Why Long Term Forward Interest Rates(Almost) Always Slope Downwards

Roger H. Brown Stephen M. Schaefer∗

Warburg Dillon Read London Business School,

May 1, 2000

∗Corresponding author: Stephen M. Schaefer, London Business School, Sussex Place, Re-gents’ Park, London NW1 4SA, UK. e-mail: sschaefer@lbs.ac.uk

Abstract

The paper documents a persistent and thus far largely overlooked empirical regu-

larity in the yield curve: the tendency for the term structure of long term forward rates

to slope downwards. The persistence of this feature is demonstrated using data on US

and UK Government conventional (nominal) bonds and UK Government index-linked

bonds. We show that the downward slope is the result of interest rate volatility. Using a

two factor Gaussian model we show that the long term forward rate curve will be down-

ward sloping whenever the volatility of the long term zero coupon yield is sufficiently

high. Using data on US Treasury STRIPs, the paper further shows that the slope of

the forward rate curve predicts the volatility of long term rates and that the implied

volatility from bond futures options explains the slope of the forward rate curve.

2

Why Long Term Forward Interest Rates(Almost) Always Slope Downwards1

1. Introduction

The term structure of long term forward interest rates is persistently downwardsloping. In nearly ten years of daily data on US Treasury STRIPs from 1985 to1994, the implied two year forward rate spanning years 24 to 26 is lower thanthe forward rate for years 14 to 16 on 98.4% of occasions. The average differencein these rates is 138 basis points. A similar “downward tilt” also appears inestimates of forward rates derived from the prices of coupon bonds in the USTreasury market and in the UK market for both real and nominal governmentbonds2.

We show that this feature of the term structure is not anomalous; nor do weneed to turn to institutional features of the market for an explanation. Rather, itarises as a consequence of the combined effect of the term structure of volatilityof long term interest rates and the greater convexity of long term bonds.

Explaining the relation between the yields on default-free bonds and theirmaturity, the “term structure”, is a classical problem in financial economics andthe subject of extensive research over many years. The shape of the yield curvevaries substantially over time: sometimes upward sloping, sometimes downwardand sometimes hump-shaped. Thus it is interesting to find one part of the curve —long term forward rates — whose shape appears to be always the same, irrespectiveof the pattern of rates for shorter maturities. It may well be the single mostpredictable feature of the term structure of interest rates. It is also interesting tonote that, even though term structure theory has focussed mainly on expectationsof future rates and risk premia to explain the shape, we find that this particularhighly persistent feature is the result of interest rate volatility alone.

Most term structure theories (e.g., Vasicek (1977), Cox, Ingersoll and Ross

1The authors gratefully acknowledge the helpful comments and suggestions from AndreaBerardi, Mark Britten-Jones, Antii Ilmanen and Davide Menini, from seminar participants atthe universities of Barcelona, Imperial College, Louvain, Lancaster, Toulouse and Verona andfrom participants at the Workshop on Mathematical Finance, Strobl, Austria 1999. The authorsare also very grateful to Mark Fisher, Douglas Nychka and David Zervos for allowing us to usetheir estimates of zero coupon yields. The usual disclaimer applies.

2The estimates for the US were obtained by Fisher M.E., Nychka D. & Zervos D (1994) andMcCulloch and Kwon (1993). Those for the UK are the authors own calculations.

3

(1985) and Longstaff and Schwartz (1992)) predict that the term structure oflong term forward rates is flat. Indeed, ever since Durand (1942) constructed hisfamous series of yields by drawing a curve (freehand) through the lower envelopeof yields-to-maturity on high grade corporate bonds, the available data has seemedto suggest that actual yield curves are also flat for long maturities3. In the case ofyields-to-maturity on coupon bearing bonds this is merely a matter of arithmeticsince the yields on all such bonds approach the yield on a perpetuity as maturitybecomes infinite. However, the shape of the zero-coupon yield curve (or theforward rate curve) is not constrained in this way, apart from the no-arbitragecondition that (nominal) forward rates are non-negative.

What our results show is that under empirically relevant conditions - essen-tially, when long term zero coupon rates experience parallel shifts - the termstructure of long term forward rates is always downward sloping.

The consistent downward slope in long term forward rates has a number ofpractical implications. First, our results have implications for attempts to extractmarket expectations from the yield curve4. Abstracting from considerations ofrisk premia, classical term structure theory identifies the market’s expectation ofa future interest rate as the forward rate. In models which make use of contingentclaims analysis (i.e., Merton (1970), Vasicek (1977) and followers) forward ratesand the expected path of the short rate will diverge even when risk premia arezero. The difference is accounted for by convexity effects; our results show that,for long maturities, these effects can be substantial. Second, some approacheswhich have been proposed for estimating the term structure of zero coupon ratesfrom the prices of coupon bonds use approximating functions which constrain thelong term forward rate curve to be flat5. Our results suggest that such methodsmay be mis-specified. Third, the relation which we derive between interest ratevolatility and the shape of long term forward rates may be useful to practitionersfacing the problem of pricing a new issue which has a longer maturity than anyexisting issue.

Using a two-factor affine model [Brown and Schaefer (1994b), Duffie and Kan(1996)], we derive a simple relation between the term structure of long term for-ward rates in a cross section and the volatility of long term zero coupon rates.Here we follow a number of authors who have attempted to use term structure

3Durand drew his curves by choosing one of five “shapes”, all of which were flat for longmaturities.

4See Deacon and Derry (1994).5For example, Vasicek and Fong (1982) and Nelson and Siegel (1987).

4

models to estimate the level of interest rate volatility implied by the term struc-ture. For example, Brown & Dybvig (1986), Barone, Cuoco & Zautzik (1991)and Brown & Schaefer (1994a) have all used the Cox, Ingersoll and Ross (1985)(CIR) model to impute interest rate volatility from cross sections of bond prices.While all three studies find that implied volatility corresponds quite well to timeseries estimates, the volatility parameter in term structure studies is estimatedjointly with the other parameters of the model and it is correspondingly difficultto identify the influence on the shape of the term structure alone.

Litterman, Scheinkman & Weiss (1991) (LSW) adopt a simpler approach andestimate the relation between the implied volatility of options on US Treasurybond futures and the level of short, medium and long term zero coupon yields.They find that the 1 month, 3-year and 10-year zero coupon yields together explainaround 70% of the variation in implied volatility. The present study can beregarded as being in the spirit of LSW, namely to identify some easily observablefeature of the yield curve which reflects the impact of interest rate volatility butto do so in the context of a model. Thus we derive a simple relation between theterm structure of interest rate volatility and the term structure of forward interestrates. Our relation is also easily inverted: in other words it is straightforward toobtain an estimate of “implied interest rate volatility” from the pattern of longterm forward rates. Caverhill (1998) has also recently investigated the relationbetween the forward curve and interest rate volatility.

The paper is organised as follows. Section 2 uses a highly simplified model ofthe yield curve to provide an intuitive explanation of the relation between forwardrates and the volatility of zero coupon yields. The main prediction of the model isthat the spread between longer and shorter term forward rates should be negativeand proportional to the variance of long term zero coupon rates. Section 3 thenshows that the same relation may be derived as an approximation within a morerealistic two-factor “affine” model. Section 4 describes our data and Section 5presents summary statistics on estimates of forward rate spreads for three markets:US Treasury bonds, UK conventional (nominal) gilts and UK index-linked gilts.We find, as the model predicts, that forward rate spreads - particularly since theearly 1980’s - are strongly and persistently negative. Section 6 contains our mainempirical results. Here we show that (a) the shape of the term structure of forwardrates predicts the volatility of zero coupon yields and (b) that the forward ratespread is itself predictable using the implied volatility of bond futures options.Section 7 concludes.

