Statistical Implications of Inflation Targeting

12 On Market-Based Measures of Inflation Expectations

Carol Carson, Claudia Dziobek, and Charles Enoch
Published Date:
September 2002
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INFLATION-LINKED financial securities can be used to infer market-based measures of expectations of future inflation, and they can also provide an indication of investors’ attitudes to inflation risk. Inflation-linked securities can be a useful alternative to surveys and econometric inflation forecasting approaches; they offer the advantage of being entirely forward looking, timely, and frequently updated for a range of maturities. This chapter considers the technical and institutional issues involved in the extraction and interpretation of market-based measures of inflation expectations for the United Kingdom, which are regularly presented to the Bank of England’s Monetary Policy Committee to inform its assessment of economic conditions.

In the United Kingdom, information on market inflation expectations can be derived from conventional and index-linked gilts, and from structured inflation-linked derivatives such as inflation swaps. By fitting real and nominal interest rates to conventional and index-linked gilts, it is possible to back out “break-even” inflation rates. Although these break-even rates are affected by institutional distortions, risk premiums (arising from investors’ aversion to inflation risk), and the mathematical effects of bond convexity, the expectations hypothesis of the term structure of interest rates suggests that break-even forward inflation rates are also driven by investors’ inflation expectations. These market forecasts of both future inflation and inflation risk premiums are useful for monetary policymakers.

The next section of this chapter briefly outlines the history of the price indexation of financial securities and looks at the inflation-linked debt and swap markets in the United Kingdom. Then it considers why investors should be concerned about inflation and outlines suggested criteria for effective inflation-proofing of financial securities. Subsequent sections explain the concept of break-even inflation rates and discuss the technical difficulties involved in extracting information about inflation expectations and risk premiums from bond prices. Complications arise from the indexation lag, the convexity of bond prices, and investors’ aversion to risk. The chapter also discusses institutional distortions that may drive prices away from economic fundamentals. Despite technical and institutional complications, break-even inflation rates do contain useful information for policymakers. Indeed, break-even inflation rates are regularly presented to the Bank of England’s Monetary Policy Committee. To gauge what incremental information can be extracted from break-even rates, this chapter compares the forecasting performance of two-year U.K. break-even inflation rates with that of two-year Basix inflation surveys. Longer-term break-even inflation forwards also provide a barometer of credibility for monetary policymakers. The last part of the chapter investigates 10-year-ahead one-year break-even forward rates for the United Kingdom since 1985 and finds that antiinflationary credibility is stronger since operational independence in monetary policy was given to the Bank of England in 1997.

The United Kingdom Index-Linked Gilt Market

A Brief History of Inflation-Linked Securities

Price indexation of financial contracts is not a new phenomenon. The idea of designing contracts to protect both parties from fluctuations in the price level dates back at least as far as 1780, when Massachusetts issued “depreciation notes” as wages to its soldiers during the American Revolution.1 In 1822 Joseph Lowe advocated that long-term contracts generally be settled in terms of an index number, or “tabular standard,” based on a “table comprising articles of general consumption to each of which is fixed the probable amount of money expended on it by the public.” This idea was taken up by Jevons (1898), who even argued the case for making indexation of long-term debts compulsory. There are four main arguments in favor of debt indexation:

  • to safeguard the real cost of borrowing and return on lending (ex ante benefiting both issuers and lenders);
  • to deliver cheaper ex ante debt funding (benefiting the issuer);
  • to provide an inflation hedge (expanding investors’ investment opportunities and generating general welfare improvements); and
  • to remove the monetary authorities’ incentives to reduce the value of the government’s debt through inflationary measures (thereby enhancing the monetary authorities’ anti-inflationary credibility).

In countries with high inflation, indexed debt may also provide access to and foster the development of long-term capital markets, though it has also been argued that debt indexation can perpetuate the inflationary process by encouraging inflation-linking of other financial contracts. Since 1980, however, indexed debt issues have largely come from countries with relatively low inflation. Recent issuers have included the United Kingdom (1981)—admittedly not a low inflation country at the time—followed, in order, by Australia (1985), Canada (1991), Sweden (1994), (significantly) the United States (1997), and France (1998).2

The U.K. Index-Linked Gilt Market

In 1980 the Chancellor of the Exchequer announced the government’s intention to issue index-linked stock. The index chosen was the general index of retail prices (RPI), the inflation measure already employed for uplifting state pensions. The first index-linked gilt was auctioned in March 1981, and although access to the index-linked market was initially restricted to pensioners and pension funds, by March 1982 it was open to all investors. Since then the index-linked gilt (ILG) market has grown steadily; by the end of 2001 the uplifted amount outstanding, at £70.5 billion, was more than 25 percent the size of the total outstanding gilt stock (£274.9 billion). Turnover, however, is much lower in the index-linked gilt market. In the fourth quarter of 2001, ILG turnover by transaction value was only £20.4 billion, around 4.2 percent of total gilt market turnover by gilt-edged market-makers.3 Bid-ask spreads are also around five times as wide.4 Nevertheless, the U.K. ILG market is special because of its maturity, size, and range of maturities. Table 12.1 allows a comparison with other major government index-linked bond markets, and shows that the U.K. market is second only to the U.S. market in terms of absolute size, though it has the most bonds (see Table 12.2). This is a major advantage because ILGs are numerous enough and distributed evenly enough along the maturity structure to permit one to fit a reasonably well-specified yield curve.

Table 12.1.Global Issuers of Inflation-Indexed Government Bonds
CanadaFranceSwedenUnited KingdomUnited States
Market value (in US$ billions)11.412.811.298.2135.4
Number of indexed bonds326118
Longest maturity (year)20312029202820302029
Weighted average real yield (percent)3.533.573.722.523.39
Source: Barclays Capital. Updated June 30, 2001.Note: Bonds of less than one year to maturity or with a market value of less than $100 million are excluded from these statistics.
Source: Barclays Capital. Updated June 30, 2001.Note: Bonds of less than one year to maturity or with a market value of less than $100 million are excluded from these statistics.
Table 12.2.United Kingdom: Outstanding Index-Linked (IL) Gilts


First Issue

Nominal Face

Value Outstanding

(£ millions)
IL Treasury Stock 20032.5005/20/0310/27/822,700
IL Treasury Stock 20044.37510/21/049/22/921,300
IL Treasury Stock 20062.0007/19/067/08/812,500
IL Treasury Stock 20092.5005/20/0910/19/822,625
IL Treasury Stock 20112.5008/23/111/28/823,475
IL Treasury Stock 20132.5008/16/132/21/854,635
IL Treasury Stock 20162.5007/26/161/19/834,965
IL Treasury Stock 20202.5004/16/2010/12/834,175
IL Treasury Stock 20242.5007/17/2412/30/864,820
IL Treasury Stock 20304.1257/22/306/12/922,600
Source: U.K. Debt Management Office,
Source: U.K. Debt Management Office,

Given the advantages of issuing index-linked debt, it is difficult to explain why the format has not been more widely adopted by private sector issuers. The sterling index-linked corporate bond market, for example, has less than £5 billion of stock outstanding. Part of the answer must be that, for many private issuers, index-linked debt does not help to match liabilities to corporate earnings. Issuing long-term index-linked debt can make little sense to a company with an earnings stream that may not be correlated with general inflation, and could merely increase uncertainty in financial planning. One exception (at least in the United Kingdom) is the various utilities sectors whose earnings are directly linked to the RPI through the price-capping formulas deployed by U.K. regulators. Indeed, most of the recent private sector index-linked sterling issues by private nonfinancial companies have been by water companies, electricity generators, and gas distribution companies. The non-gilt index-linked market, however, is not yet developed enough to allow comparisons with same-issuer conventional bonds, from which one might derive measures of market inflation expectations.

