = 9.8; p = 1; DM_{ARIMAX_OLS} = 11.0; p = 1
2.	INRGBP	-σ^Eq	∆IR, σ^Eq, ComInd	None
		Adj R² = 0.006;RMSE = 0.048	Adj R² = 0.295;RMSE = 0.751	Adj R² = 0.801;RMSE = 0.400
		DM_OLS-MARS = 0.9; p = 0. 0.8; DM_MARS-ARIMAX = 14.0; p = 1; DM_{ARIMAX_OLS} = −12.2; p = 0
3.	INREUR	None	∆IR, σ^Eq, ComInd	None
		Adj R² = −0.009;RMSE = 0.050	Adj R²=0.304;RMSE=0.823	Adj R² = 0.826;RMSE = 0.412
		DM_OLS−MARS = −13.1; p = 1; DM_MARS-ARIMAX = 10.1; p = 0; DM_{ARIMAX_OLS} = 13.4; p = 0
4.	INRJPY	None	∆IR, σ^Eq	-σ^Eq
		Adj R² = −0.006;RMSE = 0.066	Adj R²=0.287;RMSE=0.982	Adj R² = 0.844;RMSE = 0.461
		DM_OLS-MARS = −12.6; p=0; DM_MARS-ARIMAX = 9.8; p = 1; DM_{ARIMAX_OLS} = 11.8; p = 1
5.	INRAUD	−LIQ^FX	∆IR, σ^Eq	−LIQ^FX
		Adj R² = 0.459;RMSE = 0.242	Adj R² = 0.308;RMSE = 0.767	Adj R² = 0.772;RMSE = 0.441
		DM_OLS-MARS = 0.1; p = 0.5; DM_MARS-ARIMAX = 1.1; p = 0.86; DM_ARIMAX-OLS = −1.1; p = 0.13
6.	INRNZD	−∆IR, −σ^Eq,− LIQ^FX	∆IR, σ^Eq	-σ^Eq
		Adj R² = 0.295;RMSE = 0.043	Adj R² = 0.429;RMSE = 0.574	Adj R² = 0.784;RMSE = 0.353
		DM_OLS-MARS = −7.62; p = 0.0; DM_MARS-ARIMAX=4.9; p = 1; DM_{ARIMAX_OLS} = 10.6; p = 1
7.	INRCNY	-σ^Eq, LIQ^FX	∆IR, σ^Eq, LIQ^FX	None
		Adj R² = 0.060;RMSE = 0.043	Adj R² = 0.206;RMSE = 0.701	Adj R² = 0.898;RMSE = 0.252
		DM_OLS-MARS = −13.13; p=0; DM_MARS-ARIMAX = 9.4; p = 1; DM_{ARIMAX_OLS} = 10.6; p = 0
8.	INRHKD	None	∆IR	−∆IR
		Adj R² = 0.004;RMSE = 0.039	Adj R² = 0.043;RMSE = 0.511	Adj R² = 0.734;RMSE = 0.275
		DM_OLS-MARS = −7.28; p = 0; DM_MARS-ARIMAX = 5.2; p = 1; DM_{ARIMAX_OLS} = 5.5; p = 1
9.	INRRUB	None	∆IR	–σ^∆IR
		Adj R² = 0.842;RMSE = 0.717	Adj R² = 0.126;RMSE = 1.054	Adj R² = 0.796;RMSE = 0.511
		DM_OLS-MARS=2.6; p=1; DM_MARS-ARIMAX=7.6; p=1; DM_{ARIMAX_OLS}=−7.7; p=0
10	INRBRL	∆IR	∆IR, σ^Eq, ComInd	ComInd
		Adj R² = 0.085;RMSE = 0.058	Adj R² = 0.389;RMSE = 0.946	Adj R² = 0.801;RMSE = 0.541
		DM_OLS-MARS = 4.8; p = 1; DM_MARS-ARIMAX = 9.4; p = 1; DM_ARIMAX-OLS = −9.5; p = 0.0
11	INRSGD	∆IR, σ^Eq	∆IR, σ^Eq, LIQ^FX	∆IR, σ^∆IR
		Adj R² = 0.066;RMSE = 0.034	Adj R² = 0.343;RMSE = 0.499	Adj R² = 0.835;RMSE = 0.250
		DM_OLS-MARS = 5.24; p = 1; DM_MARS-ARIMAX = 7.6; p = 1; DM_{ARIMAX_OLS} = −9.3; p = 0

Source: The authors.

