Inflation, real interest rates, and the bond market

ELSEVIER Journal of Monetary Economics 39 (1997) 361-383 JOURNALOF Monetary ECONOlVIICS Inflation, real interest rates, and the bond market: A study of UK nominal and index-linked government bond prices David G. Barr a, John Y. Campbell b'c'* a Center .[or Empirical Research in Finance, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK b Department of Economics, Littauer Center, Harvard Unit,ersity, Cambridge, MA 02138, USA c National Bureau of Economic Research, Cambridge MA 02138, USA Accepted 18 March 1997 Abstract This paper estimates expected future real interest rates and inflation rates from observed prices of UK government nominal and index-linked bonds. The estimation method takes account of imperfections in the indexation of UK index-linked bonds. It assumes that expected log returns on all bonds are equal, and that expected real interest rates and inflation follow simple time-series processes whose parameters can be estimated from the cross-section of bond prices. The extracted inflation expectations forecast actual future inflation more accurately than nominal yields do. The estimated real interest rate is highly variable at short horizons, but comparatively stable at long horizons. Changes in real rates and expected inflation are strongly negatively correlated at short horizons, but not at long horizons. Keywords: Index-linked bonds; Inflation expectations; Real interest rates; Yield curves JEL classi[ication: E31; E41; E43; G12 * Correspondence address: Department of Economics, Littauer Center, Harvard University, Cambridge, MA 02138, USA. The authors are grateful to seminar participants at the Centre for Economic Performance, LSE, and the NBER for helpful comments. Campbell acknowledges the financial support of the National Science Foundation. 0304-3932/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S0304-3932(97)00027-5 362 D.G. Barr, J.Y. Campbell~Journal of Monetary Economics 39 (1997) 361~83 1. Introduction The prices of assets traded by forward-looking agents offer a rich source of information about the expected future. Financial markets summarise the disparate and largely unobservable expectations of asset holders and reveal them in the coded form of observable prices. The challenge to economists is to break this code by modelling the relationship between individuals' expectations and market prices. In this paper we use a simple asset pricing model to convert the prices of nominal and index-linked UK government bonds into implied expectations of future real interest rates and inflation. Implied market expectations are useful in many ways. Investors can use them to identify the points at which their own expectations diverge from the market consensus, adjusting the balance of their portfolios to take advantage of what they may regard as the market's errors. There is a similar role for market expectations in public-sector debt management, where the government may wish to minimise its funding costs. Another important application is to monetary policy. Information about inflation expectations offers the monetary authorities a measure of the credibility of their commitment to low inflation, and may be used to inform their decisions about the setting of short-term interest rates. The nominal yield curve alone cannot be used for this. For example, an increase in nominal yields may be due to an increase in expected inflation, in which case policy should be tightened; but it may also be due to an increase in real interest rates driven by real factors, in which case tighter monetary policy may be entirely inappropriate. To the casual observer the daily activities of govemment bond markets can be summarized by the 'yield curve'. The only real data to emerge from the markets, however, are in the form of bond prices. There are many methods, of varying degrees of sophistication, that may be used to convert these prices into a curve. 1 Similarly, there are many ways of drawing inferences about agents' expectations from the estimated curves. Our approach is to perform these two exercises simultaneously. We use simple time-series models to represent agents' expectations, and obtain bond-price equations in which the effects of changing expectations are intuitive and easily estimated. Once we have estimated market expectations of real interest rates and inflation, we study their properties over the period January 1985 through October 1994. We describe the risk and return characteristics of real and nominal bonds - important information for investors making portfolio decisions - and the ability of the bond market to forecast future inflation - important information for policymakers formulating monetary policy. The organisation of the paper is as follows. The next section briefly explains why the task of extracting implied real interest rates from