The U.K. Inflation Swaps Market

In recent years, investor demand has prompted the development of structured financial derivative products designed to deliver a hedge against price inflation.5 One of these products is the inflation swap, which is a bilateral contractual agreement requiring one party (the “inflation payer”) to make periodic floating-rate payments linked to the RPI inflation in exchange for predetermined fixed-rate “coupon” payments on the same notional principal from the “inflation receiver.” Inflation swap contracts are directly priced from the inflation forward rates implied by conventional and index-linked gilts.

Inflation payers are typically institutions with incomes linked to inflation. Examples include utility companies6 (whose incomes increase with inflation), private finance initiatives (with government-guaranteed cash flows linked to RPI), and guaranteed return products (which face higher capital gains taxes on indexed gains when inflation is low). Typical inflation receivers are investors with inflation-linked liabilities, such as pension funds and investors with liabilities on inflation-protected investment products.

Although a growing market, inflation swaps are “exotic” inflation-linked products, with structures generally tailored to the client’s particular requirements. Even though it is only a fraction of turnover in the index-linked fixed income market, inflation swap activity may well enhance that market’s liquidity by providing a hedging facility for investors.7 Market contacts report that trading is relatively infrequent and that the products are not sufficiently standardized to allow meaningful tracking and interpretation of historical prices.

Designing Inflation-Protected Debt Securities

Why Should Investors Be Concerned About Inflation Risk?

Inflation affects the current value of conventional fixed-income securities in two ways. First, anticipated inflation determines the expected real value of a fixed nominal income stream. Second, the risk of unanticipated inflation may further alter the price of a conventional bond—higher-than-anticipated inflation outturns, for example, reduce the real value of a fixed nominal income stream. Unanticipated inflation can therefore have redistributive consequences, transferring wealth between lenders and borrowers. Thus investors are concerned about both the level and the volatility of price inflation.

It is reasonable to expect markets to incorporate participants’ views of future inflation in prices payable today for conventional fixed-income securities. Investors are ultimately concerned about real returns, and therefore about the likely real value of an asset’s payoffs and the risks surrounding those payoffs. For a conventional bond held to maturity, investors measure real yield to maturity. When the holding period is shorter than the bond’s maturity, investors are concerned about expected real holding period returns.

In a certain and stable inflation climate, the nominal yield (Yn,t) on a conventional security with a given term of n at time t can be decomposed into a real yield (Rn,t) and an average inflation component n,t), to obtain the familiar compound-form Fisher equation:8

In reality, of course, the evolution of the general price level is a stochastic process. So conventional fixed-income assets leave both issuers and purchasers vulnerable to unexpected developments in the general price level. As Fisher (1911) put it, “The ideal is that neither debtor nor creditor should be worse off from having been deceived by unforeseen changes [in the price level].” In a world of unpredictable inflation, with risk-averse agents, the creation of a financial asset class that delivers an income stream of known purchasing power may deliver a previously unavailable inflation hedge, helping to complete the financial markets and generating welfare improvements for both issuers and lenders.9

As will be described later, under certain circumstances the volatility of inflation may drive a wedge between nominal yields on conventional financial instruments and yields justified by real rates and inflation expectations. An enhanced formulation of the Fisher decomposition that admits this possibility is:10

where pn, t is a broad risk premium term that captures a risk premium arising from investor aversion to inflation uncertainty, differences in liquidity between conventional and index-linked bond markets, and/or the effects of bond convexity (all of these factors are discussed more fully below).

Selecting the Reference Price Index

The choice of reference price index is critical in providing issuers and investors with real value certainty. In principle, bonds could be indexed to any of a number of variables, including price indices, commodity prices, foreign currencies, or wage or earnings measures. Price (1997) suggests that the selection of a reference index should be guided by a number of criteria:

  • The reference index should meet the hedging requirements of both issuer and investor, though often in practice these are not likely to coincide. Governments, for example, may prefer indexing debt to a broad price measure that is closely correlated to taxation and spending schedules, such as the GDP deflator. Retail investors, on the other hand, may wish to purchase protection against consumer price inflation. Institutional investors (such as pension funds) might want to match liabilities linked to earnings growth.
  • The index should be free of measurement bias. Price indices are subject to measurement and sampling errors and periodic reweighting. In the short to medium term, this may cause consumer price indices to be both an inaccurate and a sometimes upwardly biased reflection of the true cost of living.11 The upshot is that index-linked bonds might actually (on average) overprotect against inflation risk. Of course, if the biases were known and stable, bond prices could be expected to fully discount for the bias, and the distortion could be negligible. But if index measurement biases were unstable, investors might demand higher real yields on index-linked bonds to compensate.
  • The reference price index should be understood, recognized, and calculated by a body regarded as independent from the issuer (to avoid any possible conflict of interest). The bond prospectus should describe the index, allocate responsibility for its calculation, and detail the frequency and place of publication. The data behind the index should be reliable and transparent. In addition, the index should be free from regular revision, and, if such revisions occur, the procedures for dealing with payment calculations should be outlined in the prospectus.12
  • The indexation lag should be short. For price-indexed bonds to provide complete real value certainty, all cash flows would have to be corrected for changes in purchasing power right up to the moment at which they were due. In practice, however, unavoidable delays between actual movements in prices and adjustment to bond cash flows distort the inflation-proofing properties of indexed securities. Indexation lags produce a period at the end of a bond’s life when there is no inflation-proofing, counterbalanced by a period of equal length before issue for which inflation compensation is paid. Because inflation in the two periods is not likely to be the same, the real return on an indexed bond will not be fully invariant to inflation—the longer the lag and the greater the variability of inflation, the poorer the security’s inflation-proofing.13 Because real rates are then distorted, the information content from index-linked bonds will also be affected, with short- and medium-term bonds (which may be of particular interest to the monetary authorities) the worst affected.

In practice, most indexed government bonds have been linked to an index of consumer prices. The primary reasons are that consumer price indices reflect price developments faced by many bond investors, and they are generally well understood, widely disseminated, broad-based, rarely revised, and issued with a short time lag (which is important for pricing and trading in the secondary market).

