THE DYNAMICS OF LONG FORWARD RATE TERM STRUCTURES XINGGUO LUO JIN E. ZHANG*

In this article, we look at study the dynamics of forward rates with maturities longer than 14 years. We re-document the phenomenon of the downward sloping long forward rate term structure using U.S. Treasury STRIPS data over the period 1988 to 2007. By calibrating Diebold F. X. and Li C.-L.’s (2006) dynamic Nelson C. R. and Siegel A. F. (1987) and Christensen J. H. E., Diebold F. X., and Rudebusch G. D.’s (2007) arbitrage-free Nelson-Siegel models, we find that both models explain the empirical phenomenon very well. Out-of-sample comparison shows that imposing no-arbitrage restriction indeed improves the forecasting performance. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:957–982, 2010

INTRODUCTION The classical interest rate models, such as Vasicek (1977), Cox, Ingersoll, and Ross (1985), and Duffie and Kan (1996), focus on modeling the process of This study was previously circulated under the title “The Dynamics of Interest Rate Term Structure.” We are especially grateful to an anonymous referee whose helpful comments and suggestions substantially improved the study. We also acknowledge helpful comments from Bob Webb (editor), Andrew Carverhill, and seminar participants at the University of Hong Kong and the Quantitative Methods in Finance 2008 (QMF 2008) Conference in Sydney. Jin E. Zhang has been supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7549/09H). *Correspondence author, School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong, P. R. China. Tel: (852) 2859-1033, Fax: (852) 2548-1152, e-mail: [email protected] Received April 2009; Accepted November 2009

■

Xingguo Luo and Jin E. Zhang are at the School of Economics and Finance, The University of Hong Kong, Hong Kong, P. R. China.

The Journal of Futures Markets, Vol. 30, No. 10, 957–982 (2010) © 2010 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.20447

958

Luo and Zhang

short rates by specifying a system of stochastic differential equations for state variables. The maturities of bonds used in most empirical research are no longer than 10 years, and the long end of the term structure of interest rates is usually assumed to be less volatile and flat. However, our result in this study shows that long maturity interest rates do have their own characteristics that are not captured by the classical yield-factor models. With the U.S. Treasury STRIPS data over the recent 20 years from 1988 to 2007, we observe that the term structure of long forward rates is downward sloping and that the volatility of long forward rates does not decrease as maturity increases. These findings should be valuable to fixed-income investors, especially to those who invest in mortgages, pension funds, and other long-term investments.1 In fact, the information in long maturity forward rates has been explored to forecast future interest rates. Fama and Bliss (1987) provide empirical evidence that forward rates predict future spot rates for horizons beyond one year, and they interpret the predictability by using the mean reversion of spot rates towards a constant expected value. Fama (2006) updates the result with more recent data from 1988 to 2004 and reinterprets it with a time-varying expected value. Jorion and Mishkin (1991) find similar results across different countries. Boudoukh, Richardson, and Whitelaw (2005) discover that forward rates forecast both future spot rates and inflation rates and that differentials of these forward rates also forecast exchange rates movements. However, the long forward rates considered in these studies have maturities up to five years. In this study, we focus on forward rates with much longer maturities from 14 years up to 29 years. Few studies have been conducted on directly examining the properties of long maturity forward rates. Dybvig, Ingersoll, and Ross (1996) prove that, in the absence of arbitrage, asymptotic (i.e., infinite maturity) forward rates in frictionless markets can never fall over time. On the cross-sectional perspective, Brown and Schaefer (2000) document the phenomenon of a downward sloping long forward rate term structure using daily STRIPS data from April 22, 1985 to October 5, 1994. They explain the phenomenon with a two-factor Gaussian model that produces a negative correlation between the long-term forward rate slope and the long-term zero coupon yield volatility. Their empirical evidence shows that the relationship can be used to predict both factors. Carverhill (2001) documents independently the same phenomenon using weekly coupon STRIPS data from January 1, 1990 to December 31, 1999. He observes that the volatility of long forward rates does not attenuate and attributes most of the volatility in long forward rates to a single predictable factor. He develops a Vasicek (1977) type of equilibrium model that incorporates the 1

In Europe, with its aging population, insurers of long-term bonds and pension funds are eager to reduce the mismatch between their long-term liabilities and shorter-dated securities. In 2005, France and Britain issued 50-year bonds with great success.

Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

959

predictable factor to explain the observed phenonmenon. Christiansen (2005) also documents the negative correlation between the slope of long forward rates and the long yield volatility. In this study, we re-document the phenomenon of the downward sloping long forward rate2 using the STRIPS price data over the period of the most recent twenty years from January 4, 1988 to December 31, 2007, and we explore simultaneously the time-series and the cross-sectional properties of the long forward rates with the help of the recently developed Nelson and Siegel (1987) type models. Different from previous term structure models that start from specifying the process of state variables to derive bond pricing and yield curve formulae, Nelson and Siegel (1987) directly propose a type of parsimonious yield-curve functional form. The model is quite effective at capturing general shapes of the yield curve, and it has found wide applications among financial markets practitioners and central banks, see, e.g., Bank for International Settlements (2005). Recently, a dynamic version of this model was developed by Diebold and Li (2006), where three latent factors are reinterpreted as level, slope, and curvature. It turns out that the dynamic Nelson-Siegel model is easy to estimate and especially good at forecasting compared with many other models. Christensen, Diebold, and Rudebusch (2007) make further progress in this direction by combining the Nelson-Siegel factor loading structure with a no-arbitrage requirement. Their empirical evidence shows that the forecasting performance, particularly for moderate-to-long maturities, is improved by imposing absence of arbitrage. The purpose of this study is to enhance our understanding of the dynamics of long forward rates. First, we update the documentation of the downward sloping phenomenon with more recent data. Second, we examine properties of long forward rates in detail by using Diebold and Li’s (2006) dynamic Nelson-Siegel and Christensen et al.’s (2007) newly developed arbitrage-free Nelson-Siegel models. To our knowledge, we are the first to investigate in-sample fitting and outof-sample forecasting abilities of two models in the context of long forward rates. The rest of this study proceeds as follows. The next section presents empirical evidence on the downward sloping feature of long forward rate term structure. Later section briefly reviews the dynamic Nelson-Siegel and the arbitrage-free Nelson-Siegel models. Following that section the performance of in-sample fitting is studied and the downward sloping phenomenon observed in the earlier section is explained. Penultimate section examines the performance of out-ofsample forecasts. Final section concludes the study. 2

The forward rate is more sensitive to the change in maturity than is the zero-coupon bond yield, because the yield is a kind of average of forward rates with different time to maturity. Journal of Futures Markets

