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

Why Long Term Forward Interest Rates (Almost) Always Slope Downwards Roger H. Brown Warburg Dillon Read Stephen M. Schaefer London Business School, May 1, 2000 Corresponding author: Stephen M. Schaefer, London Business School, Sussex Place, Regents Park, London NW1 4SA, UK. e-mail: sschaefer@lbs.ac.uk

Abstract The paper documents a persistent and thus far largely overlooked empirical regularity 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 downward 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 Downwards 1 1. Introduction The term structure of long term forward interest rates is persistently downward sloping. In nearly ten years of daily data on US Treasury STRIPs from 1985 to 1994, the implied two year forward rate spanning years 24 to 26 is lower than the forward rate for years 14 to 16 on 98.4% of occasions. The average difference in these rates is 138 basis points. A similar downward tilt also appears in estimates of forward rates derived from the prices of coupon bonds in the US Treasury market and in the UK market for both real and nominal government bonds 2. We show that this feature of the term structure is not anomalous; nor do we need to turn to institutional features of the market for an explanation. Rather, it arises as a consequence of the combined effect of the term structure of volatility of long term interest rates and the greater convexity of long term bonds. Explaining the relation between the yields on default-free bonds and their maturity, the term structure, is a classical problem in financial economics and the subject of extensive research over many years. The shape of the yield curve varies substantially over time: sometimes upward sloping, sometimes downward and 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, irrespective of the pattern of rates for shorter maturities. It may well be the single most predictable feature of the term structure of interest rates. It is also interesting to note that, even though term structure theory has focussed mainly on expectations of future rates and risk premia to explain the shape, we find that this particular highly persistent feature is the result of interest rate volatility alone. Most term structure theories (e.g., Vasicek (1977), Cox, Ingersoll and Ross 1 The authors gratefully acknowledge the helpful comments and suggestions from Andrea Berardi, Mark Britten-Jones, Antii Ilmanen and Davide Menini, from seminar participants at the universities of Barcelona, Imperial College, Louvain, Lancaster, Toulouse and Verona and from participants at the Workshop on Mathematical Finance, Strobl, Austria 1999. The authors are also very grateful to Mark Fisher, Douglas Nychka and David Zervos for allowing us to use their estimates of zero coupon yields. The usual disclaimer applies. 2 The estimates for the US were obtained by Fisher M.E., Nychka D. & Zervos D (1994) and McCulloch and Kwon (1993). Those for the UK are the authors own calculations. 3

(1985) and Longstaff and Schwartz (1992)) predict that the term structure of long term forward rates is flat. Indeed, ever since Durand (1942) constructed his famous series of yields by drawing a curve (freehand) through the lower envelope of yields-to-maturity on high grade corporate bonds, the available data has seemed to suggest that actual yield curves are also flat for long maturities 3.Inthecaseof yields-to-maturity on coupon bearing bonds this is merely a matter of arithmetic since the yields on all such bonds approach the yield on a perpetuity as maturity becomes infinite. However, the shape of the zero-coupon yield curve (or the forward rate curve) is not constrained in this way, apart from the no-arbitrage condition that (nominal) forward rates are non-negative. What our results show is that under empirically relevant conditions - essentially, when long term zero coupon rates experience parallel shifts - the term structure of long term forward rates is always downward sloping. The consistent downward slope in long term forward rates has a number of practical implications. First, our results have implications for attempts to extract market expectations from the yield curve 4. Abstracting from considerations of risk premia, classical term structure theory identifies the market s expectation of a future interest rate as the forward rate. In models which make use of contingent claims analysis (i.e., Merton (1970), Vasicek (1977) and followers) forward rates and the expected path of the short rate will diverge even when risk premia are zero. The difference is accounted for by convexity effects; our results show that, for long maturities, these effects can be substantial. Second, some approaches which have been proposed for estimating the term structure of zero coupon rates from the prices of coupon bonds use approximating functions which constrain the long term forward rate curve to be flat 5. Our results suggest that such methods may be mis-specified. Third, the relation which we derive between interest rate volatility and the shape of long term forward rates may be useful to practitioners facing the problem of pricing a new issue which has a longer maturity than any existing 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 forward 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 3 Durand drew his curves by choosing one of five shapes, all of which were flat for long maturities. 4 See Deacon and Derry (1994). 5 For example, Vasicek and Fong (1982) and Nelson and Siegel (1987). 4

models to estimate the level of interest rate volatility implied by the term structure. 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 time series estimates, the volatility parameter in term structure studies is estimated jointly with the other parameters of the model and it is correspondingly difficult to identify the influence on the shape of the term structure alone. Litterman, Scheinkman & Weiss (1991) (LSW) adopt a simpler approach and estimate the relation between the implied volatility of options on US Treasury bond 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 explain around 70% of the variation in implied volatility. The present study can be regarded as being in the spirit of LSW, namely to identify some easily observable feature of the yield curve which reflects the impact of interest rate volatility but todosointhecontextofamodel.thuswederiveasimplerelationbetweenthe term structure of interest rate volatility and the term structure of forward interest rates. Our relation is also easily inverted: in other words it is straightforward to obtain an estimate of implied interest rate volatility from the pattern of long term forward rates. Caverhill (1998) has also recently investigated the relation between the forward curve and interest rate volatility. The paper is organised as follows. Section 2 uses a highly simplified model of the yield curve to provide an intuitive explanation of the relation between forward rates and the volatility of zero coupon yields. The main prediction of the model is that the spread between longer and shorter term forward rates should be negative and proportional to the variance of long term zero coupon rates. Section 3 then shows that the same relation may be derived as an approximation within a more realistic two-factor affine model. Section 4 describes our data and Section 5 presents 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 the early 1980 s - are strongly and persistently negative. Section 6 contains our main empirical results. Here we show that (a) the shape of the term structure of forward rates predicts the volatility of zero coupon yields and (b) that the forward rate spread is itself predictable using the implied volatility of bond futures options. Section 7 concludes. 5