Calculating Real Interest Rates and Break-Even Inflation

Conventional and index-linked bond markets allow one to derive real and nominal yield curves. From these nominal and real rates, it is then possible to calculate implied break-even inflation rates that provide a guide to market inflation expectations. This section describes how to derive real rates from index-linked bonds, from which one can then calculate break-even inflation rates.

Real, Index-Linked, and Nominal Bonds

It is useful to distinguish among three varieties of bond: nominal (or conventional), real, and index-linked (IL) bonds. A conventional bond is a bond that provides coupon and/or principal payments of prespecified nominal amounts. A real bond is a financial asset that provides coupon and/or principal cash flows with prespecified purchasing power. Index-linked bonds are financial securities whose coupon and/or principal payments are tied to a particular price index and hence are protected to some degree against general price inflation. Real (that is, perfectly indexed) bonds can therefore provide a certain real rate of return, while conventional government liabilities can provide certainty only with respect to nominal returns. Real bonds differ from index-linked bonds because of (1) measurement errors in the inflation index itself and (2) indexation lags.

Assuming no-arbitrage conditions for the prices of conventional, real, and IL bonds, the distinction between bond types can be illustrated more formally using a framework suggested by Evans (1998). Consider the price at time t of a semiannual coupon-bearing bond with n months to maturity, principal payment of £1, and a coupon rate of C. The nominal price of the conventional bond, Pn,t,N can thus be expressed as the sum of discounted future cash flows:

where Ij,t is an indicator function for coupon payments and dj,tN is the discount factor at time t for a prespecified nominal payment due j months ahead.

Now consider a world where the evolution of prices is subject to uncertainty. Let dj,tR be the discount factor applied at time t for a prespecified real payment (if uncertain in nominal terms) due at time t + j. Then if Πt represents the reference price level at time t, one can express the price of a perfectly indexed, or real, bond, Pn,tR, as:

In general, unless investors are risk-neutral with respect to inflation uncertainty, the discount factors applied to nominal payments derived from real versus conventional bonds will differ. This difference will be attributable to an inflation risk premium.

Price indices are calculated with a lag, whereas bonds should be tradable every working day. In order to allow calculation of accrued interest and thus permit trading, IL bonds are usually referenced to a lagged inflation index, rather than to the contemporaneous index level. When there is an indexation lag of l periods, and the reference price level becomes Πt-1, the nominal price at time t of an index-linked bond Pn,tIL is given by the following expression (when n > l):

The existence of an indexation lag of l periods means that, at any time t, any coupon payments due in the subsequent l months are known with certainty in nominal terms (and are therefore discounted using the same discounted factor used with conventional bonds). The present discounted value of these known nominal cash flows is represented by the first term on the right of equation (12.5). Coupon and principal payments due after period t + 1 remain uncertain in nominal terms, however, and are thus discounted the way real bonds are. The implication of an indexation lag is that in the final l months of its life, an IL bond effectively becomes a conventional bond, and that it should be priced accordingly. Evans’s (1998) insight that index-linked bond prices are a function of both real and nominal interest rates can thus be exploited to calculate a real yield curve.14

Break-Even Inflation Rates

If conventional and index-linked bond markets are efficient and are arbitraged by investors, such that both markets incorporate the same information about real interest rates, then the difference between nominal and real interest rates should contain information about investors’ expectations of future inflation. Indeed, in a world of deterministic real, nominal, and inflation rates, the difference between nominal and real rates is equal to the (certain) inflation rate over the same period—following the simple form of the Fisher identity. In reality, of course, interest rates and inflation are stochastic rather than deterministic processes. In these circumstances, implied inflation forward rates continue to be related to, but are not equal to, investors’ objective expectations of future inflation, for theoretical reasons dealt with in the next section. Implied inflation rates calculated in this way are better referred to as break-even inflation rates.

Calculating a break-even inflation spot rate for zero-coupon bonds is straightforward.15 The break-even inflation zero-coupon rate is the ratio of the zero-coupon yields on two same-maturity conventional and perfectly indexed bonds. Mathematically:

where zBEI is the break-even inflation zero-coupon, or spot, rate and zN and zR are the spot nominal and real yields. Break-even inflation is the average inflation rate that would have to occur over the life of the bonds for the uplifted index-linked bond to generate the same nominal return to maturity as the conventional bond—hence the term “break-even.” Another way to think of break-even inflation rates, though, is as scaling factors applied to future real payments to transform them into future nominal payments of equal present value. Looking at break-even inflation rates in this way suggests that for coupon bonds, break-even inflation rates should be calculated by comparing the yields to redemption on same-coupon, same-maturity index-linked and conventional bonds.

Aside from the U.K. government, no other government issuer currently has a sufficient number of outstanding index-linked bonds to permit estimation of a meaningful real yield curve. Consequently, for most countries it is not possible to estimate spot or forward break-even inflation rates, and one is limited to calculating crude break-even inflation yields from differences in redemption yields on particular conventional and index-linked bonds.16

Fitting Break-Even Inflation Rates

The United Kingdom is fortunate in having a sufficient number of index-linked government bonds to be able to fit a real yield curve. When that is combined with a nominal yield curve, one can derive break-even inflation yields. But the break-even rates obtained will be influenced by the choice of yield-curve fitting technique, and the differences between techniques will be most pronounced when there are relatively few bond prices to fit through.

When choosing a yield-curve fitting technique, the Bank of England looks for a technique that will deliver a relatively smooth forward curve, because the aim is to estimate market expectations for monetary policy purposes rather than to fit prices precisely.17 The technique should also be sufficiently flexible to capture movements in and key features of the underlying term structure. Finally, the yield curves produced should be stable, in the sense that fitted yields at one maturity should be robust to small changes in bond data at another maturity.

With reference to these criteria, Anderson and Sleath (1999,2001) set out the advantages of cubic spline approaches such as Waggoner (1997), over parametric approaches such as Nelson and Siegel (1987) or Svensson (1994, 1995). Rather than specifying a single functional form to describe forward rates, spline-based techniques fit a curve to the data that is composed of many segments, with constraints imposed to ensure that the overall curve is continuous and smooth. This is the principal advantage of spline-based techniques over parametric forms because subject to the continuity constraints, individual segments can move almost independently of one another. So while parametric approaches are often characterized by instability, particularly at longer maturities, spline-based techniques, being nonparametric, generally outperform in stability tests. Another disadvantage of parametric approaches is that the functional form can fail to capture key features of the underlying data. The Svensson (1994, 1995) technique, for example, constrains long forward rates to converge to a constant level. In reality, however, bond strips data show that the profile of long-term forward rates can often be downward sloping, a feature captured by spline-based techniques.

The Bank of England currently calculates real and nominal yield curves using a cubic spline incorporating a variable roughness penalty (VRP), which encourages forward curves to fit more closely at short maturities but promotes smoothness at longer maturities. In the index-linked gilts market, indexation lags are dealt with by incorporating a version of Evans’s (1998) analysis to extract real yields.