DOI: 10.1002/fut

960

Luo and Zhang

EMPIRICAL EVIDENCE ON THE DOWNWARD SLOPING PHENOMENON OF THE LONG FORWARD RATES CURVE In this section, we introduce our STRIPS data from which the long forward rates will be extracted and examine the cross-sectional property of long forward rates. Data The data used in this study are the daily closing STRIPS3 prices obtained from DataStream, ranging from January 4, 1988 to December 31, 2007, where the former is the earliest starting date available.4 From its inception in February 1985, the Treasury STRIPS market quickly expanded and it is now considered to be a highly successful Treasury program, because it allows investors to create and trade zero-coupon securities more easily and at a lower cost. Treasury Bonds, Treasury Inflation-Protected Securities (TIPS), and Treasury Notes are the three main securities eligible for the STRIPS program. As reported in the Monthly Statement of the Public Debt by the U.S. Treasury, maturities of the STRIPS securities are available from 5 to 30 years for Treasury Bonds, at 19 and 20 years for TIPS and from 1 month to 10 years for Treasury Notes. The market sizes of the STRIPS from the three fixed-income securities are around US$150 billion, US$30 billion, and US$170 million, respectively. Long-term STRIPS are more liquid than are short-term ones. STRIPS enable the dealer to separately sell individual coupons and principals after buying Treasury bonds, and so there is no default risk. An important distinction between coupon and principal STRIPS is the fungibility of the former in the sense that coupon STRIPS from different stripped securities with the same payment are treated as the same security, while principal STRIPS can only be created from a particular note or bond and must be used to reconstitute that original security. This fungibility may induce large differences in the prices of coupon and principal STRIPS as noted by Daves and Ehrhardt (1993) and Sack (2000). Furthermore, STRIPS can be regarded as having pure discount factors, and there is no need to estimate it from Treasury coupon bonds. Finally, Zaretsky (1995) and Brown and Schaefer (2000) notice that the term structures derived from STRIPS and coupon bonds are essentially similar. Specifically, we use only coupon STRIPS, while Brown and Schaefer (2000) do not discriminate between coupon and principal STRIPS. In addition, in contrast to Carverhill (2001) who uses the February 15 to August 15 3

STRIPS is an acronym for the Separate Trading of Registered Interest and Principal Securities. The ending date, December 31, 2007, was the most recent year end when the first draft of this study was prepared.

4

Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

961

cycle only, we make full use of May 15 to November 15 cycle whenever it is available, which may improve the accuracy of interpolation in calculating nonexistent STRIPS. The Downward Sloping Phenomenon of Long Forward Rates The long forward rates considered in this study are instantaneous forward rates with maturities beyond 14 years. Theoretically, the time t continuously compounded forward rate, f(t, T, T ⫹ ⌬T), which covers the future period (T, T ⫹ ⌬T), is defined as B(t, T ⫹ ¢T) (1) , e ⫺f(t,T,T⫹ ¢T)¢T ⫽ B(t, T) where B(t, T) is the current price of the zero-coupon bond that matures at time T. Therefore, the time t instantaneous forward rate with maturity at time T is given by f(t, T) ⫽ lim f(t, T, T ⫹ ¢T) ⫽ ⫺ ¢TS0

0 ln B(t, T) . 0T

(2)

Since instantaneous forward rates are not observable in practice, we use the average forward rates between maturities T – ⌬T and T ⫹ ⌬T to approximate the instantaneous forward rate with maturity T. Following Brown and Schaefer (2000), the instantaneous forward rate with a maturity of 20 years is calculated as f(t, t ⫹ 20) ⫽

P(t, t ⫹ 19) 365 ln , T21 ⫺ T19 P(t, t ⫹ 21)

(3)

where P(t, T) is the time t price of the coupon STRIPS with maturity T, and T19 and T21 are the exact days to maturity of 19- and 21-year STRIPS, respectively. Linear interpolation is used when there is no corresponding STRIPS. In Figure 1, we provide a three-dimensional plot of our daily forward rate curves throughout the sample period from January 1988 to December 2007 at maturities of 14–24 years. We also show the annual averages of the forward rates from 1988 to 2007 in Figure 2. Since available maturities in different years are not equally distributed, we divide the whole period into two subperiods, 1988–1995 and 1996–2007. The longest maturities of STRIPS obtained in two subperiods are 29 and 25 years, respectively. According to formula (3), the longest forward rates that can be obtained in two subperiods are at 28 and 24 years, respectively. From Figures 1 and 2, it is fair to say that the forward rate curves are decreasing with maturity, although they are not always monotonic. Journal of Futures Markets

DOI: 10.1002/fut

962

Luo and Zhang

Forward rate curves

12

Yield (Percent)

10 8 6 4 2 0 Jan08 Sep04 May01

24 22 Jan98 Sep94 Dat May91 e

20 18 16 Jan88

14

Time

to

ity Matur

) (Years

FIGURE 1

Forward rates from 1988 to 2007. We show a three-dimensional plot of the forward rates constructed from daily closing STRIPS prices using approximation formula as in Equation (3). The sample period is January 4, 1988 to December 31, 2007.

There are also some jumps in 2002 and 2003. This might be due to the effects of macroeconomic and monetary policy surprises or liquidity issues related to the longest maturities.5 Table I reports some descriptive statistics for daily forward rates at various maturities (mean, minimum, maximum, standard deviation, skewness, kurtosis, and the sample autocorrelations at displacements of 1, 21, and 126 days). From the first column of the table, we observe that the term structure of the average forward rates is downward sloping. The forward rate decreases from 7.153% for 14 years to 6.089% for 24 years. Some other stylized facts are the following: the temporal variation in the level is large; the volatility (around 1.3%) is relatively large compared with mean value of 6–7%; autocorrelations are very high for all long forward rates. The skewness is negative for maturities of 15, 17, 18, 19, and 20 years and positive for the others, but the absolute value of the skewness is small. The kurtosis is generally less than 3 except for maturity of 18 years. In general, the long forward rate is very close to normal random noise. 5

On October 31, 2001, the U.S. Treasury announced that the U.S. Government would suspend the sale of 30-year bonds and that it would not buy back any debt in January 2002.

Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

1988

1989

1990

1991

10

10

10

10

8

8

8

8

6

6

6

6

15

20

15

25

1992

20

25

15

1993 8 6 4

1996

20

15

25

25

6

4 15

20

8

6

6 25

15

1995

8

8

20

25

1994

10

15

20

963

1997

20

25

4

15

1998

20

25

1999

8

8

6

6

8

8

6

6

4 15

20

25

4

15

2000 6 4 20

4 15

25

2001

8

15

20

8

6

6

4

4 15

2004

20

15

25

2002

8

25

20

25

2003 8 6

15

25

20

2005

20

25

4

15

2006

20

25

2007

8 6

6

6

4

4

4

6 4 15

20

25

2

15

20

2

25

15

20

25

2

15

20

25

FIGURE 2

Annual average forward rates from 1988 to 2007. We show the average one-year forward rates calculated by using daily U.S. STRIPS prices. The horizontal axis is maturity while the vertical axis is annual average forward rate (percent). Note that the vertical scale is the same (500 basis points) in each case.

NELSON-SIEGEL (1987) TYPE MODELS In this section, we briefly review Diebold and Li’s (2006) dynamic Nelson-Siegel model and Christensen et al.’s (2007) arbitrage-free Nelson-Siegel model, which will be employed to explain the phenomenon observed in the last section. The Dynamic Nelson-Siegel Model In Diebold and Li’s (2006) dynamic version of the Nelson-Siegel model, at current time t, the instantaneous forward rates with time to maturity, t, are given by Journal of Futures Markets

DOI: 10.1002/fut

964

Luo and Zhang

TABLE I

Descriptive Statistics for the Forward Rates Maturity (Years)

Mean

Min.