2. The Informal Argument The term structure of long term forward rates depends on two key features of the empirical behaviour of the yield curve: the tendency of long term zero coupon rates (a) to be highly correlated and (b) to have volatilities which decline relatively slowly with maturity. 6 To understand this simple relation between forward rates and interest rate volatility, consider a zero coupon bond with time-to-maturity τ and yield at time t of y(t, τ). We assume that the yields on bonds with sufficiently long maturities are perfectly correlated with a common variance σ 2 y whichisconstantovertime and over (long) maturities. The price of such a bond is given by: and its rate of return from t to t + t by: P (y(t, τ)) = e y(t,τ)τ, (2.1) P P = 1 P [ 1 2 P 2 y 2 y2 + P ] y y + 1 P t. (2.2) P t where y denotes the change in yield from t to t + t. If the process generating y is time homogeneous it is straightforward to show that f(t, τ, y), the time t instantaneous forward rate for maturity τ, is given by: Equation (2.1) implies: f(t, τ, y) = P t P. (2.3) 1 P P y = τ and 1 2 P P y = τ 2, (2.4) 2 and, substituting into equation (2.2) and taking expectations (assuming that, for small t, E( y 2 ) σ 2 y t) the risk premium on the bond becomes: 6 Section 5 presents evidence on the term structure of volatility for long term zero coupon rates. 6

[ ] [ ] P 1 E r t t = P 2 σ2 yτ 2 τµ y + f(t, τ, y) r t t, (2.5) where r t is the short term (riskless) interest rate and E( y) µ y t. However, no arbitrage considerations imply that the risk premium on the bond is also proportional to its elasticity with respect to y, i.e.: [ ] P E r t t = τλ y t, (2.6) P where λ y is a risk premium parameter 7. Equating equations (2.5) and (2.6) we have that the forward rate is given by: f(t, τ, y) =r t + τ ( µ y λ y ) 1 2 σ2 y τ 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 given reference yield y, 1 P P y = τ and 1 P 2 P y 2 = τ 2 hold for all long maturities, then equation (2.7) shows that the shape of the long term forward rate curve depends on two terms. The first is linear in maturity and, depending on the sign of ( µ y λ y ), may be increasing or decreasing with τ. The second, however, is (a) always decreasing in maturity and (b) because it is proportional to τ 2 rather than τ, always dominates the first term for sufficiently large τ. Thus, when the term structure of yield volatility is constant for long maturities the term structure of forward rates is always downward sloping and, for long maturities neglecting the linear term we have that the difference in forward rates between two maturities τ 1 and τ 2, the forward rate spread, is given approximately by: f(t, τ 1,τ 2 ) f(t, τ 2,y) f(t, τ 1,y) (2.8) 1 [ ] 2 σ2 l τ 2 2 τ 2 1. 7 This 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 standard single factor model, e.g., Vasicek (1977) is the (implicit) assumption of zero mean reversion which supports the assumption of a flat term structure of volatility for long term rates. With non-zero mean reversion the yield volatility σ y declines with maturity and consequently equation (2.7) would not describe the term structure of long term forward rates. It is important to note, however, that in this case the yields on long bonds converge to a constant. In the next section we derive an expression for the forward rate spread using a less restrictive, i.e., more realistic, two-factor model and show that, even in the presence of non-zero mean reversion in one factor, equation (2.8) still holds as an approximation to the forward rate spread. 3. A Model of Forward Rate Spreads We consider a two factor time homogeneous affine Gaussian model of the term structure [Langetieg(1980), Brown & Schaefer (1994b), Duffie and Kan (1996)] where the state variables x and y are assumed to follow: dx = κ x (µ x x) dt + σ x dz x, (3.1) ( dy = κ y µy y ) dt + σ y dz y, where κ x, µ x, σ x, κ y, µ y and σ y are parameters and dz x and dz y are increments of standard Brownian motions with E(dz x dz y )=ρdt. Within the class of twofactor time-homogeneous Gaussian affine models Dai and Singleton (1998) show that the dynamics given in equation (3.1) are completely general. The Gaussian specification means that the probability of the short rate becoming negative in any finite interval is non-zero. Despite this the model is adequate for our purposes and, in particular, fits the term structure of yield volatility well. Since the model is affine (i.e., both the drift and diffusion coefficients are affine in x and y) thepriceofazerocouponbondwithmaturityτ is exponential affine and given by [Duffie and Kan (1996)]: P (x, y, t, τ) =exp {A(τ)+B x (τ)x(t)+b y (τ)y(t)} (3.2) where, because the model is time homogeneous, the functions A(.), B x (.) and B y (.) are functions of the parameters and time-to-maturity, τ, alone. The formulae for 8

B x (.) and B y (.) are given below; the formula for A(.) is also well know (Langetieg (1980), for example) but is not required here. The corresponding zero coupon yield, R(.), isgivenby: R(x, y, t, τ) 1 ln [P (x, y, t, τ)] (3.3) τ = A(τ) + B x(τ) x(t)+ B y(τ) y(t). (3.4) τ τ τ In equation (3.4) the functions B x (τ)/τ and B y (τ)/τ represent the sensitivities of the τ-period zero coupon yield to, respectively, x and y. For the Gaussian case giveninequation(3.1),thefunctionsb x (τ) and B y (τ) are given by: B x (τ) = 1 exp( κ xτ) κ x and B y (τ) = 1 exp( κ yτ) κ y. (3.5) We do not present detailed estimates of the parameters κ x and κ y in this paper since our result merely requires that one of the mean reversion coefficients κ x,say, is close to zero while the other is not. Appendix A describes the method we use to estimate the mean reversion parameters κ x and κ y from the covariance matrix of changes in zero coupon yields. Using data on US Treasury STRIPs for the period 1988-94, we find an estimated value for one of the mean reversion coefficients is close to zero (in the range 0.02 to 0.04) while the other is between five and ten times larger [Table A2] 8. We now turn to the implications of these mean reversion coefficients for the behaviour of long term forward rates. The valuation equation, to which equation (3.2) is the solution, is: κ x (µ x x) P x + 1 2 P 2 σ2 x x + κ y(µ 2 y y) P y + 1 2 P 2 σ2 y y + P 2 t + 2 P σ xy r(x, y)p =0. (3.6) x y 8 The US Treasury STRIP prices in the first part of our dataset, from 1985 to 1988, are noticeably noisier than in the second half. In estimating the mean reversion coefficients κ x and κ y wehavethereforeuseddatafromtheendof1998totheendofdatain1994. 9