Note: Dependent variables are carry trade returns from various currencies against INR, using spot exchange rates. Independent variables are volatility in exchange rates (σ^FX), interest rate difference (∆IR), volatility in interest rate difference (σ^∆IR), volatility in equity market (σ^Eq), Liquidity in foreign exchange (LIQ^FX) and commodity index (ComInd).

In the case of INRUSD, for example, OLS, deemed as the most accurate method by DM test, suggests that interest rate differential and liquidity in forex markets significantly affect INRUSD carry trade return. On the other hand, ARIMAX, having highest adjusted R², suggests that volatility in interest rate differential has a significant impact. Thus, there is lack of consensus in the variables that are found to be significant. However, the most commonly observed significant variable is interest rate differential. In fact, as per MARS method, for all currency pairs, interest rate difference emerges as a significant factor. As per OLS, in 3 out of 11 currency pairs, viz. INRUSD, INRBRL and INRSGD, interest rate difference significantly affects the carry trade returns. As per ARIMAX, only INRSGD returns are significantly impacted by this factor. This is in agreement with Brunnermeier et al. (2008) and Lustig et al. (2011). While all these are positive influences, we find INRNZD carry trade returns are negatively impacted by interest rate differential as per OLS, which has unequivocally been deemed superior to both MARS and ARIMA.

From Table 2, we find that in the case of MARS, the interest rate difference and volatility in equity market are two such variables, which affect he carry trade returns in 9 out of 11 cases, whereas the interest rate difference affects the returns in all the cases. From Table 2, RMSE gives the MARS method as the best.

Volatility in equity markets is also a significant factor affecting most carry trade returns as per MARS, in 4 out of 11 pairs as per OLS and in 2 pairs as per ARIMAX. But, contrary to the evidence provided by Brunnermeier et al. (2008) and Christiansen et al. (2011), MARS results indicate that equity market volatility positively influences carry trade returns from both developed economy currencies having low interest rates and also emerging economies having high interest rates. But OLS results contradict this finding; we note that equity market volatility negatively affects carry trade returns from GBP, CNY and SGD. Analogously, ARIMAX results indicate that the equity market volatility negatively affects carry trade returns from JPY and NZD. All these are a mix of high- and low-interest-rate currencies.

Other explanatory variables, such as volatility in interest rate differential ( $σ^{ΔIR}$ ), volatility in forex markets ( $σ^{FX}$ ), liquidity in forex markets ( ${LIQ}^{FX}$ ) and commodity index returns (ComInd) are mostly insignificant predictors of returns of carry trade using spot exchange rates.

The results related to spot exchange rate are presented in Appendix (Tables A1 –A11).

Table 3 provides the results of OLS, MARS and ARIMAX for determinants of carry trade undertaken with 3-month exchange rate future. Again, we first examine which method emerges as the best, based on the performance metrics of adjusted R², RMSE and DM test. As per adj R², we find MARS is the best, followed closely by ARIMAX, and OLS is the worst. As per RMSE, on the other hand, OLS is the best, with MARS and ARIMAX close to each other. Finally, DM test outcomes are similar to those from adj R², viz. there is no significant difference between MARS and ARIMAX, and both these methods are statistically superior to OLS. Thus, in this case, there is one agreement among the three performance metrics—MARS and ARIMAX perform equivalently.

Table 3.

Significant Variables for Carry Trade Using 3-month Exchange Rate Future.

Sl	Carry Trade	OLS	MARS	ARIMAX
1.	INRUSD	∆IR	∆IR	∆IR
		Adj R² = 0.169;RMSE = 0.003	Adj R² = 0.258;RMSE = 0.062	Adj R² = 0.232;RMSE = 0.063
		DM_OLS-MARS = 2.2; p = 1; DM_MARS-ARIMAX = −0.8; p = 0.2; DM_{ARIMAX_OLS} = −1.3; p = 0.09
2.	INRGBP	−∆IR, LIQ^FX	∆IR, LIQ^FX, σ^Eq, ComInd	∆IR, −LIQ^FX
		Adj R² = 0.419;RMSE = 0.006	Adj R² = 0.479;RMSE = 0.093	Adj R² = 0.443;RMSE = 0.096
		DM_OLS-MARS = 10.5; p = 1; DM_MARS-ARIMAX = −0.6; p = 0.3; DM_{ARIMAX_OLS} = −10.5; p = 0
3.	INREUR	∆IR, σ^Eq	∆IR	−∆IR, LIQ^FX
		Adj R² = 0.169;RMSE = 0.006	Adj R² = 0.250;RMSE = 0.109	Adj R² = 0.226;RMSE = 0.111
		DM_OLS-MARS = 10.5; p = 1; DM_MARS-ARIMAX = −0.6; p = 0.3; DM_{ARIMAX_OLS} = −10.5; p = 0.0
4.	INRJPY	∆IR, LIQ^FX, σ^Eq	∆IR, LIQ^FX, σ^Eq, σ^FX	∆IR, LIQ^FX
		Adj R² = 0.547;RMSE = 0.006	Adj R² = 0.594;RMSE = 0.100	Adj R² = 0.571;RMSE = 0.103
		DM_OLS-MARS = 2.8; p = 1; DM_MARS-ARIMAX = −1.2; p = 0.2; DM_{ARIMAX_OLS} = −1.4; p = 0.07

