Estimating the Term Structure of Government Bond Yields in the EU A Study Based on Parsimonious Models Moses B. Kwesiga

Estimating the Term Structure of Government Bond Yields in the EU A Study Based on Parsimonious Models

Moses B. Kwesiga∗

Thesis for Master of Science in

Econometrics, Operations Research and Actuarial Studies

Supervisor: Prof. Dr. Theo K. Dijkstra

University of Groningen July 2006

Acknowledgement

This thesis is dedicated to the memory of my parents. They were very strong people and friends working their way through many obstacles. I hope I have inherited at least some of their strength.

I am obliged to Professor Theo K. Dijkstra for the close supervisory contacts and consid-erable guidance from the start through the end of this thesis project. In fact, he suggested to me the topic of term structure forecasting. As my supervisor, he showed me the way of research step by step and helped me to build up the foundation of my independent research ability. I benefited quite a lot from his solid economic knowledge and serious attitude to-ward work.

For helpful and stimulating discussions, I am grateful to Professor Paul Bekker. He greatly contributed a lot to my understanding of interest rate models. I am impressed by this knowledgeable, industrious professor of all the time. He recommended many useful papers to me and read my drafts in a detailed way. His comments and suggestions have been of great value.

I would also like to thank the other classmates in our econometrics class of academic year 2005-06. Discussing with them refreshed my mind and their friendship relieved my homelessness in a foreign country.

Full financial support by the Eric Bleumink Fund (EBF) at the University of Gronin-gen is gratefully acknowledged. Without this scholarship, my hope for econometric studies at RUG would have remained a dream.

Finally, I would like to thank members of my family, relatives and friends who may have contributed to my life and education in one way or the other. In a deserving special way, I would like to convey my very warm thanks to Lydia. Honestly Lydia’s contribution to my work has been invaluable. To start with, she kept on encouraging me to ”go for further studies”. And eventually through this study time, her patience, care and tender love have really kept me going. I would like to repay Akiiki somehow at some point in our lives. In all this, I praise the name of the Almighty God, my Provider.

Moses B. Kwesiga Groningen,

Contents

1

Introduction

1.1 The Dataset . . . 2

2

Theoretical Framework

2.2 Interest Rate Terminology . . . 4

3

The Nelson-Siegel Yield Curve Model

3.2 Methods . . . 9

3.2.1 Fitting the Yield Curve . . . 9

3.2.2 Forecasting the Yield Curve . . . 10

3.3 Results . . . 12

4

The Cochrane-Piazzesi Model

4.2 Methods . . . 13

4.2.1 Excess Return Forecasting . . . 13

4.2.2 Yield Curve Forecasting . . . 15

4.3 Results . . . 16

5

Combining Forecasts

5.2 Methods . . . 18

5.3 Results . . . 18

6

Summary and Conclusions

7

References

Summary

This paper uses statistical techniques to estimate daily yield curves in the European Union (EU) using secondary government securities data on swap transactions from 1988 to 2005. We use two parsimonious models to estimate and forecast daily yield curves: the Nelson-Siegel three-factor model and Cochrane-Piazzesi tent-shaped single-factor model. We find a good in-sample fit of Siegel model. In terms of out-of-sample performance, Nelson-Siegel’s model is second-rate and just compares as well as standard benchmark yield fore-cast models.

Cochrane-Piazzesi’s tent-shaped single-factor model forecasts one-year excess returns with an R2 _{of up to 45%. This forecasting ability is however reduced considerably when we}

consider regressions of net excess returns (i.e, excess return above the random walk return) on the linear combination of forward rates. When used to forecast yields, the tent-shaped factor produces better out-of-sample results at the one-year horizon than alternative mod-els considered.

1

Introduction

Forecasting interest rate term structure is still one of the major challenges for the financial industry today. Characterizing interest rates as a function of maturity embodies infor-mation about future movements in interest rates and has direct implications for the real economic activity. Monetary policy is conducted by targeting rates at the short end of the curve, and longer-term yields reflect expectations of future changes in short rates. In financial markets, the term structure is crucial to the pricing of interest rate contingent claims and fixed-income derivative securities. Therefore the quest for reliable forecasts of the term structure can not be overemphasised.

Several interest rate models have been proposed in the literature. As with other econo-metric models, most term structure models have focused on estimating the term structure as parsimoniously as possible. The major advances in interest rate term structure mod-elling and estimation have been spurred by research during the last two decades and can broadly be categorised into two subfields: the first category emphasises equilibrium models whereas the second category uses statistical techniques to estimate the term structure. The equilibrium models focus on modelling the dynamics of the short-rate. These models are essentially affine models (constant-plus-linear functions of some state variable) although one may also observe quadratic models in the recent literature. Essentially, these models relate the short-rate to some underlying state variables and model these state variables as stochastic processes. Seminal papers of the equilibrium models approach include Vasicek (1977), Cox, Ingersoll and Ross (1985), and Duffee and Kan (1996). Statistical methods of measuring term structure, on the other hand, focus on obtaining a continuous yield curve from cross-sectional term structure data using curve-fitting techniques. These methods can also be broadly divided into two: the first using splines and the second using parsimonious functional forms to fit the term structure.

Spline-based models owe to the seminal paper of McCulloch (1971). McCulloch used quadratic splines to fit the term structure data. Vasicek and Fong (1981) introduced ex-ponential splines, and Shea (1984) used B-splines for this purpose. Parsimonious models, on the other hand, specify a functional form for the spot rate, the discount rate or implied forward rate. Chambers et al. (1984) proposed a simple polynomial for the spot rate. On the other hand, Nelson-Siegel (1987) used an exponential function for the implied forward rate and analytically solved for the spot rate.

Earlier studies similar to Cochrane and Piazzesi (2005) single-factor model have included among others, Robert Stambaugh (1988) who ran regressions of 2 − 6 month bond excess returns on 1−6 month forward rates and found a tent-shaped pattern of coefficients similar to Cochrane-Piazzesi’s; Lars Hansen and Robert Hodrick (1983) and Wayne Ferson and Michael Gibbons (1985) single index or latent variable models used to capture time-varying expected returns; Antti Ilmanen (1995) who ran regressions of monthly excess returns on bonds in different countries on a term spread, the real short rate, stock returns, and bond return and found statistical and economic significance in the predictability of bond returns. In this study, we concentrate on two popular parsimonious term structure estimation mod-els.The purpose of the study is to compare performance of the term structure models, viz, the Nelson-Siegel (1987) model (as factorised by Diebold and Li (2006)) and the Cochrane-Piazzesi (2005) single-factor model. We estimate each model on daily EU government bond data spanning a period of 3648 days, from 12 February 1988 to 31 May 2005. We combine and compare different specifications of the two models and choose the best method in terms of its ability to replicate market yields of bonds, by performing in-sample and out-of-sample analyses. Specifically, we compare estimated term structure forecast methods by the differ-ences between their out-of sample root-mean-squared error statistics following Bliss (1997). We proceed as follows. First we present details of the term structure dataset used in our empirical study in the next subsection. We give a theoretical background to interest rate term structure in Section 2. In Section 3, we present the theory, methods and re-sults from the Nelson-Siegel estimation model. Section 4 presents the theory, methods and results for the Cochrane-Piazzesi single factor excess return forecast model. We combine different yield forecast models and the details are presented in Section 5. Finally, Section 6 ends the study with a summary and conclusion.