Since the model is time homogeneous, the time t instantaneous forward rate for date t + τ is given, as before, by: f(x, y, t, τ) = 1 P Furthermore, from (3.2) we have that: P t. (3.7) 1 P P x = B x(τ), 1 P 1 P P y 2 P x 2 = B y (τ) and 1 P = B2 x (τ), 1 P 2 P y 2 = B2 y (τ). Substituting for P t, P x, P y etc. in (3.6) we have: 2 P x y = B x(τ)b y (τ) (3.8) f(x, y, t, τ) = r(x, y)+κ x (µ x x)b x (τ)+κ y (µ y y)b y (τ) 1 2 σ2 x B2 x 1 2 σ2 y B2 y ρσ xσ y B x B y. (3.9) Using (3.4) the variance of the τ-period zero coupon yield is: Var[R(x, y, t, τ)] = σ 2 x ( Bx (τ) τ ) 2 + σ 2 y +2ρσ x σ y ( Bx (τ)b y (τ) τ 2 and equation (3.9) for the forward rate therefore becomes 9 : ( By (τ) ) 2 τ ), (3.10) f(x, y, t, τ) = r(x, y)+κ x (µ x x)b x (τ)+κ y (µ y y)b y (τ) 1 2 Var[R(x, y, t, t + τ)] τ 2. (3.11) 9 Even though our analysis uses a Gaussian model, notice that an equation very similar to eq. (3.11) will hold for any affine model. The only difference is that, with n state variables there will be n terms which are linear in the drifts rather than two. The short rate and the term involving 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 taking the 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 ) = κ x (µ x x)[b x (τ 2 ) B x (τ 1 )] + κ y (µ y y)[b y (τ 2 ) B y (τ 1 )] 1 ( Var[R(x, y, t, t + τ 2 )] τ 2 2 2 Var[R(x, y, t, t + τ ) 1)] τ 2 1. (3.12) As mentioned earlier, our estimates of κ x and κ y indicate that κ x 0 and κ y 0 and we now show that this implies that the terms in (3.12) which are linear in B x (τ) and B y (τ) are close to zero. If κ x 0 then B x (τ) is approximately equal to τ and therefore finite and, since it is multiplied by κ x it follows that the entire term which is linear B x (τ) is also close to zero. Similarly, if κ y 0 then, since Lt B y(τ) = 1, τ κ y B y (τ 2 ) B y (τ 1 ) is close to zero for long maturities and, as before, the entire term which is linear B y (τ) is close to zero. Under these conditions, the forward rate spread for long maturities is dominated by the last term in equation (3.12), i.e., that: f(x, y, t, τ 1,τ 2 ) 1 2 ( ) Var[R(x, y, t, t + τ 2 )] τ 2 2 Var[R(x, y, t, t + τ 1 )] τ 2 1. (3.13) Table 1 shows the term structure of zero-coupon yield volatility for the data on US Treasury STRIPs used in Appendix A 10. The table shows that, for maturities beyond ten years, the volatility of long term yields attenuates only slowly. If, to a first approximation, we assume that zero coupon yields at maturities τ 1 and τ 2 have the same volatility (σ l )thentherighthandsideofequation(3.13)becomes further simplified to: f(x, y, t, τ 1,τ 2 ) 1 2 σ2 l 10 Once again, we describe our data in more detail below. [ ] τ 2 2 τ 2 1. (3.14) 11

