University of Pescara. Monetary Economics. International Finance Basic Relationships and Carry Trade. Paolo Vitale

Transcription

1 University of Pescara Monetary Economics International Finance Basic Relationships and Carry Trade Paolo Vitale Academic Year

2 To explain how carry trade operates we first need to revise a series of theoretical ad empirical results in international finance. Therefore, in Section 1 we briefly discuss the basic relationships found in international finance. In Section 2 we will discuss the methods which can be used to verify such conditions and the relevant empirical literature. Section 3 is dedicated to carry trade, the excess returns it generates and the possible explanations for its profitability. 1 International Finance Basic Relationships International finance theory is built on a number of basic relationships between interest rates, exchange rates and prices. This Section presents the most fundamental of these relationships. Some of these conditions are pure arbitrage relationships such that their violation would permit agents to make risk-free non-negative returns at zero cost. Others are not pure arbitrage relationships in that their violation does not imply the existence of risk-free profit opportunities. We will examine the Covered and Uncovered Interest Rate Parity (CIP and UIP), and the Purchasing Power Parity (PPP) as the most basic conditions. However, we will mention the Real Interest Rate Parity (RIPP), also known as the International Fisher relationship, and triangular arbitrage conditions as well. Throughout this Section we will use the following notation: S t is the spot exchange rate measured as the domestic currency price of a unit of foreign currency; F t is the one-period forward exchange rate (a k period forward rate would be denoted Ft k); i t is the one period interest rate at the beginning of period t; P t is the price level at time t. Lower case letters stand for logarithms of upper case values (aside from the interest rates). Variables with a denote foreign country values. 1.1 Covered Interest Parity Consider a domestic investor holding one unit of the home currency. She/he is faced with two investment possibilities: investment in domestic and foreign bonds. Investing in the domestic asset yields a one period return equal to R d t = (1 + i t ). Investing in the foreign bond involves three steps. The investor must change her/his cash to the foreign currency, where one unit of the domestic currency yields S 1 units of the foreign currency. This money is then invested for one period at the foreign interest rate of return i. Finally, to remove any exchange rate risk, the investor sells the investment proceeds forward for the domestic currency (at rate F t ), where a forward contract is an agreement made today for obligatory exchange of funds at some specified time in the future. 1 The return on the foreign investment (in domestic 1 Because the foreign investment is immunized against exchange rate risk it is termed as covered.

3 1.2 Uncovered Interest Parity Carry Trade currency terms) is then R f t = F t S t (1 + i t ). Both the domestic and foreign investments deliver returns which are risk-free. As they both cost 1 unit of the domestic currency, absence of arbitrage requires that R d t = R f t. This yields the Covered Interest Parity (CIP) condition F t S t = 1 + i t 1 + i t f t s t = i t i t, (1.1) where the second expression in equation (1.1) is a logarithmic approximation of the first. 2. Notice that the logarithmic version of the CIP holds as an exact non-arbitrage conditions when interest rates are continuously compounding ones, as in this case the gross return on the domestic and foreign bond investment are respectively R d t = exp(i t) and R f t = (F t /S t ) exp(it ). The logarithmic version of the CIP has a very simple reading. The term f t s t is the (percentage) forward premium. Thus, equation (1.1) tells us that forward premia and interest rate differentials should be identical. The CIP to hold requires that investors can freely move funds from one country to the other and viceversa. Such condition is referred in the international finance literature as perfect capital mobility. Perfect capital mobility rules out significant transaction costs, capital controls and generally any obstacle to capital movements. This implies that exchange rates adjust instantly to equilibrate the demand and supply for stocks of international assets. If perfect capital mobility holds, violations of the CIP lead to arbitrage opportunities, in that an investor can profit from CIP deviations without bearing any risk. 1.2 Uncovered Interest Parity The uncovered interest parity (UIP), instead, is not a pure non-arbitrage condition. It uses the fact that if an investor is risk-neutral, (s)he only cares about expected returns from investment. The assumption that investors, or a significant component of investors, are risk-neutral is at the core of the notion of perfect capital substitutability. This means that the currency denomination of assets in investors portfolios is irrelevant as long as the expected returns of foreign and domestic assets are equal when expressed in the same currency. In fact, under this condition, if two bonds, one denominated in the domestic currency and the latter denominated in a foreign currency, yields the same expected rate of return, when calculated in the same currency, they will be considered as equivalent by a risk-neutral investor. Continuing our analysis of an investor who can buy foreign and domestic bonds, we assume that the investor is considering the foreign strategy. However, instead of locking in her/his return 2 This and other logarithmic approximations will be particularly useful when testing all these relationships using financial data 3

