A Sentiment-Based Explanation of Forward Premium Puzzle

Transcription

1 A Sentiment-Based Explanation of Forward Premium Puzzle Jianfeng Yu University of Minnesota Abstract This paper presents a sentiment-based explanation for forward premium puzzle. Agents over- or underestimate the growth rate of the economy. When domestic investor sentiment is higher than foreign sentiment, the interest rate at home is higher than foreign interest rate. At the same time, when agents prefer earlier resolution of uncertainty, the home currency is expected to appreciate due to the mean reversion of investor sentiment. Thus, the model can account for forward premium puzzle. This paper further empirically compares the sentimentbased explanation with those based on habit-formation and long-run risk. The evidence in the data strongly supports for a sentiment-based explanation. Surplus ratio predicts changes in exchange rates, but with an opposite sign as implied by the model. Consumption volatility has very weak power to predict returns on foreign exchange. By contrast, investor sentiment has significant power to predict both exchange rate changes and returns on foreign exchange (relative to U.S. dollar) across 19 industrial countries. In addition, forecasts of exchange rates based on sentiment easily beat random walk forecasts - with a smaller mean square error than that of random walk forecasts for 18 out of 19 industrial countries at a 6-month horizon. JEL Classification: G12, F31, G14, G15, Keywords: investor sentiment, forward premium puzzle, long-run risk, habit-formation First Draft: November I would like to thank Terry Odean and Stavros Panageas for comments. All errors are my own. jianfeng@umn.edu Address: Department of Finance, Carlson School of Management, University of Minnesota, Minneapolis, MN Phone: (612)

2 1 Introduction There have been extensive literature examining the impact of investor irrationality on the stock market. Previous studies document that investor sentiment has most influence on the stocks which are hard to value and hard to arbitrage (e.g. Baker and Wurgler (2006)). The current paper explores the role of investor sentiment in another major market, the foreign exchange market. According to the uncovered interest rate parity (UIP), the expected change in exchange rates (foreign price of domestic currency) should equal the interest rate differential between foreign and domestic risk-free rate. Therefore, the regression coefficient of future changes in exchange rates on interest rate differentials should equals one. However, Fama (1984) and subsequent studies consistently find a negative coefficient in such regressions. That is, a raise in foreign interest rate forecasts an appreciation of foreign currency. This is referred as forward premium puzzle in the literature. The forward premium puzzle has inspired a vast empirical and theoretical work dissecting this puzzle. In this paper, I depart from rational expectation. The agents can be optimistic or pessimistic about the economic fundamental. In particular, the agent over- or underestimate the mean growth rate of consumption endowment. When the domestic agent is optimistic (i.e., the investor sentiment is high at home), the interest rate at home is higher than foreign interest rate. At the same time, the home currency is expected to depreciate because of the mean reversion of investor sentiment. As a results, through the channel of investor sentiment, the model can account for forward premium puzzle. In a complete market, the logarithm changes in exchange rates, measured in unit of foreign goods per domestic goods, equals the difference between the (log) stochastic discount factors for domestic and foreign countries. With preferences on early resolution of uncertainty, the pricing kernel is negatively related to the return on consumption claim. When sentiment is high, the consumption claim is over valued. As a result, the return on consumption claim is expected to be low due to the mean-reverting feature of investor sentiment. All together, high investor sentiment in domestic country leads to a higher future spot exchange rate. Given the positive relation between riskfree rate and investor sentiment, the model implies a negative association between interest rate differentials and exchange rate changes, i.e., the forward premium puzzle. Intuitively, in response to a rise in domestic investor sentiment, the equilibrium foreign price of the domestic currency drops immediately, so that relative to the new level today, the domestic currency is expected to appreciate tomorrow. At the same time, an increase in domestic sentiment raises the yields on domestic bonds and decreases the interest rate differentials between foreign and domestic countries. All together, when domestic sentiment is relatively high, an appreciation of the domestic currency is expected, and at the same time push down the interest rate differentials. This can qualitatively account for 2

3 the violation of the UIP condition in the data. The external habit-formation model and long-run risk model are two major workhorses in asset pricing. Verdelhan (2009) proposes an explanation of the forward premium puzzle based on habit-formation and Bansal and Shaliastovich (2009) provides an explanation based on long-run risk. I empirically compare the sentiment-based explanation with those based on habit-formation and long-run risk. In particular, I take the models one step further by investigating directly the mechanisms generating the forward premium puzzle. In habit-based model, a) the interest rate differential (between foreign and domestic countries) is negatively associated with surplus ratio in domestic country, b) changes in future exchange rates are positively correlated with home surplus ratio, and c) returns on foreign exchange is negatively associated with home surplus ratio. In longrun risk model, the three implications are a) the interest rate differential is positively associated with the volatility of consumption in domestic country, b) changes in future exchange rates are negatively correlated with home consumption volatility, and c) returns on foreign exchange is positively associated with home consumption volatility. However, the evidence in the data indicates that a) the interest rate differential is positively associated with surplus ratio in domestic country, and positively associated with the volatility of consumption in domestic country. b) changes in future exchange rates are negatively correlated with home surplus ratio and uncorrelated with home consumption volatility and c) returns on foreign exchange is positively associated with home surplus ratio and insignificantly associated with home consumption volatility. By contrast, investor sentiment has significant power to predict changes in exchange rates (relative to U.S. dollar) across 19 industrial countries. In addition, investor sentiment predicts returns on foreign exchange, even after controlling for forward premium and other predictors. Specifically, in a dataset with bilateral exchange rates between 19 industrial countries and the U.S., I show that foreign currencies appreciate significantly in subsequent months if the current U.S. investor sentiment is low. The results are strikingly consistent across all the 19 industrial countries in post Bretton-Woods periods. It is well known that forward premium (i.e. the interest rate differential) strongly predicts returns on foreign exchange. This paper documents that investor sentiment significantly improves the predictability of returns on foreign exchange. The R 2 of a regression which both forward premium and investor sentiment are included is more than doubled for 13 out of 19 countries compared with that of a regression in which forward premium is the only predictor. Meese and Rogoff (1983) show that macroeconomic variables have low power of forecasting exchange rate out of sample especially at short horizons (one year or less), though some of them can predict exchange rates in sample. While subsequent studies find evidence of exchange 3