This expression predicts that the shape of the long term forward rate curve will be quadratic and downward sloping and that the degree of concavity will be directly related to the volatility of long term zero-coupon yields. The critical condition for equation (3.14) to provide a good characterisation of the shape of the long term forward curve beyond the assumption of affine dynamics is that the term structure of volatility for long term zero-coupon yields is flat. How significant is the forward rate spread predicted by equation (3.14)? Suppose that τ 1 and τ 2 are 15 and 25 years respectively. If the volatility of long term yields is, say, 70basis points per year (consistent with the volatility given in table 1 of the 20-year zero coupon yield in the US Treasury market over the period 1988-94) then the forward rate spread predicted by the expression above is just under 100 basis points. With a volatility of 100 basis points p.a., the predicted spread rises to 200 basis points. Of course this may be offset to some extent by the first two terms in equation (3.12) which capture the effects of expectations and risk premia. Empirically, however, we find that (a) forward rate spreads are almost always negative and that (b) the size of the spread appears to be well explained by equation (3.14). 4. The Data We present data on the behaviour of forward rate spreads in three major bond markets: those for US Treasury Notes and Bonds, Conventional (i.e., fixed nominal payments) UK Government Bonds and Index-Linked UK Government Bonds 11. We have chosen to use (a) data from a variety of markets and (b) zero-coupon yields estimated via a number of different methods in order to minimise the possibility that our findings are the result of a feature of one particular market or one particular method of measuring the term structure. This section describes our data. 4.1. US Treasury Market McCulloch & Kwon (1993) (MK) estimate end-month term structures from December 1947 through to February 1991. They use the procedure described in McCulloch (1975) to estimate a tax-adjusted zero coupon yield curve from US Treasury bills, notes and bonds. This method approximates the discount function using cubic splines, accommodating non-symmetric taxation of capital gains 11 For 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 tax rates which best explain observed prices. MK have modified McCulloch s original technique, which assumed coupons arrived continuously, to recognise discrete semi-annual coupon payments. The data represent the longest available time series of forward rates of which we are aware. However, during certain periods there were no long term Treasuries in issue and, at these times, we were unable to obtain yields for the long maturities which are the focus of our study. One problem faced by MK is that until August 1985 callable bonds constituted the majority of long term issues 12. MK s estimation procedure ignores the value of this option and uses the market s simple par rule to determine maturity dates 13. 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 at 3.30pm 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. FNZ exclude Treasury bills, callable bonds and flower bonds. They also exclude all securities issued prior to 1 January 1980in order to control for the illiquidity of older issues. Furthermore, they remove the two most recently issued securities to prevent any liquidity premium in these issues from biasing their estimates. The term structure is estimated by approximating the forward rate curve with a cubic basis spline. Unlike McCulloch (1975) or Schaefer (1981) their approach ignores potential tax effects, although it might be argued that, at least for the estimates covering the post-1988 period when income and capital gains have been taxed at equal rates, tax effects will be less pronounced. Our third source of US forward rates is US Treasury STRIP 14 prices obtained from Street Software (who also provide the Wall Street Journal with US STRIP data). The dataset we use is essentially an electronic version of the STRIP prices appearing in the US Treasury Issues table in the Wall Street Journal. Because STRIP prices are direct observations of the discount factors, no estimation is required to obtain spot or forward rates. As a result estimates of forward rates obtained from STRIPs are free from the effects of any a priori restrictions on the functional form of the yield curve which are present when approximating functions, such as splines, are used. With STRIPs, forward rates are computed 12 This problem also arises in the case of the UK Goverment conventional bonds. 13 The 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. 14 STRIP 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 liquid STRIPs are recorded. From 1 June 1990onwards all prices are recorded, resulting in around 160observations of discount factors on a typical day. Both bid and offer quotes are available. It is, of course, not possible to measure instantaneous forward rates - such as those appearing in equation (3.14) - directly from STRIP prices. As a proxy for the instantaneous forward rate for maturity τ, we therefore use the average forward rate between maturities τ τ and τ + τ, where τ is,say,oneyear. Thus, to obtain an estimate of the instantaneous forward rate at 25-years, for example, we extract for each trading day the bid prices and maturity dates of STRIPs 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-year forward rate as F 25,where: F 25 = 365 τ 26 τ 24 ln ( P24 P 26 ) 100, (4.1) P 24 (P 26 ) is the bid price of the STRIP closest in maturity to 24 (26) years and τ 24 (τ 26 ) is the exact time to maturity of the 24 (26) year STRIP. We do not discriminate between coupon and principal payments though there are clear price discrepancies 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) UK Government Bonds ( gilts ) to allow reliable estimates of long term zero-coupon rates and forward rates and, for this reason, we have restricted our attention to the period from 1994 onwards. Even so, for conventional gilts, the maturity of the longest forward rate we are able to estimate consistently is just over 20-years 15. 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 for the period April 1994 to November 1998. This method utilises a linear program to identify tax efficient bonds for an investor with a pre-specified tax rate on income and capital gains. The term structure depends only on those bonds in the 15 Actually, 21.5 years. 14

efficient set, with the discount function approximated by cubic basis splines. We do not include Treasury bills but base the estimates on every gilt in issue at each cross section, excluding only callable bonds and undated bonds (e.g. Consols 2.5%) 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-linked bonds using an adaptation of Schaefer s LP procedure and Bernstein polynomials to approximate the discount function. Coupon payments and the repayment of principal for index-linked gilts are not indexed for approximately 8-months prior to the payment date; in other words the indexation in imperfect. To accommodate this feature we use an ARIMA model to forecast inflation over these 8-month periods and adjust the real cash flows accordingly. We then assume that the real cash flows are known with certainty and proceed as before. The presence of longer (non-callable) issues in the index-linked market makes it easier to generate reliable estimates of long-term forward rates than is the case in the nominal market. We use weekly cross-sections of index-linked gilts over the period 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 Evidence. 5.1. Forward Rate Spreads: Summary Statistics The simplest way to illustrate that long-term forward rates are indeed downward sloping is to look at the average difference or spread, between rates for two maturities. Table 5 gives summary statistics for estimates of the mean spread between forward rates at long maturities for the US Treasury market (for all three data sources), the UK nominal gilt market and the UK index-linked gilt market (one data source each). Each data set covers some part of the period between January 1966 and November 1998. Depending on the data source, these rates are either 16 Presently 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 one stock: Conversion 3.5%. 15

2-year forward rates (e.g., between 24 and 26 years) or estimated instantaneous forward 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 for the first and last periods. From January 1986 to December 1994 two estimates are 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 are negative. Arguably the most reliable data are those taken directly from STRIP prices: here the means are negative for each of the three sub-periods. The mean spreads 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 is negative. 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-period is between 100 and 200 basis points. Figures 1a to 1e show the time series of forward rates spreads, for all five data sources, used in deriving table 3. 18 For the McCulloch & Kwon data (Figure 1a) the estimated spreads are sometimes positive in the early part of the period. However, as we have pointed out earlier, the estimates prior to August 1985 are potentially biased because they are based in part on the prices of callable bonds and the option premium component of the price is ignored. After August 1985 the estimates use only non-callable bonds and here all the spreads are negative. Breaks in the series occur when there were no long bonds in issue and no estimates of long term yields are provided by MK. Figure 1b shows the corresponding spread for the US Treasury market estimated by Fisher, Nychka and Zervos. Here, as reported above, the spread is negative on each of the 1673 days in the sample. All our estimates of forward rate spreads, apart from one, are derived from term structure estimates obtained using splines, or some other form of curve fitting technique. It is therefore possible that the behaviour of the forward rates in these 17 Apart from the case of UK nominal gilts where, because few long term bonds were in issue over the period, we compute the difference between the forward rates at 20 years and at 10 years. 18 As mentioned earlier, in the case of UK conventional gilts (Figure 1e) the spread shown is the 20-year minus the 10-year forward rate. For all other data sources the figures show the spread between forward rates at 25 years and 15 years. 16