To demonstrate the impact that the choice of technique can have on the break-even rates derived, Figure 12.1 presents two break-even inflation forward curves for one particular date (December 20, 1999), using both the old Svensson technique and the new VRP technique.

Figure 12.1.United Kingdom: Break-Even Inflation Forward Curves


Source: Bank of England.

1VRP, variable roughness penalty (Bank of England).

Technical Complications

The expectations hypothesis of the term structure states that in a deterministic world, expected rates of return on different maturity bonds are equalized only when all forward rates equal expected short-term interest rates. Combined with the efficient market hypothesis—which has several forms, all of which require investors to use information efficiently—the pure expectations theory states that the observed market forward rate curve provides the best forecast of future spot rates.

Of course, in reality, the world is not deterministic. But in a complete and efficient market without distortions, break-even inflation forward rates should be determined by three factors: (1) inflation expectations, (2) the convexity adjustments present in conventional and index-linked bonds, and (3) the inflation risk premium. This section considers how convexity biases and risk premiums can drive a wedge between breakeven inflation forwards and true inflation expectations. Given that one is interested in the extraction of inflation expectations from conventional and index-linked bond prices, it is natural to concentrate on break-even inflation forward rates.

This section discusses in some detail the complications posed by bond convexity adjustments and time-varying inflation risk premiums. Though these concepts are referred to later, readers may wish to skim the technical details or move directly to the section below on institutional distortions.

The Convexity Adjustment

Modern theories of the term structure demonstrate that, even in the absence of risk premiums, when interest rates are stochastic the convexity of bond prices18 means that Jensen’s Inequality19 drives a wedge between forward rates and objective expectations of future short rates. Put simply, interest rate compounding means that bond prices are convex. The combination of bond convexity and interest rate volatility raises bond prices, which pushes forward rates below expectations of future short rates. This effect is known as the convexity bias, and it grows with maturity (as compounding increases) and can vary across time (as yield volatilities change).

In a risk-neutral world, the price of a zero-coupon bond may be written as an expectation of the value of the bond’s payoff, discounted at future short rates. But this expectation must be taken under the risk-neutral probability measure. Using annual compounding for ease of exposition, the bond price may be written as:20

where {r(s), sε[t,T]} is the path of future one-year rates between the current time t and the maturity of the bond T, and the expectation is taken at time t using the risk-neutral probability measure (indicated by the tilde on the expectations operator). By definition, forward rates, {f(s), sε[t,T]}, are related to bond prices in the following way:

Because the pricing function, P(.), is convex, Jensen’s Inequality implies that:

which, because all zero-coupon bonds are convex, implies that f(s)E˜t[r(s)] for all s.

So when short rates are volatile, then, even under risk-neutrality, Jensen’s Inequality pulls forward rates below expected future short rates. This effect is likely to be more pronounced at long maturities, where convexity is greatest because of higher levels of compounding.

The implication is that break-even inflation forwards may be driven away from actual inflation expectations by differences in the convexity adjustment applying to real and nominal forward rates. For example, if the convexity adjustment for the nominal forward curve dominated that for real forwards (perhaps because inflation was adding to the volatility of nominal rates), then one might expect the net convexity adjustment also to bias long-term break-even inflation forwards below true expectations.21

The Inflation Risk Premium

When held to maturity, real bonds can provide real value certainty, and conventional bonds give nominal value certainty. So the return to maturity on a conventional bond is fixed in nominal terms, but it is uncertain in real terms as a result of inflation risk. Because investors are ultimately concerned about the risks surrounding real returns, they may be willing to pay a premium for a security that provides real value certainty. In theory, the extent to which inflation risk commands a premium depends on the covariance between inflation—and thereby real returns on a conventional bond—and the discount factor the market applies to real wealth in future states of the world, the “pricing kernel.”

Perhaps the most difficult task in extracting expectations from yield curves is accounting for time-varying risk premiums. Yet one also wants risk premiums to satisfy the no-arbitrage condition. For any asset pricing model that rules out arbitrage opportunities, the price of any financial security can be expressed in terms of a common pricing variable, referred to as the real stochastic discount factor, or pricing kernel.22 The real pricing kernel attaches a real value today to payments received in each possible future state of the world, regardless of the asset type. The general no-arbitrage asset pricing condition is:

where Et is the conditional expectations operator, Ri,t+1 is the real return on some asset i, and Mt+1 is the real stochastic discount factor (pricing kernel) for payments in period t + 1. In models with utility-maximizing investors, Mt+1 measures the intertemporal marginal rate of substitution. Appendix 12.1 demonstrates that the expected return on any asset is negatively related to its covariance with the real pricing kernel as follows:

where Rf is the risk-free real rate of interest. Financial assets with real returns that are positively correlated with the pricing kernel tend to deliver high real payoffs when the discount factor applied to future consumption wealth is also high (that is, because anticipated future consumption wealth is high and therefore marginal utility is low). Such assets increase portfolio risk by tending to deliver real wealth precisely when it is least valued, and vice versa, and investors demand a premium to hold them. Conversely, assets that covary negatively with consumption, such as insurance, can offer expected rates of return that are lower than the risk-free rate.

If unanticipated inflation is (perceived to be) negatively correlated with consumption-wealth on an investor’s entire portfolio, then a risk-averse investor is willing to pay a premium for a security that is protected against inflation risk. Acquisition of conventional fixed-income securities in such circumstances contributes positively to overall consumption-wealth risk, and risk-averse investors require an incentive to hold such assets. This higher ex ante real yield priced by the market on conventional securities over maturity-matched real securities is the inflation risk premium.

The inflation risk premium (IRP) can be defined as the excess expected real rate of return on nominal over real (that is, perfectly indexed) debt. More formally, denoting real returns on conventional and real bonds by RN and RR, respectively, and the conditional expectations operator by Et, one can define the inflation risk premium as an expected excess holding period (real) return:

where n is the remaining time to maturity, t represents the current time, and h equals the holding period over which returns are calculated. Thus IRPn,t(h) is the additional compensation demanded by investors for bearing inflation risk on an n-maturity conventional bond over the period t to t + h.23

It is a given that the real returns on an n-period real pure discount bond are related to real bond prices in the simple way (1 + Rn,t+1) = Pn-1,t+1/Pn,t. Substituting this into (12.9) satisfies:

which by repeated substitution can be expressed as:

This pricing kernel approach applies naturally to fixed-income securities. When cash flows are random, the stochastic properties of the cash flows help to determine the covariance of an asset’s return with the pricing kernel. But a conventional fixed-income security has deterministic nominal cash flows, so returns vary with the pricing kernel only because there is time-variation in discount rates. Because this variation in discount rates is driven by the time-series behavior of the kernel, term structure models are equivalent to time-series models for the pricing kernel.