4 rate predictability with macroeconomic variables, most results remain fragile or quantitatively moderate. 1 However, this paper finds strong evidence that investor sentiment forecasts exchange rates out of samples. At a 6-month horizon, out-of-sample forecasts based on investor sentiment have smaller mean squared errors than that of the random walk forecasts for 18 out of 19 exchange rates. The evidence of expectation-driven fluctuations in the foreign exchange market has policy implication that is absent in the stock market. The exchange rate differs from other asset prices in that exchange rate movements can change the relative prices of imports and exports when prices are fixed in the exporter s currency in the short run. A major benefit of exchange rate flexibility is that it facilitates the adjustment of the relative price in response to a country specific real shock (Friedman, 1953). However, Devereux and Engel (2007) argue that the expectation-driven exchange rate fluctuations may conflict with this conventional argument for the flexible exchange rate regime. If exchange rate movements are mainly driven by changes of expectations, rather than current demand and supply conditions in goods market, exchange rate fluctuations in this case may cause price distortions in the goods market rather than facilitating price adjustments in response to country specific real shocks. This is especially true when exchange rates are driven by irrational changes of expectations as documented in this paper. The model has implication on the role of investor sentiment in bond yield as well. When investor sentiment is high on the stock market, investors want to borrow to invest in the stock market. Through this channel, they drive up interest rate. As a result, a lower bond yield is expected in the future when investor sentiment finally return to its steady state. This paper empirically documents that, indeed, investor sentiment affects stock market and bond market hand in hand. In particular, high investor sentiment significantly predicts a negative change in bond yields. The contribution of this paper is twofold. First, this paper presents a sentiment-based model which can account for forward premium puzzle. Compared with the explanations based on leading asset pricing models, I find strong direct support for the sentiment channel in the data. Instead, there is only weak empirical evidence on the other two channels. Second, this paper document that investor sentiment has substantial predictive power in both currency and bond markets. Therefore, this study suggests that investor sentiment is a pervasive predictor in various asset markets and provide a natural out-of-sample support for the important role of investor sentiment in stock market noticed by previous studies. There is considerable earlier work on forward premium puzzle. Apart from Bansal and Shaliastovich (2009) and Verdelhan (2009) mentioned above, numerous studies attempt to explain 1 For instance, see Mark (1995), Groen (2000), Mark and Sul (2001), Kilian and Taylor (2003), Faust, Rogers, and Wright (2003), Molodtsova and Papell (2009), Wang and Wu (2008) among others. See Cheung, Chinn and Pascual (2005) for a recent comprehensive study. 4

5 this puzzle under rational expectations. Notable recent papers include Alvarez, Atkeson, and Kehoe (2009) in a segmented market, Bacchetta and Van Wincoop (2009) in a rational inattention framework, Burnside, Eichenbaum, and Rebelo (2007) with microstructure frictions, Colacito and Croce (2005) in a Bansal and Yaron (2004) long-run risk setting, Farhi and Gabaix (2008) in a Rietz (1988) rare disaster framework, and Lustig and Verdelhan (2007) in a cross-sectional analysis of foreign exchange portfolios. Earlier equilibrium-model based efforts to explain the forward premium puzzle include Backus, Gregory, and Telmer (1993) and Bekaert (1996). A number of earlier papers have provided insightful analysis on the role of investor irrationality in foreign exchange markets. An early application of irrationality to currency markets is provided by Frankel and Froot (1990). Mark and Wu (1998) develop a model in the noise trading setting of DeLong, Shleifer, Summers, and Waldmann (1990): traders overweigh the forward premium when predicting future changes in the exchange rate. Gourinchas and Tornell (2004) provide an explanation base upon a distortion in investors beliefs about the dynamics of the forward premium. Han, Hirshleifer and Wang (2007) offer an explanation based on investor overconfidence. The rest of the paper is organized as follows. Section 2 presents the sentiment-based model. Section 3 compares the sentiment-based explanation with that based on habit-formation and longrun risk. Section 4 presents the results on the role of investor sentiment in foreign exchange market. Finally, Section 5 concludes. 2 A Model with Investor Sentiment In this section, I present a simple model of exchange rate with investor sentiment. The model is specified such that it is observationally equivalent with the Bansal-Yaron economy. As a result, the model is very tractable. In the current model, a separate pure exchange economy for the home country and the foreign country are specified. Consumption in the two countries is exogenously given To highlight the role of investor sentiment on forward premium puzzle, the model is abstract from money and inflation, and heterogenous belief is the only deviation from a standard two country model with recursive preferences. The model relies on investor sentiment alone to reproduce forward premium puzzle where financial markets are complete. 2.1 Preferences and Consumption Dynamics Assume that consumption stream, C t, follows a geometric Brownian motion under objective probability belief P : dc t C t = µdt + σdb t 5