Max.

Std. Dev.

Ske.

Kur.

r(1)

r(21)

r(126)

14 15 16 17 18 19 20 21 22 23 24

7.153 7.092 7.047 6.984 6.837 6.687 6.639 6.573 6.437 6.273 6.089

4.678 4.528 4.469 4.596 0.153 2.742 3.724 3.838 3.870 3.648 3.617

10.557 9.855 11.647 9.561 9.706 9.429 9.381 9.577 9.481 9.720 9.364

1.214 1.266 1.335 1.293 1.363 1.358 1.376 1.398 1.358 1.373 1.338

0.005 ⫺0.094 0.047 ⫺0.016 ⫺0.332 ⫺0.077 ⫺0.014 0.009 0.021 0.005 0.061

2.077 1.990 2.098 1.792 3.056 1.932 1.811 1.814 1.840 1.771 1.671

0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.998 0.998 0.997 0.996

0.962 0.967 0.937 0.964 0.926 0.966 0.968 0.973 0.975 0.973 0.968

0.841 0.864 0.814 0.843 0.797 0.852 0.879 0.885 0.877 0.886 0.887

Note. In this table we show descriptive statistics for daily forward rates (percent) at different maturities. The forward rates are constructed from daily closing STRIPS prices using an approximation formula as in Equation (3). Reported are the mean, minimum, maximum, standard deviation, skewness, kurtosis, and the sample autocorrelations at displacements of 1, 21, and 126 days. The sample period is January 4, 1988 to December 31, 2007 (5,216 observations).

Factor loadings with λ⫽0.107. 1 1: X1t Loadings 0.8 e⫺λτ: Xt Loadings Loadings

2

0.6

0.4 λτ e⫺λτ: Xt Loadings 3

0.2

0

0

5

10

15

20

25

30

35

40

τ (Maturity, in Years) FIGURE 3

Factor Loadings. We show the factor loadings in the three factor model ft (t) ⫽ X1t ⫹ e ⫺ltX t2 ⫹ lte ⫺ltXt3 with l ⫽ 0.05, where X1t , X2t , and X3t are three factors. t denotes time to maturity (in years).

ft (t) ⫽ X1t ⫹ e ⫺ltX2t ⫹ lte ⫺ltX3t ,

(4)

where Xt ⫽ (X1t , X2t , X3t ) can be interpreted as three latent dynamic factors, and the associated loadings are 1, e⫺lt, and lte⫺lt, respectively. To have better understanding of this Nelson-Siegel factor structure, we illustrate the three Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

965

factor loadings with a fixed l ⫽ 0.05 in Figure 3. The loading on X1t is a constant; hence, X1t may be viewed as a long-term level factor. The loading on X2t is a function that starts at 1 but decays monotonically and quickly to 0; hence, X2t may therefore be viewed as a short-term slope factor. The loading on X3t is a hump; hence, X3t may be viewed as a medium-term factor. Moreover, in order to construct forecasts, Diebold and Li (2006) model the three factors to be independent first-order autoregressions. It turns out that this simple model is adequate enough in forecasting future yields. We will examine whether this powerful AR(1) process is also good at forecasting long forward rates in a later section. The Arbitrage-Free Nelson-Siegel Model In Christensen et al.’s (2007) arbitrage-free Nelson-Siegel model, three factors are assumed to follow a system of stochastic differential equations under the risk-neutral measure, Q, dX1t 0 0 ° dX2t ¢ ⫽ ° 0 l dX3t 0 0

0 uQ X1t dW1,Q 1 t 2 2,Q ⫺ l ¢ £ ° uQ ¢ ⫺ ° X ¢ § dt ⫹ © ° dW ¢, 2 t t Q 3 3,Q l u3 Xt dW t

(5)

2,Q where the three incremental standard Brownian motions, dW1,Q t , dW t , and dW 3,Q t , are assumed to be independent, the variance-covariance matrix,

s11 s12 s13 © ⫽ ° s21 s22 s23 ¢ s31 s32 s33 is symmetric, i.e., sij ⫽ sji with i, j ⫽ 1, 2, 3, and the mean-reverting speed, l, is positive. The instantaneous risk-free rate, rt, is assumed to be the sum of the first two factors: rt ⫽ X1t ⫹ X2t .

(6)

Q Q Then, by fixing uQ 1 ⫽ u2 ⫽ u3 ⫽ 0 under the Q-measure, Christensen et al. (2007) show that zero-coupon bond yields are given by

y(t, T) ⫽ X1t ⫹

C(t) 1 ⫺ e ⫺lt 2 1 ⫺ e⫺lt Xt ⫹ a ⫺ e ⫺lt b X3t ⫺ , t ⬅ T ⫺ t, (7) t lt lt

where Journal of Futures Markets

DOI: 10.1002/fut

966

Luo and Zhang

C(t) ⫽ A

t3 t 1 ⫺ e ⫺lt 1 ⫺ e ⫺2lt ⫹ Ba 2 ⫺ ⫹ b 6 2l l3 4l3

⫹ Ca

t te⫺lt t2e⫺2lt 3te⫺2lt 2(1⫺ e⫺lt ) 5(1⫺ e⫺2lt ) ⫹ ⫺ ⫺ ⫺ ⫹ b 2l2 l2 4l 4l2 l3 8l3

⫹Da

t2 1 ⫺ e ⫺lt te ⫺lt ⫹ ⫺ b 2l l2 l3

⫹ Ea

3(1 ⫺ e ⫺lt ) 3te ⫺lt t2 t2e ⫺lt ⫹ b ⫹ ⫺ l2 2l l l3

⫹ Fa

3(1 ⫺ e ⫺lt ) 3(1 ⫺ e ⫺2lt ) t te ⫺lt te ⫺2lt ⫹ ⫺ ⫺ ⫹ b l2 l2 2l2 l3 4l3

(8)

and A ⫽ s211 ⫹ s212 ⫹ s213 B ⫽ s221 ⫹ s222 ⫹ s223 C ⫽ s231 ⫹ s232 ⫹ s233 D ⫽ s11s21 ⫹ s12s22 ⫹ s13s23 E ⫽ s11s31 ⫹ s12s32 ⫹ s13s33 F ⫽ s21s31 ⫹ s22s32 ⫹ s23s33. There are some important features of the model. First, the three latent factors can be identified as the level, slope, and curvature, which eliminates the troublesome local maxima. As pointed out by Diebold and Li (2006), current factor loadings are more preferred than the ones in Nelson and Siegel (1987) in the sense of estimation. Second, the parameter l is assumed to be constant over the sample, while Nelson and Siegel (1987) choose the best-fitting value of it for different data sets. Third, the crucial point that matters to empirical performance is the additional maturity-dependent term, ⫺ C(t)/t, through which the variance matrix, ⌺, affects the yield function. When shocks of three factors are assumed to be independent,6 i.e., sij ⫽ 0, for i ⫽ j, the corresponding time t instantaneous forward rates with time to maturity t in the arbitrage-free Nelson-Siegel model become 0C(t) 0t 1 ⫺lt 2 ⫺lt 3 ⫽ Xt ⫹ e Xt ⫹ lte Xt ⫹ D(t),

ft (t) ⫽ X1t ⫹ e ⫺ltX2t ⫹ lte ⫺ltX3t ⫺

6

(9)

Christensen et al. (2007) show that the independent-factor case is more preferred due to the generally poor performance and over-parameterizations of the correlated-factor model.

Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

967

where D(t) ⫽ ⫺s211

⫺lt 2 (1 ⫺ e ⫺lt ) 2 ) t2 2 (1 ⫺ (1 ⫹ lt)e ⫺ s222 ⫺ s . 33 2 2l2 2l2

(10)

Comparing Equations (4) and (9), we notice that the only difference between forward rates in dynamic and arbitrage-free versions of the NelsonSiegel model is the additional maturity-dependent term, D(t). Later, we examine the importance of this term after s11, s22, s33, and l are estimated. We use Equations (4) and (9) to fit the market data in the estimation. Before moving into the details of the estimation, let us examine theoretical properties of the long forward rates in these two models. Properties of Long Forward Rates From the cross sectional perspective, differentiating ft(t) with respect to maturity t in Equations (4) and (9), we have 0ft (t) ⫽ le ⫺lt ( ⫺ X2t ⫹ (1 ⫺ lt)X3t ), 0t

(11)

and 0ft (t) ⫽ le ⫺lt (⫺ X2t ⫹ (1 ⫺ lt)X3t ) ⫺ s211t 0t ⫺ s222

(1 ⫺ e ⫺lt )e ⫺lt ⫺ s233 (1 ⫺ (1 ⫹ lt)e ⫺lt )te ⫺lt l

(12)

respectively. Furthermore, when parameters and dynamics of latent factors are available in the estimation, we can check if both models are able to capture the previously observed downward sloping phenomenon of the long forward rate curve in the market. From the time series perspective, Dybvig et al. (1996)7 show that if the forward rate from T to T ⫹ 1 is stochastic, then absence of arbitrage implies that the infinite horizon forward and zero-coupon rates, taking the limit as t S ⬁ , may increase, but they can never decrease with time t. 7

The result is now well known as the Dybvig, Ingersoll, and Ross (DIR) theorem. McCulloch (2000) corrects an error in DIR’s proof. Hubalek, Klein, and Teichmann (2002) provide a general proof of the DIR theorem and discuss some inconsistencies in the original proof. Jordan, Jordan, Smolira, and Travis (2008) provide empirical evidence that the long forward rate did fall substantially during the period 1996–2000. Journal of Futures Markets

DOI: 10.1002/fut

968

Luo and Zhang

Currently, the forward rates in two models are f DNS (t, T, T ⫹ 1) ⫽ X1t ⫹ a

1 ⫺ e ⫺l(t⫹1) 1 ⫺ e ⫺lt 2 ⫺ b Xt l l

1 ⫺ e ⫺lt 1 ⫺ e ⫺l(t⫹1) ⫺ (t ⫹ 1)e ⫺l(t⫹1) ⫺ ⫹ te ⫺lt bX3t ⫹a l l

(13)

and f AFNS (t, T, T ⫹ 1) ⫽ f DNS (t, T, T ⫹ 1) ⫹ C(t) ⫺ C(t ⫹ 1), t ⬅ T ⫺ t,

(14)

where C(t) is given by Equation (8) with sij ⫽ 0, for i ⫽ j, and we use the relationship between forward rates and yields as in Equation (1); that is, f(t, T, T ⫹ ¢T) ⫽ ⫺ ⫽

ln B(t, T ⫹ ¢T) ⫺ ln B(t, T) ¢T

y(t, T ⫹ ¢T)(T ⫹ ¢T ⫺ t) ⫺ y(t, T)(T ⫺ t) . ¢T

Therefore, two corresponding asymptotic (i.e., infinite maturity) long forward rates are lim f DNS (t, T, T ⫹ 1) ⫽ X1t

tS⬁

(15)

and lim f AFNS (t, T, T ⫹ 1) ⫽ X1t ⫹ s211

tS⬁

⫹

s222

⫺ 3t2 ⫺ 3t ⫺ 1 6

⫺1 2 ⫺ 1 ⫽ ⫺⬁. 2 ⫹ s33 2l 2l2

(16)

Since the dynamic Nelson-Siegel model is not constructed with the no-arbitrage restriction, the fact that its asymptotic long forward rate, X1t , is normal random noise (it might fall with time t) does not contradict the DIR theorem. The asymptotic long forward rate of the arbitrage-free Nelson-Siegel model does not exist. This fact does not contradict the DIR theorem either. ESTIMATION We estimate two models by using the Kalman filter method. The well-known Kalman filter can be used to estimate parameters of a dynamic term structure model if it can be formulated into a state-space form. This approach has recently gained popularity in the affine term-structure literature, see e.g., Duan and Simonato (1999), Babbs and Nowman (1999), de Jong (2000), and De Rossi Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

969

(2004). The main advantage of the state-space model is that it allows the econometrician to exploit the information on both the cross sections and the time series of interest rates at each time instance. The inherent dynamic features of the three latent factors make it easy to transform the two current models into state-space forms, as illustrated below. The State-Space Form of the Dynamic Nelson-Siegel Model In the dynamic Nelson-Siegel model, when three factors are assumed to be independent first-order autoregressions, the state equation is given by a11 0 0 X1t⫺1 ⫺ m1 h1t X1t ⫺ m1 ° X2t ⫺ m2 ¢ ⫽ ° 0 a22 0 ¢ ° X2t⫺1 ⫺ m2 ¢ ⫹ ° h2t ¢ , X3t ⫺ m3 0 0 a33 X3t⫺1 ⫺ m3 h3t

(17)

where the error terms, ht ⫽ (h1t , h2t , h3t ) , have a conditional covariance matrix, 0 q211 0 2 Q ⫽ ° 0 q22 0 ¢ . 0 0 q233

(18)

Recall from Equation (4) that the measurement equation in matrix form is ft (t1 ) 1 e ⫺lt1 f (t ) 1 e ⫺lt2 ± t 2 ≤ ⫽ ± o o o ⫺ltN 1 e ft (tN )

lt1e ⫺lt1 et (t1 ) X1t ⫺lt2 e (t ) lt2e ≤ ° X2t ¢ ⫹ ± t 2 ≤ , o o X3t ⫺ltN ltNe et (tN )

(19)

where the measurement errors, et(ti), are assumed to be i.i.d. white noise. The State-Space Form of the Arbitrage-Free NelsonSiegel Model In order to satisfy the required structure described in Equation (5) under the Q – measure and maintain affine dynamics under the P – measure, we impose an essentially affine risk premium specification as in Duffee (2002). Particularly, dWtQ ⫽ dW tP ⫹ gt dt,

(20)

where g01 g11 gt ⫽ ° g02 ¢ ⫹ ° 0 g03 0

0 g22 0

0 X1t g32 ¢ ° X2t ¢ . g33 X3t Journal of Futures Markets

(21)