4 1.3 Forward Rate Unbiasedness Carry Trade using the one period forward rate (F t ), she/he computes an expected return using the expected one-period ahead spot rate. This yields an expected return of E t [R f t ] = E t [S t+1 ] S t (1 + i t ), where E t [S t+1 ] is the expectation of the one-period ahead spot rate formulated on the basis of the available information in period t. Assuming that all investors are risk-neutral and have homogeneous expectations, equating the expected returns from the foreign and domestic strategy yields the Uncovered Interest Parity (UIP) relationship E t [S t+1 ] S t = 1 + i t 1 + i t E t [s t+1 ] s t = i t i t. (1.2) Once again, notice that we employ a logarithmic approximation. Hence, we now get an expression which says that the expected exchange rate depreciation should be identical to the interest rate differential. In other words, the expected currency changes should offset any gain one makes on interest rates. If this is not the case, then risk neutral agents should act to profit from (and hence correct) the imbalance. Notice, however, that a violation of the UIP does not generate a risk-free profit opportunity and hence the UIP cannot be considered a non-arbitrage condition. 1.3 Forward Rate Unbiasedness Assume that all investors are risk-neutral and have homogeneous expectations. This implies that the UIP holds. Note that the CIP must always hold as a pure arbitrage condition. Equating equations (1.1) and (1.2) yields E t [s t+1 ] s t = f t s t. If we now assume rational expectations, i.e. that agents do not commit systematic errors, we can write s t+1 = E t [s t+1 ] + u t, where u t is a rational expectations error which has zero mean and which is independent of any information investors may possess when formulating their expectations. Substituting this into the prior equation yields s t+1 = ( f t s t ) + u t, (1.3) where s t+1 = s t+1 s t. Hence, use of the UIP and CIP tells us that the current one period forward premium should predict the one period change in the exchange rate (aside from a rational expectations error). If equation (1.3) were to hold, we could describe forward rates as unbiased predictors of future spot rates. Notice that equation (1.3) directly implies that s t+1 = f t + u t. 4

5 1.4 Purchasing Power Parity and Other Parity and Arbitrage Conditions Carry Trade This expression reinforces the prior point that the (one period) forward rate is an unbiased predictor of the one step ahead spot exchange rate, E t [s t+1 ] = f t. (1.4) 1.4 Purchasing Power Parity and Other Parity and Arbitrage Conditions The Purchasing Power Parity (PPP) is obtained on the assumption of arbitrage in goods markets. It assumes that there are no frictions to inter-country trade; that domestic and foreign products are perfect substitutes; and that there exists perfect information. Notice that the PPP is not a definition but an equilibrium condition, which requires that the prices of the same good in foreign and domestic markets must be identical (when measured in domestic currency terms). Implicit in this notion if that in the presence of any deviation from the PPP equilibrium condition, the exchange rate will move to clear goods market. 3 Hence, P t = S t P t p t = s t + p t. (1.5) We can also write down the PPP condition in terms of inflation rates (i.e. changes in log prices) as p t = s t + p t. (1.6) This is also known as the Relative Purchasing Power Parity (RPPP). This shows that an important implication of the PPP is that the domestic inflation rate must be identical to the foreign inflation rate adjusted for the rate of depreciation of the exchange rate. 4 Another implication pertains to the real exchange rate between two national currencies. In logarithms this is defined as q t s t + p t p t. Using the logarithmic version of the PPP it can be easily established that the real exchange rate, q t, is constant. Combining the PPP with the UIP yields another important relationship which concerns real interest rate. In particular, equating the representations for the expected exchange rate depreciation in the two countries and rearranging one gets the Real Interest Rate Parity, also known as Fisher Open, r t i t p e t+1 = i t p e t+1 r t, (1.7) where the apex e denotes an expected value. In other words, the interaction of the PPP and the 3 In fact, if P t < S t Pt, that is if domestic goods are cheaper than foreign ones, it would be convenient to purchase domestic goods and sell them in the foreign country. As funds are repatriated to the domestic country the foreign currency depreciates and the PPP is established. An inverse adjustment mechanism applies if P t > S t Pt. 4 Notice, in fact, that using a logarithmic approximation, π t+1 = p t+1 p t and πt+1 = p t+1 p t, while s t+1 is the depreciation rate for the domestic currency between time t and t

6 2.1 Basic Statistical Features of Exchange Rates Carry Trade UIP implies that the expected real interest rates must be equal across nations. 5 We complete this brief presentation of the basic international relationship by considering the triangular equilibrium condition. This is a very straightforward condition, which is best described through an example. A UK investor changes 1 to $1 using the GBP/USD exchange rate. He then changes these dollars to yen at the USD/JPY rate. Compare the amount of yen he ends up with that from directly changing sterling to yen. For absence of arbitrage, the two must be identical, GBP/USD USD/JPY = GBP/JPY. 2 Exchange Rate Parity Conditions: Empirical Evaluation The parity conditions treated in the previous Section are fundamental. Their violations in fact may conduct to arbitrage or speculative opportunities. In this Section we focus on the empirical work which evaluates the FRU and PPP conditions. As we will see, the CIP is generally regarded as holding more-or-less continuously which then implies that tests of the FRU may also be regarded as tests of the UIP. However, prior to presenting the testing frameworks and the assessment of the corresponding results, we give a basic characterization of exchange rate data in terms of their statistical features. 2.1 Basic Statistical Features of Exchange Rates We notice that the series of (post-1973) floating exchange rates invariably present the following features: i) spot and forward exchange rates are non-stationary and integrated of order one; ii) at relatively coarse sampling frequencies, exchange rate returns are uncorrelated; iii) exchange rate returns have zero-mean and present leptokurtic distributions; iv) the volatility of exchange rate is strongly autocorrelated at fine sampling frequencies. The accepted representation for finely sampled exchange rate volatility is GARCH(1,1). While the validity of some of these statements is sensitive to sampling frequency and that of others is simply doubtful, together they provide a useful benchmark representation. 5 In particular, r t r t = i t (p e t+1 p t) i t (p e t+1 p t ) = }{{} UIP = }{{} PPP s e t+1 s t (p e t+1 p e t+1 ) + (p t p t ) s e t+1 (pe t+1 p e t+1 ) }{{} 0 [s t (p t p t )]. }{{} 0 6