6 If the agent has belief ˆP. By Girsanov theorem, there exists a process xt such that d ˆP dp t = ( t exp 0 x sds 1 ) t 2 0 x2 sds. Hence, in the agent s perception, dc t C t = (µ + x t ) dt + σd ˆB t where ˆB t is a standard Brownian motion. Therefore, under the perception of the agent, the growth rate of consumption is µ+x t rather than µ. As a result, x t can be interpreted as investor sentiment. When x t is positive, the agent is optimistic, vise versa. The origin of the sentiment, x t, is not modeled. It could be derived from learning with overconfidence (e.g., Scheinkman and Xiong (2003), and Xiong and Yan (2009)). Modeling the origin of sentiment do not add any economic insight for the purpose of this study, and it only complicates the analysis. Therefore, the sentiment process is specified exogenously. For easier presentation, I consider a discrete-time economy with infinite horizons developed by Bansal and Yaron. However, the interpretation on the state variable, x t, is different here. For simplicity, I assume that the true consumption growth rate, g t log (C t ) log (C t 1 ), is an i.i.d. process. However, the representative agent in the economy is sentimental and has a subjective belief which is different with the objective belief. Under his subjective belief, the mean growth rate of the economy, x t, is time-varying and the dynamics of the consumption growth is the following 2, g t+1 = µ g + x t + σ g η t+1 (1) In Bansal and Yaron (2004), x t is the true expected consumption growth rate. However, in this paper, the true consumption growth is assumed to be i.i.d., for simplicity, and x t measures investor sentiment. The investors preferences over uncertain consumption stream C t can be described by the Epstein-Zin recursive utlity function (e.g. Epstein and Zin (1989)) [ U t = (1 δ) C 1 γ θ t + δ (Êt U 1 γ t+1 ) 1 ] θ 1 γ θ, where Ê t ( ) is the expectation under the agent s subject belief conditional information up to time t, the parameter 0 < δ < 1 is the time discount factor, γ 0 is the risk-aversion parameter, ψ 0 is the intertemporal elasticity of substitution (IES) parameter, and the parameter θ = 1 γ. The 1 1 ψ sign is θ is determined by the values of risk aversion and IES. When risk aversion is larger than the reciprocal of the IES, the agent prefers early resolution of uncertainty of consumption path. Hence, 2 An alternative assumption is that the true expected growth rate is the same as Bansal and Yaron (2004). On top of that, there is a misperception process. This way, the model has two state variables, and the model can match more moments of the data. However, it does not add more intuition for the purpose here. 6

7 these preferences allow for agent s preference for the timing of the resolution of uncertainty. As in long-run risk model of Bansal and Yaron (2004), I assume that the agent prefers early resolution of uncertainty. 2.2 The Stochastic Discount Factor As shown in Epstein and Zin (1989), the logarithm of the intertemporal marginal rate of substitution (IMRS) is given by log (m t+1 ) = θ log δ θ ψ g t+1 + (θ 1) r a,t+1 (2) where r a,t+1 is the logarithm of the gross return on as asset that delivers aggregare consumption as its dividends each period. For any continuous return r t+1 = log (R t+1 ), including the one on wealth portfolio Ê t [exp (m t+1 + r t+1 )] = 1 (3) As shown below, with log-linear approximation, the return on wealth is solved explicitly by replacing r t+1 with r a,t+1 in equation (3). Hence, the IMRS is obtained explicitly. 2.3 Foreign Exchange Market A foreign country is introduced into the model. A similar setup is used in Colacito and Croce (2005) and Bansal and Shaliastovich (2009). For tractability, complete symmetry is imposed and all the model parameters are identical across countries. Under the foreign agent s subjective belief, mean growth rate of the foreign economy, x t, is time-varying and the dynamics of the consumption growth, g t, is the following, g t+1 = µ g + x t + σ gη t+1 The discount factor used to price assets denominated in foreign currency is given by, log ( m ) θ t+1 = θ log δ ψ g t+1 + (θ 1) ra,t+1 (4) the where the preference parameter are assumed to be the same at home and foreign countries, ra,t+1 is the logarithm of return on wealth. Let E t be the level of spot rate at the end of month t for the foreign country and it is defined as the foreign price per U.S. dollar. For notational convenience, denote e t = log(e t ). Under frictionless complete market, the exchange rate e t are linked to domestic and foreign discount factors m t and 7