Source: The authors.

Note: Dependent variables carry trade returns from various currencies against INR, using a 3-month exchange rate futures. Independent variables are volatility in exchange rates (σ^FX), interest rate difference (∆IR), volatility in interest rate difference (σ^∆IR), volatility in equity market (σ^Eq), liquidity in foreign exchange (LIQ^FX) and commodity index (ComInd).

Among the explanatory variables, all three methods indicate that interest rate differential has significant predictive power for carry trades from all four currencies. This result is in congruence with Hattori and Shin (2007), Brunnermeier et al. (2008) and Lustig et al. (2011). Liquidity in foreign exchange markets is also a significant predictor of carry trade returns for all currencies, except USD. This is in accordance with the findings of Christiansen et al. (2011), but contrary to those reported by Menkhoff et al. (2012). Finally, volatility of equity markets positively affects carry trade returns of GBP, EUR and JPY, but not USD, as reported by Brunnermeier et al. (2008) and Christiansen et al. (2011). Other variables are predominantly insignificant.

Table 4.

Significant Variables for Carry Trade Using 1-month Exchange Rate Future.

Sl	Carry Trade	OLS	MARS	ARIMAX
1.	INRUSD	None	σ^Eq, σ^FX	LIQ^FX
		Adj R² = −0.028;RMSE = 0.003	Adj R² = 0.263;RMSE = 0.024	Adj R² = 0.201;RMSE = 0.025
		DM_OLS-MARS = 2.7; p = 0.99; DM_MARS-ARIMAX = −0.6; p = 0.3; DM_{ARIMAX_OLS} = −2.8; p = 0
2.	INRGBP	−∆IR, σ^Eq	σ ^Eq	-σ^Eq, σ^∆IR
		Adj R²=0.034;RMSE=0.005	Adj R²=0.091;RMSE=0.097	Adj R²=0.076;RMSE=0.098
		DM_OLS-MARS = 2.0; p = 0.98; DM_MARS-ARIMAX = −0.4; p = 0.3; DM_{ARIMAX_OLS} = −1.2; p = 0.1
3.	INREUR	None	∆IR, LIQ^FX, σ^FX	∆IR,−σ^FX
		Adj R² = 0.007;RMSE = 0.005	Adj R² = 0.015;RMSE = 0.088	Adj R² = 0.026;RMSE = 0.088
		DM_OLS-MARS = −11.96; p = 0.0; DM_MARS-ARIMAX = 0.3; p = 0.6; DM_{ARIMAX_OLS} = 11.8; p = 1
4.	INRJPY	None	∆IR, σ^Eq	σ ^Eq
		Adj R² = −0.006;RMSE = 0.065	Adj R² = 0.003;RMSE = 0.094	Adj R² = 0.041;RMSE = 0.092
		DM_OLS-MARS = 9.5; p = 1; DM_MARS-ARIMAX == −10.2; p = 0; DM_{ARIMAX_OLS} = −1.6; p=0.1

Source: The authors.

Note: Dependent variables are carry trade returns from various currencies against INR, using 1-month exchange rate futures. Independent variables are Volatility in exchange rates (σ^FX), interest rate difference (∆IR), volatility in interest rate difference (σ^∆IR), volatility in equity market (σ^Eq), liquidity in foreign exchange (LIQ^FX) and commodity index (ComInd).

The results related to 3-month exchange rate future are presented in Appendix from Table A16 through Table A19.