DOI: 10.1002/fut

970

Luo and Zhang

Due to the flexible risk premium specification of gt, we are free to choose the dynamics of the state variables under the P – measure.8 Therefore, in the independent-factor, arbitrage-free Nelson-Siegel model, the state equation becomes dX1t kP11 0 0 uP1 X1t s11 ° dX2t ¢ ⫽ ° 0 kP22 0 ¢ £ ° uP2 ¢ ⫺ ° X2t ¢ § dt ⫹ ° 0 0 0 kP33 0 dX3t uP3 X3t

0 0 s22 0 ¢ dW tP, 0 s33 (22)

and the measurement equation from Equation (9) is ft (t1 ) D(t1 ) 1 e ⫺lt1 f (t ) D(t2 ) 1 e ⫺lt2 ± t 2 ≤ ⫽ ± ≤ ⫹ ± o o o o ⫺lt3 1 e ft (tN ) D(tN )

et (t1 ) lt1e ⫺lt1 X1t ⫺lt2 lt2e e (t ) ≤ ° X2t ¢ ⫹ ± t 2 ≤ , (23) o o X3t ⫺ltN lt3e et (tN )

where the measurement errors, et(ti), are also assumed to be i.i.d. white noise. The Kalman Filter Since forward rates (see Equation (9)) are determined under the risk-neutral Q – measure, it seems that imposing restrictions on the drift terms Q Q (uQ 1 ⫽ u2 ⫽ u3 ⫽ 0) could limit the ability of the model to fit observed forward rates. Proposition 2 in Christensen et al. (2007) fixes this problem, and it gives a simple condition under which two types of identifying restrictions under the two measures are equivalent. Therefore, we are able to let the mean in the arbitrage-free Nelson-Siegal model be 0 under the Q – measure and let it be estimated under the P – measure. In order to implement Kalman filter estimation, we first transform continuous time state equations of two models into discrete-time versions. For the dynamic Nelson-Siegel model, we rewrite the state Equation (17) as Xt ⫽ (I ⫺ A)m ⫹ AXt⫺1 ⫹ ht ,

(24)

where I is the (3 ⫻ 3) identity matrix and m ⫽ (m1, m2, m3)⬘,

8

a11 0 0 A ⫽ ° 0 a22 0 ¢ , 0 0 a33

ht ⬃ N(0, Q).

(25)

Christensen et al. (2007) point out that a more general specification of the risk premium is required in the correlated-factor, arbitrage-free Nelson-Siegel model.

Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

971

For the arbitrage-free Nelson-Siegel model, the state Equation (22) is transformed into9 Xt ⫽ (I ⫺ e ⫺K

)uP ⫹ e ⫺K ¢tXt⫺1 ⫹ ht ,

(26)

kP11 0 0 P P P P P P u ⫽ (u1 , u2 , u3 )⬘, K ⫽ ° 0 k22 0 ¢ . 0 0 kP33

(27)

P

¢t

P

where

The conditional covariance matrix for the shock terms is given by Q⫽

冮

¢t

0

e

⫺KPs

©©⬘e

⫺ (KP)⬘s

ds,

s11 ©⫽ ° 0 0

0 0 s22 0 ¢ . 0 s33

(28)

Following Diebold, Rudebusch, and Aruoba (2006), for both types of models, the Kalman filter is started at the unconditional mean and covariance, and the measurement error covariance matrices are assumed to be diagonal. Estimation Results One data sample is selected every four weeks, starting from the second Wednesday10 in January 1988 until December 2007, and 11 long maturities, namely 14-year to 24-year maturities, are selected in the estimation. There are 260 observations. Note that treatment of l is different in Diebold and Li (2006) and Christensen et al. (2007). The former presumes it to be a constant whereas the latter estimates it at the same time as other parameters. Because of this, we estimate two versions of the dynamic Nelson-Siegel model (DNS hereafter) and the arbitrage-free Nelson-Siegel model (AFNS hereafter) for comparison. At first, we estimate parameters by fixing l ⫽ 0.05, which maximizes the loading on the curvature factor at 20 years. Then, we allow l to be estimated as well. We denote DNS when l is fixed and DNS-l when l is also estimated. Table II reports the estimates of the DNS model. In both versions of the DNS model, the estimated mean-reversion speeds and volatilities are very close, while the curvature factor (a33) has the highest mean-reverting speed. However, the differences are substantial with opposite signs for the estimated mean-reverting levels, m1 and m2, of the term-structure level and slope factors. Actually, in an unreported exercise, we find that the two estimated factors, The matrix exponential, eA, is defined as eA ⫽ 1 ⫹ A2/2! ⫹ A3/3! +. . . , where A is a square matrix. In particular, the second Wednesday of every four weeks is selected and there is not much difference when other Wednesdays are used. 9

10

Journal of Futures Markets

DOI: 10.1002/fut

972

Luo and Zhang

TABLE II

Parameter Estimates of the Independent-Factor DNS Model Parameters a11 a22 a33 m1 m2 m3 q211 q222 q233 l

DNS

DNS-l

0.990328 (0.007847) 0.926389 (0.026086) 0.994878 (0.005650) ⫺0.013452 (0.258496) 0.064821 (0.006522) 0.142872 (0.272751) 0.000021 (0.000004) 0.000057 (0.000007) 0.000134 (0.000025) – –

0.978240 (0.011350) 0.984248 (0.010109) 0.991067 (0.007683) 0.032980 (0.007490) ⫺0.023020 (0.027266) 0.128516 (0.051917) 0.000009 (0.000001) 0.000104 (0.000017) 0.000104 (0.00015) 0.093941 (0.003976)

Note. This table presents the estimation results for the independent-factor DNS model with monthly data on 14- to 24-year forward rates. The sample period is 1988:01 to 2007:12. The statespace form of the independent-factor DNS model is given by X 1t ⫺ m1 a11 0 0 X 1t⫺1 ⫺ m1 h1t ° X 2t ⫺ m2 ¢ ⫽ ° 0 a22 0 ¢ ° X 2t⫺1 ⫺ m2 ¢ ⫹ ° h2t ¢ , X 3t ⫺ m3 0 0 a33 X 3t⫺1 ⫺ m3 h 3t where h1t , h2t , and h3t are independent normal random noises with variance q211, q222, and q233, respectively. Parameters are estimated by the Kalman filter method. Estimated standard deviations are given in parentheses. DNS, l is fixed to be 0.05; DNS-l, l is estimated. The maximum log-likelihood values are 7750.02 and 7838.09.