UK index-linked bond J A useful review of estimation methods is provided by Deacon and Derry (1994b). D.G. Barr, J.Y. Campbell~Journal of Monetary Economics 39 (1997) 361~83 363 prices is not as simple as it may seem: Index-linked bonds have significant in- dexation lags, and so they are not pure real bonds. To handle this problem, in Section 3 we lay out a general framework that relates nominal and index-linked bond prices to expected future inflation rates and bond returns. We then impose the log pure expectations hypothesis of the term structure to obtain a bond pricing model that is suitable for estimation. Section 4 discusses our data and economet- ric methodology, summarizes the recent history of nominal and real interest rates in the UK, and describes the risk and return characteristics of real and nominal bonds in our sample period. Section 5 studies the ability of the bond market to forecast future inflation, and tests the expectations hypothesis as applied to the nominal, real, and expected-inflation term structures. Section 6 concludes. 2. lndexation lags in UK index-linked bonds Economists seek to construct yield curves from real bonds in order to measure the term structure of real interest rates. Unfortunately, UK index-linked bonds are not pure real bonds. A perfectly indexed bond would pay a nominal coupon equal to the coupon rate announced at the time of issue multiplied by the pro- portionate increase in the general price index between the issue date and the time of payment. UK index-linked bonds, however, pay nominal coupons equal to the coupon rate announced at the time of issue multiplied by the proportionate increase in the price index from a 'reference level' dated eight months before the bond's issue date to a date eight months before the coupon payment occurs. The same indexation lag applies to the repayment of principal. Thus nominal pay- ments on UK index-linked bonds are left unprotected against inflation occurring in the last eight months before the payments are made. 2 This feature of index-linked bonds creates technical difficulties in extracting implied real interest rates from index-linked bond prices. Observed changes in the price of an index-linked bond may reflect changes in inflation expectations, albeit with a sensitivity well below that of a purely nominal bond (see Barr and Pesaran, 1995). It is common practice to calculate the yield to maturity on an index-linked bond conditional on a profile of inflation throughout its remaining life. Quoted index- linked yields typically assume a constant 5% inflation rate and are presented as a 'real' rate. This creates a temptation to subtract this real rate from the nominal yield on a nominal, or conventional, bond of equivalent maturity (or duration) to generate a figure for average expected inflation over the remaining life of the bonds. The potential inconsistency between the derived inflation profile and that assumed at the outset is obvious. This conflict can, however, be corrected by an iterative process whereby the generated expected inflation is used to recompute 2 See Bootie (1991) for a description of institutional features of the U K index-linked bond market. 364 D.G. Barr, J.Y. Campbell/Journal of Monetary Economics 39 (1997) 361J83 the real yield on the index-linked bonds, from which a new figure for inflation can be obtained, and so on. This approach, which originates in papers by Arak and Kreicher (1985) and Woodward ( 1988, 1990), generates 'break-even inflation rates', so called because these are the rates that equate the yields on conventional and indexed bonds. The break-even method suffers from two problems. First, it does not generate a complete term structure of inflation, since it can be applied only to those maturities where there are equivalent pairs of real and nominal bonds. Recent research at the Bank of England has attempted to address this problem (Deacon and Derry, 1994a). Second, the break-even method takes no account of risk or liquidity premia on real or nominal bonds. This problem is hard to handle without specifying a complete equilibrium model of the term structure such as the Cox, Ingersoll, and Ross (1985) model used by Brown and Schaefer (1994) in a study of the index- linked bond market. Brown and Schaefer estimate a real yield curve allowing for real term premia, but do not look at nominal bonds and therefore do not model inflation risk premia. They also assume that UK index-linked bonds are perfectly indexed. In future work we plan to fit an equilibrium model of both real interest rates and inflation to UK nominal and index-linked bond prices, but in this paper we follow most of the literature and assume that risk premia on all bonds are zero. 3. A pricing model for nominal and index-linked bonds In this section we develop a framework relating the prices of govemment bonds to expected future log real retums and inflation rates. We then use a specific model of expected bond returns, the log pure expectations hypothesis, to derive an empirically implementable bond pricing model. 