cases is, to a greater or lesser extent, dependent on the estimation methodology used. After all, long term discount factors will usually be the most difficult to estimate because they depend on the prices of a relatively small number of the longest bonds in the sample and the forward rate spread then depends on the ratio of these discount factors. The forward rate spreads we obtain from STRIP prices are, therefore, particularly significant because these are obtained directly from prices. Figure 1c shows all the daily estimates of the spread obtained from STRIP prices. The figure shows that they are negative almost all the time (only 37 out of 2363 observations, or 1.2%, are positive) and that the size of the spread is quite substantial, around -100 basis points to -200 basis points for much of the time. Figure 1d shows the 20-year minus 10-year forward rate spread for the UK conventional gilts over the period April 1994 to December 1998. Of the 243 weekly 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, from the beginning of 1985 the estimated spread is also mainly negative. The spread is negative for every week between November 1984 and October 1992 and, between October 1992 and July 1998 fluctuates around zero. Table 3 and Figures 1a-1e use forward rates at two particular maturities, in most cases at 15 and 25 years. This raises the question of whether this choice of maturities is special in some sense: perhaps the term structure of forward rates at other long maturities behaves differently. In Figure 2 we show the average oneyear forward rate curve, derived from STRIP prices, for maturities between 13 years and 26. Each of the ten charts shows the term structure of average forward rates for a calendar year and, while these are not always monotonic, the shape of the curve is, in each case, clearly downward sloping. Overall, the evidence shows that the term structure of long term forward rates is persistently downward sloping. This is true not only in the US market but also in the UK market for both conventional and index-linked bonds. The next section examines whether, as we have suggested in Section 3, the downward tilt can be explained 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 the volatility of zero coupon rates and the spread in long term forward rates. Our analysis is restricted to the data on US Treasury STRIPs since these allow us to observe forward rates directly; the other estimates (McCulloch and Kwon, Fisher, Nychka and Zervos and our own calculations for UK gilts) are all potentially subject to the influence of the estimation methodology. Equation (3.14) predicts that the shape of the long term forward rate curve is quadratic in maturity and this suggests the following cross sectional regression to obtain an estimate of volatility from the shape of the forward rate curve: f(t, τ j )=β 0 + β 1 τ 2 j + ε j τ j τ 0 (6.1) where τ 0 is a long maturity, e.g., 15 years and ε j is an error term capturing the effects of the missing terms in equation (3.14) and also estimation error in the forward rates. In equation (6.1) the predicted value of the coefficient β 1 is negative and equal to 1/2σ 2 l. In this section we give the results of estimating the implied zero coupon yield volatility, σ l, from equation (6.1) and then using this estimate as a predictor of time series volatility. Estimates of β 1 obtained from individual term structures are likely to be imprecise as a result of observation error in the forward rates and this will lead to bias in subsequent regressions of time series volatility on implied volatility. There are a number of approaches which could be pursued at this point but the one we have chosen to adopt is simply to average equation (6.1) over periods of between two to four months and then to regress average forward rates on squared maturity 19. Figure 3 shows the results of estimating equation (6.1) in this manner. We use 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 of the forward rates over three month intervals and estimate equation (6.1) using 19 Because the regressors are the same for each cross section, we would obtain exactly the same results if the regressions were carried out on individual cross-sections of forward rates and the regression coefficients averaged over time. 18

OLS. The figure shows the time series of estimates of the volatility parameter, σ l, derived from the estimate of the slope coefficient in (6.1) using: σ l = 2 β 1. (6.2) By way of comparison, the figure also shows the time series of the volatility of two-day changes in the 20-year zero coupon rate, also derived from STRIPs data, for the same three month intervals. The estimated value of β 1 was negative in each of the 37 three month intervals and the mean value of the implied volatility was 83 basis points compared with the mean time series estimate for the 20year zero-coupon rate of 93 basis points. As with implied volatilities from option prices, the implied volatility derived from the cross sectional regression (6.1) can be regarded as a predictor of time series volatility. Accordingly we have regressed the time series volatility for the k th M-month interval on the implied volatility for the (k 1) st interval. The results are shown in Table 4 which also includes, for purposes of comparison, regressions of period k time series volatility on contemporaneous (period k) implied volatility. In the table the first column gives M, the number of months data used in computing both the time series volatility and average forward rates. Panel (a) gives results for the entire sample period (April 1985 to October 1994). The third row, for example, shows the results, for M =3, of regressing period-k time series volatility on period-k implied volatility. The difference between the coefficient on implied volatility, 0.929, and unity is small and not statistically significant and, although these observations are derived from the same period in calendar time, it should be noted that the estimate of volatility derived from cross sections of forward rates is independent of the temporal sequence of the data. The regression reported in the fourth row of Table 4 examines the predictive power of volatility estimates derived from forward rates for future volatility. Here the time series volatility for period k is regressed on the implied volatility for period k 1. The coefficient on implied volatility is lower (0.825) but still less than one standard error from unity 20. The remaining regressions in panel (a) examine the effect of averaging the forward rates over shorter (2 months) and longer (4 month) periods. Overall this does not make a substantial difference to the coefficients (although the problems 20 The table shows Newey-West corrected standard errors with one lag. 19