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2. The Informal Argument

The term structure of long term forward rates depends on two key features ofthe empirical behaviour of the yield curve: the tendency of long term zero couponrates (a) to be highly correlated and (b) to have volatilities which decline relativelyslowly with maturity.6

To understand this simple relation between forward rates and interest ratevolatility, consider a zero coupon bond with time-to-maturity τ and yield at timet of y(t, τ ). We assume that the yields on bonds with sufficiently long maturitiesare perfectly correlated with a common variance σ2y which is constant over timeand over (long) maturities. The price of such a bond is given by:

P (y(t, τ )) = e−y(t,τ)τ , (2.1)

and its rate of return from t to t+∆t by:

∆P

P=

1

P

[1

2

∂2P

∂y2∆y2 +

∂P

∂y∆y

]+

1

P

∂P

∂t∆t. (2.2)

where ∆y denotes the change in yield from t to t+∆t. If the process generatingy is time homogeneous it is straightforward to show that f(t, τ , y), the time tinstantaneous forward rate for maturity τ , is given by:

f(t, τ , y) =PtP. (2.3)

Equation (2.1) implies:

1

P

∂P

∂y= −τ and

1

P

∂2P

∂y2= τ 2, (2.4)

and, substituting into equation (2.2) and taking expectations (assuming that, forsmall ∆t, E(∆y2) ≈ σ2y∆t) the risk premium on the bond becomes:

6Section 5 presents evidence on the term structure of volatility for long term zero couponrates.

6

E

[∆P

P

]− rt∆t =

[1

2σ2yτ

2 − τµy + f(t, τ , y)− rt

]∆t, (2.5)

where rt is the short term (riskless) interest rate and E(∆y) ≡ µy∆t. However,no arbitrage considerations imply that the risk premium on the bond is alsoproportional to its elasticity with respect to y, i.e.:

E

[∆P

P

]− rt∆t = τλy∆t, (2.6)

where λy is a risk premium parameter7. Equating equations (2.5) and (2.6) wehave that the forward rate is given by:

f(t, τ , y) = rt + τ(µy − λy

)−

1

2σ2yτ

2. (2.7)

If we assume that the long end of the term structure experiences parallel shifts,i.e., that the term structure of zero coupon yield volatility is flat, then, for a givenreference yield y∗,

1

P

∂P

∂y∗= −τ and

1

P

∂2P

∂y∗2= τ 2

hold for all “long” maturities, then equation (2.7) shows that the shape of thelong term forward rate curve depends on two terms. The first is linear in maturityand, depending on the sign of

(µy − λy

), may be increasing or decreasing with τ .

The second, however, is (a) always decreasing in maturity and (b) because it isproportional to τ 2 rather than τ , always dominates the first term for sufficientlylarge τ . Thus, when the term structure of yield volatility is constant for longmaturities the term structure of forward rates is always downward sloping and,for long maturities — neglecting the linear term — we have that the difference inforward rates between two maturities τ 1 and τ 2, the “forward rate spread”, isgiven approximately by:

∆f(t, τ 1, τ 2) ≡ f(t, τ 2, y)− f(t, τ 1, y) (2.8)

≈ −1

2σ2�

[τ 22 − τ

21

].

7This model is essentially identical to Merton (1970) with a zero rate of mean reversion.

7

The only significant difference between the analysis in this section and a stan-dard single factor model, e.g., Vasicek (1977) is the (implicit) assumption of zeromean reversion which supports the assumption of a flat term structure of volatilityfor long term rates. With non-zero mean reversion the yield volatility σy declineswith maturity and consequently equation (2.7) would not describe the term struc-ture of long term forward rates. It is important to note, however, that in this casethe yields on long bonds converge to a constant.

In the next section we derive an expression for the forward rate spread usinga less restrictive, i.e., more realistic, two-factor model and show that, even in thepresence of non-zero mean reversion in one factor, equation (2.8) still holds as anapproximation to the forward rate spread.

3. A Model of Forward Rate Spreads

We consider a two factor time homogeneous affine Gaussian model of the termstructure [Langetieg(1980), Brown & Schaefer (1994b), Duffie and Kan (1996)]where the state variables x and y are assumed to follow:

dx = κx (µx − x) dt+ σxdzx, (3.1)

dy = κy(µy − y

)dt+ σydzy,

where κx, µx, σx, κy, µy and σy are parameters and dzx and dzy are incrementsof standard Brownian motions with E(dzxdzy) = ρdt. Within the class of two-factor time-homogeneous Gaussian affine models Dai and Singleton (1998) showthat the dynamics given in equation (3.1) are completely general. The Gaussianspecification means that the probability of the short rate becoming negative inany finite interval is non-zero. Despite this the model is adequate for our purposesand, in particular, fits the term structure of yield volatility well.

Since the model is affine (i.e., both the drift and diffusion coefficients are affinein x and y) the price of a zero coupon bond with maturity τ is exponential affineand given by [Duffie and Kan (1996)]:

P (x, y, t, τ ) = exp−{A(τ) +Bx(τ )x(t) +By(τ)y(t)} (3.2)

where, because the model is time homogeneous, the functions A(.), Bx(.) andBy(.)are functions of the parameters and time-to-maturity, τ , alone. The formulae for

8

Bx(.) and By(.) are given below; the formula for A(.) is also well know (Langetieg(1980), for example) but is not required here. The corresponding zero couponyield, R(.), is given by:

R(x, y, t, τ) ≡ −1

τln [P (x, y, t, τ)] (3.3)

=A(τ)

τ+Bx(τ )

τx(t) +

By(τ)

τy(t). (3.4)

In equation (3.4) the functions Bx(τ)/τ and By(τ )/τ represent the sensitivities ofthe τ -period zero coupon yield to, respectively, x and y. For the Gaussian casegiven in equation (3.1), the functions Bx(τ ) and By(τ ) are given by:

Bx(τ ) =1− exp(−κxτ)

κxand By(τ) =

1− exp(−κyτ)

κy. (3.5)

We do not present detailed estimates of the parameters κx and κy in this papersince our result merely requires that one of the mean reversion coefficients κx, say,is close to zero while the other is not. Appendix A describes the method we use toestimate the mean reversion parameters κx and κy from the covariance matrix ofchanges in zero coupon yields. Using data on US Treasury STRIPs for the period1988-94, we find an estimated value for one of the mean reversion coefficients isclose to zero (in the range 0.02 to 0.04) while the other is between five and tentimes larger [Table A2]8.

We now turn to the implications of these mean reversion coefficients for thebehaviour of long term forward rates. The valuation equation, to which equation(3.2) is the solution, is:

κx(µx − x)∂P

∂x+

1

2σ2x∂2P

∂x2+ κy(µy − y)

∂P

∂y+

1

2σ2y∂2P

∂y2+∂P

∂t+

σxy∂2P

∂x∂y− r(x, y)P = 0. (3.6)

8The US Treasury STRIP prices in the first part of our dataset, from 1985 to 1988, arenoticeably noisier than in the second half. In estimating the mean reversion coefficients κx andκy we have therefore used data from the end of 1998 to the end of data in 1994.

9

Since the model is time homogeneous, the time t instantaneous forward rate fordate t+ τ is given, as before, by:

f(x, y, t, τ) =1

P

∂P

∂t. (3.7)

Furthermore, from (3.2) we have that:

1

P

∂P

∂x= −Bx(τ),

1

P

∂2P

∂x2= B2

x(τ),1

P

∂2P

∂x∂y= Bx(τ)By(τ ) (3.8)

1

P

∂P

∂y= −By(τ ) and

1

P

∂2P

∂y2= B2

y(τ).