8 m t by the following equation 3 e t+1 e t = log m t+1 log m t+1 (5) To understand the relation between investor sentiment and exchange rate, more structure on the pricing kernels is needed. Recall that the pricing kernel is given by log(m t+1 ) = θ log δ θ ψ g t+1 + (θ 1) r a,t+1 (6) When domestic sentiment is temporarily high at time t, the claim on consumption is over-valued currently. Therefore, the expected return on wealth is low since sentiment eventually returns to its long-run mean. This effect should be stronger at longer horizons since investor sentiment is persistent. Notice that θ 1 < 0 when risk aversion and IES are larger than 1, the expected value of domestic discount factor m t+1 is relatively high. From the above equation (5) and (6), holding else equal, the value of e t+1 e t is expected to be high at time t. That is, a high investor sentiment predicts an appreciation of U.S. dollar after controlling for other factor which might influence exchange rates. These arguments lead to the following prediction: Prediction 1: Domestic investor sentiment S t is positively associated with exchange rate changes, e t+1 e t. That is, high domestic sentiment forecasts an appreciation of domestic currency. Prediction 1 holds under very general conditions. No assumption needs to made on the dynamics of investor sentiment except the mean-reverting feature. I can go one step further by calculating the exactly relation between exchange rate changes and investor sentiment under this simple economy. To this end, I assume AR(1) processes for investor sentiment in both domestic and foreign countries: x t+1 = ρx t + ϕ e σ g ε t+1 x t+1 = ρx t + ϕ e σ g ε t+1 To solve for the unobserved wealth-to-consumption ratio in the model, I substitute r a,t+1 into the above Euler equation. From Campbell-Shiller log-linear approximation, r a,t+1 κ 0 + κ 1 z t+1 z t + g t+1, (7) z t A 0 + A 1 x t, (8) 3 Here, the agents at home and foreign have the same belief although their beliefs can be wrong. See Backus, Foresi, and Telmer (2001) for a rigorous argument. Among others, Backus, Foresi, and Telmer (2001) and Brandt, Cochrane and Santa-Clara (2006) exploit this equation to relate the dynamics of the exchange rate to the dynamics of the domestic and foreign discount factors. 8

9 where A 0 and A 1 are constants. Substituting equation (7) and (8) back into the Euler equation (3), it yields A 1 = 1 1 ψ 1 κ 1 ρ. Therefore, given the equilibrium solution to the IMRS, I can write down the solution to exchange rate changes in the following way: ( e t+1 e t = γgt+1 γg t+1 + γ 1 ) (x t x ( t ) + (θ 1) A 1 κ 1 ϕ e σ e εt+1 ε ) t+1 ψ Taking the conditional expectation, it yields the following key implication on the predictive ability of investor sentiment on exchange rate changes: E t (e t+1 e t ) = (9) ( γ 1 ) (x t x t ) (10) ψ Here, E t ( ) is the expectation under the objective belief. Notice that when the agent prefers earlier resolution of consumption uncertainty, γ 1 ψ > 0. Hence, as long as the domestic sentiment x t and foreign sentiment x f t are not perfectly correlation, there is a positive relation between domestic investor sentiment and expected exchange rate changes. The return on foreign exchange is defined as the excess return of a domestic investor who borrows at riskfree rate (in logs), r f,t, at home, converts them to a foreign currency, lends at foreign risk-free rate (in logs), rf,t, and then converts his earnings back to domestic currency. Hence, in logs, the return on foreign exchange, rx F X t is given by r F X t+1 = r f,t r f,t + e t e t+1. (11) The interest rate differentials are r f,t r f,t = 1 ψ ( ) x t x f t (12) Taking together, it leads to, ( ) ) E t r F X t+1 = γ (x t x f t (13) The above argument leads to the following two predictions: Prediction 2: Interest rate differentials are negatively associated with domestic investor sentiment x t, and positively associated with foreign investor sentiment, x t. Prediction 3: Returns on foreign exchange is negatively associated with domestic investor sentiment x t, and positively associated with foreign investor sentiment, x t. 9

10 Notice that in true Bansal-Yaron economy where x t is the expected growth rate under true belief, the expected returns on foreign exchange is a constant and independent of x t. This is because that in equation (9), the expected value of g t+1 is x t, rather than a constant, which is the case in the economy under objective belief. It follows from equation (10), (12) and (13) that E t (e t+1 e t ) = (1 ψγ) (r f,t r ) f,t ( ) ( ) E t r F X t+1 = γψ r f,t r f,t (14) (15) Equation (14) and (15) are the key results from the model. From equation (14), the UIP coefficient is 1 ψγ, which is negative as long as the agent prefers an earlier resolution of uncertainty. From equation (15), the regression coefficient of returns on foreign exchange on forward premium is γψ, which is larger than one when the agent prefers an earlier resolution of uncertainty. These are consistent with the empirically findings of Fama (1984). Therefore, this simple model with investor sentiment can generate forward premium puzzle. Intuitively, in response to a rise in domestic investor sentiment, the equilibrium foreign price of the domestic currency drops immediately, so that relative to the new level today, the domestic currency is expected to appreciate tomorrow due to the mean reversion in investor sentiment, and the dollar return on investments abroad is expected to be low as well. At the same time, a higher domestic sentiment raises the yields on domestic bonds and decreases the interest rate differentials between foreign and domestic countries. All together, when domestic sentiment is relatively high, an appreciation of the domestic currency and a lower return on foreign exchange are expected, and at the same time the interest rate differential is pushed down. Thus it results in a negative coefficient in UIP regressions. The model also has implications on the role of investor sentiment in bond yield and returns on consumption claim. In particular, in the model, the riskfree rate is given by ( ) r f,t = log Et m d t+1 = C ψ x t (16) where C 1 is a proper constant. Indeed, in the data, the correlation between real interest rate and investor sentiment is 27%. As a result, the following key implication on the predictive ability of 10