7 2.2 Tests of the Covered Interest Parity Carry Trade 2.2 Tests of the Covered Interest Parity These tests are generally hugely supportive for developing countries with mature financial markets where exchange rates are free to float and there are no restrictions to capital movements, as one might expect given the pure arbitrage nature of the condition. Slight deviations from the parity condition can be associated to small transaction costs or other market frictions. Thus, Frenkel and Levich (1975, 1977) examine departures from the CIP using both euro-deposit and T-bill rates for the major currency pairs in the 1970s. Allowing for transaction costs they find that very few departures exist. However, with the Euro-deposit data although evidence is less strong for the T-bill rates. Taylor (1987, 1989) uses a high-frequency data base and confirms prior evidence. He finds very few profitable departures from the CIP. Only in periods of market turbulence do significant opportunities appear. More recently, Akram et al. (2008) use very high frequency data and show that deviations from CIP are tiny and immediately removed by arbitragers. This evidence is unsurprising, in that most dealers in the foreign exchange market will construct their forward quotes by adding the interest rate differentials to the spot quotes they advertise. 2.3 Tests of the Forward Rate Unbiasedness A first generation of tests dedicated to verifying the Forward Rate Unbiasedness (FRU) were based on the relationship between the levels of the spot and forward rates. In practice, the following regression specification is considered s t+1 = α + β f t + γ X t + u t+1. (2.1) Then, if the FRU were correct, we should have that α = 0, β = 1, γ = 0, and u t u t j with t = t j, where the symbol denotes probabilistic independence. Early work assumed that γ was zero and tested other parts of the hypothesis. It was usually found that α and β were statistically indistinguishable from their hypothesized values. However statistical theory suggests that equation (2.1) is not appropriate. In fact, classical inference does not not work for non-stationary series and hence first generation tests are mis-specified. By moving to the difference representation of the FRU hypothesis we hopefully remove any problems associated with non-stationarity. The regression specification in this context is s t+1 = α + β ( f t s t ) + γ X t + u t+1. (2.2) For γ = 0 this collapses to the famous Fama s regression. Now, the FRU implies precisely the same conditions for equation (2.2) as it did for equation (2.1). The FRU is strongly rejected. In fact, 7

8 2.3 Tests of the Forward Rate Unbiasedness Carry Trade Table 1 Fama Regression: Realized Returns The Table presents OLS estimates of α and β from the regression s t+1 s t = α + β ( f t s t ) + ɛ t+1, where s t+1 s t is the spot return over the next month, f t s t is the corresponding forward premium, while f t and s t are the log of the forward rate (for maturity of one month) and the spot rate. Coefficient values indicated by are significant at the 5%-level. Sample: Jun Dec GBP is the base currency in all cases. Rate Constant (α) Slope (β) R 2 GBP/USD GBP/JPY in the estimation of equation (2.2) we usually find that β is closer to -1 than 1! In Table 1 we report the results of the estimation of Fama s regression using monthly data on GBP/USD and GBP/JPY, for the period from June 1978 to December Table 2 Fama Regression: Realized Returns The Table presents OLS estimates of β k from the regression s t+k s t = α k + β k ( f k t s t ) + ɛ t+k, where s t+k s t is the spot return over the next k months, f k t s t is the corresponding forward premium, while f k t and s t are the log of the forward rate (for maturity k) and the spot rate. The maturity k is equal to 1, 3, 6 and 12, while t-statistics, based on robust standard errors, are reported in parenthesis below coefficient estimates. Coefficient values indicated by are significantly smaller than 1 at the 5%-level. Sample: Jan Apr USD is the base currency in all cases. 1 Month 3 Month 6 Month 12 Month USD/EUR ( 2.59) ( 3.13) ( 4.29) ( 6.02) USD/JPY ( 1.19) ( 1.09) ( 1.48) ( 2.34) USD/GBP ( 1.30) ( 1.23) ( 1.36) ( 1.90) Results suggest that the constant terms tend to be negative, one significantly so. In addition, neither slope term is anywhere near its theoretical value of +1. Indeed, both terms are very negative (one significantly so). Notice the very low value of the coefficient of multiple determination. This is typical of this sort of regressions, as it is very difficult to pin down the determinants of exchange rate dynamics. Results like those reported in Table 1 are not unusual. In Table 2 we have those from Breedon 8

9 2.4 Irrationality or Risk Premia Carry Trade et al. (2015). All in all, these results appear to point towards dramatic departures from the UIP. We find that the forward premium does forecast subsequent exchange rate returns but with the wrong sign. So, when the forward premium is positive we see subsequent falls in the spot rate and vice versa. Where should we go to explain these results? The logical place to start looking is in the assumptions we have made. To recap, our main assumptions are as follows: market participants are risk neutral; market participants have rational expectations. If either of these assumptions are violated this will generate departures from the FRU. In what follows we will investigate the ability of departures from each to account for our results. Accounting for departures from risk-neutrality will require us to think about time-varying risk premia. To assess the importance of our rational expectations assumption we will need to get an idea of the expectations market participants actually develop. More importantly this result (i.e. the result that forward premia predict subsequent exchange rate changes with the wrong sign) is as yet unexplained. It remains an empirical anomaly and justifies carry trade. 2.4 Irrationality or Risk Premia We now discuss some research which evaluates the contributions of both risk premia and irrational expectations to the forward premium puzzle. This research agenda was initiated by Froot and Frankel (1989). They employ market forecasts of exchange rates. The forecasts given in the surveys they consider allow them to assess the formation of expectations against a rational benchmark and to investigate whether it is non-rational expectation formation rather than risk premia which generates the predictability we see in expected excess returns. First, take the expression for the slope coefficient in Fama s regression. Note that the market expectation of the exchange rate change must be equal to the actual change in the rate plus a forecast error (i.e. s t+1 = E t [ s t+1 ] + η t+1 ) so we can write the slope as follows β = Cov [E t [ s t+1 ], f t s t ] Var [ f t s t ] + Cov [η t+1, f t s t ] Var [ f t s t ]. Now, denote with ρ t the risk premium paid by the domestic currency, ρ t E t [s t+1 ] f t. Hence, the expected change in the exchange rate is equal to the forward premium plus the risk premium, E t [ s t+1 ] = ( f t s t ) + ρ t. 9