This pricing-kernel framework allows Anderson and Talbot (2000) to derive expressions for the inflation risk premium, conditional on holding period (h) and bond maturity (n). When the holding period is equal to a bond’s time to maturity (h = n), inflation risk consists entirely of uncertainty over the real redemption value of the nominal bond. But when the holding period is shorter than the time to maturity (h < n), investors at time t are concerned about the impact of inflation on (1) payment streams before sale and (2) the real value of the bond at sale, in period t + h. This second component is determined both by actual inflation over the period t to t + h, and by revisions to expectations of inflation for the remaining life of the bond.

Under the assumption that the real pricing kernel and the future price level are individually and jointly lognormal, one can derive an expression for the one-period inflation risk premium on a one-period conventional bond (h = n = 1):24

where mt+1 = ln(Mt+1) and πt+1= lnt+1). Again, if inflation is positively correlated to the kernel, then real returns on the nominal bond (which in the one-period case are inversely related to inflation) are low precisely when consumption is already low and the investor most values real wealth. Purchase of a nominal bond, and its associated inflation risk, therefore increases the overall risk of future real wealth, for which the risk-averse investor demands compensation, by means of an inflation risk premium.

However, it is worth noting that consumption-based asset pricing models have generally produced unrealistic investor risk preferences when calibrated to prices. Intuitively, it seems sensible to expect the market price of inflation risk to be positive. From a bond investor’s viewpoint, unanticipated price inflation erodes real wealth, and it is not obvious that this loss is offset by an association between high inflation outturns and times of strong economic growth when consumption income from other sources will be high. Risk-averse fixed-income investors should therefore be willing to pay for the consumption-smoothing properties of indexed debt. The government (as a major debt issuer), on the other hand, may be far less concerned about inflation uncertainty than bond investors are, and may therefore be relatively more willing to include conventional bonds in its balance sheet. First, government taxation income stream is closely tied to general prices and second, government debt managers (unlike individuals) do not face a finite investment horizon. Similarly, most corporate debt issuers evidently prefer to issue conventional rather than indexed liabilities to match their asset structure. The net effect is that investors are likely to pay a premium on index-linked bonds over conventionals.

Modeling the Inflation Risk Premium

The pricing-kernel analysis outlined above suggests that, in theory, the asset pricing problem reduces to one of identifying the correct pricing kernel for the economy. An estimate for the inflation risk premium (IRP) could then be obtained by modeling the conditional covariance between the estimated pricing kernel and inflation.

Several approaches suggest themselves. The first would be to derive estimates for the kernel from first principles using a consumption-based capital asset pricing model (CCAPM). In the CCAPM, for instance, the pricing kernel is the product of the coefficient of relative risk aversion and the growth of aggregate consumption. This method requires explicit specification of a “representative” time-separable Von-Neumann Morgernstern expected utility function defined over consumption-state-space, which could then be calibrated to obtain parameter estimates. This is the approach taken by Chan (1994), who uses a power utility function version of the CCAPM to obtain small but significant estimates of the IRP for the United States.

The second approach, rather than trying to estimate the pricing kernel directly, is to identify a set of state variables that are assumed to be proportional to the kernel, and that could then be used to measure the IRP indirectly. This is the approach taken by Campbell and Shiller (1996) for the United States, where variables suggested by asset pricing theory (aggregate consumption growth and value-weighted “market portfolios”) are used in CCAPM and CAPM frameworks to obtain estimates of the risk premiums on five-year nominal bond returns, which they assume to consist mostly of IRPs.

Though the advantage of these asset pricing approaches is that IRP estimates should have a clear economic interpretation, being derived from explicitly specified asset pricing models, the major disadvantage is that these models may easily be misspecified. In addition, CCAPM- and CAPM-based measures require reliable estimates for the coefficient of relative risk aversion, aggregate consumption growth, and the equity risk premium. In addition, a general weakness of these calibrated models is that both parameters and conditional covariances are derived from historical data, which might not deliver risk premium estimates appropriate for current and expected future economic conditions.

A third way of deriving IRP estimates is to model the time-series behavior of both inflation, πt, and the real stochastic discount factor, mt, using econometric methods to identify the time-series processes, with particular emphasis on covariances. Term structure models in which suitably rich stochastic processes for mt and πt are specified are equivalent to time-series models of the (log) pricing kernel and log rate of inflation. The main difficulties are identification of relevant state variables, and modeling of their stochastic processes. The great advantage of this approach, however, is that ex ante explicit modeling of stochastic processes allows direct ex ante pricing of inflation risk. The main difficulty with term structure models, though, is maintaining analytical tractability at the same time one is attempting to allow time-varying conditional covariance between possibly unspecified factors whose stochastic processes also need to be estimated.

Campbell and Viceira (2001) apply a two-factor no-arbitrage affine yield model of the term structure, where innovations to expected inflation are correlated with innovations in the log pricing kernel through a common shock. Using a generalization of the discrete-time version of the Vasicek (1977) model, they allow for nonzero correlation between innovations in the real interest rate and inflation factors. However, in order to keep their model in the class of affine yield models for which there exist analytical solutions, Campbell and Viceira (2001) assume that the inflation risk premium is constant. This permits estimation of the underlying parameters by a Kalman filter, with another advantage being that the pricing kernel does not need to be specified. Unfortunately, the model uses actual inflation data rather than information from the real term structure.

Gong, Remolona, and Wickens (1998) propose an alternative two-factor affine yield model (a generalization of the discrete-time Cox and Ingersoll (1985) model) that uses information from both the nominal and real term structures. No-arbitrage conditions are imposed to obtain simultaneous estimates of forward-looking expectations, risk premiums, and convexity adjustments for both real interest rates and inflation. Unfortunately, though their specification allows for time variation of risk premiums through the use of autoregressive conditional het-eroscedasticity (ARCH) processes for factor shocks, the inflation risk and real interest rate risk factors are assumed by construction to be independent. The direct implication of assuming zero correlation between inflation and real interest rates is that, by the pricing kernel definition used here, the IRP is therefore actually assumed to be zero.

Using a foreign exchange analogy that allows one to consider the price level as an exchange rate that transforms real prices to nominal prices, Panigirtzoglou (2001) presents an arbitrage-free affine yield model with three latent factors. The real and nominal pricing kernels are each driven by two unobserved factors, where one factor is common to both kernels. The factors follow a discrete-time version of the Cox and Ingersoll (1985) diffusion process. The price level serves as the exchange rate that ensures the two pricing kernels are identical when measured in a common “currency.” Inclusion of a common factor allows for time-varying risk premiums and nonzero time-varying correlations. Panigirtzoglou’s is the Bank’s favored approach, because it uses price information from both the index-linked and conventional bond markets in a framework that permits commonality between the factors driving inflation and real interest rate risk, and it allows risk premiums to vary with time.