1.1

The Dataset

We use daily swap rates on EU Treasury bonds downloaded from Bloomberg2_{. These are}

closing rates that cover a period from beginning of June 1991 to the end of May 2005, a total of 3648 observations. German Deutschmark-denominated swap rates were used in the period prior to the introduction of the Euro. The Euro denominated data begins only in 1999 when the Euro was first launched. Prior to that period, the German market was considered as the main European bond market and therefore representative of the region’s bond market activity. A total of fourteen bonds in maturities of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20 and 30 years is included. Additionally, we include the 3-month EURIBOR3 _to

represent the short end of the maturity spectrum. For the 1-, 2-, 3-, 4- and 5-year maturity bonds, data is available from June 7, 1991 till May 31, 2005. Full data across the rest of the other maturities is only available for the period from February 12, 1998 to May 31,

2

I am grateful to Maarten van Ravenswaaij of SNS Asset Management for providing the data.

3

2005 for a total of 1903 observations.

Table 1 presents the summary statistics of the yield curve data, while Figure 1 plots the 3-month, 12-year and 30-year yields over the period under study. The yield curve is a normal one with an upward sloping shape. In fact, the EU term structure is on a downward trend overall under the study sample data period. And as can be seen from both the data graphics and sample autocorrelation functions(ACF), the yields have strong persistence over the sample period.

2

Theoretical Framework

2.1

Term Structure Theory

The term structure of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relation-ship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market’s anticipations of future events. An explanation of the term structure gives a way to extract this information and to predict how changes in the underlying variables will affect the yield curve.

In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, most theories of term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalisations of the certainty equilibrium relationships. It is common to identify much of term structure work as belonging to one of the following four strands of thought.

The Market Expectations (Pure Expectations) Hypothesis

First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holding period returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that forward rates are equal to the expected spot rates. Generally, this approach means: (a) the return on holding a long-term bond until maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities.

The Liquidity Preference Theory

term premium is the increment required to induce investors to hold longer-term (”riskier”) securities.

The Market Segmentation Theory

Third, there is the Market Segmentation Theory of Culbertson and others, which offers a different explanation of term premiums. Here it is asserted that individuals have strong maturity preferences and that bonds of different maturities trade in separate and distinct markets. The demand and supply of bonds of a particular maturity are supposedly little affected by the prices of bonds of neighbouring maturities i.e., in this theory, financial instruments of different terms are not substitutable.

The Preferred Habitat Theory

In their preferred habitat theory, Modigliani and Sutch use some arguments similar to those of the market segmentation theory. Another variation on the Pure Expectations Theory, the Preferred Habitat Theory states that in addition to interest rate expectations, investors have distinct investment horizons and require a meaningful premium to buy bonds with maturities outside their preferred maturity, or habitat. Proponents of this theory believe that short-term investors are more prevalent in the fixed-income market and therefore, longer-term rates tend to be higher than short-term rates.

2.2

Interest Rate Terminology

In this section the terminology that will be used for modeling the term structure in the subsequent sections is introduced. We exclude bonds with coupon payments from the definitions since their treatment follows directly from zero-coupon bonds.

Yield to Maturity

We first introduce the concept of yield, technically termed as Yield to Maturity. Yield is the interest rate at which the present value of a bond’s cash flow at maturity is equal to its current price. In the literature two kinds of yields are always used. These are simple yield and continuously compounded yield. We denote simple yield by Y and continuously compounded yield4 _{by y.}

Definition 1: Suppose the current price of a zero-coupon bond with face value 1 EUR maturing in m periods is P . Then simple yield Y satisfies

P = 1

(1 + Y )m (1)

4

Definition 2: Continuously compounded yield also called gross yield is the natural loga-rithm of one plus the simple yield, i.e, y = log(1 + Y ). For a zero-coupon bond with face value 1 EUR maturing in m periods, continuously compounded yield, y, satisfies

P = e−m·y ₍₂₎

It is often more convenient to work with continuously compounded yields, although we will use both simple and continuously compounded yields in this paper (all our results will be transformed and reported on a simple-yield basis, however).

Spot rate, Implied Forward rate and Discount function5

After defining yield we next introduce the spot rate, implied forward rate, and the discount rates.

The yields defined so far are also referred to as spot rates. An analysis based on spot rates (yields), for different maturities, gives information about the term structure of inter-est rates. When we plot the spot rates against maturities, we get the spot yield curve. We next turn to the issue of defining the forward rate. We first differentiate implied forward rate from market forward rate. Implied forward rate is derived theoretically from spot rates. Market forward rate, on the other hand, is the actual rate that is realised in a forward or a futures contract.

To clarify the concept of implied forward rate, consider the following example. Suppose that y(1) and y(2) are spot rates for one- and two-year bonds respectively. Also consider an investor wishing to hold money for two years. We consider the following two alternatives. The investor could buy a two-year pure discount bond. The ending value per Euro invested would be

e2·y(2)

Alternatively, the investor could buy a one-year pure discount bond and simultaneously agree to invest the proceeds at year one at the forward rate from year one to year two. Investing in one year bond today and receiving

ey(1)

at the end of the year and reinvesting this amount in the one year bond having yield y(1)2_,

5

which is the year two spot yield for the one year bond. Since the forward rate is known at time zero and the commitment is made at the same time, the investor can analyse which alternative is better at time zero. For there not to be arbitrage opportunities (buying the more attractive and financing this by issuing the less attractive), the return must be the same from the two alternatives or

e−2·y(2)_{= e}−y(1)_{· e}−y(1)2_{. (3)}

The yield satisfying this condition is also referred to as the implied forward rate of a one year bond belonging to the next period, f (2). Since y(1) and y(2) are available, one can easily calculate f (2). When we extend this argument to an m-period case, the formula becomes

e−m·y(m) _{= e}−y(1)_{· e}−f (2)_{· e}−f (3)_{· · · e}−f (m) ₍₄₎

By solving this equation recursively, one can calculate implied forward rates for all m pe-riods, given the spot rates. An important result is that spot rate is the average of forward rates. Taking the periods infinitesimally closer we obtain the instantaneous implied for-ward rate, f (m). Plotting the instantaneous implied forfor-ward rates against maturity gives us the implied forward rate curve. The present value of the future cash flow to be received in m periods can be computed by multiplying cash flow, F V , by the discount function, δ(m):

P V = δ(m) · F V (5)

The discount function used in evaluating continuously compounded yields for example,is given by

δ(m) = e−m·y ₍₆₎

We note therefore that δ(m) is a continuous function of time. Plotting the discount function against time gives the discount curve. We also note the important fact that by construction, the discount function ranges between zero and one, starting from one at m = 0 and monotonically declining to zero as m grows without bound.