Substituting for Pt, Px, Py etc. in (3.6) we have:

f(x, y, t, τ ) = r(x, y) + κx(µx − x)Bx(τ ) + κy(µy − y)By(τ)

−12σ2xB

2x −

12σ2yB

2y − ρσxσyBxBy. (3.9)

Using (3.4) the variance of the τ -period zero coupon yield is:

V ar [R(x, y, t, τ)] = σ2x

(Bx(τ )

τ

)2

+ σ2y

(By(τ )

τ

)2

+2ρσxσy

(Bx(τ )By(τ )

τ 2

), (3.10)

and equation (3.9) for the forward rate therefore becomes9:

f(x, y, t, τ ) = r(x, y) + κx(µx − x)Bx(τ ) + κy(µy − y)By(τ)

−1

2V ar [R(x, y, t, t+ τ )] τ 2. (3.11)

9Even though our analysis uses a Gaussian model, notice that an equation very similar toeq. (3.11) will hold for any affine model. The only difference is that, with n state variablesthere will be n terms which are linear in the drifts rather than two. The short rate and the terminvolving the variance of the τ -period zero coupon rate will be unchanged.

10

Finally, computing equation (3.11) at maturities τ 2 and τ 1 (τ 2 > τ 1) and takingthe difference we obtain an expression for the forward rate spread :

∆f(x, y, t, τ 1, τ 2) ≡ f(x, y, t, t + τ 2)− f(x, y, t, t+ τ 1)

= κx(µx − x) [Bx(τ 2)−Bx(τ 1)] + κy(µy − y) [By(τ 2)−By(τ 1)]

−1

2

(V ar [R(x, y, t, t+ τ 2)] τ

22 − V ar [R(x, y, t, t+ τ 1)] τ

21

). (3.12)

As mentioned earlier, our estimates of κx and κy indicate that κx ≈ 0 and κy � 0and we now show that this implies that the terms in (3.12) which are linear inBx(τ ) and By(τ) are close to zero. If κx ≈ 0 then Bx(τ) is approximately equalto τ and therefore finite and, since it is multiplied by κx it follows that the entireterm which is linear Bx(τ ) is also close to zero. Similarly, if κy � 0 then, since

Ltτ→∞

By(τ) =1

κy,

By(τ 2)−By(τ 1) is close to zero for long maturities and, as before, the entire termwhich is linear By(τ) is close to zero. Under these conditions, the forward ratespread for long maturities is dominated by the last term in equation (3.12), i.e.,that:

∆f(x, y, t, τ 1, τ 2) ≈

− 12

(V ar [R(x, y, t, t + τ 2)] τ

22 − V ar [R(x, y, t, t+ τ 1)] τ

21

).

(3.13)

Table 1 shows the term structure of zero-coupon yield volatility for the data onUS Treasury STRIPs used in Appendix A10. The table shows that, for maturitiesbeyond ten years, the volatility of long term yields attenuates only slowly. If, toa first approximation, we assume that zero coupon yields at maturities τ 1 and τ 2have the same volatility (σ�) then the right hand side of equation (3.13) becomesfurther simplified to:

∆f(x, y, t, τ 1, τ 2) ≈ −1

2σ2�

[τ 22 − τ

21

]. (3.14)

10Once again, we describe our data in more detail below.

11

This expression predicts that the shape of the long term forward rate curve will bequadratic and downward sloping and that the degree of concavity will be directlyrelated to the volatility of long term zero-coupon yields. The critical conditionfor equation (3.14) to provide a good characterisation of the shape of the longterm forward curve — beyond the assumption of affine dynamics — is that the termstructure of volatility for long term zero-coupon yields is flat.

How significant is the forward rate spread predicted by equation (3.14)? Sup-pose that τ 1 and τ 2 are 15 and 25 years respectively. If the volatility of long termyields is, say, 70 basis points per year (consistent with the volatility given in table1 of the 20-year zero coupon yield in the US Treasury market over the period1988-94) then the forward rate spread predicted by the expression above is justunder 100 basis points. With a volatility of 100 basis points p.a., the predictedspread rises to 200 basis points. Of course this may be offset to some extent bythe first two terms in equation (3.12) which capture the effects of expectationsand risk premia. Empirically, however, we find that (a) forward rate spreads arealmost always negative and that (b) the size of the spread appears to be wellexplained by equation (3.14).

4. The Data

We present data on the behaviour of forward rate spreads in three major bondmar-kets: those for US Treasury Notes and Bonds, Conventional (i.e., fixed nominalpayments) UK Government Bonds and Index-Linked UK Government Bonds11.We have chosen to use (a) data from a variety of markets and (b) zero-couponyields estimated via a number of different methods in order to minimise the pos-sibility that our findings are the result of a feature of one particular market orone particular method of measuring the term structure. This section describesour data.

4.1. US Treasury Market

McCulloch & Kwon (1993) (MK) estimate end-month term structures from De-cember 1947 through to February 1991. They use the procedure described inMcCulloch (1975) to estimate a “tax-adjusted” zero coupon yield curve from USTreasury bills, notes and bonds. This method approximates the discount func-tion using cubic splines, accommodating non-symmetric taxation of capital gains

11For a full description of the UK Index Linked Bond Market, see Bootle (1991).

12

and losses, and identifies for each month implicit income and capital gains taxrates which best explain observed prices. MK have modified McCulloch’s origi-nal technique, which assumed coupons arrived continuously, to recognise discretesemi-annual coupon payments. The data represent the longest available time se-ries of forward rates of which we are aware. However, during certain periods therewere no long term Treasuries in issue and, at these times, we were unable to obtainyields for the long maturities which are the focus of our study.

One problem faced by MK is that until August 1985 callable bonds constitutedthe majority of long term issues12. MK’s estimation procedure ignores the value ofthis option and uses the market’s simple “par rule” to determine maturity dates13.After August 1985 MK exclude all callable bonds.

A second set of US Treasury curves comes from Fisher, Nychka and Zervos(1994) (FNZ) who use the closing bid prices of US Treasury securities collected at3.30pm by the Federal Reserve Bank of New York’s Domestic Open Market desk.The data are daily covering the period 1 December 1987 to 23 August 1994. FNZexclude Treasury bills, callable bonds and flower bonds. They also exclude allsecurities issued prior to 1 January 1980 in order to control for the illiquidity ofolder issues. Furthermore, they remove the two most recently issued securities toprevent any “liquidity premium” in these issues from biasing their estimates. Theterm structure is estimated by approximating the forward rate curve with a cubicbasis spline. Unlike McCulloch (1975) or Schaefer (1981) their approach ignorespotential tax effects, although it might be argued that, at least for the estimatescovering the post-1988 period when income and capital gains have been taxed atequal rates, tax effects will be less pronounced.

Our third source of US forward rates is US Treasury STRIP14 prices obtainedfrom Street Software (who also provide the Wall Street Journal with US STRIPdata). The dataset we use is essentially an electronic version of the STRIP pricesappearing in the US Treasury Issues table in the Wall Street Journal. BecauseSTRIP prices are direct observations of the discount factors, no “estimation” isrequired to obtain spot or forward rates. As a result estimates of forward ratesobtained from STRIPs are free from the effects of any a priori restrictions onthe functional form of the yield curve which are present when approximatingfunctions, such as splines, are used. With STRIPs, forward rates are computed

12This problem also arises in the case of the UK Goverment conventional bonds.13The “par rule” assumes that a bond will be called at the earliest possible date if its current

price is above par and at the latest possbile date if the current price is below par.14STRIP is an acronym for the Separate Trading of Registered Interest and Principal.

13

directly from the ratio of prices.The STRIP data is daily and covers the period 22 April 1985 to 5 October

1994. For the years 1985 through to 1989 the prices of only the thirty most liquidSTRIPs are recorded. From 1 June 1990 onwards all prices are recorded, resultingin around 160 observations of discount factors on a typical day. Both bid and offerquotes are available.