of serial correlation appear somewhat less severe with a four month estimation period). 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 liquidity in the earlier years of the market) and (b) that the period including the crash gives rise to a large outlier in many of the regressions. We were concerned that the results in panel (a) might have been affected by these factors and we have therefore re-estimated the results presented in panel (a) excluding the data up to and including the crash. However, the results, shown in panel (b), are little changed. For the three regressions on lagged implied volatility, the coefficient is closer to unity for M =2months and a little further away for M =3months and M =4months. 6.2. Does Interest Rate Volatility Explain the Forward Rate Spread? The previous section looked at the question of whether the shape of the term structure - implied volatility derived from forward rates - predicts the volatility of long term zero coupon yields. Here we reverse the question and ask whether a prediction 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 conventional standard deviation or, perhaps an ARCH/GARCH, estimate) as our forecast. But it seems preferable to use a forward looking estimate which reflects agents expectations and we have therefore used the implied volatility on bond futures options. The implied volatilities are those on a near-maturity at-the-money option. As the time-to-maturity becomes short the option for which the implied volatility is calculated switches to the next shortest option. The implied standard deviation (ISD) is computed using a Black-Scholes-like formula and, ignoring features such as delivery options and so forth, may be regarded as the anticipated volatility of proportional price changes of the cheapestto-deliver (CTD) bond. This is to be contrasted with the arithmetic yield changes required by equation (3.14). If the CTD bond underlying the futures contract has modified duration D, then we may compute the volatility of arithmetic changes in the yield-to-maturity of the CTD bond as: σ y = 1 D σ p. (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 simply use the futures options ISD as an index of anticipated bond market volatility 21. We proceed in two stages. First, we regress time-series volatility in period k on futures 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 and forward rates at 15 years. Finally, we regress the actual average forward rate spread for period k on the period k-1 forecasts of the spread. Table 5 gives the estimated prediction rule for yield volatility: the option ISD is always significant and, overall, explains around 15 to 20percent of the variability in next period s volatility. When we regress time series volatility on option 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 that with a forecast horizon of three months much of the variability of volatility is unpredictable. In Table 6 we regress the period k average forward rate spread between 25 and 15 years on the expected spread. The latter variable is computed as follows. We use the forecast for volatility given by the second row of Table 5, i.e., we set M to three months and, using the average ISD for the last ten days of period k 1, we compute the expected volatility of the 20-year zero coupon rate for period k. We then use equation (3.14) to compute the expected forward rate spread for period k. In the Table 6 the coefficient on the expected spread is 0.506 with a standard error of 0.14 and, although over two standard errors below the expected value of one, 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 future interest rates and (b) are consistent with the no-arbitrage condition, the term structure of yields depends on the anticipated level of volatility even in the absence of risk premia. 22 After all, one has only to look at the pricing formulae in models such as Vasicek (1977) and Cox, Ingersoll and Ross (1985). What has, perhaps not 21 Evenifwedidknow the duration of the CTD bond, we could not safely assume that the volatility of its yield to maturity is equal to the volatility of long term zero coupon yields. 22 Or, equivaliently, under risk-neutral dynamics. 21

been properly appreciated before, is that (a) the effects of interest rate volatility are to be found most strongly in the yields on long term bonds, (b) these effects result in a strong regularity in the term structure of forward rates, the downward tilt and (c) that estimates of implied zero coupon yield volatility can be easily obtained from the size of the tilt. In this paper we have first attempted to document the persistence of the downward tilt in long term forward rates. A number of data sources on both US and UK bond markets show that forward spreads are habitually negative. Using a simple two-factor Gaussian model we have derived a simple relation between the difference in long term forward rates at two maturities (the forward rate spread ). Finally, we have tested this theory by looking both at the ability of forward rate spreads to predict interest rate volatility and at the extent to which the forward rate may be predicted using a measure of agents expectations of volatility. Both tests broadly support the theory. 22

References 1. Barone E., Cuoco D. & Zautzik E. (1991). Term structure estimation using the 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, second edition, (Woodhead-Faulkner, Cambridge, UK). 3. Brown R.H. & Schaefer S.M. (1994a). The term structure of real interest rates and the Cox, Ingersoll and Ross model, Journal of Financial Economics, Vol. 35, pp. 3-42. 4. Brown R.H. & Schaefer S.M. (1994b). Interest rate volatility and the shape of 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 of Finance, Vol. 41, pp. 617-630. 6. Caverhill, Andrew (1998), Modelling Long Term Forward Rates via the Kalman Filter, Working Paper, Hong Kong University of Science and Technology. 7. Cox J.C., Ingersoll J.E. & Ross S.A. (1985). A theory of the term structure of interest rates, Econometrica, Vol. 53, pp. 385-407. 8. Dia, Q. and Singleton K., (1998). Specification Analysis of Affine Term Structure Models, Working Paper, Graduate School of Business, Stanford University. 9. Deacon, M. & Derry A., (1994). Deriving EStimates of Inflation Expectations from the Prices of UK Government Bonds. Bank of England working paper no. 23. (July) 10. Duffie D. & Kan R. (1996). A yield-factor model of interest rates, Mathematical Finance, Vol. 6, 379-406. 11. Fisher M.E., Nychka D. & Zervos D (1994). Fitting the term structure of interest rates with smoothing splines, unpublished working paper Federal Reserve Board, January. 23