Calculating Spot rates, Implied Forward rates and Discount functions

f (m) = −δ ′

(m)

δ(m) (7)

where δ′

(m) is the derivative of the discount function. Since spot rates are the averages of forward rates, it follows immediately that the spot rate function is:

y(m) = 1 m

Z m

0 f (v)dv (8)

Using the relationship between the discount function and the instantaneous implied for-ward rate, the spot rate function is given as

y(m) = −1 m Z m 0 δ′ (v) δ(v)dv (9)

And upon taking the integral one readily obtains:

y(m) = −ln(δ(m))

m (10)

The previous three equations show that knowledge of any one of these functions is sufficient to solve for the other two.

3

The Nelson-Siegel Yield Curve Model

3.1

Introduction

Nelson and Siegel (1987) introduced a model for yield curves which explained 96 percent of the variation of the yield curve across maturities using US zero-coupon government se-curities market data. The model’s popularity is due its ability to describe variations in the yield curve using only few parameters. Later similar work by Litterman and Scheinkman (1991) recognised that the level, the steepness and curvature factors can explain 98% of the variation in yields of US coupon bonds.

and Siegel (1987) propose that if spot rates are generated by a differential equation, then implied forward rates will be the solutions to this equation. Assuming that spot rates are generated by a second order differential equation, the instantaneous implied forward rate function at any time t can be written as

ft(m) = β0,t+ β1,t.e−m/τ1,t+ β2,t.

m τ2,t

.e−m/τ2,t ₍₁₁₎

where τ1 and τ2 are the real roots of the differential equation. However, Nelson and Siegel

concluded that the model is over-parameterised and decided to use the equal-roots solu-tion. Hence the instantaneous implied forward rate function becomes

ft(m) = β0,t + β1,t.e−m/τt + β2,t.

m τt

.e−m/τt ₍₁₂₎

And using the relation

y(m) = 1 m

Z m

0 f (v)dv,

the spot rate function can be written as yt(m) = β0,t+ β1,t à 1 − e−m/τt m/τt ! + β2,t à 1 − e−m/τt m/τt − e−m/τt ! (13)

As demonstrated by Diebold and Li (2006), this spot rate curve specification is flexible enough to accommodate monotonic, S-shaped and humped spot rate curves, which are shapes usually associated with empirical yield curves.

Nelson and Siegel show that another merit of the model is the ease with which its pa-rameters can be interpreted. Taking the limits of the spot rate curve as m approaches zero and infinity, Nelson and Siegel found that the contribution of the long term component is β0,t and the contribution of the short term component is β0,t+ β1,t. Also −β1,t can be

inter-preted as a term premium factor, since it is the difference between long term and short term yields. The parameter τ is the time constant measuring the point of the beginning of decay. Diebold and Li (2006), interpret the parameters β0,t, β1,t, and β2,t as three latent

dy-namic factors. The loading on β0,t is 1, a constant that does not decay to zero in the limit;

hence it may be viewed as a long-term factor. The loading on β1,t is

³

1−e−m/τ m/τ

´

, a function that starts at 1 but decays monotonically and quickly to 0; hence it may be viewed as a short-term factor. The loading on β2,t is

³

1−e−m/τ

m/τ − e−m/τ

´

it may be viewed as a medium-term factor. Figure 2 gives a plot of the three factor loadings. Diebold and Li interpret the three factors of long-term, short-term and medium-term in terms of level, slope and curvature. The long-term factor β0,t is interpreted as the yield

curve level, since an increase in β0,t increases yields at all maturities equally. The

short-term factor β1,t is closely related to the yield curve slope, as an increase in β1,t increases

short yields more than long yields, because the short rates load on β1,t more heavily, thereby

changing the slope of the yield curve. Finally, β2,t can be interpreted as curvature of the

yield curve. An increase in β2,t has little effect on very short and very long yields but

increases the medium term yields.

Diebold and Li’s three factor model parameter vector {β0,t, β1,t, β2,t} is estimated by

lin-ear regression using the ordinary least squares procedure. Using the fixed value of 1/τ = 0.06096_{, we compute the values of the two regressors} ³1−e−m/τ

m/τ

´

and ³1−e_m/τ−m/τ − e−m/τ´ _for

each observation day across the fifteen maturities of the dataset. According to Diebold and Li, estimating only three parameters with 1/τ fixed in advance not only provides sim-plicity and convenience but as well enhances numerical trustworthiness of the results as the number of optimization steps is considerably reduced.

3.2

Methods

3.2.1 Fitting the Yield Curve

We fit the m-period yield at time t, using the three factor model,

yt(m) = β0,t+ β1,t à 1 − e−m/τt m/τt ! + β2,t à 1 − e−m/τt m/τt − e−m/τt ! (14)

We apply ordinary least squares to the yield data for each day to obtain a time series of estimates nβˆ0,t, ˆβ1,t, ˆβ2,t

o

and thereby giving rise to a time series of yield curve residuals. Table 2 gives a summary of basic statistics of the estimated factors that are as well plotted in Figure 3. The level factor, ˆβ0,t is clearly governing the movement of the term structure

of the period 1998 to 2005. It is persistent as shown by the autocorrelation at various lags and it is on a declining trend in line with the yield curve trend depicted in Figure 1. Furthermore, as reported in Table 2, we observe that the ADF7 _{at three lags fails to reject}

the null of a unit-root in each of the factors.

6

The constant value of 1/τt = 0.609 corresponds to m = 30 months which is the practice with the

Nelson-Siegel approach. However, two or three years is the common maturity range used for this purpose so we simply picked the average (30 months). The loading on the medium term (curvature) factor is then maximised using this predetermined maturity.

7

3.2.2 Forecasting the Yield Curve

We model and forecast the yield curves by forecasting the Nelson-Siegel factors as univari-ate AR(1) processes. We use a sample of data from June 1991 to May 2005 for a total of 1903 observations. We estimate the forecast h−step-ahead yield ˆyt+h/t(m) at time t with

a univariate factor specification of the form:

ˆ yt+h(m) = ˆβ0,t+h/t+ ˆβ1,t+h/t à 1 − e−m/τt m/τt ! + ˆβ2,t+h/t à 1 − e−m/τt m/τt − e−m/τt ! (15)

where ˆβi,t+h/t is the h-step-ahead forecast of factor i made at time t. Forecasting the yield

curve is equivalent to forecasting the factorsnβˆ0,t, ˆβ1,t, ˆβ2,t

o

as the yield curve is dependent on only these three factors. We therefore subsequently model and forecast the three factors with random walk8_{, AR(1) and VAR(1) models.}

A) Random Walk on Factors: ˆ βi,t+h/t= ˆβi,t (16) for i = 0, 1, 2. B) AR(1) on Factors: ˆ βi,t+h/t= ˆµi+ ˆδiβˆi,t (17)

for i = 0, 1, 2, and ˆµ and ˆδ are regression coefficients obtained from OLS regression of ˆβi,t

on an intercept and ˆβi,t−h.

