Predictability of Interest Rates and Interest-Rate Portfolios - PowerPoint PPT Presentation

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Predictability of Interest Rates and Interest-Rate Portfolios Liuren Wu Zicklin School of Business, Baruch College Joint work with Turan Bali and Massoud Heidari July 7, 2007 The Bank of Canada - Rotman School of Management Workshop: Advances in Portfolio Management Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts The evolution of interest-rate forecasting • History: Expectation hypothesis regressions (since 1970s): • The current term structure contains useful information about future interest-rate movement. • Recent: Multifactor dynamic term structure models (DTSM) • Affine (Duffie, Kan, 96; Duffie, Pan, Singleton, 2000) • Quadratic (Leippold, Wu, 2002; Ahn, Dittmar, Gallant, 2002) • Now: Use DTSM to explain expectation hypothesis regression results. • Backus, Foresi, Mozumdar, Wu (2001), Dai, Singleton (2002), Duffee (2002), Leippold, Wu (2003), ... • Question: • Why don’t we directly use DTSM to forecast interest rate movements? Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Let’s try We estimate several three-factor affine DTSM models using 12 interest-rate series. • The forecasting performances of these models are no better than the random walk hypothesis! • This result is not particularly dependent on model design. • The models fit the term structure well on a given day. • All three factors are highly persistent and hence difficult to forecast. • The pricing errors are much more transient (predictable) than the factors or the raw interest rates. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts What do we do now? • We use the DTSM not as a forecasting vehicle, but as a decomposition tool. y τ t = f ( X t , τ ) + e τ t • What the DTSM captures ( f ( X t , τ )) is the persistent component, which is difficult to forecast. • What the model misses (the pricing error e ) is the more transient and hence more predictable component. • We propose to form interest-rate portfolios that • neutralize their first-order dependence on the persistent factors. • only vary with the transient residual movements. • Result: The portfolios are strongly predictable, even though the individual interest-rate series are not. ⇒ What is left out from the factors can also be economically significant. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Three-factor affine DTSMs • Affine specifications: • Risk-neutral factor dynamics: dX t = κ ∗ ( θ ∗ − X t ) dt + √ S t dW ∗ [ S t ] ii = α i + β ⊤ t , i X t . • Short rate function: r ( X t ) = a r + b ⊤ r X t • Bond pricing: Zero-coupon bond prices : � − a ( τ ) − b ( τ ) ⊤ X t � P ( X t , τ ) = exp . • Affine forecasting dynamics: γ ( X t ) = √ S t λ 1 + � S − t λ 2 X t . • Does not matter for bond pricing. • Specification is up to identification. • Dai, Singleton (2000): A m (3) classification with m = 0 , 1 , 2 , 3. • We estimate all four generic specifications. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Data • Data: 12 interest rate series on U.S. dollar: 1, 2, 3, 6, and 12-month LIBOR; 2, 3, 5, 7, 10, 15, and 30-year swap rates. • Sample periods: Weekly sample (Wednesday), May 11, 1994 – December 10, 2003 (501 observations for each series). • Quoting conventions: actual/360 for LIBOR; 30/360 with semi-annual payment for swaps. ! 100 1 1 − P ( X t , τ ) LIBOR ( X t , τ ) = − 1 , SWAP ( X t , τ ) = 200 × . P 2 τ τ P ( X t , τ ) i =1 P ( X t , i / 2) • Average weekly autocorrelation ( φ ) is 0.991: Half-life = ln φ/ 2 / ln φ ≈ 78 weeks (1 . 5 years ) Interest rates are highly persistent; forecasting is difficult. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Estimation: Maximum likelihood with UKF • State propagation (discretization of the forecasting dynamics): � X t +1 = A + Φ X t + Q t ε t +1 . • Measurement equation: � LIBOR ( X t , i ) � i = 1 , 2 , 3 , 6 , 12 months y t = + e t , SWAP ( X t , j ) j = 2 , 3 , 5 , 7 , 10 , 15 , 30 years . • Unscented Kalman Filter (UKF) generates conditional forecasts of the mean and covariance of the state vector and observations. • Likelihood is built on the forecasting errors: l t +1 (Θ) = �� � − 1 � �� � ⊤ � − 1 � − 1 � � 2 log � A t +1 y t +1 − y t +1 A t +1 y t +1 − y t +1 . 2 Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Estimated factor dynamics: A 0 (3) P ∗ : dX t = ( − b γ − κ ∗ X t ) dt + dW ∗ P : dX t = − κ X t dt + dW Risk-neutral dynamics κ ∗ Forecasting dynamics κ  0 . 002 0 0   0 . 014 0 0  (0 . 02) (11 . 6) −− −− −− −−         − 0 . 186 0 . 480 0 0 . 068 0 . 707 0         (0 . 42) (1 . 19) (1 . 92) (20 . 0) −− −−         − 0 . 749 − 2 . 628 0 . 586 − 2 . 418 − 3 . 544 1 . 110     (1 . 80) (3 . 40) (2 . 55) (10 . 7) (12 . 0) (20 . 0) • The t -values are smaller for κ than for κ ∗ . • The largest eigenvalue of κ is 0.586 ⇒ Weekly autocorrelation 0.989, half life 62 weeks. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Summary statistics of the pricing errors (bps) R 2 Maturity Mean MAE Std Max Auto 1 m 1 . 82 6 . 89 10 . 53 60 . 50 0 . 80 99 . 65 3 m 0 . 35 1 . 87 3 . 70 31 . 96 0 . 73 99 . 96 12 m − 9 . 79 10 . 91 10 . 22 55 . 12 0 . 79 99 . 70 2 y − 0 . 89 2 . 93 4 . 16 23 . 03 0 . 87 99 . 94 5 y 0 . 20 1 . 30 1 . 80 10 . 12 0 . 56 99 . 98 10 y 0 . 07 2 . 42 3 . 12 12 . 34 0 . 70 99 . 91 15 y 2 . 16 5 . 79 7 . 07 22 . 29 0 . 85 99 . 40 30 y − 0 . 53 8 . 74 11 . 07 34 . 58 0 . 90 98 . 31 Average − 0 . 79 4 . 29 5 . 48 27 . 06 0.69 99 . 71 • The errors are small. The 3 factors explain over 99%. • The average persistence of the pricing errors (0.69, half life 3 weeks) is much smaller than that of the interest rates (0.991, 1.5 years). Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts 4-week ahead forecasting Three strategies: (1) random walk (RW); (2) AR(1) regression (OLS); (3) DTSM. Explained Variation = 100 × [1 − var ( Err ) / var (∆ R )] Maturity RW OLS DTSM 6 m 0.00 0.53 -31.71 2 y 0.00 0.02 -7.87 3 y 0.00 0.13 -0.88 5 y 0.00 0.44 0.81 10 y 0.00 1.07 -3.87 30 y 0.00 1.53 -36.64 • OLS is not that much better than RW, due to high persistence (max 1.5%). • DTSM is the worst! DTSM can be used to fit the term structure (99%), but not forecast interest rates. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Use DTSM as a decomposition tool • We linearly decompose the LIBOR/swap rates ( y ) as H i = ∂ y i � y i i X t + e i t t ≈ H ⊤ � t , � ∂ X t � X t =0 • We form a portfolio ( m = [ m 1 , m 2 , m 3 , m 4 ] ⊤ ) of 4 LIBOR/swap rates so that 4 4 4 4 � � � � m i y i m i H ⊤ m i e i m i e i p t = i X t + t = t . t ≈ i =1 i =1 i =1 i =1 • We choose the portfolio weights to hedge away its dependence on the three factors: Hm = 0. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Example: A 4-rate portfolio (2-5-10-30) Portfolio weights: m = [0 . 0277 , − 0 . 4276 , 1 . 0000 , − 0 . 6388]. Long 10-yr swap, use 2, 5, and 30-yr swaps to hedge. 8 −15 7.5 Interest Rate Portfolio, Bps −20 7 10−Year Swap, % 6.5 −25 6 −30 5.5 −35 5 4.5 −40 4 −45 3.5 Jan96 Jan98 Jan00 Jan02 Jan96 Jan98 Jan00 Jan02 Hedged 10-yr swap Unhedged 10-yr swap φ (half life): 0.816 (one month) vs. 0.987 (one year). R 2 = 0 . 14 , ∆ R t +1 = − 0 . 0849 0 . 2754 R t + e t +1 , − (0 . 0096) (0 . 0306) R 2 = 1 . 07% for the unhedged 10-year swap rate. Liuren Wu @ Advances in Portfolio Management

Background Overview Affine DTSM Data & Estimation Results Predictability Profitability After thoughts Predictability of 4-rate portfolios Four−Instrument Portfolios 100 90 80 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 Percentage Explained Variance, % • 12 rates can generate 495 4-instrument portfolios. • Robust: Improved predictability for all portfolios (against unhedged single rates) Liuren Wu @ Advances in Portfolio Management