It is, of course, not possible to measure instantaneous forward rates - suchas those appearing in equation (3.14) - directly from STRIP prices. As a proxyfor the instantaneous forward rate for maturity τ , we therefore use the averageforward rate between maturities τ −∆τ and τ +∆τ , where ∆τ is, say, one year.Thus, to obtain an estimate of the instantaneous forward rate at 25-years, forexample, we extract for each trading day the bid prices and maturity dates ofSTRIPs maturing closest to dates 24 and 26-years from the trade date. Then,taking account of each STRIP’s exact maturity date, we approximate the 25-yearforward rate as F25, where:

F25 =365

τ 26 − τ 24ln

(P24P26

)100, (4.1)

P24 (P26) is the bid price of the STRIP closest in maturity to 24 (26) years andτ 24 (τ 26) is the exact time to maturity of the 24 (26) year STRIP. We do notdiscriminate between coupon and principal payments though there are clear pricediscrepancies between the two types of STRIP having the same maturity date.

4.2. UK Nominal Government Securities

Prior to 1994, there were insufficient non-callable conventional (nominal) UKGov-ernment Bonds (“gilts”) to allow reliable estimates of long term zero-coupon ratesand forward rates and, for this reason, we have restricted our attention to the pe-riod from 1994 onwards. Even so, for conventional gilts, the maturity of thelongest forward rate we are able to estimate consistently is just over 20-years15.

To accommodate tax effects in the market for UK gilts we use Schaefer’s(1981) “tax-specific” approach to estimate weekly term structures from gilts forthe period April 1994 to November 1998. This method utilises a linear programto identify tax efficient bonds for an investor with a pre-specified tax rate onincome and capital gains. The term structure depends only on those bonds in the

15Actually, 21.5 years.

14

efficient set, with the discount function approximated by cubic basis splines. Wedo not include Treasury bills but base the estimates on every gilt in issue at eachcross section, excluding only callable bonds and “undated” bonds (e.g. Consols2.5%)16. The estimates we quote are for a tax-exempt investor.

4.3. UK Index-linked Government Securities

Here we estimate the real term structure from UK Government inflation-linkedbonds using an adaptation of Schaefer’s LP procedure and Bernstein polynomialsto approximate the discount function. Coupon payments and the repayment ofprincipal for index-linked gilts are not indexed for approximately 8-months prior tothe payment date; in other words the indexation in imperfect. To accommodatethis feature we use an ARIMA model to forecast inflation over these 8-monthperiods and adjust the real cash flows accordingly. We then assume that the realcash flows are known with certainty and proceed as before.

The presence of longer (non-callable) issues in the index-linked market makesit easier to generate reliable estimates of long-term forward rates than is the casein the nominal market. We use weekly cross-sections of index-linked gilts over theperiod January 1984 to November 1998. For more details see Brown & Schaefer(1994a). Once again, the estimates are calculated for a tax-exempt investor.

The different data sources are summarised in table 2.

5. The Behaviour of Forward Rate Spreads: Empirical Ev-

idence.

5.1. Forward Rate Spreads: Summary Statistics

The simplest way to illustrate that long-term forward rates are indeed downwardsloping is to look at the average difference or spread, between rates for two matu-rities. Table 5 gives summary statistics for estimates of the mean spread betweenforward rates at long maturities for the US Treasury market (for all three datasources), the UK nominal gilt market and the UK index-linked gilt market (onedata source each). Each data set covers some part of the period between January1966 and November 1998. Depending on the data source, these rates are either

16Presently there are six such gilts which are often erroneously referred to as “irredeemable”.They are, in fact, all redeemable at the governments option and a sinking fund operates in onestock: Conversion 3.5%.

15

2-year forward rates (e.g., between 24 and 26 years) or estimated instantaneousforward rates.17 If forward rates are downward sloping the spread will be negative.

Results are shown for eight sub-periods, each five years in length except forthe first and last periods. From January 1986 to December 1994 two estimatesare available for the US market: the FNZ estimates and those from STRIP prices.

In panel (a) of table 3 callable bonds are excluded from all the estimates.These data should be the more reliable. McCulloch and Kwon’s earlier estimates(up to August 1985) in panel (b) include callable Treasuries

The results are striking: thirteen of the fifteen five-year mean spreads arenegative. Arguably the most reliable data are those taken directly from STRIPprices: here the means are negative for each of the three sub-periods. The meanspreads are also negative in the case of the FNZ data for the US Treasury market;indeed here every single daily estimate of the spread in the sample 1673 days isnegative. The downward tilt is not only persistent but it is also of a significant size.For the FNZ and STRIP estimates, the average downward tilt in each sub-periodis between 100 and 200 basis points.

Figures 1a to 1e show the time series of forward rates spreads, for all five datasources, used in deriving table 3.18

For the McCulloch & Kwon data (Figure 1a) the estimated spreads are some-times positive in the early part of the period. However, as we have pointed outearlier, the estimates prior to August 1985 are potentially biased because theyare based in part on the prices of callable bonds and the option premium compo-nent of the price is ignored. After August 1985 the estimates use only non-callablebonds and here all the spreads are negative. Breaks in the series occur when therewere no long bonds in issue and no estimates of long term yields are provided byMK.

Figure 1b shows the corresponding spread for the US Treasury market esti-mated by Fisher, Nychka and Zervos. Here, as reported above, the spread isnegative on each of the 1673 days in the sample.

All our estimates of forward rate spreads, apart from one, are derived fromterm structure estimates obtained using splines, or some other form of curve fittingtechnique. It is therefore possible that the behaviour of the forward rates in these

17Apart from the case of UK nominal gilts where, because few long term bonds were in issueover the period, we compute the difference between the forward rates at 20 years and at 10years.

18As mentioned earlier, in the case of UK conventional gilts (Figure 1e) the spread shownis the 20-year minus the 10-year forward rate. For all other data sources the figures show thespread between forward rates at 25 years and 15 years.

16

cases is, to a greater or lesser extent, dependent on the estimation methodologyused. After all, long term discount factors will usually be the most difficult toestimate — because they depend on the prices of a relatively small number of thelongest bonds in the sample — and the forward rate spread then depends on theratio of these discount factors. The forward rate spreads we obtain from STRIPprices are, therefore, particularly significant because these are obtained directlyfrom prices. Figure 1c shows all the daily estimates of the spread obtained fromSTRIP prices. The figure shows that they are negative almost all the time (only37 out of 2363 observations, or 1.2%, are positive) and that the size of the spreadis quite substantial, around -100 basis points to -200 basis points for much of thetime.

Figure 1d shows the 20-year minus 10-year forward rate spread for the UKconventional gilts over the period April 1994 to December 1998. Of the 243weekly observations, only 8 are positive and the largest of these is 14 basis points.

Finally, figure 1e gives estimates of the spread for the UK index-linked market.Here, the spread is often positive for the first few months of 1984. However, fromthe beginning of 1985 the estimated spread is also mainly negative. The spread isnegative for every week between November 1984 and October 1992 and, betweenOctober 1992 and July 1998 fluctuates around zero.

Table 3 and Figures 1a-1e use forward rates at two particular maturities, inmost cases at 15 and 25 years. This raises the question of whether this choice ofmaturities is “special” in some sense: perhaps the term structure of forward ratesat other long maturities behaves differently. In Figure 2 we show the average one-year forward rate curve, derived from STRIP prices, for maturities between 13years and 26. Each of the ten charts shows the term structure of average forwardrates for a calendar year and, while these are not always monotonic, the shape ofthe curve is, in each case, clearly downward sloping.

Overall, the evidence shows that the term structure of long term forward ratesis persistently downward sloping. This is true not only in the US market but alsoin the UK market for both conventional and index-linked bonds. The next sectionexamines whether, as we have suggested in Section 3, the “downward tilt” can beexplained in terms of interest rate volatility.