C) VAR(1) on Factors:

ˆ

βt+h/t= ˆµ + ˆ∆ ˆβt (18)

Comparing With Other Forecast Methods

We compare out-of-sample performance of the Nelson-Siegel method with other yield fore-cast benchmark models including the following.

8

D) Random walk on Yields: ˆ

yt+h/t(m) = yt(m) (19)

The best forecast of ”tomorrow’s” yield is ”today’s” yield. E) AR(1) on Yield Levels:

ˆ

yt+h/t(m) = ˆc(m) + ˆλyt(m) (20)

F) VAR(1) on Yield Levels: ˆ

yt+h/t = ˆc + ˆΛyt (21)

where

yt = [yt(3), yt(12), yt(24), yt(60), yt(120), yt(240)]T

for 3M, 1Y, 2Y, 5Y, 10Y and 20Y tenure rates respectively. We define the forecast error εt+h at time t + h as

εt+h= yt+h(m) − ˆyt+h/t(m) (22)

that is, for each day, εt+h is the gap between the h-step-ahead actual and fitted yields for

a given maturity bond. Essentially, we base our comparison of relative performance of the models using the root-mean-squared yield error (RMSE) criterion:

RM SE = v u u t 1 N X i ε2 i (23)

new yield vector. The procedure is repeated for each observation vector in the forecast subsample and the RMSE and other statistics computed for each maturity.

3.3

Results

In Figure 4, we give an overview of the fit of the model by plotting the average yield curve together with the average fitted yield curve. As can be seen, the Nelson-Siegel method pro-vides a good fit. Moreover, the in-sample residual statistics reported in Table 3 indicate a good fit although there are persistent effects as depicted by the high autocorrelations. Bliss (1997b) argued that autocorrelations can only be minimised but not eliminated in any term structure estimation method due to persistent tax and/or liquidity effects. Our data too, being on a daily basis might exaggerate the observed autocorrelations.

In Tables 4 − 7, we provide the root mean squared, standard deviation and mean pric-ing errors(in percentage points) for the out-of-sample fitted swap rates with maturities of 3 months, 1 , 2, 5, 10 and 20 years at forecast horizons h = 1, 3, 6 and 12 months. Results from Tables 4 − 7 indicate that the simple random walk on yields and random walk on the Nelson-Siegel factors outperform all the other models based on RMSE statistics. As we can observe at all forecast horizons, the two random walk specifications are just competing and one cannot easily identify which one is superior over the other. The results here are quite different from Diebold and Li (2006)’s finding on US bond data in which the simple AR(1) model on factors had superior out-of-sample performance especially at the 12-month-ahead forecast horizon. The result is not a surprising one though; when we make a closer look at the mean errors of all the forecast models used, we recognise that each model does give an ambitious forecast at all horizons overall. We can therefore attribute this phenomenon to the nature of the EU term structure that is on a continuous declining trend over the sample period considered. It is one of the weaknesses of most forecast models in the literature that they are always not quick and/or flexible enough to respond to peculiar changes in the variables they forecast.

4

The Cochrane-Piazzesi Model

4.1

Introduction

Although the expectations hypothesis (EH) (that the forward rate is an unbiased predictor of the future spot rate) remains one of the most important explanations of interest rate term structure behavior, the question of whether it is true has occupied a central place in research on the term structure, particularly empirical research. There is no clear consensus empirically, but in fact, recent research concludes that it is a simple matter to prove that it cannot be true theoretically and almost surely can not be true in reality.

and Robert Bliss (1987). Fama and Bliss run regressions of one-year excess returns on long term bonds against the forward-spot rate. Fama and Bliss find that the spread between the m-year forward rate and the one-year yield predicts the 1-year excess return of the m-year bond, with an R2 _{of up to 18%. Campbell and Shiller (1991) find that excess bond}

returns are forecastable by Treasury yield spreads.

Yet the EH implies that excess returns should not be predictable. In another ground-breaking evidence against EH, John Cochrane and Monika Piazzesi (2005) run regressions of one-year excess returns on all forward rates, and find that, a single tent-shaped linear combination of forward rates predicts excess returns on 1-5 year maturity bonds with an R2 _{up to 45%. It is the subject of this section to study in detail Cochrane and Piazzesi’s}

bond return forecasting approach.

4.2

Methods

4.2.1 Excess Return Forecasting Some Notation

We treat the daily market swap rates data as simple yields. Following Cochrane and Pi-azzesi (2005), we adopt the following notation. Let yt(m) denote the yield level of an

m-year discount bond at time t. We define the forward rate at time t for loans between time t + m − 1 and t + m as

ft(m) ≡ (1 + yt(m))m/(1 + yt(m − 1))m−1− 1

for m = 2, 3, 4, 5 years, and we write the one-year holding period return from buying an m-year bond at time t and selling it as an (m − 1)-year bond at time t + 1 as

hprt+1(m) ≡ (1 + yt(m))m/(1 + yt+1(m − 1))m−1− 1

And accordingly, we denote one year excess returns by

hprxt+1(m) ≡ hprt+1(m) − yt(1)

We use the same letters without maturity indicator (m) to denote vectors across maturity, e.g.

hprxt+1 ≡ [hprxt+1(2), hprxt+1(3), hprxt+1(4), hprxt+1(5)]T

yt≡ [1, yt(1), yt(2), yt(3), yt(4), yt(5)]T

ft ≡ [1, yt(1), ft(2), ft(3), ft(4), ft(5)]T

Following Cochrane and Piazzesi(2005), we run regressions of bond excess returns at time t + 1 on all forward rates at time t. The regressions are of the form:

hprxt+1(m) = β0,m+ β1,myt(1) + β2,mft(2) + . . . + β5,mft(5) + εt+1(m) (24)

for m = 2, 3, 4, 5.

In order to determine the estimates of the return forecasting factor γ, we run regressions of the form:

hprxt+1(m) = am+ bm(γ0+ γ1yt(1) + γ2ft(2) + . . . + γ5ft(5)) + εt+1(m) (25)

Cochrane and Piazzesi argue that the coefficients bm and γm are not separately identified,

since one can double all the b and half all the γ. In order to uniquely identify bm and γm ,

we therefore impose the restrictions P5

m=2am = 0 and 1₄P5m=2bm = 1.