11 investor sentiment on bond yield changes is obtained: E t (r f,t+1 r f,t ) = 1 ψ (ρ 1) x t (17) This leads to the following prediction of the model: Prediction 4: There is a negative relation between investor sentiment and changes on bond yields. As it will become clear later, Section 4 provides strong evidence to support prediction 1-4. So far, the results are for real exchange rates and interest rate. The results can be extended for nominal terms by modeling inflation process. The implications on the role of investor sentiment on exchange rate remains the same. The results are omitted here for brevity. In the empirical test, following the literature, I focus on nominal terms, and use real exchange rates as robustness checks. 3 Comparison with Habit-Formation and Long-run Risk Explanations Within the class of rational, representative-agents frameworks, two workable and influential models are the external-habit model of Campbell and Cochrane (1999) and the long run risk model of Bansal and Yaron (2004). Previous studies show that these two models can account for a set of asset pricing puzzle, such as high equity premium, low risk free rate, high volatility of stock return, low volatility of riskfree rate, the failure of expectation hypothesis, and the forward premium puzzle. In particular, Verdelhan (2009) proposes an explanation of the forward premium puzzle based on habit-formation and Bansal and Shaliastovich (2009) provides an explanation based on long-run risk. Both models can successfully account for the failure of UIP. Before documenting empirically the role of investor sentiment on foreign exchange market, I directly compare the channels in sentiment-based model with the the channels in these two models. The key state variables in habit formation based explanation, in the long-run risk based explanation, and in the sentiment-based explanation are the surplus ratio, sp t, the time-varying consumption volatility, σ gt, and the investor sentiment S t, respectively. For these three explanations, they all work in the following manner: The interest rate differential is approximately a linear function of some state variable, (i.e., surplus ratio, consumption volatility or investor sentiment), at the same time, the expected changes in exchange rates and the expected return on foreign exchange are also approximately a linear function of the same state variable. If the coefficients on the state variable of interest rate differentials and expected changes in exchange rate are opposite, then the model yields a negative coefficient in UIP regression. To see which 11

12 explanation is more relevant empirically. I take one step further by examining the relation between the state variable and the interest rate differential, changes in exchange rates, and returns on foreign exchange. With consumption data, surplus ratio and consumption volatility can be constructed. Baker and Wurgler (2006) create investor sentiment index, and the university of Michigan provides consumer sentiment data as well. Therefore, the three models can be tested directly by linking surplus ratio, consumption volatility, and investor sentiment to changes in exchange rates and returns on foreign exchange. Below, I provide a brief review on the key mechanisms in these three models. For habit-formation model by Verdelhan (2009), the relation between the interest rate differential and expected changes in spot rate and the expected returns on foreign exchange are given by E t (e t+1 e t ) = γ 1 φ B E t ( r F X t+1 ) = γ 2 σ 2 g 2B ( r f,t r f,t ) ( r f,t r f,t ) (18) (19) where sp t and sp t are the surplus ratio in home country and foreign country, respectively, φ < 1 is the habit persistence, σ g is the constant consumption volatility in the habit model, constant B is a proper constant determined by preference parameters and consumption volatility,and γ is the risk aversion. Hence, to generate a negative coefficient in UIP regression, Verdelhan (2009) requires B < 0. However, the key mechanisms for the model to account for forward premium puzzle are the following three equations: r f,t r f,t = B (sp t sp t ) (20) E t (e t+1 e t ) = γ (1 φ) (sp t sp t ) (21) ( ) γ E t r F X 2 σg 2 t+1 = (sp t sp t ) (22) 2 Notice that B < 0, γ (1 φ) > 0, and γ2 σ 2 g 2 > 0. As a results, the habit-formation model implies that interest rate differentials are negatively correlated with domestic surplus ratio, changes in exchange rate are positively related to domestic surplus ratio, and the return on foreign exchange is negatively associated with surplus ratio. For long-run risk model by Bansal and Shaliastovich (2009), the relation between the interest rate differential and expected changes in spot rate and the expected returns on foreign exchange 12

13 are given by E t (e t+1 e t ) = m x (x t x t ) + ( 1 2 m gs γ + γ 1 ψ ( ) E t r F X ψγ 2 ( ) t+1 = r γψ + γ 1 f,t r f,t ) ( rf,t r ( f,t) + mxs σ 2 xt σxt 2 ) (23) (24) where m x = 1 ψ, m gs = 1 2 ( γ 1 ) (γ 1), m xs = 1 ( γ 1 ) ( ) 2 κ1 (γ 1). ψ 2 ψ 1 κ 1 ρ For the parameter values in the long-run risk model, m x, m gs, and m xs are all negative. Thus, the UIP regression coefficient is negative under proper parameter values. However, the mechanisms behind the failure of UIP are the following three equations identifying the relation between consumption volatility, σ gt, and interest rate differential, expected changes in spot rate, and the expected returns on foreign exchange: rf,t r f,t = 1 ( γ + γ 1 ) (σ 2 gt σ 2 ) gt 2 ψ E t (e t+1 e t ) = m x (x t x ( t ) + m gs σ 2 gt σgt 2 ) ( + mxs σ 2 xt σgt 2 ) ( ) E t r F X 1 ( t+1 = 2 γ2 σgt 2 σgt 2 ) ( ) For the parameter values in the long-run risk model, 1 2 γ + γ 1 ψ > 0, m gs < 0, and 1 2 γ2 > 0. Therefore, long-run risk model implies that interested rate differential is positively related to domestic consumption volatility, that changes in exchange rate are negatively related to domestic consumption volatility, and that the return on foreign exchange is positively related to domestic consumption volatility. Finally, for the sentiment-based explanation, (25) (26) (27) E t (e t+1 e t ) = (1 ψγ) (r f,t r ) f,t ( ) ( ) E t r F X t+1 = γψ r f,t r f,t (28) (29) The model can account for the forward premium puzzle when 1 < ψγ. Again, the key mechanisms 13