10 2.4 Irrationality or Risk Premia Carry Trade Substituting this into the above and rearranging slightly gives β = 1 + Cov [ρ t, f t s t ] Var [ f t s t ] + Cov [η t+1, f t s t ] Var [ f t s t ]. Finally, using once more the result that the forward premium is the difference between the expected change in the exchange rate and the risk premium we get the following Hence, by defining β = 1 Var [ρ t] Cov [ρ t, E t [ s t+1 ]] Var [ f t s t ] + Cov [η t+1, f t s t ] Var [ f t s t ]. β rp Var [ρ t] Cov [ρ t, E t [ s t+1 ]], Var [ f t s t ] β er Cov [η t+1, f t s t ] Var [ f t s t ] we get the following decomposition for the slope coefficient, β = 1 β rp β er. (2.3) From decomposition (2.3) we conclude that if risk premia vary over time, then β rp is non-zero and the slope coefficient will deviate from one. In addition, if market forecast errors are correlated with forward premia, then β er will be non-zero and again the slope coefficient will deviate from unity. Of course, both rational expectations and risk neutrality could fail, meaning that both β RE and β RP are non-zero. Froot and Frankel use market survey data which yield exchange rate forecasts. The use of survey data means that one can empirically identify η t = s t+1 E t [ s t+1 ] and armed with this information it is easy to decompose the slope coefficient as above. Using monthly survey data from The Economist from June 1981 to December 1985, the following results, which merge evidence from British pound, French franc, Swiss franc, yen and mark against the US dollar, can be obtained. Results in Table 3 are clear. All the β coefficients are less than zero as in the earlier results. Note, however, that in all cases this negativity for the slope coefficient is largely due to β er being large and positive. Hence these results suggest that, according to this decomposition, the forward premium puzzle is driven by the fact that β er is large and non-zero. The interpretation of this conclusion is that the puzzle is mostly caused by departures from fully rational expectation formation. In particular, the forecast errors of market participants appear to be strongly negatively correlated with the current forward premium. Of course, this violates rational expectation formation as the hypothesis of rational expectations (RE) requires that forecast errors be uncorrelated with any information in the market information set. On the contrary, little of the departure of the slope coefficient from unity is driven by the existence of a time-varying risk premium. Hence, one interpretation of these results is that 10

11 2.4 Irrationality or Risk Premia Carry Trade Table 3 Froot and Frankel s Decomposition The Table presents estimates of the coefficient β, alongside the coefficients β rp and β er from Froot and Frankel s decomposition from pooled monthly survey data from The Economist on GBP/USD, USD/FF, USD/CHF and USD/DEM rates from June 1981 to December Forecast horizon β β rp β er 1 month month month they indicate irrationality in aggregate market expectations. Table 4 Fama Regression: Expected Return The Table presents OLS estimates of 1 β k rp from the regression s k t,e s t = α k rp + (1 β k rp) ( f k t s t ) + ɛ rp t,k, where s k t,e s t is the expected return over the next k months the interval (t, t + k) as s k t,e denotes the median value in month t of the k months ahead exchange rate forecasts contained in the Reuters survey, ft k s t is the corresponding forward premium, while ft k and s t are the log of the forward rate (for maturity k) and the spot rate. The maturity k is equal to 1, 3, 6 and 12, while t-statistics, based on robust standard errors, are reported in parenthesis below coefficient estimates. Coefficient values indicated by are significantly smaller than 1 at the 5%-level. Sample: Jan Apr USD is the base currency in all cases. 1 Month 3 Month 6 Month 12 Month USD/EUR ( 1.87) ( 1.10) (0.74) (1.86) USD/JPY ( 1.80) ( 1.75) ( 0.72) ( 0.09) USD/GBP ( 0.64) (0.01) (0.76) (1.47) Notice, however, that more recent studies have only partially confirmed this conclusion. Consider Table 4 which contains results from Breedon et al. (2015). From this Table emerges, in particular for the USD/JPY and the USD/JPY rates, that a time-varying risk-premium contributes significantly to the bias in the Fama s regression, as it appears that β rp is larger than zero and significantly so (specifically at all horizons for the USD/JPY rate and for the monthly and quarterly horizon for the USD/EUR). Incidentally, it should be noted that these are typically carry trade crosses. 11