12. Langetieg, T. 1980. A Multivariate Model of the Term Structure, Journal of Finance, 35, 71-97. 13. Litterman R., Scheinkman J. & Weiss L. (1991) Volatility and the yield curve, Journal of Fixed Income, Vol. 1, No. 1, June, pp. 49-53. 14. Litterman R. & Scheinkman J. (1991). Common factors affecting bond returns, 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 term structure: 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 structure data, 1947-1991, unpublished working paper 93-6, Ohio State University. 18. Merton, R.C. (1970) A Dynamic General Equilibrium Model of the Asset Market and its Application to the Pricing of the Capital Structure of the 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 Yield Curves, Journal of Business, Vol. 60, pp. 473-89. 20. Schaefer S. M. (1981). Measuring a tax-specific term structure of interest 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 Exponential 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 parameters κ x and κ y. Since ours is a two-factor linear model the covariance matrix of zero-coupon yields has rank two. It is simple to show that the two eigenvectors associated with the covariance matrix of zero coupon yields are linearly related to the functions B x (τ) and B y (τ). The latter depend only on the mean reversion coefficients, the correlation coefficient and maturity and our method estimates the mean reversion coefficients, and the correlation coefficient, from the eigenvectors. We have carried out a principal components analysis on weekly changes in zero coupon rates, derived from the prices of US Treasury STRIPs, with annual maturities between 2 years and 26 years using data for the period December 1988 to October 1994. Table A1 shows the eigenvalues and the percentage of total variance explained by the first eight principal components. Around 86% of total variance is accounted for by the first principal component and 95.5% by the first two. Thus the assumption of that prices are determined by two factors seems reasonable and we use the first two eigenvectors to infer the mean reversion coefficients κ x and κ y. Using the notation used above, the vector of innovations in zero coupon yields, R t, is related to the innovations in the state variables x and y by: R t = [ B x /τ B y /τ ] ( x y ), (A.1) where { } B x(y) = B j x(y)(τ j ). Denoting the eigenvectors of the covariance matrix of R t as EV 1 and EV 2 we may also write R t in terms of the principal components p: R t = [ EV 1 EV 2 ] p. (A.2) The state variables {x, y} and the principal components are linearly related (the principal components are simply the state variables of the same model rewritten in terms of orthogonal state variables) and so, for some (2 by 2) weighting matrix, W,wemaywrite: 25

( x p = W y ). (A.3) From equations (A.1), (A.2) and (A.3) it is straightforward to show that [ Bx /τ B y /τ ] W 1 = [ EV 1 EV 2 ]. (A.4) For given maturities, the vectors B x /τ and B y /τ depend only on κ x and κ y. Thus, given estimates of EV 1 and EV 2 we estimate κ x, κ y and ρ xy, the imputed value of the correlation coefficient, along with the elements of W,byfindingthe least squares fit between the directly estimated and fitted eigenvectors, i.e., the right-hand side and left hand side, respectively, of equation (A.4) subject to the constraint that Corr( p 1, p 2 ) 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 eigenvectors for the period December 1988 to October 1994 24. Table A2 gives the estimates of κ x and κ y and ρ xy for the same period and for two sub-periods. For the whole period (1988-94) the estimates for κ y are 0.23 using the unadjusted covariance matrix and 0.21 with an adjusted covariance matrix. The corresponding estimates κ x are ten times smaller in the first case and 5 times smaller in the second case (0.021 and 0.039). These values are consistent with a relatively flat term structure of volatility for long maturities. 23 In fact we have found that it was easier to find the global minimum of the sum of squares by rewriting the model in terms of orthogonal state variables. If the orthogonalised state variables are x and y (only one state variable need change) then the eigenvectors are simply B x /τ and B y /τ, although the latter is now a more complicated expression than before. 24 Thecovariancematrix from which the eigenvectors were computed was estimated using the observation error adjustment method described in Appendix B. 26

B. Adjusting for Observation Error in the Estimation of Variances and Covariances of Zero Coupon Yields In estimating both the term structure of volatilities and the covariance matrix of yield changes, we have found that the results are potentially affected by observation error in the yields. We have attempted to correct for this in the following way. Let Rt denote the n-vector of observed zero-coupon yields at time t and assume that this differs from the true value R t by a zero mean n-dimensional iid variate ṽ t : R t = R t + ṽ t. (B.1) Further assume that R t follows a random walk and that the increment between time t and t +1is another n-dimensional iid variate ũ t+1 : R t+1 = R t + ũ t+1. (B.2) From equations (B.1) and (B.2) it is simple to show that the covariance matrix of changes in observed yields, Σ R, is related to the covariance matrix of true yields, Σ R,by: Σ R =Σ R +2Σ v, (B.3) where Σ v is the covariance matrix of the observation errors v. The first order serial covariance in R t, changes in the vector of observed yields in R t,isgiven by Σ v. We therefore estimate the covariance matrix of true changes as: Σ R +2Cov( R t+1, R t ). (B.4) 27

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 adjusted 1 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. 28

Table 2 Summary of Data Estimate Estimation Method Type of Bond Frequen cy US Treasury Markets Period Covered Comments McCulloch & Kwon Fisher, Nychka & Zervos Spline approximation to discount function Spline approximation to forward rate curve coupon bonds monthly 12/1947-2/1991 Period before 8/1986 includes callable bonds 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 --- 29

Table 3 Summary Statistics on Forward Rate Spreads: 25-year Forward minus 15-year Forward US Treasury Bonds UK Gilts a UK Index-Linked b 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. 30

Table 4 Implied Volatility from Forward Rates as Predictors of Time Series Volatility The table shows estimates of the following regression: TSVOL( R 20 ) k, M = α 0 + α 1ISDk, M + α 2 ISDk 1, M + ε k, where TSVOL(R 20 ) k,m is the annualised time series volatility of 2-day changes in the 20-year zero coupon rate during the k th sub-period of length M months. ISD k,m is the implied interest rate volatility (from regression equation (6.1)) for the k th 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-bar 2 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. 31

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: TS VOL( R 20 ) k, M = α 0 + α 1 Option ISD k 1, M + ε k, where TSVOL(R 20 ) k, M is the annualised time series volatility of 2-day changes in the 20- year zero coupon rate during the k th 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 ISD k-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) 32

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 = α 0 + α 1 E Spread k 1 + ε k, 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) 33

Figure 1a Spread Between 25-year and 15-year Forward Rates in US Treasury Market (Source: McCulloch & Kwon) 6 4 2 0-2 -4-6 -8-10 -12 Figure 1b Spread Between 25 Year and 15 Year Forward Rates, Dec 1987 to July 1994. Source: Fisher, Nychka nd Zervos 0.00-0.50-1.00-1.50-2.00-2.50-3.00-3.50-4.00-4.50-5.00 Dec-87 Aug-88 May-89 Jan-90 Oct-90 Jul-91 Mar-92 Dec-92 Sep-93 Jun-94 Jan-66 Jan-69 Jan-72 Jan-75 Jan-78 spread % p.a. Jan-81 Jan-84 Jan-87 Jan-90 rate (p.a.)