14 for this model are: rf,t r f,t = 1 ψ (x t x t ) (30) ( E t (e t+1 e t ) = γ 1 ) (x t x t ) (31) ψ E t ( r F X t+1 ) = γ (xt x t ) (32) As a result, the sentiment-based model implies that forward premium is negatively correlated with domestic sentiment, changes in exchange rate are positively related to domestic surplus ratio, and the return on foreign exchange is negatively associated with surplus ratio. In a nutshell, all the three models can generate forward premium puzzle with properly chosen parameter values. The sentiment-based model works through the channel of investor sentiment. The habit-formation model works through the channel of time-varying risk aversion (i.e. inversion of surplus ratio), while the long-run risk model functions through the channel of time-varying consumption volatility. With empirical measures on investor sentiment, surplus ratio, and consumption volatility, I can investigate these three channels directly. In particular, empirically tests are performed for equation (25), (26), (27), (20), (21), (22), (30), (31) and (32) in the next section. 4 Empirical Tests In this section, I first introduce notations on predictive variables; then I present the econometric models and the data used in this section. The empirical specifications are standard long-horizon predictive regressions (e.g. Fama and French (1988) and Hodrick (1992)). 4.1 Definition of Variables First, define changes in spot rate as e t = e t e t 1. and define E 52,t is the maximum spot rates in the past 52-weeks at the end of month t. Similarly, E min,t is defined as the historical minimum until the end of month t. Now two proxies for underand overreaction can be computed as 4 : 4 Both psychological and statistical evidence indicates that individuals tend to underreact to sporadic news (e.g. conservatism), such as quarterly earnings announcement, but overreact to a prolonged series of news, regardless good or bad (e.g. representativeness). See Barberis, Shleifer, and Vishny (1998) and Kanneman, Slovic, and Tversky (1982) for details. Li and Yu (2009) and Wang, Yu, and Yuan (2009) argue that nearness to the 52-week high captures the degree of underreaction to recent news, while nearness to the historical low capture the degree of overreaction to 14

15 x 52 = E t E 52,t, x max = E t E min,t Empirical studies find that macro fundamentals have very weak power predicting exchange rate changes. However, Wang, Yu, and Yuan (2009) show that x 52 and x min have strong power to predict exchange rate changes and returns on foreign exchange. Therefore x 52 and x min is used as control variables in predicting exchange rate changes. In empirical test that follows, the 3-month forward exchange rate F t is observed at the end of month t for a delivery at the end of month t + 3. Let f t = log(f t ). The 3-month forward premium is defined as fp t = f t e t. By covered interest parity, fp t = rf,t r f,t, where rf,t is the 3-month riskfree rate (from end of month t to the end of month t + 3) in foreign countries at the end of month t and r f,t the 3-month domestic riskfree rate the end of month t. The return on foreign exchange is therefore, r F X t+3 = f t e t+3. Finally, following Wachter (2006), surplus ratio is approximated by sp t = 40 j=1 φ j g t j. and following Bansal and Shaliastovich (2009), consumption volatility, σ gt, is measured as a 4.5 year sum of absolute residual from AR(3) projections of consumption growth rates. 4.2 Empirical Design The empirical specifications are standard long-horizon predictive regressions (e.g. Fama and French (1988) and Hodrick (1992)). All variables are monthly observed. Future spot rate changes and returns on foreign exchange rates are regressed onto investor sentiment, S t, surplus ratio, sp t, and consumption volatility, σ gt. The ratio of current spot rate to the 52-week high spot rate, x 52, and the ratio of current spot rate to historical minimum spot rate, x min are used as control variables in various regressions. Thereofore, the dependent variable is either log changes in foreign exchange rates, e t+h e t with different horizons or 3-month log returns on foreign exchange r F X t+3. The independent variables are investor sentiment, S t, surplus ratio, sp t, and consumption volatility, σ gt. prolonged extreme record. 15