Cochrane and Piazzesi define model (25) as a restricted version of (24) since one can iden-tify the coefficients in the models as β = bγT _{(by suppressing the a}

ms) where β is a 4 X 6

matrix. We then estimate the model in equation (25) in two steps. First, we estimate the γs by running a single regression of the average (across maturity) excess returns on all forward rates, 1 4 5 X m=2 hprxt+1(m) = γ0+ γ1yt(1) + γ2ft(2) + . . . + γ5ft(5) + εt+1 (26) or for brevity, hprxt+1= γTft+ εt+1 (27)

Then the parameter estimates for am and bm are obtained by running the four regressions

hprxt+1(m) = am+ bmγTft+ εt+1(m), (28)

4.2.2 Yield Curve Forecasting

We use the Cochrane-Piazzesi single factor model in (28) to forecast one-year-ahead yields on 1-, 2-, 3-, and 4-year maturity bonds. First, we redefine yields, forward rates, and holding period return (and hence excess return) in terms of their log values. We then run regressions similar to the simple yield/returns in the previous section. Without changing notation, we run log holding period return regressions of the form:

hprxt+1(m) = am+ bmγTft+ εt+1(m), (29)

for m = 2, 3, 4, 5. It is easy to verify that in the logarithmic form, the one-year holding period excess return on an m−year bond is given by

hprxt+1(m) = m.yt(m) − (m − 1).yt+1(m − 1) − yt(1), (30)

where yt+1(m − 1) is the 1-year-ahead log yield forecast at time t on an m − 1 bond, for

m = 2, 3, 4, 5. We proceed to forecast the yield curve as follows. Using the expected value of excess return in equation (29) together with equation(30), one readily obtains

yt+1(m − 1) = −

1

m − 1(am+ bmγ

T_f

t− m.yt(m) + yt(1)) (31)

for m = 2, 3, 4, 5, corresponding to yield forecasts in maturities of 1, 2, 3, and 4 years respectively. Therefore we use the 1 − 5 year forward rate vector together with 1− and m−year maturity bond yields at time t to forecast an (m − 1)−year yield a year ahead at time t + 1.

Yield Curve Forecasting with Stock Index

The role played by stock markets in an economy is so significant that the strength of an economy can be measured by the performance of the country’s stock markets. The well established connection between interest rates and stock market prices is a good starting point to link forecastability of bond returns with stock returns. In this subsection, we set out to forecast bond returns at t + 1 by including stock market information avail-able at time t. We include the German DAX annual excess return denoted by DAXt at

time t as a right hand variable in the forecast equation of 1-year bond excess return at t+1: hprxt+1(m) = am+ bmγTft+ αDAXt+ εt+1(m), (32)

accordingly forecast (m − 1) yields at time t + 1 using information available at time t as yt+1(m − 1) = − 1 m − 1(am+ bmγ T_f t+ αDAXt− m.yt(m) + yt(1)) (33)

for 1-, 2-, 3- and 4-year tenure yields.

4.3

Results

Excess Return Regressions

We now turn to the discussion of results from excess return regressions. First we report on the computed excess returns and forward rates. Table 8 presents summary statistics of the computed forward rates and one-year excess returns on the four maturities. As we observe, the computed mean forward rate curve is a normal one, that is, an upward sloping curve in the four maturities considered. The average one-year excess returns too are increasing with maturity. The strong persistence depicted by the high autocorrelations in forward rates and excess returns are strongly explained by the fact that we are using daily yields data.

Table 9 presents the estimated coefficients, standard errors (corrected for overlap by the Newey-West (1987) method), t-statistics and R2 _{for 12-month excess returns forecast for}

the period from June 1991 to May 2005. The results are from regressing individual ma-turity excess returns on all forward rates for the unrestricted model in (24). As can be observed, the coefficients are all statistically significant. We plot the coefficients against maturity in the top panel of Figure 6. The forward rate vector forecasts excess returns with an R2 _{of up to 45 percent. Table 10 presents coefficient estimates of the return}

fore-casting factor γ that are obtained by regressing average (across maturity) excess returns on a constant, the one-year yield and four forward rates. As we can see from the t-statistics, the γ estimates are significant overall.

In Table 11, we report the coefficients of the return-forecasting factor in the restricted model (25). The loadings bm of expected returns on the return-forecasting factor γTf

reported in column 5 of Table 11 increase smoothly with maturity . The bottom panel of Figure 6 plots the coefficients of individual-bond expected returns on forward rates, as implied by the restricted model, that is, for each m, it includes [bmγ1. . . bmγ5].

Com-paring this plot with the unrestricted estimates of the top panel, we can clearly observe that the single-factor captures the unrestricted parameter estimates. To show precisely the fit of the single-factor model, consider for example 5-year maturity bond and com-pare [β1,5 β2,5, . . . , β5,5] i.e, [−0.046, −2.393, −0.628, 8.78, −2.193, −2.178] in Table 9 with

b5γ = 1.51 ∗ [−0.026, −1.557, −0.577, 6.171, −1.346, −1.821], or equivalently,

In order to test the fit of the return forecasting model, we plot the excess return residuals in Figure 5. Overall, the residual plot indicates a good fit as there are no clear patterns in the return forecast errors. We take a more keen look at the high values of R2 _reported

in Tables 9 − 11. It appears that the R2_{s are perhaps exaggerated by the fact one-year}

excess returns include the obvious one-year ”random walk” return obtained by assuming yields do not change in the next period. To test the impact of ignoring such consideration, we also ran similar regressions with one-year net excess return as the left hand variable in equation (25). We define one-year net excess return,N ethprxt+1(m), as the excess return

as earlier defined less the one-year return obtained by assuming random walk on yields, that is, N ethprxt+1(m) ≡ (1 + yt(m))m (1 + yt+1(m − 1))m−1 − (1 + yt(m)) m (1 + yt(m − 1))m−1 − yt(1) (34)

The results shown in Table 12 are interesting. There is a substantial difference in the results especially on the reported R2 _{between excess return and net excess return}

regres-sions. Although the return-forecasting factor retains its beautiful tent-shaped pattern (not included in here) as in the original excess return forecasts, the highest R2 _{now turns out to}

be only 30%.It would appear therefore the single-factor forecasting ability is questionable when one-year random walk returns are netted out of the holding period return equation. Out-of-Sample Yield Forecast Error Analysis

Table 13 presents out-of-sample forecasting error statistics of one-year-ahead yields using the return forecasting factor. We also report yield forecast results that include the stock index return in the forecasting equation. First, we note that there is a big improvement in terms of mean error and RMSE reductions compared to other forecast models at the one-year horizon. More specifically, the Nelson-Siegel model is clearly outcompeted when compared to Cochrane-Piazzesi yield method. Furthermore, as we can see from the RMSE values of the two yield forecast models in Table 13, there are substantial improvements in the forecasts gained by including a stock return in the forecasting equation. We might therefore conclude that an investor incorporating stock market information in his yield curve forecast is likely to have a better forecast than one using only bond information.

5

Combining Forecasts

5.1

Introduction

which method is most accurate, and when one wants to avoid large errors. In this section we undertake such a forecast combination exercise.

5.2

Methods

In this subsection,we show how forecast combination is implemented. Consider any two yield curve forecasting models, ya

t+h,t and yt+h,tb . We define a combined yield forecast yct+h,t

as9

yc_t+h,t= ωy_t+h,tc + (1 − ω)yc_t+h,t (35)

where ω is a combining weight parameter that can be analytically computed(Either through variance-covariance or regression methods, the procedure involves finding a weight that minimises the variance of the resulting combined forecast error). However, we use a fixed value of ω = 0.5 throughout this exercise following Clemen (1989)10_.