Figure 1c Spread Between Average Forward Rates for 24 to 26 year and 14 to 16 years. STRIP's Data 2% 1% 0% rate (% p.a.) -1% -2% -3% -4% -5% 19850422 19860422 19870422 19880421 19890421 19900420 19910419 19920420 19930422 19940426 Figure 1d Forward Rate Spreads in the UK Nominal Gilt Market: Spread between the 6-month forward rates at 20 years and at 10 years. 0.25 0.00-0.25 rate (% p.a.) -0.50-0.75-1.00-1.25-1.50-1.75 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

Figure 1e 1.0 0.5 0.0 Spread between 25 Year and 15 Year Forward Rates, January 1984 to November 1998, UK Goverment Index-Linked Bonds rate (% p.a.) -0.5-1.0-1.5-2.0-2.5 Jan-84 Jul-85 Dec-86 Jun-88 Dec-89 Jun-91 Dec-92 Jun-94 Dec-95 Jun-97

Figure 2 Average Forward Rates from STRIP Prices The figures show the average one-year forward rate computed from STRIP prices for annual maturities between 13 and 26 years. The data are weekly and the averages are shown for each calendar year. Note that the vertical scale is the same (300 basis points) in each case. 1985 1986 12 10 average forward rate (% p.a.) 11 10 average forward rate (% p.a.) 9 8 9 12 14 16 18 20 22 24 26 28 7 12 14 16 18 20 22 24 26 28 maturity maturity 1987 1988 10 10 average forward rate (% p.a.) 9 8 average forward rate (% p.a.) 9 8 7 12 14 16 18 20 22 24 26 28 maturity 7 12 14 16 18 20 22 24 26 28 maturity 1989 1990 9 10 average forward rate (% p.a.) 8 7 average forward rate (% p.a.) 9 8 6 12 14 16 18 20 22 24 26 28 maturity 7 12 14 16 18 20 22 24 26 28 maturity

Figure 2, continued 1991 1992 10 10 average forward rate (% p.a.) 9 8 average forward rate (% p.a.) 9 8 7 12 14 16 18 20 22 24 26 28 maturity 7 12 14 16 18 20 22 24 26 28 maturity 1993 1994 9 9 average forward rate (% p.a.) 8 7 average forward rate (% p.a.) 8 7 6 12 14 16 18 20 22 24 26 28 maturity 6 12 14 16 18 20 22 24 26 28 maturity

Figure 3 Estimates of Interest Rate Volatility from Cross Sectional Regressions of Forward Rates on squared maturity. Each regression uses average forward rates over a three month period and includes annual maturities from 15 to 26 years. The graph also shows the contemporaneous time series volatility of two-day changes in the 20-year zero-coupon rate. All interest rate data are derived from daily STRIP prices over the period April 1985 to October 1994. 2.25 2.00 1.75 Implied volatility from cross-sectional regressions Time series volatility of 20-year zero-coupon rate 1.50 1.25 1.00 0.75 0.50 0.25 Jul-85 Apr-86 Jan-87 Oct-87 Jul-88 Apr-89 Jan-90 Oct-90 Jul-91 Apr-92 Jan-93 Oct-93 Jul-94 Volatility, % p.a.

Table A1 Eigenvalues and Percentage of Variance Explained The table shows the first eight eigenvalues of the estimated covariance matrix of weekly changes in zero coupon rates with annual maturities of 2 to 26 years derived from the prices of US Treasury STRIPS over the period December 1988 to October 1994. The covariance matrix is adjusted for observation error in the yields using the procedure described in the Appendix. Eigenvalues Fraction of Variance Explained Cumulative Fraction of Variance Explained 1st PC 2nd PC 3rd PC 4th PC 5th PC 6th PC 7th PC 8th PC 0.275 0.031 0.0058 0.0024 0.0018 0.0009 0.0006 0.0006 85.8% 9.7% 1.8% 0.7% 0.6% 0.3% 0.2% 0.2% 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 reversion parameters κ x and κ y and the correlation coefficient ρ xy estimated from the eigenvector of the covariance matrix of zero coupon yields calculated from the prices of US Treasury STRIPs. For details see the Appendix. Period Observation Error κ x κ y ρ xy Correction 1 88-94 Unadjusted 0.0212 0.232-0.254 88-94 Adjusted 0.0393 0.206-0.336 88-91 Unadjusted 0.0089 0.354-0.103 88-91 Adjusted 0.0328 0.334-0.163 91-94 Unadjusted 0.0386 0.163-0.402 91-94 Adjusted 0.0417 0.126-0.556 1 Note: The term adjusted refers to a correction for measurement error in computing the 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 Covariance Matrix of Changes in Zero-Coupon Yields The figure shows (a) the estimated eigenvectors calculated from the estimated covariance matrix of zero-coupon yields and (b) the fitted eigenvectors computed using the mean reversion coefficients estimated from the estimated eigenvectors. The data used are weekly STRIP prices from December 1988 to October 1994. The estimated covariance matrix is adjusted for observation error in the STRIP yields using the method described in the Appendix. 0.30 0.20 0.10 0.00-0.10-0.20-0.30-0.40-0.50-0.60 Fitted First EV Fitted Second EV Directly Estimated First EV Directly Estimated Second EV 0 5 10 15 20 25 30 File: Tables and Fig for Appendix Date: 09/11/99