17

6. Forward Rate Spreads and Yield Volatility: Empirical

Analysis

6.1. Do Forward Rate Spreads Predict Interest Rate Volatility?

In this section we use equation (3.14) to investigate the relation between thevolatility of zero coupon rates and the spread in long term forward rates. Ouranalysis is restricted to the data on US Treasury STRIPs since these allow us toobserve forward rates directly; the other estimates (McCulloch and Kwon, Fisher,Nychka and Zervos and our own calculations for UK gilts) are all potentiallysubject to the influence of the estimation methodology.

Equation (3.14) predicts that the shape of the long term forward rate curve isquadratic in maturity and this suggests the following cross sectional regression toobtain an estimate of volatility from the shape of the forward rate curve:

f(t, τ j) = β0 + β1τ2j + εj τ j ≥ τ 0 (6.1)

where τ 0 is a “long maturity”, e.g., 15 years and εj is an “error” term capturingthe effects of the missing terms in equation (3.14) and also estimation error inthe forward rates. In equation (6.1) the predicted value of the coefficient β1 isnegative and equal to −1/2σ2� . In this section we give the results of estimatingthe implied zero coupon yield volatility, σ�, from equation (6.1) and then usingthis estimate as a predictor of time series volatility.

Estimates of β1 obtained from individual term structures are likely to be im-precise as a result of observation error in the forward rates and this will lead tobias in subsequent regressions of time series volatility on implied volatility. Thereare a number of approaches which could be pursued at this point but the onewe have chosen to adopt is simply to average equation (6.1) over periods of be-tween two to four months and then to regress average forward rates on squaredmaturity19.

Figure 3 shows the results of estimating equation (6.1) in this manner. Weuse estimates of forward rates for annual maturities of between 15 and 26 years,derived from daily STRIP prices using equation (4.1). We then form averages ofthe forward rates over three month intervals and estimate equation (6.1) using

19Because the regressors are the same for each cross section, we would obtain exactly the sameresults if the regressions were carried out on individual cross-sections of forward rates and theregression coefficients averaged over time.

18

OLS. The figure shows the time series of estimates of the volatility parameter, σ�,derived from the estimate of the slope coefficient in (6.1) using:

σ� =

√−2β1. (6.2)

By way of comparison, the figure also shows the time series of the volatility oftwo-day changes in the 20-year zero coupon rate, also derived from STRIPs data,for the same three month intervals.

The estimated value of β1 was negative in each of the 37 three month intervalsand the mean value of the implied volatility was 83 basis points compared withthe mean time series estimate for the 20 year zero-coupon rate of 93 basis points.

As with implied volatilities from option prices, the implied volatility derivedfrom the cross sectional regression (6.1) can be regarded as a predictor of timeseries volatility. Accordingly we have regressed the time series volatility for the kth

M -month interval on the implied volatility for the (k−1)st interval. The results areshown in Table 4 which also includes, for purposes of comparison, regressions ofperiod k time series volatility on contemporaneous (period k) implied volatility. Inthe table the first column gives M, the number of months data used in computingboth the time series volatility and average forward rates.

Panel (a) gives results for the entire sample period (April 1985 to October1994). The third row, for example, shows the results, for M = 3, of regress-ing period-k time series volatility on period-k implied volatility. The differencebetween the coefficient on implied volatility, 0.929, and unity is small and not sta-tistically significant and, although these observations are derived from the sameperiod in calendar time, it should be noted that the estimate of volatility derivedfrom cross sections of forward rates is independent of the temporal sequence ofthe data.

The regression reported in the fourth row of Table 4 examines the predictivepower of volatility estimates derived from forward rates for future volatility. Herethe time series volatility for period k is regressed on the implied volatility forperiod k − 1. The coefficient on implied volatility is lower (0.825) but still lessthan one standard error from unity20.

The remaining regressions in panel (a) examine the effect of averaging theforward rates over shorter (2 months) and longer (4 month) periods. Overall thisdoes not make a substantial difference to the coefficients (although the problems

20The table shows Newey-West corrected standard errors with one lag.

19

of serial correlation appear somewhat less severe with a four month estimationperiod).

Close examination of the data shows that (a) the STRIP data appear much“noisier” in the early part of the sample (possibly as a result of limited liquidityin the earlier years of the market) and (b) that the period including the crashgives rise to a large outlier in many of the regressions. We were concerned thatthe results in panel (a) might have been affected by these factors and we havetherefore re-estimated the results presented in panel (a) excluding the data upto and including the crash. However, the results, shown in panel (b), are littlechanged. For the three regressions on lagged implied volatility, the coefficient iscloser to unity forM = 2 months and a little further away forM = 3 months andM = 4 months.

6.2. Does Interest Rate Volatility Explain the Forward Rate Spread?

The previous section looked at the question of whether the shape of the termstructure - implied volatility derived from forward rates - predicts the volatilityof long term zero coupon yields. Here we reverse the question and ask whether aprediction of volatility can explain the shape of the forward curve.

It would be possible to use a time series estimate of volatility (e.g., a con-ventional standard deviation or, perhaps an ARCH/GARCH, estimate) as ourforecast. But it seems preferable to use a forward looking estimate which reflectsagents’ expectations and we have therefore used the implied volatility on bond fu-tures options. The implied volatilities are those on a near-maturity at-the-moneyoption. As the time-to-maturity becomes short the option for which the impliedvolatility is calculated switches to the next shortest option.

The implied standard deviation (ISD) is computed using a Black-Scholes-likeformula and, ignoring features such as delivery options and so forth, may beregarded as the anticipated volatility of proportional price changes of the cheapest-to-deliver (CTD) bond. This is to be contrasted with the arithmetic yield changesrequired by equation (3.14). If the CTD bond underlying the futures contract hasmodified duration D, then we may compute the volatility of arithmetic changesin the yield-to-maturity of the CTD bond as:

σy =1

Dσp. (6.3)

In fact our data do not allow us to determine the modified duration of the CTD

20

bond and we therefore assume that this is constant; in other words, we simplyuse the futures options ISD as an “index” of anticipated bond market volatility21.We proceed in two stages. First, we regress time-series volatility in period k onfutures options ISD in period k− 1 to derive a forecasting rule for yield volatility.Next, we use this rule to generate a time series of forecasts of yield volatility and,using equation (3.14), the expected spread between forward rates at 25 years andforward rates at 15 years. Finally, we regress the actual average forward ratespread for period k on the period k-1 forecasts of the spread.

Table 5 gives the estimated prediction rule for yield volatility: the optionISD is always significant and, overall, explains around 15 to 20 percent of thevariability in next period’s volatility. When we regress time series volatility onoption ISD for the same period (not reported), the R-bar squared is almost 70%.Thus it appears that option ISD is strongly related to realised volatility but thatwith a forecast horizon of three months much of the variability of volatility isunpredictable.

In Table 6 we regress the period k average forward rate spread between 25 and15 years on the expected spread. The latter variable is computed as follows. Weuse the forecast for volatility given by the second row of Table 5, i.e., we setM tothree months and, using the average ISD for the last ten days of period k− 1, wecompute the expectedvolatility of the 20-year zero coupon rate for period k. Wethen use equation (3.14) to compute the expected forward rate spread for periodk. In the Table 6 the coefficient on the expected spread is 0.506 with a standarderror of 0.14 and, although over two standard errors below the expected value ofone, nonetheless much different from zero. The R-bar squared in this case is 0.25.When a dummy for the crash is included, the coefficient moves closer to unity(0.68) and the R-bar squared rises marginally.