12 2.5 Expectations, Learning and Peso Problem Carry Trade 2.5 Expectations, Learning and Peso Problem Researchers have suggested that there may be other potential reasons for expectations appearing to be irrational in finite samples of data. Specifically, expectations formation will look irrational in finite samples of data from economies in which agents are learning about (past) changes in structure. Similar effects may occur when agents are worried about the possibility of future regime switches that are not represented in a given sample. According to the first hypothesis, a learning mechanism drives the empirical findings outlined above. According to the mechanism, an economy at some (known) past time may have switched from being in an original regime to being in a new regime. Though, agents are unsure about whether the switch actually occurred. After the switch-date, they will use new data generated by the economy to update beliefs about the state they are in. The fact that, in general, probabilities agents attach to the two states are both positive when clearly only one state can prevail means that systematic forecast errors occur until learning is complete. According to the second hypothesis, a peso problem is at work, as agents may worry about a potential future switch in the economic regime. The possibility of such a switch will affect their current expectations. If in a given sample, the switch never materializes, then in-sample expectations will appear to be systematically biased. These appeared to have been the case for the Mexican currency, the peso, in the 1980s. Lewis (1989) and Evans and Lewis (1995) demonstrate that learning effects and peso problems can generate results much like those we see from Fama s regressions. 2.6 Testing the PPP Consider the following regression equation relevant to the PPP hypothesis s t = α + β (p t p t ) + ɛ t. (2.4) Here, focus in the running of such regressions is the null: H 0 : β = 1. Results from this type of regressions tend to fall into two categories. During hyper-inflationary episodes, there is pretty good evidence that β is close to unity (Frenkel (1978)); however, using other data sets: The PPP is strongly rejected (see Frenkel (1981) for evidence on 1970s industrial countries). Equation (2.4) poses some unanswered econometric and economic issues. In particular, one my wonder whether the equation (2.4) is well specified, in that: i) exchange rates and prices are simultaneously determined; ii) the spot rate and the price levels may be non-stationary; and iii) the PPP may not hold continuously. In fact, one may actually assume that the PPP might only hold as a long run condition, so that the short run dis-equilibrium and dynamics of exchange rates and 12

13 2.7 Non-linear Adjustments to Equilibrium Carry Trade prices reflects this. These considerations suggest using a different econometric approach to verify the PPP. We have already defined the real exchange rate as the deviation from the PPP, q t = s t p t + p t. Then, a more recent strand of empirical investigation over the PPP examines the long-run evidence by examining the stationarity of the real exchange rate. The employed tests impose the null hypothesis of random walk behaviour for q t against an alternative of mean-reversion. In general, researchers employ the DF or ADF procedures (i.e. unit root tests) to assess covariance stationarity. Others use the variance ratio approach. Still, further research permits long memory in the real exchange rate dynamics. Looking mostly at post-1973 data there is very little evidence against the random walk model, so that the assumption of stationarity is rejected. See for example Meese and Rogoff (1988). Evidence is less clear cut for fixed exchange rate data. An alternative more sophisticated approach applies cointegration tests as it concentrates on the modeling of long-run economic relationships. However, the examined conditions are weaker than those of the PPP, as cointegration only requires that some linear combination of exchange rate and prices is stationary. Hence, cointegration tests amount to the relaxation of symmetry/proportionality conditions, which may occur in reality due to inclusion of non-tradeables in the price indices. The methodology employed by this type of tests is simple. We first have to confirm the order of integration of the nominal exchange rate and price indices. Then, given we have shown that they are I(1), we need to confirm that s t + µ p t + µ p t is a stationary process for some appropriate choice of the coefficients µ and µ. There are plenty of applications of these types of test. See, amongst others, Corbae and Ouliaris (1988), Kim (1990) and Mark (1990). Broadly speaking, they conclude that: i) cointegration seems more tenable during fixed exchange rate periods; ii) tests based on CPI s lead to fewer rejections of the null of no cointegration than tests based on WPIs; iii) rejections of the null hypothesis of no cointegration is most frequent for totally unrestricted systems (i.e. for free µ and µ ); and iv) constraining the cointegrating coefficients on price series to be identical in magnitude but of different sign leads to fewer rejections (although more than for stage 2 tests). All in all there is still little definitive evidence. One might say that evidence of stationarity is however questionable. 2.7 Non-linear Adjustments to Equilibrium Some models introduce transactions costs to international trade. They lead to non-linear dynamics for the real exchange rate. Close to the equilibrium (i.e. for q t = 0) little mean-reverting tendencies 13

14 2.7 Non-linear Adjustments to Equilibrium Carry Trade are observed in the real exchange rate as arbitrage opportunities fall within the transactions costs band. Outside such a band, however, mean reversion is much stronger due to the profitable of goods arbitrage. To test this thesis we need a non-linear model of real exchange rate dynamics that adequately captures real variations. Recent literature has explored this idea using various non-linear models (Obstfeld and Rogoff (1998), Michael, Nobay and Peel (1997)). This empirical analysis concludes that: i) non-linearities are generally present; ii) for small deviations from their unconditional mean, real rates behave like non-stationary processes; and iii) for large deviations, they are strongly mean-reverting. Hence, non-linear models allow us to refine prior evidence. In particular, the preponderance of observations close to the equilibrium implies that linear models generally indicate non-stationarity. In fact, real exchange rates are globally stable. 3 Carry Trade Let us consider an investor taking a long-foreign/short-domestic position. This entails shorting domestic bonds, i.e. borrowing in the domestic currency, and longing the foreign ones. In particular, assume you borrow one unit of the domestic currency to purchase 1/S t units of the foreign one. This yields at the end of the year (1 + i t )(1/S t) units of the foreign currency. As the foreign investment is not covered the gross return on the long position is (1 + i l )(S t+1/s t ), while the cost of the short position is 1 + i t. The excess return, X t+1, on the long-foreign/short-domestic position is X t+1 = (1 + i t ) S t+1 S t (1 + i t ), which using a logarithmic approximation can be written as x t+1 = (i t i t ) + (s t+1 s t ). (3.1) Using the CIP and Fama s regression, this can also be written as x t+1 = }{{} CIP ( f t s t ) + (s t+1 s t ) }{{} = α + (β 1) ( f t s t ) + u t+1. Fama For E t [u t+1 ] = 0, the expected value of the excess return on the foreign currency is E t [x t+1 ] = (β 1) ( f t s t ). Suppose, as found for the USD/JPY cross in Table 1, that α = 0 and β = 3.5. In addition, assume the interest rate in the funding (domestic) currency, the Japanese yen, is on average lower than that for the investing (foreign) currency, the US dollar. Say that i t = 0.01 and that it = 0.03, so that the interest rate differential between the US and Japan is 2%, i.e. it i t = Hence, the forward premium is negative and equal to minus 2% ( f t s t = i t it = 0.02). Then, E t [x t+1 ] = 0.09, 14