By fitting the model to data from the Bank of England’s conventional and real yield curves, Panigirtzoglou obtains estimates for the inflation risk premium, estimating the ex ante two-year inflation risk premium at around 100 basis points for most of the period from 1987 to 1997, but finding that by 1998 it had fallen to around zero. It is reassuring, also, that all of the approaches outlined above generate inflation risk premium estimates of comparable magnitude—though there are of course differences in country, maturity, and sampling. Table 12.3 summarizes the inflation risk premium estimates provided by each of these modeling approaches.

Table 12.3.Comparison of Inflation Risk Premium (IRP) Estimates
ApproachInflation Risk Premium (IRP) Estimate1
Campbell and Shiller (1996)(U.S.) 5-year IRP: range 60–150 bp.
Sample: 1953–1994.
Gong, Remolona, and Wickens (1998)(U.K.) 2-year IRP:
100 bp between 1982 and 1997;
70 bp between Sept. 1992 and 1997.
Campbell and Viceira (2001)(U.S.) 10-yr IRP: 60 bp.
Sample: 1952Q1–1996Q3.
Panigirtzoglou (2001)(U.K.) 2-year IRP:
100 bp between 1990Q1 and 1997Q1;
Falling to 0 bp between 1997Q1 and 1998.

IRP estimates are denoted in basis points (bp). One basis point is one hundredth of a percentage point.

IRP estimates are denoted in basis points (bp). One basis point is one hundredth of a percentage point.

Institutional Distortions to Break-Even Inflation Rates

In theory, break-even inflation rates derived from conventional and index-linked government bonds should reflect rational expectations of future inflation plus an adjustment for inflation risk in the way outlined in the previous section. Under certain conditions, however, the break-even inflation rates can be distorted by institutional considerations. The first of these is the potential impact on relative prices of differences in tax treatment between conventional and index-linked bonds. Second, institutional distortions may create price-inelastic demand for gilts, which, when combined with a constrained supply, might drive break-even forward rates away from levels consistent with economic fundamentals.


Because investors are concerned about net-of-tax real cash flows, taxation can produce differences in the prices that investors with identical views and risk preferences are willing to pay for the same fixed-income security, depending on their tax status. By influencing the prices investors will pay for bonds, taxes complicate the calculation of net-of-tax yields and break-even inflation rates. Furthermore, the variety of possible investor tax profiles means that it may not be clear what assumptions to make when trying to calculate net-of-tax yields for a representative investor.

Tax authorities have to decide whether income and capital gains taxes should be applied to nominal or real cash flows—in other words, whether taxes should be levied on the inflation uplift for coupon and principal payments. Because real value certainty is the most important characteristic of indexed bonds, though, a tax system that taxes that inflation uplift effectively reintroduces inflation risk. Under such a system, even if pretax real yields remain constant, an increase in inflation that raises the nominal yield on indexed bonds increases the tax liability and lowers the post-tax real yield.

In the United Kingdom, for tax purposes the inflation uplift on the principal is considered a capital gain (note that gilts are exempt from capital gains tax). But the uplift on coupon payments is treated as income, and taxed accordingly. The implication is that the post-tax real returns on index-linked gilts are not entirely protected from erosion by high inflation outturns.

Of course, calculation of net-of-tax real yields faced by the marginal investor (who sets the price) requires one to make assumptions about the marginal tax rates faced by that investor. In the United Kingdom, most of the conventional and index-linked gilt stocks are held by largely tax-exempt institutional investors. So if one assumes these tax-exempt institutional investors to be the marginal purchasers of gilts (and being largely tax-exempt they should be willing to pay the most), then it is not unreasonable to set aside tax considerations when looking at implied break-even rates, at least in the United Kingdom.

Other Institutional Considerations

At the end of 2000, U.K. life assurance and pension funds (LAPFs) held around 65 percent of the £320 billion of the outstanding gilt stock. Consequently, the portfolio allocation decisions of these institutions could have significant effects on gilt prices. In the United Kingdom, a number of factors may have helped generate price-inelastic demand for gilts from LAPFs. In particular, pension funds have raised their holdings of gilts in response to (1) demographic aging of the U.K. population (actually a fundamental factor), (2) the introduction of minimum funding requirement legislation, (3) the need to hedge old policies with (previously unhedged) guaranteed annuity rates, and (4) the practice of appraising pension fund and bond portfolio managers’ performance against either industry peer group or gilt yield benchmarks, thereby providing an incentive to hold gilts.

In April 1997 government legislation designed to ensure that defined benefit pension funds would protect fund members in the event of the employer becoming insolvent came into force.25 The minimum funding requirement (MFR) was designed to ensure that a scheme would have sufficient assets to be able to fully protect pensions already in payment and provide younger members with a transfer value that would give them a reasonable expectation of replicating scheme benefits if they transferred to another pension scheme.

The MFR values a fund’s assets at current prices by marking to market. However, to ensure that defined benefit schemes hold sufficient assets to meet their liabilities, the MFR applies a set of liability valuation rules linked to yields on a set of gilt indices.26 Although it does not actually require pension funds to purchase gilts, legislation that requires the use of 15-year conventional gilt and 5-year index-linked gilt indices as discount factors for valuing liability also generates strong incentives for defined benefit pension funds to hold these gilts on the asset side.27 Matching assets and liabilities in this way, by making the same discount rates common to both, diminishes the likelihood that fluctuations in financial prices will result in the fund becoming underfunded. Furthermore, work at the Bank of England has uncovered some evidence that the widespread use of FTSE gilt indices can also prompt gilt prices to respond to compositional changes.28 By influencing the demand for gilts in this way, it is possible that the MFR and the use of FTSE gilt indices may have distorted (and continues to distort) implied break-even inflation rates at certain points along the yield curve.29

The distortionary impact of price-inelastic demand from the pension fund industry has arguably been aggravated by concerns about the outlook for future new supply and the outstanding stock of government debt. In the United Kingdom, net debt issuance as a percentage of GDP has been shrinking since the first quarter of 1996 and has been negative since 1997. A diminishing supply of government debt in the United Kingdom, together with a shortage of alternative high-quality long-dated fixed-income sterling securities (such as supranational or high-grade corporate paper), combined with strong inelastic demand from institutional investors, may have driven prices out of line with economic fundamentals.30

Indications of institutional distortions might be obtained from (1) international comparisons of break-even inflation rate levels, and (2) break-even inflation forward curve profiles for sterling. Figure 12.2 provides an international comparison of break-even inflation rates on selected index-linked government bonds since 1994.31 In fact, it is not clear, when one looks at absolute levels, that the sterling break-even inflation yield for the 2011 index-linked gilt is out of line with break-even rates for other economies at similar maturities. Furthermore, any divergence could be attributed to economic fundamentals and investor preferences rather than to institutional distortions.

Figure 12.2.International Break-Even Inflation Yields Derived from Government Securities


Source: Bloomberg.