7. Conclusion

It has long been known that, in models which (a) admit uncertainty in futureinterest rates and (b) are consistent with the no-arbitrage condition, the termstructure of yields depends on the anticipated level of volatility even in the absenceof risk premia.22 After all, one has only to look at the pricing formulae in modelssuch as Vasicek (1977) and Cox, Ingersoll and Ross (1985). What has, perhaps not

21Even if we did know the duration of the CTD bond, we could not safely assume that thevolatility of its yield to maturity is equal to the volatility of long term zero coupon yields.

22Or, equivaliently, under risk-neutral dynamics.

21

been properly appreciated before, is that (a) the effects of interest rate volatilityare to be found most strongly in the yields on long term bonds, (b) these effectsresult in a strong regularity in the term structure of forward rates, the “downwardtilt” and (c) that estimates of implied zero coupon yield volatility can be easilyobtained from the size of the tilt.

In this paper we have first attempted to document the persistence of thedownward tilt in long term forward rates. A number of data sources on both USand UK bond markets show that forward spreads are habitually negative.

Using a simple two-factor Gaussian model we have derived a simple relationbetween the difference in long term forward rates at two maturities (the “forwardrate spread”). Finally, we have tested this theory by looking both at the abilityof forward rate spreads to predict interest rate volatility and at the extent towhich the forward rate may be predicted using a measure of agents’ expectationsof volatility. Both tests broadly support the theory.

22

References

1. Barone E., Cuoco D. & Zautzik E. (1991). “Term structure estimation usingthe Cox, Ingersoll and Ross model: The case of Italian Treasury bonds”,Journal of Fixed Income 1, pp.87-95.

2. Bootle R. (1991). Index-linked Gilts: a practical investment guide, secondedition, (Woodhead-Faulkner, Cambridge, UK).

3. Brown R.H. & Schaefer S.M. (1994a). “The term structure of real interestrates and the Cox, Ingersoll and Ross model”, Journal of Financial Eco-nomics, Vol. 35, pp. 3-42.

4. Brown R.H. & Schaefer S.M. (1994b). “Interest rate volatility and the shapeof the term structure”, Philosophical Transactions of the Royal Society A,Vol. 347, pp. 563-576.

5. Brown S.J. & Dybvig P.H. (1986). “The empirical implications of the Cox,Ingersoll, Ross theory of the term structure of interest rates”, Journal ofFinance, Vol. 41, pp. 617-630.

6. Caverhill, Andrew (1998), “Modelling Long Term Forward Rates via theKalman Filter”, Working Paper, Hong Kong University of Science and Tech-nology.

7. Cox J.C., Ingersoll J.E. & Ross S.A. (1985). “A theory of the term structureof interest rates”, Econometrica, Vol. 53, pp. 385-407.

8. Dia, Q. and Singleton K., (1998).“Specification Analysis of Affine TermStructure Models”, Working Paper, Graduate School of Business, StanfordUniversity.

9. Deacon, M. & Derry A., (1994). “Deriving EStimates of Inflation Expecta-tions from the Prices of UK Government Bonds”. Bank of England workingpaper no. 23. (July)

10. Duffie D. & Kan R. (1996). “A yield-factor model of interest rates”, Math-

ematical Finance, Vol. 6, 379-406.

11. Fisher M.E., Nychka D. & Zervos D (1994). “Fitting the term structure ofinterest rates with smoothing splines”, unpublished working paper FederalReserve Board, January.

23

12. Langetieg, T. 1980. “A Multivariate Model of the Term Structure”, Journalof Finance, 35, 71-97.

13. Litterman R., Scheinkman J. & Weiss L. (1991) “Volatility and the yieldcurve”, Journal of Fixed Income, Vol. 1, No. 1, June, pp. 49-53.

14. Litterman R. & Scheinkman J. (1991). “Common factors affecting bondreturns”, Journal of Fixed Income, Vol. 1, No. 1, June, pp. 54-61.

15. Longstaff F.A. & Schwartz E.S. (1992). “Interest rate volatility and the termstructure: a two-factor general equilibrium model”, Journal of Finance, Vol.47, No. 4, pp. 1259-1282.

16. McCulloch J. Huston, (1975). “The tax adjusted yield curve” Journal of

Finance, Vol. 30, pp. 811-830.

17. McCulloch, J. Huston & Heon-Chul Kwon (1993). “U.S. term structuredata, 1947-1991”, unpublished working paper 93-6, Ohio State University.

18. Merton, R.C. (1970) “A Dynamic General Equilibrium Model of the As-set Market and its Application to the Pricing of the Capital Structure ofthe Firm”, Working Paper No. 497-70, A.P. Sloan School of Management,Massachusetts Institute of Technology, MA.

19. Nelson, C.R. and A.F. Siegel (1987), “Parsimonious Modeling of YieldCurves”, Journal of Business, Vol. 60, pp. 473-89.

20. Schaefer S. M. (1981). “Measuring a tax-specific term structure of inter-est rates in the market for British government securities”, The Economic

Journal, Vol. 91, pp. 415-438.

21. Vasicek O. (1977). “An equilibrium characterization of the term structure”,Journal of Financial Economics 5, pp. 177-188.

22. Vasicek, O.A. and H.G. Fong (1982), “Term Structure Modeling Using Ex-ponential Splines”, Journal of Finance, Vol. 37, pp. 339-48.

24

Appendices

A. Estimation of Mean Reversion Parameters

This appendix describes our method of estimating the mean reversion parame-ters κx and κy. Since ours is a two-factor linear model the covariance matrix ofzero-coupon yields has rank two. It is simple to show that the two eigenvectorsassociated with the covariance matrix of zero coupon yields are linearly relatedto the functions Bx(τ) and By(τ ). The latter depend only on the mean reversioncoefficients, the correlation coefficient and maturity and our method estimates themean reversion coefficients, and the correlation coefficient, from the eigenvectors.

We have carried out a principal components analysis on weekly changes inzero coupon rates, derived from the prices of US Treasury STRIPs, with annualmaturities between 2 years and 26 years using data for the period December1988 to October 1994. Table A1 shows the eigenvalues and the percentage oftotal variance explained by the first eight principal components. Around 86%of total variance is accounted for by the first principal component and 95.5% bythe first two. Thus the assumption of that prices are determined by two factorsseems reasonable and we use the first two eigenvectors to infer the mean reversioncoefficients κxand κy.

Using the notation used above, the vector of innovations in zero coupon yields,∆Rt, is related to the innovations in the state variables x and y by:

∆Rt =[Bx/τ By/τ

]( ∆x∆y

), (A.1)

where{B′

x(y)

}j= Bx(y)(τ j). Denoting the eigenvectors of the covariance matrix of

∆Rt as EV1 and EV2 we may also write ∆Rt in terms of the principal components∆p:

∆Rt =[EV1 EV2

]∆p. (A.2)

The state variables {x, y} and the principal components are linearly related (theprincipal components are simply the state variables of the same model rewrittenin terms of orthogonal state variables) and so, for some (2 by 2) weighting matrix,W , we may write:

25

∆p =W

(∆x∆y

). (A.3)

From equations (A.1), (A.2) and (A.3) it is straightforward to show that

[Bx/τ By/τ

]W−1 =

[EV1 EV2

]. (A.4)

For given maturities, the vectors Bx/τ and By/τ depend only on κx and κy.Thus, given estimates of EV1and EV2 we estimate κx, κy and ρxy, the imputedvalue of the correlation coefficient, along with the elements of W , by finding theleast squares fit between the directly estimated and fitted eigenvectors, i.e., theright-hand side and left hand side, respectively, of equation (A.4) subject to theconstraint that Corr(∆p1,∆p2) is zero

23.Overall we have found a good correspondence between the directly estimated

and fitted eigenvectors. Figure A1 shows the fitted and directly estimated eigen-vectors for the period December 1988 to October 199424.