X1t and X2t , are also very different as well. In addition, as can be seen from Table II, estimated l (0.093941) is almost twice as large as the presumed one (0.05). Our exercise shows that the parameter l is crucial in interpreting the three latent factors of level, slope, and curvature in the case of the DNS model. The maximum log-likelihood values for DNS-l (7838.09) is larger than that for DNS (7750.02). This is consistent with the additional flexibility in the optimization when l is also estimated. Table III reports the estimates of the AFNS model. In contrast to the results in the DNS model, there is not much difference between estimated parameters in two estimations, except for the magnitude of mean-reverting speed (kP22) and the level (uP2 ) of the term-structure slope factor. Similarly, more flexible estimation achieves a larger maximum log-likelihood value (7761.81 vs. 7877.18). To compare with the DNS model, we calculate the one-month Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

973

TABLE III

Parameter Estimates of the Independent-Factor AFNS Model Parameters kP11 kP22 kP33 uP1 uP2 uP3 s211 s222 s233 l

AFNS

AFNS-l

0.100110 (0.066468) 0.892973 (0.192669) 0.044431 (0.071812) 0.533331 (0.049637) ⫺0.420635 (0.040860) ⫺0.565920 (0.111666) 0.000173 (0.000019) 0.000703 (0.000089) 0.001090 (0.000137) – –

0.254459 (0.145281) 0.166631 (0.107415) 0.118191 (0.097068) 0.228360 (0.018887) ⫺0.109764 (0.045323) ⫺0.198242 (0.055558) 0.000096 (0.000009) 0.001711 (0.000333) 0.001284 (0.000180) 0.107442 (0.005164)

Note. This table presents the estimation results for the independent-factor AFNS model with monthly data on 14- to 24-year forward rates. The sample period is 1988:01 to 2007:12. The statespace form of the independent-factor AFNS model is given by P dX 1t k11 ° dX t2 ¢ ⫽ ° 0 dX t3 0

0 P k22 0

0 uP1 X 1t s11 0 ¢ £ ° uP2 ¢ ⫺ ° X 2t ¢ § dt ⫹ ° 0 P uP3 X t3 k 33 0

0 0 dW 1,P t s22 0 ¢ ° dW t2,P ¢ , dW t3,P 0 s33

2,P where dW 1,P and dW t3,P are independent incremental standard Brownian motions. t , dW t Parameters are estimated by the Kalman filter method. Estimated standard deviations are given in parentheses. AFNS, l is fixed to be 0.05; AFNS-l, l is estimated. The maximum log-likelihood values are 7761.81 and 7877.18.

conditional mean-reversion matrix in the AFNS model by using the transform ~ P1 K ⫽ e⫺K 12 and have ~ KAFNS ⫽ °

0.9917 0 0 0.9790 0 0 ~ 0 0.9276 0 ¢ , KAFNS⫺l ⫽ ° 0 0.9862 0 ¢. 0 0 0.9963 0 0 0.9902 (29)

These results are about the same as those reported for the DNS model. It can be concluded that the AFNS model is slightly better than the DNS model in terms of fitting performance, which is indicated by the root mean squared errors (RMSEs) in Table IV. In the table, we report summary statistics Journal of Futures Markets

DOI: 10.1002/fut

974

Luo and Zhang

TABLE IV

Summary Statistics of In-Sample Fit DNS

DNS-l

AFNS

AFNS-l

Maturity

Mean

RMSE

Mean

RMSE

Mean

RMSE

Mean

RMSE

14 15 16 17 18 19 20 21 22 23 24

1.04 ⫺0.96 0.74 2.99 ⫺2.48 ⫺7.14 ⫺0.17 4.40 3.19 1.71 ⫺3.34

19.54 18.79 20.53 20.97 18.22 31.55 32.56 12.74 28.91 31.81 17.02

1.62 ⫺1.36 0.08 2.53 ⫺2.53 ⫺6.74 0.54 5.18 3.70 1.54 ⫺4.65

19.62 18.73 20.43 20.91 18.16 31.49 32.43 12.70 28.91 31.85 16.92

⫺0.92 ⫺0.30 2.40 4.53 ⫺1.74 ⫺7.45 ⫺1.42 2.66 1.71 1.50 ⫺1.01

19.48 18.86 20.63 21.03 18.22 31.68 32.76 12.73 28.88 31.80 17.11

⫺0.38 ⫺0.54 1.86 4.05 ⫺1.92 ⫺7.23 ⫺0.86 3.38 2.28 1.47 ⫺2.22

19.44 18.71 20.42 20.86 18.03 31.50 32.51 12.47 28.83 31.83 16.92

Note. This table presents the means and the root mean squared errors (RMSEs) for 11 different maturities. The DNS model is given by equations (17) and (19). The AFNS model is given by equations (22) and (23). The DNS and AFNS mean l is fixed to be 0.05 in the estimation, and the DNS-l and AFNS-l mean l is also estimated in the corresponding models. All numbers are measured in basis points. Note that the results are comparable with those in Christensen, Diebold and Rudebusch (2007).

of fitted errors defined as the difference between market-implied and modelimplied (see Equations (4) and (9)) forward rates. Generally, both models achieve very small RMSEs for the in-sample fit. Note that the RMSEs are of the same magnitude as those reported by Christensen et al. (2007). More importantly, the two models are both good at capturing the aforementioned downward sloping characteristic of the long forward rates. This becomes clear when we look at Figures 4 and 5, where we plot the average forward rates obtained by the AFNS model and compare them with market data. Note that, in Figure 5, we only select maturities at 14 and 24 years to demonstrate the model’s fitting performance. In addition, as mentioned before, we are able to investigate the property of long forward rates in more detail when parameters are estimated (see Equations (11) and (12)). Figure 6 displays this by showing the time series of partial derivatives of forward rates with respect to maturity. It can be seen that the AFNS model can generate a negative relationship between the long forward rate and the maturity. The results for the DNS model are similar to those for the AFNS model presented in Figures 4–6. In fact it is easy to understand that the two models have similar fits to the market data because the same dynamics are assumed under the P – measure (see Equations (24) and (26)), and the only difference is the additional maturitydependent term, D(t), in the AFNS model (see Equation (10)). This additional term is plotted in Figure 7 together with its three components that come from the volatilities of the term-structure level, slope, and curvature. For nearly every maturity shown in the figure, the absolute value of the maturity-dependent Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

975

Average forward rate by monthly data 7.5

Forward rate (Percent)

Actual AFNS AFNS−λ

7

6.5

6 14

15

16

17 18 19 Maturity (Years)

20

21

22

23

24

FIGURE 4

Market and fitted average forward rates by the AFNS Model. We show the market average forward rates and the fitted average forward rates obtained by the AFNS model. The sample period is 1988:01 to 2007:12. AFNS means l is fixed to be 0.05 in the estimation, and AFNS-l means l is also estimated in the AFNS model.