16 For example, to examine the predictive ability of investor sentiment on changes in spot rates, the main regression is e t+h e t = α h + β 1,h x 52,t + β 2,h x min,t + β 3,h S t + ɛ t,t+h (33) The error term ɛ t,t+h is an element of the time t + h, and is autocorrelated due to overlapping observations when the forecast horizon h > 1. Both Newey and West (1987) standard errors and Hodrick (1992) standard errors are used to adjust for this autocorrelation and potential heteroscedasticity. The t-statistics resulting from both adjusted standard errors test the null hypothesis that a given slope coefficient equals zero. Hodrick (1992) t-statistics retain the correct size in small sample (e.g. Ang and Bekaert (2007). For small h, these two methods give a similar t-statistics. However, the difference can be substantial when forecast horizon h is large, such as 2- to 4-year. 4.3 Data The main analysis uses investor sentiment data from the website of Jeffrey Wurger. Michigan consumer sentiment index is also used for robustness checks. The monthly sentiment index spans from January 1966 to December 2005, a total of 40 years. Baker and Wurgler (2006) form a composite sentiment index that is the first principal component of six measures of investor sentiment. The principal component analysis filters out idiosyncratic noise in the six measures and captures their common component. The six measures are the number and the average first-day returns of IPOs, the dividend premium, the closed-end fund discount, the NYSE turnover, and the equity share in new issues. The monthly sentiment data spans from 1966m1 to 2005m12. Annual sentiment measure is used for robustness checks as well, and the results, omitted for brevity, are very similar with the results reported here. The dataset on foreign exchange includes monthly spot and 3-month forward exchange rates between the US dollar and currencies in 19 industrial countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, and the UK. Data sample is between 1973m1 and 2009m08 for most countries and may vary in some countries due to the data availability. Monthly CPI for these 19 countries and U.S.A. from 1973m1-2009m8 is obtained from IMF. Annual real consumption per capita from is obtained from OECD. Spot exchange rates are from the Federal Reserve and 3-month forward exchange rates are from the International Financial Statistics (IFS). The spot and forward exchange rates are used to calculate interest rate differentials between the US dollar and other currencies. For several 16

17 countries, part of the forward exchange rates are missing in the IFS. In this case, the forward rate is obtained from Bloomberg, or calculated from the 3-month interest rate obtained from G-10 dataset. The choice of the data source is dictated by the availability of the data. Finally, I obtain monthly prices for one to five year zero-coupon bonds from June, 1952 to December 2008 from CRSP. 4.4 Summary Statistics Investor sentiment is standardized with mean zero and standard deviation one. Table 1 reports the summary statistics for monthly changes on spot rates ( e t ), the ratio of current spot rate to the 52-week high spot rate, x 52, and the ratio of current spot rate to historical minimum spot rate, x min. The means and standard deviations of variables are on a percent basis. The autocorrelation for x 52 and x min is the annualized autocorrelation. The monthly volatility of changes in spot rate is about 3% for most of the countries, which is consistent with previous studies. The autocorrelation of changes on spot rates is very low, which suggests that spot rates behave like a random walk. Nearness to the 52-week high, x 52, is close to one, and moderately volatile, the autocorrelation is reasonably low. However, nearness to the historical low is much more volatile and very persistent. On average, it is as persistent as the traditional predictors, dividend yield, in stock market. The last two column reports the correlation between nearness to the 52-week high and investor sentiment and the correlation between nearness to the historical low and investor sentiment. The correlation between nearness to the historical low and investor sentiment is about 50% across 19 countries. It is therefore important to control for nearness to the historical low in the predictive regression. From Table 1, the currencies of Japan, Swiss, Austria, Germany, and Netherlands appreciate on average against U.S. dollar from 1973 to (Denmark and Belgium also appreciate slightly against U.S. dollar in this period). Table 2 shows that exactly these currencies have lower interests than that in U.S. during this period (or equivalently a negative forward premium). This implies that the expectation hypothesis in currency markets seems hold on average or in the long-run. Table 2 also indicates that the countries with a low average interest rate also yield a negative return on foreign exchange. This implies that sorting on interest rate differential can yield a significant spread (e.g., Hustig and Verdelhan (2007)). The volatility of returns on foreign exchange is much more volatile than forward premium, consistent with the finding with Fama (1984). 4.5 Comparison of Three Models This section provides empirical results on the comparison of the three explanations. In particular, statistical tests are performed for equation (25), (26), (27), (20), (21), (22), (30), (31) and (32). 17

18 According to Section 3, interested rate differential (i.e. forward premium) is positively related to domestic consumption volatility, negatively correlated with domestic surplus ratio, and negatively correlated with domestic sentiment. The last three columns of Table 2 show that this is exactly the opposite for habit-based explanation. There is strong support for both long-run risk based and sentiment-based explanation. For example, the correlation between forward premium and domestic sentiment is negative for all the 19 countries, consistent with the prediction 2 in Section 2. Similarly, according to Section 3, returns on foreign exchange are positively related to domestic consumption volatility, negatively correlated with domestic surplus ratio, and negatively correlated with domestic sentiment. Table 4 shows that surplus ratio is strongly positively correlated with future changes in exchange rates, the opposite of that of the habit model. The results from Table 4 is consistent with the sentiment-based explanation. For long-run risk model, consumption volatility has little power to predict returns on foreign exchange and the signs of the coefficients flip across countries. Here, I focus on 1-year changes on exchange rate rather than one-month changes. All the three state variables are persistent, hence it is easier to identified their predictability in longer horizons. Finally, according to Section 3, changes in exchange rates are negatively related to domestic consumption volatility, positively correlated with domestic surplus ratio, and positively correlated with domestic sentiment. Table 3 shows that surplus ratio is negatively correlated with future changes in exchange rates, the opposite of the implications from the habit model. Again, the results from Table 3 is consistent with the sentiment-based explanation. For long-run risk model, consumption volatility has very weak power to predict changes in exchange rates and the sign flips across countries. The data for U.S. consumption is long, hence the surplus ratio and consumption volatility can be more precisely estimated. Thus, up to this point, the analysis is based on nominal exchange rate and US consumption data only. To test the models directly, real exchange rates and foreign consumption data are needed. The longest available annual real per capita consumption data from (obtained from OECD) is used to construct surplus ratio and consumption volatility for all the 20 countries. Annual surplus ratio is defined as the smoothed average of the past 10-year consumption growth. Annual consumption volatility is defined as a 3 year sum of absolute residuals from AR(3) projections of consumption growth rates. I then convert the annual observation to monthly observation by assigning the value at the end of this year to each month in the following year. Panel A of Table 5 reports the forecast of changes in real exchange rates by surplus ratio. It is seen that the predictive power of the surplus ratio is again the opposite with that implied by the model. The model implies a positive relation between domestic surplus ratio and changes 18