Table A2 gives the estimates of κxand κy and ρxy for the same period andfor two sub-periods. For the whole period (1988-94) the estimates for κy are0.23 using the unadjusted covariance matrix and 0.21 with an adjusted covariancematrix. The corresponding estimates κx are ten times smaller in the first case and5 times smaller in the second case (0.021 and 0.039). These values are consistentwith a relatively flat term structure of volatility for long maturities.

23In fact we have found that it was easier to find the global minimum of the sum of squares byrewriting the model in terms of orthogonal state variables. If the orthogonalised state variablesare x and y′ (only one state variable need change) then the eigenvectors are simply Bx/τ andBy′/τ , although the latter is now a more complicated expression than before.

24The covariance matrix from which the eigenvectors were computed was estimated using theobservation error adjustment method described in Appendix B.

26

B. Adjusting for Observation Error in the Esti-

mation of Variances and Covariances of Zero

Coupon Yields

In estimating both the term structure of volatilities and the covariance matrix ofyield changes, we have found that the results are potentially affected by observa-tion error in the yields. We have attempted to correct for this in the followingway. Let R∗

t denote the n-vector of observed zero-coupon yields at time t andassume that this differs from the true value Rt by a zero mean n-dimensional iidvariate vt:

R∗

t = Rt + vt. (B.1)

Further assume that Rt follows a random walk and that the increment betweentime t and t+ 1 is another n-dimensional iid variate ut+1:

Rt+1 = Rt + ut+1. (B.2)

From equations (B.1) and (B.2) it is simple to show that the covariance matrix ofchanges in observed yields, Σ∗

R, is related to the covariance matrix of true yields,ΣR, by:

Σ∗

R = ΣR + 2Σv , (B.3)

where Σv is the covariance matrix of the observation errors v. The first orderserial covariance in ∆R∗

t , changes in the vector of observed yields in R∗

t , is givenby −Σv. We therefore estimate the covariance matrix of true changes as:

Σ∗

R + 2Cov(∆R∗

t+1,∆R∗

t ). (B.4)

27

28

Table 1

Volatility of Changes in Long Term Zero Coupon Rates

Standard Deviation of weekly changes (% p.a.) in zero coupon yields derived from US Treasury STRIP prices.1

Maturity (years)

Period 2 3 5 7 10 15 20 25 12/88 - 10/94 adjusted1 0.991 1.030 0.994 0.904 0.857 0.733 0.686 0.684

11/88 - 10/94 unadjusted 1.043 1.065 1.045 0.999 0.971 0.878 0.837 0.825 12/88 - 12/91 adjusted 1.002 1.001 0.958 0.871 0.845 0.719 0.694 0.722 12/91 - 10/94 adjusted 0.927 1.015 1.012 0.932 0.878 0.755 0.684 0.649

1 The standard deviations and correlations are adjusted for observation error as described in Appendix B.

29

Table 2

Summary of Data

Estimate Estimation Method Type of Bond Frequency

Period Covered Comments

US Treasury Markets

McCulloch & Kwon Spline approximation to discount function

coupon bonds monthly 12/1947 - 2/1991 Period before 8/1986 includes callable bonds

Fisher, Nychka & Zervos

Spline approximation to forward rate curve

coupon bonds daily 12/1987 - 8/1994 ---

STRIPS Direct calculation from STRIP prices

Zero coupon STRIPs

daily 4/1985 - 10/1994 ---

UK Government Bonds (“GILTS”

Authors’ own calculations

Spline approximation of discount function

Fixed nominal coupon bonds

monthly 4/1994 - 11/1998

Authors’ own calculations

Bernstein polynomial approximation of discount function

Inflation linked (“Index-Linked”) coupon bonds

weekly 1/1984- 11/1998 ---

30

Table 3

Summary Statistics on Forward Rate Spreads: 25-year Forward minus 15-year Forward

US Treasury Bonds UK Gilts a UK Index-Linkedb

McCulloch & Kwon

Fisher, Nychka & Zervos (FNZ)

STRIP prices

Mean St Dev Mean St Dev Mean St Dev Mean St Dev Mean St Dev Panel (a): Data Excludes Callable Bonds

Apr 1994 Nov 1998 -0.54 0.34 .03 .16

Jan 1991 Dec 1994 -1.74* .59 -1.16 .43 -.18 .32

Jan 1986 Dec 1990 -3.00 1.59 -2.02* .73 -1.52 .75 -.89* .29

Jan 1981 Dec 1985 -1.55 1.14 -.20 .33

Panel (b): Data Includes Callable Bonds

Jan 1981 Dec 1985 -3.63 2.01

Jan 1976 Dec 1980 -1.25 1.26

Jan 1971 Dec 1975 .46 1.72

Jan 1966 Dec 1970 -.37 1.00

Notes: a In the case of UK Nominal Gilts the spread is between the forward rates at 20 and 10 years. b The data on UK Index-Linked Gilts start in January 19984. * Indicates that 100% of the observations on which the mean is based are negative.

31

Table 4

Implied Volatility from Forward Rates as Predictors of Time Series Volatility

The table shows estimates of the following regression:

TSVOL R ISD ISDk M k M k M k( ) ,, , ,20 0 1 2 1= + + +−α α α ε

where TSVOL(R20)k,M is the annualised time series volatility of 2-day changes in the 20-year zero coupon rate during the kth sub-period of length M months. ISDk,M is the implied interest rate volatility (from regression equation (6.1)) for the kth sub-period of length M months. Newey-West corrected standard errors are in parentheses. The data are derived from the prices of US STRIPs for the period April 1985 to October 1994.

M Constant ISDk,M ISDk-1,M R-bar2 DW Num Obs

Panel (a) April 1985 - October 1994

2 0.003 0.770 0.119 1.270 56 (0.003) (0.325) 2 0.004 0.646 0.084 1.311 55 (0.003) (0.374) 3 0.002 0.929 -- 0.177 1.462 37 (0.003) (0.349) 3 0.002 -- 0.825 0.149 1.346 36 (0.004) (0.425) 4 0.000 1.088 0.225 1.716 28 (0.003) (0.369) 4 0.002 0.882 0.148 1.731 27 (0.005) (0.513)

Panel (b) “Post Crash” (to October 1994)2

2 0.002 0.756 0.178 1.297 41 (0.002) (0.253) 2 0.002 0.733 0.193 1.535 40 (0.001) (0.168) 3 0.001 0.890 0.213 1.307 27 (0.002) (0.273) 3 0.002 0.711 0.157 1.387 26 (0.001) (0.187) 4 0.001 0.914 0.170 1.832 20 (0.002) (0.273) 4 -0.002 1.227 0.359 1.818 19 (0.002) (0.294)

2 In Panel (b), for each value of M, the first observation used is the first M-month interval that does not include observations for October 1987.

32

Table 5

Using the Implied Volatility on Options on Bond Futures to Construct a Predictor of Zero Coupon Yield Volatility

The table shows estimates of the following regression:

TSVOL R Option ISDk M k M k( ) ,, ,20 0 1 1= + +−α α ε

where TSVOL(R20)k , M is the annualised time series volatility of 2-day changes in the 20-year zero coupon rate during the kth sub-period of length M months, Option ISD k-1, M is the average implied volatility taken from the last 10-days in the previous M-month interval. Newey-West corrected standard errors are in parentheses. The zero coupon yield data are derived from the prices of US Treasury STRIPs and the implied standard deviations from at-the-money call options on the front US Treasury Long Bond futures contract, for the period April 1985 to October 1994.