But it is also worth looking at the profile of break-even inflation forward curves.32 During the period covered by the MFR, one might expect to see conventional gilts at and around 15-years’ maturity trading at relatively expensive levels, driving down nominal spot and forward rates. At the same time, one might also observe episodes with price discontinuities between index-linked gilts either side of the five-year maturity mark translating into “humped” real forward curves. So nominal and real interest rate and break-even inflation forward profiles, such as for December 20,1999,33 (Figure 12.3) suggest that the MFR was affecting the conventional and index-linked markets. Note that forward curves such as those for December 20, 1999, were not atypical during the period 1998 to 2000.

Figure 12.3.United Kingdom: Forward Curves for December 20, 1999


Source: Bank of England.

Note: Break-even inflation, nominal and real one-year forward curves are calculated using the Bank of England’s VRP spline-based technique (see Anderson and Sleath, 2001).

Looking at Figure 12.3, one has to question whether investors could really have had sufficient information to foresee inflation following the path indicated. It is difficult to believe that investors anticipated inflation 15 years ahead to be lower than in 20 or 25 years’ time. Arguably, break-even inflation forward curves such as the one presented in Figure 12.3, taken during a period of low and stable inflation, are difficult to reconcile with rational expectations of future inflation.34 More likely, inflation forward profiles such as that for December 20, 1999, reflect the various distortions in the gilt markets, and they provide a salutary lesson for those wishing to extract inflation expectations from break-even inflation rates. The reality is that it is difficult to isolate and quantify the distortions that can affect break-even inflation rates.

Extracting Information from Break-Even Inflation Rates

Break-Even Inflation Rates as Forecasts of Inflation

Subject to the caveats noted above, an advantage of break-even inflation rates is that they can provide an indication, unavailable elsewhere, of investors’ views of the longer-term inflation outlook. But monetary policymakers are also interested in inflation over the short-to-medium term. So, despite the theoretical and institutional complications that drive a wedge between break-even inflation rates and true inflation expectations, it is interesting to compare the forecasting performance of break-even inflation rates with survey-based measures of inflation expectations.35

Break-even inflation rates can be compared against the Barclays Basix surveys of expectations for RPI inflation over the next two years. Comparisons at shorter maturity horizons are not possible because the absence of bond data points makes the real yield curve too unreliable below two years. The Basix survey samples a number of groups, including business economists, investment analysts, academic economists, trade union economists, and the general public. This study considers only the measure that excludes the general public.36

Previous work at the Bank of England has investigated the information content of break-even inflation rates. Breedon and Chadha (1997) found that the Bank of England’s inflation term structure (ITS) derived from Svensson curves gave a somewhat better indication of the bond market’s inflation expectations than could be derived using either the nominal term structure or a variant employing strong assumptions about real interest rate behavior. The inflation forecasts of the ITS also seemed at least as good at forecasting future changes in inflation as forecasts derived from macroeconometric models.37 The authors concluded that these characteristics, and the timeliness of break-even rates, made break-even inflation forward rates a useful addition to policy analysis.

Break-even forward rates therefore seem useful. But one could also ask whether short-term break-even inflation rates (despite institutional distortions, inflation risk premiums, and convexity biases) represent rational inflation forecasts. “Weak” rationality, as defined by Brown and Maital (1981), requires that inflation expectations should be unbiased and efficient predictors of actual inflation. “Strong” rationality requires that the forecasting error should be orthogonal to other variables in investors’ information set.

This section investigates the comparative forecasting performance of break-even inflation rates and inflation surveys. A major practical complication, however, arises from overlapping forecasting horizons, which produce a moving average forecast error process. If agents make forecasts for h periods ahead, forecast errors are only discovered with a lag. At time t, the last fully revealed forecasting error is for the forecast made at t - h, and this error is likely to be less relevant or important for current forecasting purposes than the not yet fully realized error made at t - 1. The result is an MA(h - 1) forecast error process, which might cause a rational expectations formation process to resemble an adaptive expectations formation process. This may give the impression that investors are not responding to consistent errors—although in fact investors simply do not learn of these errors until much later. If agents are rational, then although forecasting errors may be serially correlated over short time windows, the long-run error process could be accepted as stationary. With a small sample and a long-run moving average error process, however, it is possible that the forecasting error process might be mistaken as nonstationary.

Figure 12.4 plots (monthly) two-year zero-coupon break-even inflation rates and (quarterly) Basix survey inflation forecasts against the actual (monthly) two-year RPI inflation outturn. As set out above in the sections on break-even inflation rates, the two-year break-even inflation spot rates are calculated from the Bank of England’s fitted yield curves, with the two-year break-even spot being the ratio between two-year nominal and real spot rates. Mathematically:

where z2,tBEI is the fitted break-even inflation spot rate and z2,tN and z2,tR are the fitted nominal and real spot rates at time t. Note that survey and break-even inflation forecasts for the two years ahead are plotted against the RPI values for the two years just ended.

Figure 12.4.United Kingdom: Two-Year Break-Even Inflation and Basix Survey Spot Rates Against Retail Price Index (RPI) Inflation Outturns


Source: Bank of England; Barclays; and Office for National Statistics.

Note: Two-year break-even inflation zero-coupon rates are calculated by the Bank of England (see Anderson and Sleath, 2001; and Deacon and Derry, 1998). The two-year variable roughness penalty (VRP) break-even inflation series is broken due to occasional absence of suffiently short index-linked bonds. Break-even and survey rates are for two years ahead, whereas two-year RPI values are for RPI over the two years ending.

Certain features of the data are worth mentioning. First, both the survey and break-even series underpredicted actual RPI inflation outturns during 1989–91 but generally overpredicted inflation after 1991. Second, two-year break-even inflation rates track current two-year RPI inflation far more closely than survey forecasts. Third, both break-even inflation and survey forecasts have been falling since 1990, though the adjustment process appears to have been lagged (and slow) when compared with actual RPI inflation. This apparent delay in the forecast error correction process is consistent with an overlapping forecast horizon problem (that is, a slow error-discovery process). Fourth, two-year spot break-even inflation and survey rates have differed often quite considerably during the sample period. Finally, revisions to survey expectations have been less volatile than those of break-even inflation rates.

An important feature of the data is the possible structural break in the differential between the break-even and survey inflation series; this is best seen in Figure 12.5. The difference between surveys of two-year inflation expectations and the break-even inflation rate implied from bond prices can be used as a proxy for the inflation risk premium. Before the third quarter in 1992, break-even inflation rates were consistently above survey expectations (on average by 1.89 percentage points). After this date, however, this differential became negative, though smaller in absolute size (on average -0.42 percentage points), as survey respondents raised their forecasts of two-year inflation after the third quarter of 1992. This apparent structural break roughly coincides, of course, with sterling’s ejection from the exchange rate mechanism (ERM).38

Figure 12.5.United Kingdom: Difference Between Two-Year Break-Even and Survey-Based Inflation Expectations

(Percentage points)

Source: Bank of England; and Barclays.

Note: The differential is the two-year break-even spot rate minus the (quarterly) two-year spot Basix survey (excluding general public).