19 in exchange rate, but the data suggest the opposite. The model also implies a negative relation between foreign surplus ratio and changes in exchange rate, but the evidence from the data is mixed. Panel B of Table 5 reports the forecast of changes in real exchange rates by consumption volatility. The empirical results indicate that the predictive power of consumption volatility is very weak and the sign flips across 19 countries. Table 6 reports results from the predictive regressions of 3-month returns on foreign exchange on surplus ratios and consumption volatility. The results confirms the findings from regression of changes in spot rates on surplus ratio and consumption volatility in Table 5. These results are very similar with the analysis based on U.S. consumption data alone and nominal exchange rates, which is of course expected given the low correlation between U.S. consumption and foreign consumption. In a nutshell, the empirical evidence in this section provides strong support for the sentiment channel. The habit-based explanation has difficulty reconciling the evidence in this section given the exactly opposite results with the predictions from the model. For long-run risk model, the support is at most modest for the time-varying consumption volatility channel. 4.6 Forecasting Changes in Exchange Rates by Sentiment Previous subsection provides support for the sentiment-based explanation. Although in the simple model presented in Section 2, there is an exact relation between forward premium. In more realistic setting with additional shocks, this won t hold. Therefore, investor sentiment may contain information which is not captured by forward premium. In subsequent analysis, I empirically explore in more detail on the predictive ability of investor sentiment in currency markets. Because a majority of foreign exchange transaction involves U.S. dollar (about 86%), U.S. dollar is used as domestic currency. Table 7 reports results from the monthly overlapping predictive regression of changes in exchange rates on investor sentiment together with control variables, for nearness to the 52-week high and nearness to the historical low 5. The forecast horizons are 1- month, 3-month, 12-month, 24-month, and 48-month. For each regression, this table reports the slopes, the adjusted R 2, and two sets of t-statistics: t-statistics calculated from standard errors corrected for autocorrelations per Newey and West (1987) and t-statistics calculated from standard errors adjusted for the use of overlapping observations in long-horizon regression per Hodrick (1992). Noticeably, investor sentiment serves as a statistically significant predictor of exchange rate changes across all the 19 countries at most horizons less than 2-year. It becomes less significant at a very long horizon, such as a 4-year horizon. In addition, the predictive power of nearness to the 52-week high is stronger at shorter horizons, while the predictive power of nearness to the historical low is 5 The is a concern that the autocorrelation of nearness to the historical low is very high. I replaced nearness to the historical low with nearness to 5-year low and the results are very similar. The persistence of nearness to the 5-year low is much lower. These results are omitted for brevity and available upon request. 19

20 stronger at longer horizons, typically at 2-3 years (as measured by Hodrick (1992) t-statistics). These results confirm prediction 1 in foreign exchange market. One striking feature of the results is the consistency across the 19 industrial countries. Adding investor sentiment into the predictive regression in generally produce twice as high R 2 as that with only nearness to the 52-week high and the historical low. Together, these three predictors can account for about 40% of variation in exchange rate changes at one-year horizon. Given the fact that it is very hard to predict exchange rate, this is an intriguing result. Traditional macroeconomic variable have very limited power to predict exchange rate changes, especially at horizons less than 1 year. By contrast, investor sentiment, which are motivated by traders behavioral biases, has significant predictive power at both short and longer horizons. The results are especially interesting because foreign exchange market are dominated by professional investors and is regard as the most efficient market in the world. In summary, this section documents a robust empirical fact that investor sentiment positively predicts changes in future spot rates. The predictive power of investor sentiment is very robust across 19 industrial countries. 4.7 Forecasting Returns on Foreign Exchange by Sentiment Forward premium predicts the future returns on foreign exchange (e.g. Fama(1984), Hansen and Hodrick (1980)). Prediction 3 states that investor sentiment serve as a contrarian predictor of returns on foreign exchange. This subsection empirically tests this prediction by controlling known predictors. In Table 8, returns on foreign exchange are first regressed on 3-month forward premium. This table confirms the finding from previous studies on the predictive ability of forward premium. The high values of Newey-West t-statistics indicate that forward premium is indeed a significant predictor of returns on foreign exchange. In the second predictive regression, investor sentiment, S t, x 52, and x min are also included into the regression. The results show that investor sentiment is a statistically significant predictor of returns on foreign exchange. The economic significance of the predictability is rather large. A one-standard-deviation increase in investor sentiment is associated with about 8% decrease in annualized future returns on foreign exchange. In addition, nearness to the 52-week high significantly negatively predicts the return on foreign exchange while nearness to historical high is significantly positively associated with the future returns on foreign exchange. This is of course, consistent with the negative predictive power of nearness to the historical low and the positive power of nearness to the 52-week high on changes on future spot rates. Finally, the interaction between forward premium and investor sentiment is generally positive and significant for 7 out of 19 countries. This indicates that forward premium has 20