3.1. A ,qeneral equat ion fo r bond pr ices We consider a claim to a single real payment to be made at time t + n. We write the log of the real payment as vt+n, the log real price of the claim at time t as Pt, and the log real retum on the claim from t to t + 1 as rt+ 1 . Log price and log return are related by rt+l = Pt+l -- Pt. (1) Inverting this equation, and taking expectations conditional on information at time t, gives a first-order difference equation which can be solved forward to time t + n to give n Pt = -- ~ Et rt+s + Et vt+n. (2) s--I D.G. Barr, J. E Campbell~Journal of Monetary Economics 39 (1997) 361 383 365 This equation relates the log price of the claim at time t to expected future log returns and the expected future real payment on the claim. It applies directly to zero-coupon bonds, while for coupon-bearing bonds one can calculate the prices for each coupon payment and the repayment of principal, and then add up across payments. Eq. (2) illustrates two problems that must be overcome before any empirical analysis can proceed. First, asset prices depend on expected values of v and r, for which we have no data. We deal with this by assuming that both variables follow simple time-series processes. Second, the expectations may be asset-specific if the dividend and return processes are unique to each asset. Since this generates more coefficients than can be separately identified, we assume that there are common factors that drive the relevant movements in expectations. 3.2. Rea l va lues o f nomina l and index- l inked payments In contrast to equities, real payments on bonds are driven by a single common f;actor, inflation. The precise way in which the real payment depends on inflation is determined by the extent to which the bond is indexed. This distinction be- tween bonds requires some new notation; we denote claims to individual nominal payments by a subscript j = 1 . . . . . J , and claims to individual index-linked pay- ments by a subscript i = 1 . . . . . I. For simplicity we normalize the declared bond payments to one, so the log of the declared bond payment is zero. In the case of nominal bonds, since there is no indexation, the real payment is the nominal payment deflated by the general price index. Working in logs, and writing z for the log price index, we have v j, t+~ = -zt+n, (3) since the declared nominal payment has log zero. For index-linked bonds, we define an indexation lag parameter, l, which is 8 months for UK government index-linked bonds. The log nominal payment on an index-linked bond is the log declared payment adjusted for the difference between the log price level l months before payment, zt+n-z , and a reference log price level :?i which is specific to each particular bond and is determined before the bond is issued. The real payment is again the nominal payment deflated by the general price index: l vi.t+n = (z,+, l - z i ) - z,+~ = -Z i - ~ 7~,+~+1-~, (4) S--I where ~t =zt -z t - l is inflation from time t - 1 to t. Hence whenever indexation is imperfect ( l>0) , the real value of the payment depends on inflation during the period of the indexation lag. 366 D.G. Barr, J.Y. Campbell~Journal of Monetary Economics 39 (1997) 361-383 3.3. Real prices of nominal and index-linked zero-coupon bonds Substituting (3) into (2), the real price of a nominal payment becomes P jn ,=-z t -E t~r j , t+s-Et~rtt+n+l_s, (5) s=l s= l while substituting (4) into (2) the real price of an index-linked payment is n l Pint = -- Zi -- Et ~ ri, t+s -- E1 ~ lrt+n+l-s. (6) s : l s : l When there is an indexation lag ( l>0) , the last l months of inflation affect the real value of the bond payment. If l < n only expected inflation is relevant whereas if l > n some of the last l months of inflation has already occurred at time t. Eq. (6) gives the same price as Eq. (5) if we set the indexation lag equal to the length of time since the reference price index was set. 3.4. Term premia and the inflation risk premium Eqs. (5) and (6) incorporate expected returns that may be asset-specific. Since this gives rise to more coefficients than can be identified, we assume that ex- pected log returns on nominal and index-linked bonds of all maturities equal the one-period interest rate. The assumption that expected long-term bond returns are equal to the short-term interest rate is known as the log pure expectations hy- pothesis, while the assumption that expected nominal bond returns equal expected real bond returns is an assumption that the inflation risk premium is zero. In the conclusion we discuss alternative assumptions that we plan to apply in future work. 3.5. Expectations processes for inflation and returns In order to obtain equations suitable for estimation we have to replace all of the expected future values in Eqs. (5) and (6) by functions of information available at time t. To do this we assume that log expected inflation follows a trend-stationary AR( 1 ) process: Etlzt+s = gO,~ + gl,~(S -- 1) + Y2 ,~ .