M Constant Option ISDk-1 R-bar sq. DW Num obs. 2 0.003 0.052 0.195 1.801 55

(0.001) (0.009)

3 0.004 0.042 0.169 1.839 36

(0.001) (0.009)

4 0.004 0.051 0.159 2.221 27

(0.001) (0.009)

33

Table 6

Predicting the Average Forward Rate Spread Between 25 and 15 Years using a Forecast of Interest Rate Volatility Based on Bond

Futures Options Implied Volatility The table shows an estimate of the following regression:

Av Spread E Spread k k= + +−α α ε0 1 1 ,

where AV Spread is the average daily spread in interval k of length 3 months between the forward rate at 25 years and the forward rate at 15 years and E Spreadk-1 is a measure of the expected spread computed at the end of the k-1st interval using (i) the results of the regression in Table 5 for M=3 months, (ii) the average ISD from bond futures options over the last 10-days in period k-1 and (iii) equation (3.13). Newey-West corrected standard errors are in parentheses. The data are derived from the prices of US Treasury STRIPs for the period April 1985 to October 1994.

Constant E-Spreadk-1 R-bar sq. DW

-0.0049 0.506 0.253 1.309

(0.0022) (0.128)

Figure 1a

Figure 1b

Spread Between 25 Year and 15 Year Forward Rates, Dec 1987 to July 1994. Source: Fisher, Nychka nd Zervos

-5.00

-4.50

-4.00

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

De

c-8

7

Aug

-88

May

-89

Jan-9

0

Oct

-90

Jul-9

1

Ma

r-9

2

De

c-9

2

Sep

-93

Jun-9

4

rate

(p

.a.)

Spread Between 25-year and 15-year Forward Rates in US Treasury Market (Source: McCulloch & Kwon)

-12

-10

-8

-6

-4

-2

0

2

4

6

Jan

-66

Jan

-69

Jan

-72

Jan

-75

Jan

-78

Jan

-81

Jan

-84

Jan

-87

Jan

-90

spre

ad

% p

.a.

Figure 1c

Spread Between Average Forward Rates for 24 to 26 year and 14 to 16 years. STRIP's Data

-5%

-4%

-3%

-2%

-1%

0%

1%

2%1

98

50

42

2

19

86

04

22

19

87

04

22

19

88

04

21

19

89

04

21

19

90

04

20

19

91

04

19

19

92

04

20

19

93

04

22

19

94

04

26

rate

(%

p.a

.)

Figure 1d

Forward Rate Spreads in the UK Nominal Gilt Market: Spread betweenthe 6-month forward rates at 20 years and at 10 years.

-1.75

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

Apr-94

Aug-94

Jan-95

May-95

Oct-95

Mar-96

Jul-96

Dec-96

Apr-97

Sep-97

Feb-98

Jun-98

Nov-98

rate

(%

p.a

.)

Figure 1e

Spread between 25 Year and 15 Year Forward Rates, January 1984 to November 1998, UK Goverment Index-Linked Bonds

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Jan-

84

Jul-8

5

Dec

-86

Jun-

88

Dec

-89

Jun-

91

Dec

-92

Jun-

94

Dec

-95

Jun-

97

rate

(%

p.a

.)

Figure 2

Average Forward Rates from STRIP Prices

The figures show the average one-year forward rate computed from STRIP prices for annual maturitiesbetween 13 and 26 years . The data are weekly and the averages are shown for each calendar year. Notethat the vertical scale is the same (300 basis points) in each case.

1985 1986

1987 1988

1989 1990

9

10

11

12

12 14 16 18 20 22 24 26 28

maturity

ave

rage

fo

rwar

d r

ate

(% p

.a.)

7

8

9

10

12 14 16 18 20 22 24 26 28

maturity

ave

rag

e f

orw

ard

rat

e (

% p

.a.)

7

8

9

10

12 14 16 18 20 22 24 26 28

maturity

aver

age

forw

ard

rate

(% p

.a.)

7

8

9

10

12 14 16 18 20 22 24 26 28

maturity

ave

rag

e f

orw

ard

rat

e (

% p

.a.)

6

7

8

9

12 14 16 18 20 22 24 26 28

maturity

ave

rage

forw

ard

rat

e (%

p.a

.)

7

8

9

10

12 14 16 18 20 22 24 26 28

maturity

ave

rag

e f

orw

ard

rat

e (

% p

.a.)

Figure 2, continued

1991 1992

1993 1994

7

8

9

10

12 14 16 18 20 22 24 26 28

maturity

aver

age

forw

ard

rat

e (%

p.a

.)

7

8

9

10

12 14 16 18 20 22 24 26 28

maturityav

erag

e fo

rwar

d ra

te (%

p.a

.)

6

7

8

9

12 14 16 18 20 22 24 26 28

maturity

aver

age

forw

ard

rat

e (%

p.a

.)

6

7

8

9

12 14 16 18 20 22 24 26 28

maturity

ave

rag

e f

orw

ard

rat

e (

% p

.a.)

Figure 3

Estimates of Interest Rate Volatility from Cross Sectional Regressions of ForwardRates on squared maturity. Each regression uses average forward rates over a threemonth period and includes annual maturities from 15 to 26 years. The graph alsoshows the contemporaneous time series volatility of two-day changes in the 20-yearzero-coupon rate. All interest rate data are derived from daily STRIP prices over theperiod April 1985 to October 1994.

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Jul-8

5

Apr

-86

Jan-

87

Oct

-87

Jul-8

8

Apr

-89

Jan-

90

Oct

-90

Jul-9

1

Apr

-92

Jan-

93

Oct

-93

Jul-9

4

Vo

lati

lity,

% p

.a.

Implied volatility from cross-sectional regressions

Time series volatility of 20-year zero-coupon rate

Table A1

Eigenvalues and Percentage of Variance Explained

The table shows the first eight eigenvalues of the estimated covariance matrix ofweekly changes in zero coupon rates with annual maturities of 2 to 26 yearsderived from the prices of US Treasury STRIPS over the period December 1988to October 1994. The covariance matrix is adjusted for observation error in theyields using the procedure described in the Appendix.

1st PC 2nd PC 3rd PC 4th PC 5th PC 6th PC 7th PC 8th PC

Eigenvalues0.275 0.031 0.0058 0.0024 0.0018 0.0009 0.0006 0.0006

Fraction ofVarianceExplained

85.8% 9.7% 1.8% 0.7% 0.6% 0.3% 0.2% 0.2%

CumulativeFraction ofVarianceExplained

85.8% 95.5% 97.3% 98.1% 98.6% 98.9% 99.1% 99.3%

Table A2

Estimated Mean Reversion Coefficients and Correlation Coefficient

The table shows estimates of the mean reversionparameters κx and κy and the correlation coefficient ρxy

estimated from the eigenvector of the covariance matrixof zero coupon yields calculated from the prices of USTreasury STRIPs. For details see the Appendix.

Period Observation ErrorCorrection1

κx κy ρxy

88-94 Unadjusted 0.0212 0.232 -0.25488-94 Adjusted 0.0393 0.206 -0.336

88-91 Unadjusted 0.0089 0.354 -0.10388-91 Adjusted 0.0328 0.334 -0.163

91-94 Unadjusted 0.0386 0.163 -0.40291-94 Adjusted 0.0417 0.126 -0.556

1Note: The term “adjusted” refers to a correction for measurement error in computingthe covariance matrix from which the eigenvectors are calculated.Unadjusted” estimates do not reflect this correction. See Appendix A.

Fig A1

Fitted and Estimated Eigenvectors from CovarianceMatrix of Changes in Zero-Coupon Yields

The figure shows (a) the “estimated eigenvectors” calculatedfrom the estimated covariance matrix of zero-coupon yieldsand (b) the fitted eigenvectors computed using the meanreversion coefficients estimated from the “estimatedeigenvectors”. The data used are weekly STRIP prices fromDecember 1988 to October 1994. The estimated covariancematrix is adjusted for observation error in the STRIP yieldsusing the method described in the Appendix.

File: Tables and Fig for AppendixDate: 09/11/99

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 5 10 15 20 25 30

Fitted First EV

Fitted Second EV

Directly Estimated First EV

Directly Estimated Second EV