5.3

Results

Table 14 presents results from the combined forecasts. We report combined forecast yields on the 1−, 2−, 3−, and 4−year bonds. Furthermore, we only combine two forecasts at a time as our attempts did not yield any benefits from combining more than two. The combined methods are Cochrane-Piazzesi and random walk on yields,Cochrane-Piazzesi (with stock return) and random walk on yields, Cochrane-Piazzesi and Nelson-Siegel (with random on factors), and Cochrane-Piazzesi(with stock return) and Nelson-Siegel (with random on factors). We concentrate on 1-year ahead forecasts. The benefits of the forecast combinations are very apparent. As we can see from Table 14, there is a major reduction in the size of the forecast error statistics relative to the individual forecasts at the one-year horizon. We register percentage RMSE reductions of up to 23.3% in relative terms when we compare the combined forecasts over the Nelson-Siegel method (with randon walk on factors).

6

Summary and Conclusions

This paper presented two parsimonious models of the term structure of interest rates. The Nelson-Siegel model (as factorised by Diebold and Li (2006)) fits the data quite well and is able to forecast reasonably well out-of-sample. The model does not however show any superiority over competitor benchmark models on the sample period considered. We at-tribute the model’s second-rate performance to the inherent continuous declining trend of

9

See Diebold(1998) for more details

10

the EU term structure over the sample period that results in overly ambitious forecasts. This result is only exaggerated in the Nelson-Siegel method but is not a surprise as the other competitor models do produce ambitious forecasts as well.

Cochrane and Piazzesi excess return regressions produce a sizeable R2 _{comparable to that}

found with the US bond data. The coefficient R2 _{is however reduced when we use net}

excess returns obtained by netting out the simple random walk excess return from the one-year holding period return.

When used to forecast the yield curve, the tent-shaped factor produces better out-of-sample results than all the models considered. The forecasting power is further improved on inclusion of a stock return as a right hand variable in the forecasting equation. The influence of stock market activity on economic activity forecasts seems to be apparent in term structure forecasts as well.

In a situation like the one in this paper,where there is no dominating forecasting model, forecast combination seems a well thought alternative. We register great benefits in terms of forecast error reductions when we combine two forecast methods at the one-year hori-zon.The combination of Cochrane-Piazzesi (with stock return) and Nelson-Siegel (with random walk on factors) produced the least root-mean-squared error values among all models considered.

7

References

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Bliss, R. (1997b). Testing Term Structure Estimation Methods. Advances in Futures and Options Research, 9: 97-231.

Campbell, J. Y. and R. J. Shiller(1991). Yield Spreads and Interest Rate Movements: A Bird’s Eye View. Review of Economic Studies, 58: 495-514.

Clemen, R. T. (1989). Combining Forecasts: A review and Annotated Bibliography. In-ternational Journal of Forecasting, 5: 559-583.

Cochrane, J.H., Piazzesi, M. (2005). Bond Risk Premia. The American Economic Review, 95: 138-160

Cox, J.C., Ingersoll, J.E., Ross, S.A. (1985). A Theory of the Term Structure of In-terest Rates. Econometrica,53: 385-407.

Diebold, F.X. (1998). Elements of Forecasting. South-Western college publishing, Cincin-nati.

Diebold, F.X. and Li, C. (2006). Forecasting the Term Structure of Government Bond Yields. Journal of Econometrics, 130: 337-364.

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Elton E. J and M. J. Gruber (1995). Modern Portfolio Theory and Investment Analy-sis, John Wiley & sons, New York.

Fama, E. F. and R. R. Bliss (1987). The Information in Long-Maturity Forward Rates. American Economic Review, 77: 680-692.

Ferson, W., and M. R. Gibbons (1985). Testing Asset Pricing Models with Changing Expectations and an Unobservable Market Portfolio. Journal of Financial Economics, 14: 216-236.

Journal of Finance, 50: 481-506.

Litterman, R., J. Scheinkman (1991). Common Factors Affecting Bond Returns. Journal of Fixed Income,1: 54-61.

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McCulloch, J.H. (1975). The Tax Adjusted Yield Curve. Journal of Finance, 30: 811-830.

Nelson, C.R., A.F. Siegel (1987). Parsimonious Modelling of Yield Curves. Journal of Business, 60: 473-489.

Newey, W. K., and K. D. West. (1987). A Simple, Positive Semi-Definite, Heteroskedas-ticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55: 703-708. Shea, G.S. (1992). Benchmarking the Expectations Hypothesis of the Interest-Rate Term Structure: An Analysis of Cointegration Vectors. Journal of Business and Economic Statis-tics, 10: 347-366.

Stambaugh, R. F. (1988). The information in Forward Rates: Implications for Models of the Term Structure. Journal of Financial Economics, 22: 3-25.

Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5: 177-188.

8

Appendix

Maturity Mean St.dev Min Max ACF(1) ACF(12) ACF(25) 3M 3.206 0.905 1.957 5.140 0.999 0.993 0.981 1Y 3.375 0.937 1.913 5.418 0.999 0.990 0.976 2Y 3.631 0.910 1.999 5.577 0.999 0.985 0.965 3Y 3.866 0.866 2.224 5.653 0.999 0.982 0.959 4Y 4.069 0.824 2.480 5.707 0.998 0.981 0.956 5Y 4.242 0.787 2.719 5.775 0.998 0.980 0.954 6Y 4.396 0.757 2.933 5.850 0.998 0.979 0.953 7Y 4.534 0.732 3.079 5.911 0.998 0.979 0.953 8Y 4.652 0.708 3.198 5.971 0.998 0.979 0.953 9Y 4.749 0.686 3.300 6.018 0.998 0.979 0.953 10Y 4.827 0.667 3.382 6.051 0.998 0.978 0.953 12Y 4.959 0.642 3.526 6.145 0.998 0.977 0.952 15Y 5.109 0.616 3.671 6.288 0.998 0.977 0.952 20Y 5.252 0.582 3.810 6.383 0.998 0.975 0.950 30Y 5.328 0.547 3.899 6.395 0.997 0.973 0.946

Table 1: Summary Statistics for the Yield Curve. The data summarises daily swap rates (yields) observations for a period from February 1998 to May 2005

Factor Mean St.dev Min Max ACF(1) ACF(12) ACF(25) ADF(3) β0t 5.516 0.529 4.048 6.634 0.997 0.969 0.939 -1.757

β1t -2.240 0.688 -3.843 -0.806 0.997 0.966 0.925 -1.603

β2t -2.643 1.338 -5.638 0.373 0.995 0.942 0.874 -2.166

Maturity 3M 1Y 2Y 5Y 10Y 20Y (A) Nelson-Siegel with

Random Walk on Factor Dynamics

RMSE 0.144 0.182 0.217 0.218 0.177 0.155 St.Dev 0.123 0.182 0.217 0.204 0.167 0.152 Mean -0.075 -0.013 0.009 -0.076 -0.059 0.032

(B) Nelson-Siegel with

AR(1) on Factor Dynamics

RMSE 0.165 0.195 0.217 0.226 0.185 0.153 St.Dev 0.126 0.190 0.216 0.202 0.167 0.152 Mean -0.106 -0.046 -0.022 -0.103 -0.080 0.014