Time series of fitted forward rates by the AFNS−λ model 11 10 9 8 7 6 5 4 3

Jun90

Mar93

Dec95

Sep98

May01

Feb04

Nov06

FIGURE 5

Market and fitted forward rates by the AFNS-l Model. We show the market forward rates (dotted line) and the fitted forward rates (solid line) at maturities of 14 years (upper) and 24 years (lower) obtained by the AFNS-l model (percent). The sample period is 1988:01 to 2007:12. AFNS-l means l is also estimated in the AFNS model. Journal of Futures Markets

DOI: 10.1002/fut

976

Luo and Zhang

⫻ 10⫺3

Time series of partial derivative of forward rates w.r.t. maturity by the AFNS−λ model

0 20⫺y 22⫺y 24⫺y

⫺0.5

⫺1

⫺1.5

⫺2

⫺2.5

⫺3

Jun90

Mar93

Dec95

Sep98

May01

Feb04

Nov06

FIGURE 6

Time series of partial derivative of forward rates w.r.t. maturity by the AFNS-l Model. We show time series of partial derivative of forward rates w.r.t. maturity by the AFNS-l Model in Equation (12) with estimated parameters in Table III.

term is larger than the corresponding mean value presented in Table I. In general, all three components are negative regardless of different values of l. It also turns out that the component from the volatility of the term-structure slope is the largest in terms of the absolute value. As discussed before, compared with the DNS model, this additional term comes from the no-arbitrage restriction in the AFNS model. We have seen from Figure 7 that the term is relatively large and will be important in estimating parameters. Furthermore, it also affects the dynamics of three latent factors, especially X1t and X3t . We illustrate this point in Figure 8, where we plot filtered factors in the two models when l is also estimated. We find that with this term, D(t), X1t shifts upward by 0.2 in the AFNS model compared with that in the DNS model, and X3t becomes negative in the AFNS model while being positive in the DNS model. FORECASTS In this section, we compare two models in terms of their out-of-sample forecast performance. Following Christensen et al. (2007), we construct 1-, 6-, and Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

0

977

The Adjustment Term for the AFNS model with λ⫽0.107.

⫺0.02

Loadings

⫺0.04 ⫺0.06 ⫺0.08 ⫺0.1 Total Level Slope Curvature Level and Slope

⫺0.12 ⫺0.14 14

15

16

17

18 19 20 τ (Maturity, in Years)

21

22

23

24

FIGURE 7

The adjustment term for the⫺AFNS Model. We show the adjustment term in the AFNS model, 2 (1 ⫺ e lt ) 2 (1 ⫺ (1 ⫹ lt )e ⫺ lt ) 2 D(t) ⫽ ⫺ s211 t2 ⫺ s222 ⫺ s233 , which is the key difference between the 2 2 2l

2l

(1 ⫺ e ⫺ lt ) 2

2 DNS and the AFNS models. The contributions of the three components, ⫺ s211 t2 , ⫺ s222 , 2l ⫺ lt 2 and ⫺ s233 (1 ⫺ (1 ⫹ 2lt )e ) , that come from the volatilities of term structure level, slope, and curvature 2l are also provided. 2

12-month-ahead forecasts by a recursive method with an expanding data window. For example, to forecast one-month-ahead forward rates in January 2007, we estimate two models by using data up to December 2006; then, we re-estimate the models by adding one month of data, and forecasts of forward rates in February 2007 can be constructed, and so on. Totally, there are 36, 31, and 25 forecasts for the 1-, 6-, and 12-month horizons, respectively. Forecast Procedure In practice, due to the addition of the Nelson-Siegel factor loadings, the forecast of the t – maturity forward rate h periods ahead at current time t is easily obtained. For the DNS model, the forecasted forward rates are DNS DNS fˆt⫹h (t) ⬅ EPt冤f t⫹h (t)冥 ⫽ EPt 冤X1t⫹h冥 ⫹ EPt 冤X2t⫹h冥e ⫺lt ⫹ EPt冤X3t⫹h冥lte⫺lt,

(30)

where the conditional expectation of the state variables in period t ⫹ h can be derived from state Equation (17) as follows: Journal of Futures Markets

DOI: 10.1002/fut

978

Luo and Zhang

Filtered X1t by the DNS−λ and the AFNS−λ model

0.4 0.3 0.2 0.1 0

Jun90

Mar93

Filtered 0.4

Dec95

X2t by

Sep98

May01

Feb04

Nov06

Feb04

Nov06

Feb04

Nov06

the DNS−λ and the AFNS−λ model

0.2 0 −0.2 −0.4

Jun90

Mar93

Dec95

Sep98

May01

Filtered Xt3 by the DNS−λ and the AFNS−λ model

0.4 0.2 0 −0.2 −0.4

Jun90

Mar93

Dec95

Sep98

May01

FIGURE 8

Filtered factors by the DNS-l and AFNS-l Models. We show the dynamics of three estimated factors by the DNS-l (dashed lines) and AFNS-l (solid lines) models. The sample period is 1988:01 to 2007:12. Note that l is also estimated in the two models.

h⫺1

EPt冤Xt⫹h冥 ⫽ a a Ai b (I ⫺ A)m ⫹ AhX t ,

(31)

i⫽0

where Xt ⫽ (X1t , X2t , X3t ) . Then, it is straightforward to compute the forecasted forward rates when parameters for A, m, and l are estimated from a sample up to time t. The time series of the optimally filtered three factors are also easily obtained. Similarly, for the AFNS model, the forecasted forward rates are AFNS AFNS fˆ t⫹h (t) ⬅ EPt冤 f t⫹h (t)冥 ⫽ EPt冤X1t⫹h冥 ⫹ EPt冤X2t⫹h冥e ⫺lt ⫹ EPt冤X3t⫹h冥lte ⫺lt ⫹ D(t),

(32) where D(t) is given by Equation (10) and the conditional expectation of the state variables in period t ⫹ h can be derived from state Equation (22) as follows: Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

EPt冤Xt⫹h冥 ⫽ (I ⫺ e ⫺K h )uP ⫹ e ⫺K hXt , P

P

979

(33)

where Xt ⫽ (X1t , X2t , X3t ) . Then, it is easy to compute the forecasted forward rates when parameters for KP, uP, l, and ⌺ are estimated. Also, the time series of the optimally filtered three factors in the AFNS can be obtained. Forecast Evaluation The forecast performance of the two models is measured in terms of the root mean squared error (RMSE) of the forecast error defined by fˆt⫹h (t) ⫺ ft⫹h (t) . In particular, we select t ⫽ 14, 16, 18, 20, 22, and 24 years, and h ⫽ 1, 6, and 12 months. These RMSEs are reported in Table V. For all the combinations of maturity and forecast horizon, the AFNS model is better than the DNS model. For example, with the AFNS model, the RMSEs are very small even with longer forecasting horizons (less than 50 basis points). However, with the DNS model, the RMSEs are slightly larger. Specifically, the differences in RMSEs are about 5, 7, and 10 basis points for the 1-, 6-, and 12-month horizons, respectively. These results are consistent with those observed in Christensen et al. (2007) where they find that the AFNS model outperforms the DNS model especially with longer horizons and longer maturities. In the literature, the role of no-arbitrage on forecasting is ambiguous. From the theoretical perspective, the absence of arbitrage is required for a term structure model due to very liquid bond markets. For example, Ang and Piazzesi (2003) notice that the forecasting ability is improved by imposing the no-arbitrage restriction on a VAR model with macroeconomic variables and Almeida and Vicente (2008) show that the no-arbitrage constraint in a parametric polynomial model significantly improves forecasts. However, Duffee (2002) finds that even the simplest random walk forecasts are better than the affine models in Dai and Singleton (2000). Coroneo, Nyholm, and VidovaKoleva (2008) conclude that imposing the absence of arbitrage restriction does not impact the empirical performance at all by comparing Nelson and Siegel (1987) and its no-arbitrage counterpart. Moreover, Duffee (2009) provides a comprehensive study on the role of no-arbitrage restrictions in forecasting future bond yields and macroeconomic activity and finds that the restrictions have no practical effect on forecast accuracy of a three-factor, discrete-time Gaussian model. CONCLUSION The parametric Nelson-Siegel model is widely used in practice for its parsimony and good performance in fitting. Recently, Diebold and Li (2006) found that the Journal of Futures Markets