15 3.1 The Variability of the Expected Excess Returns Carry Trade i.e. 9%! This implies that the standard carry trade position which prescribes shorting the low interest rate currency, the Japanese yen, and longing the high interest rate currency, the US dollar, yields a fairly large expected return. In general, this simple numerical example suggests that whenever Fama s beta is negative, so that the forward rate is a biased predictor of the future spot rate and the forward premium indicates the wrong direction for the future variation in the spot rate (when f t s t < 0 the foreign currency appreciates rather than depreciates!), carry trade is on average profitable. In fact, the expected profit from the carry trade position is quite large, in that the speculator gains on both the positive interest rate differential that exists between the investing and the funding currency and from the appreciation of the high interest rate currency between time t and t + 1. In the example above the investor borrows in yen at 1% to invest in dollar at 3%, while according to the Fama s regression the US dollar is expected to appreciate by 7% = β( f t s t ) = 3.5 2%, E t [x t+1 ] = (i t i t ) }{{} 2% + (E t [s t+1 s t ]). }{{} 7% 3.1 The Variability of the Expected Excess Returns The magnitude of the slope coefficient, β, in Fama s regression tells us something about the variability of the expected excess return. In fact, the slope coefficient in Fama s regression can be written as follows β = Cov [ s t+1, f t s t ] Var [ f t s t ]. (3.2) Moreover, notice that the excess return to a foreign investment over a domestic one can also be written as x t+1 = s t+1 f t, where the equality follows by substituting the CIP condition into equation (3.1). Now, ex-ante an investor does not know the excess return and hence will focus on its expected value, which is x t+1 E t [x t+1 ] = E t [s t+1 ] f t = E t [ s t+1 ] ( f t s t ). This implies that Var [ x t+1 ] = Var [E t [ s t+1 ]] + Var [ f t s t ] 2 Cov [E t [ s t+1 ], f t s t ]. If we combine the expression for the slope coefficient in Fama s regression and the expression for the variance of the expected excess return we get the following Var [ x t+1 ] = Var [E t E t [ s t+1 ]] + (1 2 β) Var [ f t s t ]. 15

16 3.2 Empirical Evidence of Carry Trade Profitability Carry Trade Therefore, when β < 1 2 (as the preceding regressions suggest strongly), we find that Var [ x t+1 ] > Var [E t [ s t+1 ]]. Hence, to compound the puzzle, not only are expected excess returns non-zero and predictable (so that carry trade is profitable), but they are also very volatile when compared to expected exchange rate changes. 3.2 Empirical Evidence of Carry Trade Profitability In general, the large profits associated to carry trade are possible because the UIP is violated. If this held, the larger interest rate in the US should compensate the average depreciation of the US dollar vis-a-vis the Japanese yen. In addition, notice that carry trade is a speculative activity not an arbitrage one, as it is not risk-free. The actual excess return on the carry trade position depends on the future spot rate (the excess return on the foreign currency in equation (3.1) depends on s t+1 ), which is uncertain. Table 5 Carry Trade: US Investor s Excess Returns Table reports the mean of the real excess returns (in percentage points) and the Sharpe ratio for a US investor. The portfolios are constructed by sorting currencies into eight groups at time t based on the nominal interest rate differential at the end of period t 1. Portfolio 1 contains currencies with the lowest interest rates. Portfolio 8 contains currencies with the highest interest rates. The Table reports annual returns for annually rebalanced portfolios. Source: Lusting and Verdelham (2007). Port mean SR mean SR In Table 5 we report results from Lustig and Verdelham (2007) on carry trade. They calculate the mean excess return and the Sharpe ratio for 8 different currency portfolios sorted from the lowest to the highest interest rates. The portfolios are not stable overtime, as they are rebalanced monthly to accommodate changes in interest rates. Shorting portfolio 1 and longing portfolio 8 in the years from 1970 to 2002 should have generated an average yearly return of 4.47% (while replacing portfolio 8 with portfolio 7 would have generated an average excess return of 6.93%!). Notice, however, that the Sharpe ratios are positive but not particularly large. This is a remainder of the risks associated with carry trade. 16