(C) Nelson-Siegel with

VAR(1) on Factor Dynamics

RMSE 0.131 0.198 0.238 0.229 0.194 0.192 St.Dev 0.117 0.198 0.237 0.219 0.189 0.187 Mean -0.059 0.011 0.020 -0.066 -0.047 0.044

(D)Random Walk on Yields

RMSE 0.131 0.183 0.218 0.207 0.173 0.153 St.Dev 0.126 0.177 0.213 0.203 0.169 0.150 Mean -0.039 -0.046 -0.047 -0.043 -0.038 -0.034

(E)Univariate AR(1) on Yields

RMSE 0.137 0.191 0.224 0.211 0.177 0.157 St.Dev 0.128 0.181 0.216 0.204 0.170 0.150 Mean -0.050 -0.061 -0.060 -0.055 -0.049 -0.046 (F) VAR(1) on Yields RMSE 0.123 0.190 0.236 0.231 0.201 0.184 St.Dev 0.123 0.187 0.233 0.230 0.201 0.183 Mean -0.005 -0.033 -0.041 -0.017 -0.013 -0.012

Maturity 3M 1Y 2Y 5Y 10Y 20Y (A) Nelson-Siegel with

Random Walk on Factor Dynamics

RMSE 0.299 0.355 0.407 0.402 0.316 0.250 St.Dev 0.252 0.336 0.398 0.368 0.287 0.248 Mean -0.161 -0.115 -0.090 -0.162 -0.132 -0.034

(B) Nelson-Siegel with

AR(1) on Factor Dynamics

RMSE 0.396 0.420 0.430 0.425 0.342 0.260 St.Dev 0.317 0.368 0.390 0.349 0.279 0.244 Mean -0.238 -0.204 -0.182 -0.243 -0.198 -0.091

(C) Nelson-Siegel with

VAR(1) on Factor Dynamics

RMSE 0.293 0.433 0.496 0.477 0.410 0.377 St.Dev 0.275 0.427 0.492 0.454 0.392 0.377 Mean -0.100 -0.074 -0.064 -0.147 -0.118 -0.019

(D)Random Walk on Yields

RMSE 0.288 0.370 0.419 0.388 0.313 0.265 St.Dev 0.259 0.340 0.394 0.365 0.292 0.246 Mean -0.126 -0.147 -0.145 -0.130 -0.112 -0.100

(E)Univariate AR(1) on Yields

RMSE 0.340 0.423 0.449 0.403 0.328 0.281 St.Dev 0.297 0.362 0.398 0.358 0.286 0.241 Mean -0.164 -0.218 -0.208 -0.185 -0.160 -0.146 (F) VAR(1) on Yields RMSE 0.296 0.462 0.548 0.541 0.475 0.429 St.Dev 0.297 0.449 0.537 0.538 0.473 0.427 Mean -0.002 -0.110 -0.109 -0.057 -0.040 -0.035

Maturity 3M 1Y 2Y 5Y 10Y 20Y (A) Nelson-Siegel with

Random Walk on Factor Dynamics

RMSE 0.502 0.552 0.608 0.583 0.454 0.348 St.Dev 0.389 0.475 0.555 0.500 0.376 0.316 Mean -0.317 -0.282 -0.249 -0.301 -0.254 -0.146

(B) Nelson-Siegel with

AR(1) on Factor Dynamics

RMSE 0.824 0.787 0.741 0.679 0.557 0.433 St.Dev 0.669 0.621 0.576 0.466 0.375 0.329 Mean -0.481 -0.485 -0.466 -0.494 -0.412 -0.281

(C) Nelson-Siegel with

VAR(1) on Factor Dynamics

RMSE 0.557 0.715 0.788 0.791 0.723 0.681 St.Dev 0.487 0.680 0.766 0.750 0.688 0.670 Mean -0.270 -0.222 -0.186 -0.252 -0.222 -0.125

(D)Random Walk on Yields

RMSE 0.483 0.580 0.629 0.565 0.449 0.376 St.Dev 0.391 0.489 0.552 0.496 0.383 0.311 Mean -0.284 -0.312 -0.303 -0.270 -0.235 -0.211

(E)Univariate AR(1) on Yields

RMSE 0.684 0.685 0.681 0.603 0.507 0.448 St.Dev 0.598 0.514 0.513 0.444 0.357 0.309 Mean -0.334 -0.453 -0.448 -0.408 -0.361 -0.324 (F) VAR(1) on Yields RMSE 0.497 0.738 0.856 0.890 0.815 0.725 St.Dev 0.484 0.678 0.822 0.879 0.805 0.715 Mean -0.112 -0.292 -0.241 -0.141 -0.128 -0.123

Maturity 3M 1Y 2Y 5Y 10Y 20Y (A) Nelson-Siegel with

Random Walk on Factor Dynamics

RMSE 0.861 0.860 0.869 0.792 0.621 0.481 St.Dev 0.547 0.619 0.693 0.606 0.451 0.371 Mean -0.665 -0.598 -0.524 -0.510 -0.428 -0.307

(B) Nelson-Siegel with

AR(1) on Factor Dynamics

RMSE 1.679 1.647 1.561 1.327 1.074 0.868 St.Dev 1.463 1.386 1.283 0.995 0.775 0.639 Mean -0.824 -0.890 -0.889 -0.879 -0.744 -0.587

(C) Nelson-Siegel with

VAR(1) on Factor Dynamics

RMSE 1.424 1.527 1.545 1.375 1.145 0.972 St.Dev 1.253 1.387 1.423 1.200 0.946 0.796 Mean -0.678 -0.640 -0.604 -0.673 -0.646 -0.558

(D)Random Walk on Yields

RMSE 0.835 0.890 0.896 0.771 0.616 0.519 St.Dev 0.543 0.634 0.688 0.604 0.460 0.364 Mean -0.635 -0.625 -0.574 -0.480 -0.409 -0.370

(E)Univariate AR(1) on Yields

RMSE 1.632 1.673 1.493 1.231 1.024 0.863 St.Dev 1.569 1.557 1.290 0.984 0.769 0.599 Mean -0.449 -0.613 -0.752 -0.741 -0.677 -0.621 (F) VAR(1) on Yields RMSE 1.389 1.340 1.252 1.020 0.858 0.780 St.Dev 1.353 1.265 1.168 0.899 0.675 0.527 Mean -0.316 -0.442 -0.451 -0.483 -0.531 -0.575

Maturity (in years) Mean St.dev Min Max ACF(1) ACF(12) ACF(25) Forward Rates 2 5.072 1.689 2.081 9.463 0.999 0.990 0.977 3 5.538 1.460 2.676 8.911 0.999 0.990 0.976 4 5.902 1.343 3.250 8.884 0.998 0.988 0.976 5 6.149 1.256 3.680 8.780 0.998 0.989 0.979 Excess Returns 2 0.935 1.108 -1.822 4.175 0.996 0.965 0.913 3 1.880 2.332 -4.229 8.276 0.997 0.967 0.916 4 2.664 3.386 -6.668 11.444 0.998 0.966 0.915 5 3.292 4.270 -8.929 13.501 0.997 0.966 0.914

Table 8: Summary Statistics for the Computed Forward Rates and Excess Returns. The data covers a period from 7th June 1991 to 31st May 2005, a total of 3648 observations.

