Forecasting Interest Rates and Exchange Rates under Multi-Currency Quadratic Models

Transcription

1 Forecasting Interest Rates and Exchange Rates under Multi-Currency Quadratic Models Markus Leippold Swiss Banking Institute, University of Zurich Liuren Wu Graduate School of Business, Fordham University October 25th, 22 CRIF, Fordham University

2 Overview Introduction Multi-Currency Quadratic m + n Models Property Analysis Data and Estimation Estimation Results Summary 2

3 Introduction Two Strands of Literature: 1. Interest Rate Forecasting Theme: Current term structure contains information on future short rates. A simple regression: f n 1 t+1 r t = a n + b n (ft n r t ) + e n t+1; r t+n r t = a n + b n (ft n r t ) + e n t+1; y n t+1 y n+1 t = a n + 1 n b n ( y n+1 t r t ) + e n t+1. Based on different expectation hypotheses (U-EH, L-EH, RTM-EH, YTM- EH). Literature includes e.g.: Backus, Foresi, Mozumdar, and Wu (21), Bekaert, Hodrick, and Marshall (1997), Fama (1984b), Roll (197). 3

4 2. Foreign Exchange Rate Forecasting Theme: Short rates contain information about future currency movements. A simple regression (UIP), based on martingale hypothesis s t+n s t = α n + β n (f(s) n t s t ) + e t+n = α n + β n (r t r t ) + e t+n. Literature includes: Backus, Foresi, and Telmer (21), Cheung (1993), Engle (1996) Fama (1984a). Regression has been studied with improved econometric techniques (e.g. Wu and Zhang (1997)). Are deviations from EH merely a coincidence of a few empirical artifacts (Lothian and Wu (22))? Failing of EH tests may be due to existence of time varying risk premium. Fama (1984a): risk premium on currency must be negatively correlated with expected depreciation and must have greater variance. 4

5 Our Contribution We combine interest rate and foreign exchange rate forecasting in an internally consistent way. We propose an extension of quadratic term structure models to a multicountry setup (MCQM(m+n)). We investigate 1. whether (whole) term structures contain information on exchange rates. 2. whether term structures of different countries share common factors, and may improve upon forecasting interest rates. 3. whether forecasting based on model outperforms forecasting based on EH. 5

6 Executive Summary 1. Theory In the MCQM(m+n), exchange rates can have independent movements, i.e. dynamics are not solely determined by interest rate differential. The MCQM(m+n) can generate more volatile exchange rate dynamics than implied by differences in term structure risk premia. 2. PC Analysis Co-movement of interest rates between U.S. and Japan is very small. U.S. and Germany is larger. Exchange rate and term structure movements are almost independent for U.S. and Japan....and almost independent for U.S. and Germany. 6

7 3. Estimation Results MCQM(6+1) model yields superb fitting for both term structures and currencies. Prediction of interest rates for both U.S. and Japan is excellent. Prediction of exchange rate between U.S. dollar and Japanese yen is difficult. Prediction of exchange rate between U.S. dollar and German mark/euro is... provided soon! 7

8 Multi-Currency Quadratic m + n Models Basic Notation The Economy: N countries with each M N time series of term structure data. US investor is the domestic investor. Complete markets and no-arbitrage assumption imply: [ ] MT P (, τ) = E t, M t where P (, τ) : default free zero-coupon bond with expiration time T = t + τ. M t : state-price deflator. E t [ ] : expectation operator under physical probability measure P. 8

9 Assumptions Assumption 1 ((m+n) Vector of State Variables). Let Z t [X t, Y t ] R m+n some vector Markov process that describes the state of the economy, with X t R m, Y t R n, and Xt, i Y j t = for all i = 1, 2,, m and j = 1, 2,, n. Assumption 2 (Orthogonal Decomposition). The state-price deflator is multiplicative decomposable, M t = ξ(x t )ζ(y t ), where ξ( ), ζ( ) C 2 (R) such that ξ, ζ =. The process ζ t is an exponential martingale under P. Remarks: 1. Under Assumptions 1 and 2, ζ(y t ) does not enter the pricing of zero-coupon bonds: [ ] [ ] [ ] ξt+τ ζt+τ ξt+τ P (Z, τ) = E t E t = E t = P (X t, τ). (1) ξ t ζ t ξ t 2. We label ξ as the term structure state-price density. 9

10 Assumption 3 (State Vector Dynamics). X t R m and Y t R n follow dx t = κ x X t dt + dw x t, dy t = κ y Y t dt + dw y t, (2) where Wt x R m, W y t R n are adapted standard Brownian motions with W x, W y = and κ x R m m, κ y R n n. Further, let V E(XX ). No-Arbitrage Condition The term structure state-price density, ξ, must satisfy (Duffie (1992)), where dξ t ξ t r(x t ) γ ξ (X t ) R m = r(x t )dt γ ξ (X t ) dw x t. (3) : domestic instantaneous interest rate. : domestic term structure risk premium. 1