17 3.3 Carry Trade Profitability, Reversal Risk and Other Risk Factors Carry Trade There is plenty of empirical evidence that carry trade returns present negative skewness, suggesting that such activity is subject to reversal risk, i.e. to the risk that sudden deep reversals in returns occur. In this respect see Table 6 derived from Brunnermeier et al. (2008). Here carry trade portfolios involve a limited number of currencies and are rebalanced either quarterly or weekly. In terms of average returns and Sharpe ratios this more recent dataset confirms the Lustig and Verdelham s and others findings on carry trade profitability. In particular, the Sharpe ratios reported by Brunnermeier et al. (2008) are even more favorable that those observed in Table 5. This is because they pertain to a different and shorter dataset corresponding to a golden era for carry trade. However, a crucial characteristic of carry trade which emerges from Table 6 is the large and negative skewness of its returns. Table 6 Summary Statistics for Carry Trade Portfolio Returns Table reports the mean, standard deviation, skewness, kurtosis and Sharpe ratios of portfolio long into 1, 2, and 3 currencies with the highest interest rates in the beginning of each week/quarter and short in the 1, 2, and 3 currencies with the lowest interest rates. Weekly data are from 1992 to 2006, while quarterly data are from 1986 to Source: Brunnermeier et al. (2008). 1 Long, 1 Short 2 Long, 2 Short 3 Long, 3 Short Weekly Quarterly Weekly Quarterly Weekly Quarterly Average return Standard deviation Skewness Kurtosis Annualized Sharpe ratio Carry Trade Profitability, Reversal Risk and Other Risk Factors The negative skewness of carry trade returns is the consequence of currency reversals, as manifest in Figure 1. Here the Deutsche Bank carry-trade index is plotted alongside the S&P 500 index. The DB carry-trade index tracks the performance of a global carry trade portfolio. As it is pretty evident the index is subject to sudden and deep reversals after long periods of appreciation (in the FX jargon carry trade returns are said to go up the escalator and come down in the lift ). A clear example of this phenomenon is evident in the aftermath of the financial crisis in 2008, as investors flighted to safety and unwound their carry trade positions. Breedon et al. (2015) observe that carry trade should generate a substantial proportion of trading in FX markets. In fact, if the US interest rate is larger than the Japanese one we should observe an excess of dollar-buy orders over dollar-sell ones in the USD/JPY market. In general, if carry trade motivates trading activity in FX markets the flow of orders, o t+1, in the interval (t, t + 1) (i.e. the difference between buy- and sell-orders in the market for the foreign currency) should be 17

18 Subscribe to News the market. This is as natural a cycle of human behavior as can be found. However, like all systems, this one can be distorted. Currently, the Federal Reserve and other global central banks are fully engaged in stimulus efforts that are indirectly targeting the fear element of the equation. With a large and expanding safety net, traders are encouraged to increase their exposure to risky trades and leverage. Benchmarks like the S&P 500 have reaped the reward. Yet, the support cannot last forever Investors top concern moving forward is establishing when the support ends. And, as a forward looking market; a shift in optimism won t occur simply when the first rate hikes and asset sales (Treasuries, government bonds, mortgage-backed debt, shares, etc) begin, but rather when speculators taking 3.3 Carry Trade Profitability, Reversal Risk and Other Risk Factors Carry Trade advantage of this unique market participant s presence move to exit ahead of the central banks. That is the current market concern and the interest in the Taper. With the S&P 500 driving to fresh record highs, it would seem that there is little to no concern, but what should we make of the chart below? This is a chart of the S&P 500 (on the left axis) and the Deutsche Bank Carry Trade Harvest Index (right axis). What is the carry trade? It is a trade in the Foreign Exchange market whereby you go long a highyielding currency and Figure short a low-yielding 1: The DB counterpart Carry-Trade like going tracking long AUDUSD Index or NZDJPY in an effort to collect the differential between the two. That daily roll is steady income, but traders are still exposed to the rise and fall in the exchange rate itself. The carry trade is essentially the buy-and-hold trade approach of the FX world. Thereby, it should rise and fall under the same conditions and at the same time as other clear risk barometers like US equity benchmark indexes. As we can see in the chart, the correlation over time is exceptionally strong as would be expected. However, as of late, there has been a strong deviation between the two. That stands to brings us to the all important question: which of these market measures is giving the more accurate gauge of investor sentiment? function of the forward premium, f t s t. In particular, for f t s t negative (so that the foreign interest rate is larger than the domestic one, i t > i t ), we should expect to observe a positive order flow (as carry trade entails buying the foreign currency for the domestic one there will be more buy- than sell-orders). 6 Analytically, we could assume that a negative coefficient µ exists such that /Which_is_Correct_Carry_Trade_Reversal_or_SP_500_Continuation.html o t+1 = µ ( f t s t ) + n o t+1, Pagina 1 di 6 where n o t is a component of order flow independent of carry trading. Breedon et al. (2015) present the results of the regression of the flow of orders on the forward premium in the in USD/EUR, USD/JPY and USD/GBP markets. They are reported in Table 7. The Table shows particularly strong evidence of carry trade activity in the USD/EUR market, as the coefficient µ is significantly smaller than zero at all horizons. In addition, the coefficients of multiple determination suggest that for this cross carry trade motivates a substantial portion of FX transactions. There is also limited evidence of carry trade activity in the USD/JPY market, but not for the USD/GBP one. Indeed, while the former is a well-known carry trade cross, the latter is not typically associated with carry trade. one. 6 One could observe that order flow is a measure of the excess demand for the foreign currency vis-a-vis the domestic 18