Table 4 exhibits the output of OLS, MARS and ARIMAX for predictors of returns of carry trades, undertaken with 1-month exchange rate futures. We, first, assess the supremacy of methods using the three performance measures. As per adjusted R², OLS performs worst; MARS is marginally better than ARIMAX in two cases of carry trades with USD and GBP, while, in the other two cases, ARIMAX is better than MARS. With respect to RMSE, OLS is consistently the best, with MARS and ARIMAX being very close to each other. DM test suggests different methods as the best for different currency carry trades; MARS and ARIMAX perform equally but significantly better than OLS in two cases of carry trades with USD and GBP; for carry trade with EUR, OLS is the best, whereas MARS and ARIMAX are equivalent; and, finally, for JPY carry trade, MARS performs the best, with OLS and ARIMAX being equivalent. One common theme that we obtain is that MARS and ARIMAX perform equivalently as per all three performance measures for most of the currency carry trades considered.

We, now, consider the explanatory variables that emerge significant. We find that volatility of equity markets comes out significant in carry trades with USD, GBP and JPY, as per MARS, in accordance with Lustig et al. (2011) and Christiansen et al. (2011). OLS does not show significance for most currencies. And, finally, ARIMAX does not throw any common explanatory variable as significant.

Furthermore, the results related to 1-month exchange rate future are presented in Appendix from Table A11 through Table A15.

To summarize, the best method to find explanatory variables for carry trades remain unresolved. However, irrespective of the method used, interest rate differential and liquidity in foreign exchange markets emerge as strongly significant explanatory variables for carry trade returns of all currencies, undertaken using spot exchange rates. Volatility of equity markets are observed to be significant in the case of fewer currencies. Thus, these variables can be argued to be robust to the choice of method employed. Analogously, for carry trade using 3-month exchange rate futures, these variables are largely significant. However, no variable emerges as a significant predictor of returns, from carry trade undertaken with 1-month interest rate futures, that is common to all methods used.

Conclusions

In this study, we explore the explanatory variables for carry trade returns using both spot exchange rates and exchange rate futures. For this, we employ three methods, namely OLS), MARS and ARIMAX. While OLS is the most commonly used method in the literature, it is able to capture only linear relationships, and it is not suitable for time-series data. So, we employ MARS that can capture non-linear relations and ARIMAX, which is most suited for capturing time-series characteristics of the data. We also employ three performance metrics to identify the best method—adjusted R², RMSE and DM Test.

We do not find much agreement between the performance metrics on the issue of which is the best method to determine explanatory variables for carry trade returns. However, irrespective of the method, we find that interest rate differential and volatility in equity market significantly affect carry trade returns with spot exchange rates. Interest rate differential affects positively, while liquidity in foreign exchange affects negatively. These variables are also significant for carry trades using 3-month exchange rate futures. But no variable emerged significant for carry trades using 1-month exchange rate futures, which was robust to all the methods employed in this study. Interest rate difference has been found to be determinant of carry trade by Lustig et al. (2011), Hattori and Shin (2007), and others. Here, we find that the volatility in equity market is predictor, for the carry trade return, using spot exchange rate. Christiansen et al. (2011) show that the volatility in equity market is directly related to the returns of carry trade. Hoffman (2012) shows that carry trade returns are high when VIX is high.

Thus, this study illustrates that traders can benefit from the volatility in equity market, as it is a clear predictor for carry trade returns. Carry trade returns over 3 months can be explained by higher interest rate differentials and lower liquidities in foreign exchange markets, but carry trade over 1 month using 1-month exchange rate futures does not have any robust explanatory factor. Managerial implication are as follows: high volatility implies high -returns from carry trade using spot exchange; lower liquidity implies high return for the traders; and interest rate differential can be a good indicator for the carry trade.

Footnotes

Acknowledgement

The authors are grateful to the anonymous referees of the journal for their extremely useful suggestions to improve the quality of the article. Usual disclaimers apply.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Appendix A

Table A19. Results of OLS, MARS and ARIMAX in Carry Trade Using Future Exchange Rate for 3 Months Between INR and USD.

Method	OLS	Probability	ARIMAX	SE	MARS	Coefficients
Bid–ask spread	−5.783	0.823	−18.812	26.349	h (interest rate difference-0.016)	0.744
Interest rate difference	0.396	<0.000	0.663	0.173
Innovation	−0.255	0.35	−0.322	0.227
Commodity index	0.108	0.174	0.036	0.074
VIX	0	0.116	0	0
I.R.D.V.	0.032	0.92	−0.001	0.272

Source: The authors’ own calculations.

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