Maturity const y(1) f (2) f (3) f (4) f (5) R2

2 -0.007 -0.676 -0.082 2.408 -0.515 -0.833 0.426 (0.002) (0.070) (0.312) (0.460) (0.330) (0.276) [-3.035] [-9.676] [-0.263] [5.240] [-1.560] [-3.019] 3 -0.020 -1.295 -0.717 5.863 -1.405 -1.731 0.459 (0.005) (0.161) (0.658) (0.951) (0.707) (0.582) [-4.352] [-8.051] [-1.090] [6.164] [-1.989] [-2.974] 4 -0.033 -1.864 -0.879 7.633 -1.271 -2.539 0.447 (0.007) (0.256) (0.949) (1.349) (1.046) (0.856) [-5.024] [-7.278] [-0.926] [5.657] [-1.215] [-2.967] 5 -0.046 -2.393 -0.628 8.780 -2.193 -2.178 0.421 (0.008) (0.345) (1.191) (1.679) (1.349) (1.094) [-5.526] [-6.931] [-0.527] [5.231] [-1.625] [-1.991]

Table 9: Regression of Excess Returns on all Forward Rates. This Table shows the coefficients and R2 _{from regressions of (overlapping) one year excess returns for 2- to}

5-year forward rates. The sample period is 7th June 1991 to 31st May 2005. The return regressions are of the form:

hprxt+1(m) = β0,m+ β1,myt(1) + β2,mft(2) + . . . + β5,mft(5) + εt+1(m)

γ0 γ1 γ2 γ3 γ4 γ5 R2

Estimate -0.026 -1.557 -0.577 6.171 -1.346 -1.821 0.444 std.error 0.005 0.204 0.767 1.095 0.846 0.692

t-stat -4.835 -7.624 -0.752 5.638 -1.591 -2.631

Table 10: Estimates of the return-forecasting factor. The regression equation is:

1 4

P5

m=2hprxt+1(m) = γ0 + γ1yt(1) + γ2ft(2) + . . . + γ5ft(5) + εt+1. Regressions are

based on overlapping data and standard errors use the Newey-West (1987) method for correcting for overlap.

Maturity am std.error t-stat bm std.error t-stat R2

2 0.0007 0.001 1.311 0.393 0.022 17.556 0.423 3 -0.0001 0.001 -0.0801 0.861 0.045 18.995 0.458 4 -0.0005 0.002 -0.253 1.235 0.067 18.332 0.447 5 -0.0002 0.003 -0.074 1.510 0.089 17.049 0.420

Table 11: Out-of-Sample Predictability of Excess Returns. This Table presents coefficients and R2_{s obtained from overlapping daily projections of one year excess returns, for}

different maturities, on out of-sample model implied returns. The four regressions are of the form:

hprxt+1(m) = am + bmγTft + εt+1(m), for m = 2,...,5. Regressions are based on

overlapping data and standard errors use the Newey-West (1987) method for correcting for overlap.

Maturity am std.error t-stat bm std.error t-stat R2

2 -0.021 0.002 -12.127 0.584 0.050 11.643 0.232 3 -0.007 0.002 -2.834 0.850 0.069 12.278 0.301 4 0.008 0.004 2.009 1.160 0.117 9.935 0.269 5 0.020 0.005 3.914 1.406 0.159 8.824 0.236

Maturity 1Y 2Y 3Y 4Y RMSE 0.807 0.796 0.743 0.697 ST.Dev 0.445 0.424 0.395 0.371 Mean -0.673 -0.674 -0.630 -0.589 vs. N-S 9.3% 11.1% 13.7% 14.4% Cochrane-With Stock Return

RMSE 0.660 0.689 0.690 0.679 ST.Dev 0.396 0.379 0.360 0.344 Mean -0.529 -0.576 -0.589 -0.586 vs. N-S 25.8% 23.1% 19.8% 16.6%

Table 13: Out-of-Sample Error Analysis. The Table presents Cochrane-Piazzesi 1-year-ahead return forecasts with single tent-shaped factor. The Table also reports in the bottom panel regressions that include the DAX return as RHS variable, i.e, the four regressions are of the form:

hprxt+1(m) = am+ bmγTft+ αDAXt+ εt+1(m),

where DAXt is the one-year excess return on the DAX stock index calculated as

Maturity 1Y 2Y 3Y 4Y RW-yields and C-P

RMSE 0.807 0.796 0.743 0.697

ST.Dev 0.540 0.549 0.525 0.497 Mean -0.660 -0.636 -0.594 -0.558 RW-yields and C-P with Stock

RMSE 0.660 0.689 0.690 0.679 ST.Dev 0.482 0.497 0.490 0.473 Mean -0.585 -0.585 -0.573 -0.556 N-S(RW-Factors) and C-P RMSE 0.834 0.824 0.771 0.724 ST.Dev 0.534 0.553 0.532 0.502 Mean -0.647 -0.611 -0.579 -0.560 vs.N-S 3.1% 5.2% 9.7% 12.3%

N-S(RW-Factors) and C-P(Stock)

RMSE 0.660 0.689 0.690 0.679

St.Dev 0.468 0.496 0.494 0.475 Mean -0.542 -0.524 -0.517 -0.513 vs.N-S 23.3% 20.7% 19.2% 17.7%

May ’98 May ’99 May ’00 May ’01 May ’02 May ’03 May ’04 May ’05 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Time Yield 30−Year 3−Month 10−Year

50 100 150 200 250 300 350 0 0.2 0.4 0.6 0.8 1 m (Maturity, in months) Yield (percent) ß0 Loadings ß1 Loadings ß2 Loadings

May ’98 May ’99 May ’00 May ’01 May ’02 May ’03 May ’04 May ’05 −6 −4 −2 0 2 4 6 8 Time Factor Weights ß0 ß1 ß2

50 100 150 200 250 300 350 3 3.5 4 4.5 5 5.5

Maturity (in months)

Yield (percent)

Actual Yield Curve Fitted Nelson−Siegel

May’93 May’95 May’97 May’99 May’01 May’03 May’05 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Time Residuals(Percent)

1 2 3 4 5 −4 −2 0 2 4 6 8 10 Unrestricted Model

Maturity (in years)

Coefficient loading 2−yr 3−yr 4−yr 5−yr 1 2 3 4 5 −4 −2 0 2 4 6 8 10 Restricted Model

Maturity (in years)

Coefficient loading

2 3 4 5

Figure 6: Coefficients of Excess Return Regressions on all Forward Rates. The unrestricted model coefficients are given by

hprxt+1(m) = β0,m+ β1,myt(1) + β2,mft(2) + . . . + β5,mft(5) + εt+1(m),