19 3.3 Carry Trade Profitability, Reversal Risk and Other Risk Factors Carry Trade Table 7 The Impact of the Forward Premium on Order Flow This Table reports OLS-estimates of a linear regression of order flow o t+k,k on the forward premium, f k t s t, o t+k,k = α k o + µ k ( f k t s t ) + n o t+k,k, with k = 1, 3, 6, 12 months. The order flow variable o t+k,k is cumulate order flow between month t and t + k; the forward premium is f k t s t, where f k t and s t are the log of the forward rate (for maturity k) and the spot rate observed at the beginning of month t. t-statistics are based on robust standard errors. Sample: Jan Apr USD is the base currency in all cases. Source: Breedon et al. (2015). µ k t-stat R 2 µ k t-stat R 2 µ k t-stat R 2 Horizon USD/EUR USD/JPY USD/GBP 1 month months months months Table 8 The Impact of Order Flow on the Skewness of FX Returns The Table reports the OLS-estimates from the regression of the average skewness of daily FX returns in the period (t k, t), ζt k, on order flow, the forward premium and implied volatility, ζ k t = α k sk + γk sk o t k,k + β k sk ( f k t k s t k) + δ k sk ImpVolk t k + ɛsk t,k, with k = 1,3,6, 12 months. The order flow variable o t k,k is cumulate order flow between month t k, and t; the forward premium is ft k s t, where ft k and s t are the log of the forward rate (for maturity k) and the spot rate observed at the beginning of month t; ImpVolt k denotes the k months ahead implied volatility at the end of month t. Robust t-statistics are in parenthesis. Sample: Jan Apr USD is the base currency in all cases. Source: Breedon et al. (2015). USD/EUR USD/JPY γ k sk β k sk δ k sk γ k sk β k sk δ k sk 1 Month (-3.14) (0.61) (0.14) (-0.94) (2.11) (0.11) 3 Month (-2.61) (0.46) (2.22) (-1.30) (3.60) (1.61) 6 Month (-3.00) (0.79) (1.21) (-3.15) (5.91) (4.27) 12 Month (-4.26) (0.94) (-0.03) (-5.51) (6.44) (4.21) 19

20 3.3 Carry Trade Profitability, Reversal Risk and Other Risk Factors Carry Trade Breedon et al. (2015) then argue that the trading activity in FX (the flow of orders) induced by carry trade per se could augment the risk of currency reversals. A way to test this thesis consists of regressing the skewness of FX returns on lagged order flow. In particular, if carry trade increases currency reversal risk, then, as speculators accumulate dollars vis-a-vis the yen (as they purchase more dollar than they sell against yen) this should translate into a larger probability of negative skewness for the corresponding dollar return. Therefore, regressing the skewness of the FX returns on the flow of order should yield a significantly negative slope coefficient. Breedon et al. (2015) run this regression for the two carry trade crosses identified in Table 7. The implied volatility and the forward premium are inserted among the regressors as control. The results of Breedon and his coauthors are reported in Table 8. The coefficient on order flow is negatively signed and significant for all horizons except the 1- and 3-month horizons for USD/JPY. This is consistent with the thesis that carry trade increases currency reversal risk and generates the negative skewness which plagues its returns. Other researchers, notably Burnside et al. (2006) and Menkhoff et al. (2012), have identified several sources of risk, such as liquidity and volatility, which may explain carry trade returns. Thus, Menkhoff et al. (2012) argue that a global FX-volatility risk factor can explain the cross-section of returns of carry portfolios. Burnside et al. (2006) instead claim that among several alternative risk factors the most successful one in explaining carry trade returns is associated with currency skewness. All in all these contributions suggest that in FX markets there is no free lunch, in that the excess returns associated with carry trade are justified by some priced source of risk. Exercises Exercise 1 (Synthetic Forward) Suppose you want to sell forward $100 million you receive in one year time, but no market for forward contracts exists. How, can you replicate it? Exercise 2 (Revision Question) 1. Define the concept of perfect capital mobility. 2. Consider two countries, the home and foreign countries. Let S t denote the spot rate in year t, i.e. the number of units of the domestic country to purchase one unit of the foreign one. Let i t and it denote the domestic and foreign one-year interest rates in year t. Let F t denote the forward spot rate for the one year maturity in year t. Define the covered interest rate parity, CIP). Prove that it holds under perfect capital mobility. Exercise 3 (Challenging Question) Consider the notation in Exercise 2. Assume frictions exist in international capital markets, such 20

21 3.3 Carry Trade Profitability, Reversal Risk and Other Risk Factors Carry Trade as transaction costs. Specifically assume a domestic investor, i.e. resident in the home country, in year t can purchase (sell) the foreign currency for the domestic at the ask (bid) price St a (Sb t ), where because of transaction costs in the spot FX market St a > St b. Similarly in the forward markets, in year t she can purchase (sell) at the ask (bid) price Ft a (Ft b), where Fa t > Ft b. In the home capital market the investor can borrow (lend) at the ask (bid) interest rates i a t (ib t ), where it a > ib t. Similarly, in the foreign capital market she can borrow (lend) at the ask (bid) interest rates i a t (it b ), where it a > it b. Explain under which condition(s) the investor will not find it profitable to take a short position in the home country to take a covered long position in the foreign country. In other words how the CIP is modified when you need to take into account transaction costs you observe in capital and FX markets? Exercise 4 (Revision Question) Consider the following Table, which reports the results from the following regression S t+k S t S t = α + β Fk t S t S t + U t+k, where S t is the spot rate in month t, F k t is the corresponding forward rate for delivery in month. Data cover the period between for the following crosses : GBP/JPY, GBP/CAD, GBP/CHF, GBP/USD. Spot and Forward Rates Maturity (k) 1 month 3 months α β R 2 α β R 2 GBP/JPY (0.005) (0.924) (0.014) (1.017) GBP/CAD (0.002) (0.803) (0.005) (0.858) GBP/CHF (0.003) (0.533) (0.008) (0.546) GBP/USD (0.002) (0.880) (0.006) (0.865) In parentheses the coefficient s standard errors. Source: Burnside et al. (2006), The Returns to Currency Speculation, NBER Working Paper Explain the logic of the regression model. 2. Which conclusione can you draw from the table about the forward rate unbiasedness hypothesis? 21