11 Multi-Currency Quadratic m + n Models Assumption 4 (MCQM(m+n)). Domestic interest rate and term structure risk premium are given by r(x t ) = X t A r X t + b r X t + c r, (4) γ ξ (X t ) = A ξ X t + b ξ, where A r, A ξ R m m, b r, b ξ R m, c r { R. The same functional } forms hold for the N 1 foreign countries, with A j r, A j ξ, bj r, b j ξ, cj r, j = 2,..., N. The independent martingale component of the pricing kernel is given by dζ t ζ t = γ ζ (Y t )dw y t, (5) where γ ζ (Y t ) the currency risk premium and Y the currency risk factor. Moreover, γ ζ (Y t ) = A ζ Y t + b ζ, (6) where A ζ R n n, b ζ { R n. The } same functional forms hold for the N 1 foreign countries, with A j ζ, bj ζ, j = 2,..., N. 11

12 Interest Rates and Foreign Exchange Rates Proposition 1 (Bond Prices). Under the quadratic class, prices of zerocoupon bonds are exponential-quadratic functions of the Markov process X t, [ ] P (X t, τ) = exp Xt A (τ) X t b (τ) X t c (τ), (7) where the coefficients A(τ) R m m, b(τ) R m, and c(τ) R are determined by the following ordinary differential equations (see Leippold and Wu (22a)). Proposition 2 (FX Rate Dynamics). Under Assumptions 1-3, exchange rate dynamics have a quadratic drift and an affine diffusion term in Z t, ds j t S j t = ( r(x t ) r j (X t ) + γ(z t ) ( γ(z t ) γ j (Z t ) )) dt + ( γ(z t ) γ j (Z t ) ) dwt = ( Z t A j µz t + b j µ Z t + c j µ) dt + ( A j σ Z t + b j σ) dwt, (8) where γ(z t ) [γ ξ (X t ), γ ζ (Y t )], and W t [W x t, W y t ], both of which have dimension m + n. 12

13 Property Analysis 1. Co-movement of Interest Rates Across Countries Denote by y h t the h-period yield. Then, y h t = X t A h X t + b h X t + c h, where A h = A(h )/(h ), b h = b(h )/(h ), c h = c(h )/(h ). with denoting the length (in years) of the observation interval. Cross-correlation between interest rates across countries: Corr(y h 1 t, y h 2 t ) = 2tr ( A h1 V A h 2 V ) + b h 1 V b h 2. (2tr[(A h1 V ) 2 ] + b h 1 V b h1 )(2tr[(A h 2 V ) 2 ] + b h 2 V b h 2 ) Remarks: 1. A one-factor MCQM can generate negative correlation. 2. A CIR model has difficulties in reproducing negative correlation. 13

14 2. Forecasting Interest Rates Time Series View A simple test for Expectation Hypothesis: where f h t f h 1 t+1 r t = a h + b h ( f h t r t ) + e h t+1, (9) is the forward rate at time t valid between t + h and t + (h + 1). Remarks: 1. Estimation of (9) can give regression slopes b h < Estimation of (9) is a purely econometric view. 3. Additional information gained by linking time series and cross-sectional characteristics in an internally consistent way, i.e. with an arbitrage-free model? Requires a model with enough flexibility!!! 14

15 Model View In our model, forward rate ft h b h is is a quadratic form. The forward regression slope b h = 2tr [( ) Φ A h 1 Φ A r V (Ah A r ) V ] + (b h b r ) V (Φb h 1 b r ) 2tr ((A h A r ) V ) 2 + (b h b r ), V (b h b r ) with {A h, b h } the quadratic coefficients for the forward rate f h t, and Φ = exp( κ ) the weekly autocorrelation matrix for X t. Remarks: 1. A one-factor quadratic model can generate b h < Compared to affine models, quadratic models offer much more flexibility to generate b h < 1 (see Leippold and Wu (22b)). 3. Quadratic models exploit no-arbitrage information and may improve forecasts compared to the purely statistical view. 15

16 3. Forecasting Exchange Rates Time Series View Forward premium regression: s t+h s t = α h + β h ( f h s,t s t ), with f h s,t : log of h -period forward exchange rate at time t. The forward premium regression can be written as β h = Cov ( ) p h t + qt h, qt h ( h)v ar ( ). p h t + qt h p h t = fs,t h E t (s t+h ) : forward risk premium, qt h = E t (s t+h ) s t : expected deprecation rate. Fama (1984a) s necessary and sufficient conditions that generate β h < : 1. Cov ( p h t, q h t ) <, 2. V ar ( p h t ) > V ar ( q h t ). 16