~s-l. (7) Similarly, we assume that the expected real interest rate, which equals the ex- pected real return on any bond of any maturity, also follows a trend-stationary AR(1 ) process: q_ - - s -1 Etrt+s=9O, r+gl,r(S-1) 92,rq~r • (8) At time t there are four parameters to be estimated for each process: 9o,9~, 92 and the adjustment parameter ~b. D.G. Barr, J.Y. Campbell~Journal of Monetary Economics 39 (1997) 361383 367 In principle, one might expect the real interest rate and inflation processes to be stationary around a fixed mean rather than a trend; this would imply g~ = 0. However we found that in some periods the real or inflation term structures have a significant slope even at very long horizons. This causes numerical problems for a model with gl = 0, since such a model can fit the data only by setting 0 extremely close to (but not equal to) one. Accordingly our general specification is the one given in Eqs. (7) and (8). 3.6. Implied nominal prices of nominal and index-linked zero-coupon bonds The expectations terms in Eq. (5) can be replaced by repeated substitution from Eqs. (7) and (8). We can add the current log price index to get the log nominal price of a claim to a log-zero nominal payment: =_n(go, r q_go,~z)_(gl,r q_gt,n)(n(n; 1)) (9) The same process can be applied to Eq. (6) to yield a pair of equations for the nominal prices of claims to index-linked payments (again normalized to have log zero). For payments that are of a sufficiently long maturity that their nominal value has still to be determined, i.e. for n > l, - - i ) ~gt--zi--ngo'r--loo'n--gl'r(n(n2 1))-Ql'n(l(12 ) - -O2 ' r~) -o2"rt4)n \1 -qS=) (lo) For index-linked payments that have exhausted their indexation, i.e. for which n < 1, the equation becomes pT /n(n- 1)0 = Zt+n--l -- Zi -- n(go, r + gO,~)-- (gl,r + gl,~z)~ f l -- ( l -- (11) Since indexed payments become nominal when their maturity falls below that of the indexation lag, Eq. (11) is identical to Eq, (9) for nominal bonds, except for the indexation that has already taken place by time t. 368 D.G. Barr, J. K Campbell~Journal of Monetary Economics 39 (1997) 361~83 3.7. Nominal prices o f nominal and index-linked coupon-bearing bonds We calculate nominal prices of coupon-bearing bonds by adding up the nominal prices of their coupon payments and final repayments of principal. For a nominal bond with log coupons cj and nominal principal normalized to zero, we have pn,~.% = log[ ,~.=~1 exp(p)n'~m + Ci)+ exp(p)n~m)] + ~jt. (12) For an index-linked bond with log declared coupon rate ei and log declared redemption payment normalized to zero, we have ] Pc, int = log exp(pin°tm + ci) + exp(p~n~ m) + Eit. s=l The error terms ejt and eit represent pricing errors, since our model will not fit all observed nominal bond prices perfectly. These equations ignore some practical issues that complicate the pricing of coupon-bearing bonds. Coupons on UK government bonds are paid at six-monthly intervals. We assume instead that they are paid in equal instalments each month, to match the frequency of our observations. This creates a small bias because monthly payment would make each coupon more valuable, since the early pay- ments could be reinvested. The effect of this is to bias our estimated rates up by around 30 basis points. Another issue we ignore is the tax treatment of UK government bonds. Capital gains on these bonds are tax-exempt for nearly all holders. Coupon income is taxed at a range of different rates. For a significant proportion of holders the rate is zero, and the coupon tax rate for the marginal investor is unknown. We assume a rate of zero despite the fact that the prices of some of the bonds in our sample may be influenced by investors paying a higher rate. Thus here again we probably overstate both the value of the coupons actually received by investors and the yields available on both nominal and index-linked bonds. For a discussion of tax effects see Deacon and Derry (1994a, b) and Schaefer (1981), Overall, our calculated expectations of nominal and real interest rates are likely to have a small upward bias. This bias is probably fairly constant over time, how- ever, so the movements of rates over time are likely to reflect the true movements of expectations. There is less bias in expected inflation, which is the difference between two upwardly biased interest-rate estimates. 4. Data and estimation Eqs. (9)-(11) can be substituted into Eqs, (12) and (13) to get estimable equations for observed bond prices. We can then use the cross-section of bond prices to estimate the parameter