DOI: 10.1002/fut

980

Luo and Zhang

TABLE V

Out-of-sample Forecasting RMSEs for the Two Models 1-Month-Ahead

6-Month-Ahead

12-Month-Ahead

Maturity

DNS

AFNS

DNS

AFNS

DNS

AFNS

14 16 18 20 22 24

19.37 23.96 17.24 22.69 23.36 21.57

14.34 17.49 12.77 18.60 18.34 16.22

35.76 44.81 29.10 42.07 39.55 24.73

28.41 34.22 23.26 35.36 32.30 18.91

42.59 55.71 45.66 56.30 51.45 30.24

35.36 43.05 36.86 46.52 41.24 24.85

Note. This table presents the root mean squared error of the forecasted errors defined by fˆt⫹h (t) ⫺ ft⫹h (t) where fˆt⫹h (t) are forecasted forward rates and ft⫹h (t) are market implied forward rates, for t 14, 16, 18, 20, 22 and 24 years, and h ⫽ 1, 6 and 12 months. All numbers are measured in basis points.

dynamic version of the Nelson-Siegel model is also good at forecasting. A more recent development in this direction is the arbitrage-free Nelson-Siegel model by Christensen et al. (2007), in which they combine the Nelson-Siegel specification with the no-arbitrage restriction and maintain both the empirical and theoretical advantages of the two parts. In this study, after re-documenting the downward sloping phenomenon of the long forward rates with maturities beyond 14 years by using data up to recent years, we investigate abilities of the dynamic Nelson-Siegel model and the arbitrage-free Nelson-Siegel model in explaining these empirical findings. We find that both models can capture the empirical phenomenon very well, while the arbitrage-free Nelson-Siegel model is slightly better. Furthermore, out-of-sample forecasting analysis indicates that imposing no-arbitrage restrictions does increase forecast accuracy. BIBLIOGRAPHY Almeida, C., & Vicente, J. (2008). The role of no-arbitrage on forecasting: Lessons from a parametric term structure model. Journal of Banking and Finance, 32, 2695–2705. Ang, A., & Piazzesi, M. (2003). A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables. Journal of Monetary Economics, 50, 745–787. Babbs, S. H., & Nowman, K. B. (1999). Kalman filtering of generalized Vasicek term structure models. Journal of Financial and Quantitative Analysis, 34, 115–130. Bank for International Settlements. (2005). Zero-coupon yield curve: Technical documentation (BIS Paper No. 25). Boudoukh, J., Richardson, M., & Whitelaw, R. (2005). The information in long-maturity forward rates: Implications for exchange rates and the forward premium anomaly (Working Paper No. 11840). National Bureau of Economic Research. Journal of Futures Markets

DOI: 10.1002/fut

Dynamics of Long Forward Rate Term Structures

981

Brown, R. H., & Schaefer, S. M. (2000). Why long term forward interest rates (almost) always slope downwards (working paper). London Business School. Carverhill, A. P. (2001). Predictability and the dynamics of long forward rates (working paper). Hong Kong University of Science and Technology. Christensen, J. H. E., Diebold, F. X., & Rudebusch, G. D. (2007). The affine arbitragefree class of Nelson-Siegel term structure models (Working Paper No. 13611). National Bureau of Economic Research. Christiansen, C. (2005). Variance-in-mean effects of the long forward-rate slope. Applied Financial Economics, 15(11), 753–755. Coroneo, L., Nyholm, K., & Vidova-Koleva, R. (2008). How arbitrage-free is the Nelson-Siegel model? (Working Paper No. 874). European Central Bank. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–407. Dai, Q., & Singleton, K. J. (2000). Specification analysis of affine term structure models. Journal of Finance, 55, 1943–1978. Daves, P. R., & Ehrhardt, M. C. (1993) Liquidity, reconstitution, and the value of U.S. treasury strips. Journal of Finance, 48, 315–329. de Jong, F. (2000). Time series and cross-section information in affine term-structure models. Journal of Business and Economics Statistics, 18, 300–314. De Rossi, G. (2004). Kalman filtering of consistent forward rate curves: A tool to estimate and model dynamically the term structure. Journal of Empirical Finance, 11, 277–308. Diebold, F. X., & Li, C.-L. (2006). Forecasting the term structure of government bond yields. Journal of Econometrics, 130, 337–364. Diebold, F. X., Rudebusch, G. D., & Aruoba, S. B. (2006). The macroeconomy and the yield curve: A dynamic latent factor approach. Journal of Econometrics, 131, 309–338. Duan, J.-C., & Simonato, J.-G. (1999). Estimating and testing exponential-affine term structure models by Kalman filter. Review of Quantitative Finance and Accounting, 13, 111–135. Duffee, G. R. (2002). Term premia and interest rate forecasts in affine models. Journal of Finance, 57, 405–443. Duffee, G. R. (2009). Forecasting with the term structure: The role of no-arbitrage restrictions (working paper). Johns Hopkins University. Duffie, D., & Kan, R. (1996). A yield-factor model of interest rates. Mathematical Finance, 6, 379–406. Dybvig, P. H., Ingersoll, J. E., & Ross, S. A. (1996). Long forward and zero-coupon rates can never fall. Journal of Business, 69, 1–25. Fama, E. F. (2006). The behavior of interest rates. Review of Financial Studies, 19, 359–379. Fama, E. F., & Bliss, R. R. (1987). The information in long-maturity forward rates. American Economic Review, 77, 680–692. Hubalek, F., Klein, I., & Teichmann, J. (2002). A general proof of the Dybvig-IngersollRoss theorem: Long forward rates can never fall. Mathematical Finance, 12, 447–451. Jordan, B. D., Jordan, S. D., Smolira, J. C., & Travis, D. H. (2008). Do long interest rates ever fall? Advances in Financial Planning and Forecasting, 3, 21–36.

Journal of Futures Markets

DOI: 10.1002/fut

982

Luo and Zhang

Jorion, P., & Mishkin, F. (1991). A multicountry comparison of term-structure forecasts at long horizons. Journal of Financial Economics, 29, 59–80. Luo, X.-G., & Zhang, J. E. (2008). The dynamics of interest rate term structure. Paper presented at Quantitative Methods in Finance Conference (QMF) 2008, December 17–20, 2008, Sydney, Australia. McCulloch, J. H. (2000). Long forward and zero-coupon rates indeed can never fall, but are indeterminate: A comment on Dybvig, Ingersoll and Ross (working paper). Ohio State University. Nelson, C. R., & Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business, 60, 473–489. Sack, B. (2000). Using Treasury STRIPS to measure the yield curve (FEDS Working Paper No. 2000-42). Board of Governors of the Federal Reserve. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. Zaretsky, M. (1995). Generation of a smooth forward curve for U.S. Treasuries. Journal of Fixed Income, 5, 65–77.

Journal of Futures Markets

DOI: 10.1002/fut