17 Model View In the MCQM model, for h=1: ] 2tr [Âξ V (A r A r) V + (b r b r) V ˆb ξ β 1 = 1 + 2tr ((A r A r) V ) 2 + (b r b r) V (b r b r), where Remarks: Â ξ = 1 2 ( ) A ξ A ξ A ξ A ξ ; ˆb ξ = 1 ( A 2 ξ b ξ A ξ bξ). 1. The slope of forward premium regression only depends upon the term structure risk premium, but not upon the currency risk premium. Nevertheless, the n currency risk factors may play an important role in the FX dynamics. 2. An MCQM(1+n) can generate negative slope coefficients. 3. Gaussian models with proportional or constant price of risk cannot account for the anomaly. After all, accounting for the EH-anomalies based on simple regressions is only a minimum requirement for model design. 17

18 Data and Estimation Data Description Eight years for weekly data of LIBOR/swap rates and exchange rates for U.S. Dollar and Japanese Yen (the two most actively traded currencies in the swap market). Weekly (Wednesday) closing mid quotes from April 6th, 1994 to April 17th, 22 (42 observations), provided by Lehman Brothers. LIBOR with maturities 1, 2, 3, 6 and 12 months. US dollar swap rates 2, 3, 5, 7, 1, 15, and 3 years. Japanese Yen swap rates 2, 5, 7, 1, 2, and 3 years. Exchange rate is represented in U.S. dollar prices per unit of Japanese yen. In total: 24 time series of interest rates and exchange rates. Estimation based on data before 2 (3 observations). Data after 2 (12 observations) for out-of-sample tests. 18

19 Table 1: Summary Statistics of Interest Rates and Exchange Rate Levels Differences Maturity Mean Std. Dev. Skewness Kurtosis Auto Mean Std. Dev. Skewness Kurtosis Auto A. LIBOR and Swap Rates on US Dollars 1 m m m m y y y y y y y y B. LIBOR and Swap Rates on Japanese Yen 1 m m m m y y y y y y y C. Dollar Price of Yen

20 Factor Analysis Table 2: PCA for US and Japanese Interest Rates and Exchange Rate Factor Panel A: PCA on Levels Panel B: PCA on Differences USD JPY UJR UJF USD JPY UJR UJF

21 Principal Components I US Short End US Long End Japanese Short End.4 Japanese Long End US dollar/japanese yen Figure 1: Reconstruction of original time series using the first four PC. The plots show the relative deviation from the original time series. 21

22 Principal Components II US Short End US Long End Japanese Short End Japanese Long End US dollar/japanese yen Figure 2: Reconstruction of original time series using the first six to nine PCs. deviation from the original time series. The plots show the relative 22

23 Co-movement of short and long rates 1 month LIBOR 3 month LIBOR 7 year Swap rates rates rates 2 4 US JP month timelibor 2 4 US JP month timelibor 2 4 US JP year time swap rates rates rates 2 4 US DEM month timelibor 2 4 US DEM month timelibor 2 4 US DEM year time swap rates rates rates 2 4 JP DEM time 2 4 JP DEM time 2 4 JP DEM time Figure 3: Co-movement of US, Japanese and German interest rates. 23

24 Table 3: Variance Decomposition on US and Japanese Interest Rates and Exchange Rate Factors US 1m m m m m y y y y y y y US aggregated JP 1m m m m m y y y y y y JP aggregated FX Total

25 Table 4: Correlations of the Mimicking PCA Portfolios with US and Japanese Interest Rates and Exchange Rate Factors US 1m m m m m y y y y y y y US average JP 1m m m m m y y y y y y JP average FX Total average

26 Conclusion from PC Analysis Upward sloping mean yield curve, higher kurtosis at the short end. US: higher volatility and negative skewness at the short end; Japan: higher volatility at the long end, positive skewness at the short end. PCA for single countries: 3 factors explain more than 9% variation. For PCA based on UJF (differences), we need at least 5 factors to explain more than 9% variation. No substantial change for PCA by adding exchange rate. More detailed PCA analysis reveals: Term structure movements do not have much in common. A common factor is the curvature factor. Predominant part of FX movements is not related to term structure factors. 26

27 Estimation Methodology We use QML with (Scaled) Unscented Kalman Filter (UKF). Duffee and Stanton (21) exemplify the superiority of extended Kalman filter (EKF) estimation over EMM (with SNP auxiliary model). UKF is a relatively new filtering technique more powerful than EKF. Builds on the principle that it is easier to approximate probability distribution than to approximate nonlinear function. Two main advantages over EKF: 1. More accurate results. 2. More efficient implementation. 27

28 Estimation Results Table 5: Parameter Estimation (Preliminary) Data used are weekly from April 6th, 1994 to April 17th, 22 (42 observations for each time series). A. Common Dynamics m n 1 κ x κ y.1 U.S. B. Interest Rates m A r b r c r κ x + A ξ b ξ Japan U.S. C. Foreign Exchange Rates Japan n 1 1 A ζ b ζ

29 Table 6: Estimation Results (Preliminary) Data used are weekly from April 6th, 1994 to April 17th, 22 (42 observations for each time series). Forecasting Error Fitting Error MCQM (3+1) (4+1) (5+1) (6+1) (3+1) (4+1) (5+1) (6+1) US 1m m m m m y y y y y y y JP 1m m m m m y y y y y y FX Maximum Likelihood

30 Forward Rate Regression 2 6 Forward Rate Regression Slopes Forward Rate Regression Slopes Maturity, Years Maturity, Years Figure 4: Forward Rate Regressions The solid lines are the regression slope estimates of the following regression f n t+1 r t = a n + b n (f n t r t ) + e t+1 where ft n denotes time-t forward rates between time t + n and t + n + and r is the short rate. The dashed dotted lines are the 95 percent confidence interval constructed based on the standard deviation estimates of the regression slopes. The forward rates are extracted from the data based on the (6 + 1) model. 3

31 Forward Rate Regression - Simulation Forward Rate Regression Slopes Maturity, Years Forward Rate Regression Slopes Maturity, Years Figure 5: Forward Rate Regressions Lines are the median (solid line) and 5 percent and 95 percent percentiles of the forward rate regression based on 1, simulated paths of forward rates. Each path has an weekly frequency and 3 observations. The paths are simulated based on the MCQM (6 + 1) model with parameter estimates in Table 5. 31

32 Forecasting Exchange Rate 1.3 Dollar Price of 1 Yen Jan95 Jan96 Jan97 Jan98 Jan99 Figure 6: Fitting and Forecasting of Exchange Rate Fitting (solid line) and forecasting (dashed line) of observed foreign exchange rates (circles). EH regression yields ( ) s t+1 = y t y j t + e, R 2 =.2, (.2664) (5.6917) where y t and y j t denotes the one-week spot rates on US dollar and Japanese yen, respectively. Monte Carlo simulation based on the model parameter estimates, mean slope estimated at 8.16, but with standard deviation 22.48! 32

33 Summary The MCQM(m+n) enables us to integrate Gaussian state variables, affine market price of risk, and rich nonlinear dynamics for interest rates and exchange rates into a consistent framework. The (m+n) factor structure helps to explain the, empirically observed, independent movements in term structures and exchange rates within an arbitrage-free setup. The MCQM(m+n) can readily explain documented evidence on EH based forecasting relations. PCA shows little evidence of co-movements in term structure. A predominant independent factor of exchange rate movements exists. MCQM(6+1) fits interest rates and exchange rates. Forecasting of interest rates is excellent. Forecasting of exchange rates is difficult. Future research: analyze and integrate different currencies and term structures. 33

34 References Backus, D., S. Foresi, A. Mozumdar, and L. Wu, 21, Predictable Changes in Yields and Forward Rates, Journal of Financial Economics, 59(3), Backus, D., S. Foresi, and C. Telmer, 21, Affine Term Structure Models and the Forward Premium Anomaly, Journal of Finance, 56(1), Bekaert, G., R. Hodrick, and D. Marshall, 1997, On Biases In Tests of the Expectation Hypothesis of the Term Structure of Interest Rates, Journal of Financial Economics, 44(3), Cheung, Y., 1993, Exchange Rate Risk Premiums, Journal of International Money and Finance, 12, Duffee, G., and R. Stanton, 21, Estimation of Dynamic Term Structure Models, manuscript, UC Berkley. Duffie, D., 1992, Dynamic Asset Pricing Theory. Princeton University Press, Princeton, New Jersey, second edn. Engle, C., 1996, The Forward Discount Anomaly and the Risk Premium: Survey of Recent Evidence, Journal of Empirical Finance, 3, Fama, E., 1984a, Forward and Spot Exchange Rates, Journal of Monetary Economics, 14, Fama, E. F., 1984b, The Information in the Term Structure, Journal of Financial Economics, 13, Leippold, M., and L. Wu, 22a, Asset Pricing under the Quadratic Class, Journal of Financial and Quantitative Analysis, 37(2), , 22b, Design and Estimation of Quadratic Term Structure Models, European Finance Review, forthcoming. Lothian, J. R., and L. Wu, 22, Uncovered Interest Rate Parity Over Past Two Centuries, Working paper, Graduate School of Business Administration, Fordham University. Roll, R., 197, The Behavior of Interest Rates. Basic Books, New York. Wu, Y., and H. Zhang, 1997, Forward Premiums as Unbiased Predictors of Future Currency Depreciation, Journal of International Money and Finance, 16(4),