This break in the break-even/survey differential series also poses a puzzle, because sterling’s ejection from the ERM and the associated loss of policy credibility could have been expected to drive up the inflation risk premium and thus to have widened rather than narrowed the differential, at least until the inflation-targeting framework had become established. An alternative explanation, of course, is that the United Kingdom’s abandonment of exchange rate targeting in favor of an inflation-targeting policy could have been expected to lower short-term inflation volatility, and therefore to immediately reduce the short-term inflation risk premium (even if investors had become more averse to a given level of inflation risk and the market price of inflation risk had risen). This argument allows for a simultaneous fall in the short-term inflation risk premium and a reduction in long-term policy credibility.

Formally, the rational expectations hypothesis (REH) states that actual and expected inflation should be related such that

where πt+h is the actual inflation rate at time t + h;πt,t+he denotes expected inflation at time t + h, conditional on the information set at time t; ut+h is a white-noise error term, ut+h ~ IN(0, σu); and α and β are parameters. Weak rationality requires both that α = 0 and β = 1 (that is, forecasts are, on average, unbiased) and that corr(ut+h, ut+h-s) = 0 if s > h and corr(ut+h, ut+h-s) ≠ 0 if s≤h-1 (that is, forecast errors that do not overlap are uncorrelated).

Weak rationality requires that, at least in the long run, the regression parameters are α = 0 and β = 1. However, inference on long-run estimates of α and β is complicated by the moving average error processes produced by the overlapping forecast horizons.

Furthermore, this approach is perhaps too simplistic; one would ideally model the expectations formation process as an error correction model with lags. In addition, the sample is perhaps not long enough to allow this simple approach to accurately capture the long-run relationship.39

Table 12.4 looks at the very simple ordinary least squares regressions of break-even and survey-based forecasts against actual inflation, with Newey-West standard errors to allow for the serial correlation caused by overlapping forecast horizons. It considers a sample from September 1982 to November 2001. RPI data are released monthly, and break-even rates can also be calculated monthly. Basix survey data, on the other hand, are only available quarterly. Regressing two-year break-even inflation on two-year RPI inflation produces a significant constant value of 2.192 and a significant slope coefficient of 0.772. By comparison, a regression of survey-based expectations on RPI inflation produces a significant constant term value of 3.130 and a slope coefficient of 0.319. Note that Wald tests on the null hypothesis that α = 0 and β = 1 conclusively rejected the null hypothesis for both regressions, suggesting that neither break-even inflation rates nor surveys are rational forecasts.

Table 12.4.Ordinary Least Squares Regressions of Break-Even and Survey-Based Forecasts Against Actual Inflation (U.K.)
Regression: πt,t+he = α + βπt+h + ut+h
Dependent variable, πt,t+heBreak-even inflation 2-year spotBasix survey 2-year spot
Independent variable, πt+hRPI inflation 2-year spotRPI inflation 2-year spot
Lag, h24 months8 quarters
Newey-West s.e.(0.503)(0.374)
Newey-West s.e.(0.105)(0.067)
Adjusted R20.5420.480
Observations207 (monthly)59 (quarterly)
Wald tests ofF = 20.0**F = 79.3**
linear restrictions:χ2 = 40.0**χ2 = 158.6**
H0: α = 0 and β = 1
Note: αt+h, = Actual inflation rate at time t + h; πt,t+he = expected inflation at time t + h; ut+h, = white-noise error term; ut+h, α, β are parameters.

* Indicates significance at the 5 percent level, and ** indicates rejection of the null hypothesis at the 5 percent significance level. Newey-West standard errors are computed with a uniform window size equal to the length of the moving average error process. Sample: September 1982-November 2001.

Note: αt+h, = Actual inflation rate at time t + h; πt,t+he = expected inflation at time t + h; ut+h, = white-noise error term; ut+h, α, β are parameters.

* Indicates significance at the 5 percent level, and ** indicates rejection of the null hypothesis at the 5 percent significance level. Newey-West standard errors are computed with a uniform window size equal to the length of the moving average error process. Sample: September 1982-November 2001.

But, of course, there are some good reasons to expect such results. First, given the developments in the U.K.. monetary policy framework and the move from relatively high inflation to low inflation during the sample period, it is hardly surprising that regressions of survey and break-even inflation rates on actual inflation produce ex post upward biases in the constant and downward biases in the slope coefficients. It is quite possible that investors are simply taking time to learn about changes in the inflation regime, and that these biases merely reflect the small size of the sample. A second explanation for the positive constant term in regression [A] in Table 12.4 is the possible presence of a (time-varying) positive inflation risk premium.

Furthermore, although short-term break-even inflation rates are not perfect forecasts of inflation (because of time-varying inflation risk premiums and lags in error correction), the Bank’s analysis does indicate that break-even inflation rates are superior to Basix surveys in terms of forecasting performance, and therefore they may be a useful source of information on short-term inflation expectations for monetary policymakers.

Break-Even Inflation Rates as a Measure of Central Bank Credibility

Investors’ longer-term expectations of inflation depend on their confidence in the ability and determination of the monetary authorities to control inflation. Break-even inflation rates may not be easily decomposed into inflation expectations, inflation risk premiums, and convexity biases, but these components are linked to investors’ views and preferences about the level and volatility of future inflation. As King (1995) notes, “Both the government and private sector have subjective distributions over the possible outturns for inflation at any future date. Credibility is a measure of how close are these two distributions.” One can summarize the private sector’s distribution by its mean—the expected inflation rate—and the spread of possible outturns around the mean, as proxied by the inflation risk premium. Because break-even inflation rates capture each of these components, they are a potentially useful indicator of antiinflationary credibility.

For the purpose of assessing monetary conditions, forward inflation rates are more informative than average rates of inflation, because they allow policymakers to assess both the expected average rate of inflation and its evolution over time. Implied break-even forward rates can be used to assess the impact of monetary policy on inflation credibility. To illustrate this point, Figure 12.6 presents one-year break-even inflation forwards 10 years ahead since 1982. Note the impact of two major developments in monetary policy over the period, notably the United Kingdom’s exit from the European exchange rate mechanism (ERM) in September 1992 and the concession of operational independence to the Bank of England in May 1997, together with the establishment of a 2.5 percent RPIX inflation target.40

Figure 12.6.United Kingdom: One-Year Break-Even Inflation Forward Rate Ten Years Out Since 1985


Source: Bank of England.

Note: Break-even inflation forward rate derived using the Bank of England’s VRP technique (see Anderson and Sleath, 2001).

One can see that the United Kingdom’s exit from the ERM in 1992 had a dramatic impact on market confidence, driving up break-even inflation forwards by 190 basis points. This indicates that the loss of the ERM’s external discipline on policy had a serious negative impact on the credibility of U.K. monetary policy with the financial markets. Also note, however, that 10-year-ahead break-even forwards had returned to their early 1992 levels by late 1993, as the new inflation-targeting policy became established.

In May 1997 the Chancellor of the Exchequer announced that the Bank of England would be given operational independence in the conduct of monetary policy, with a remit to achieve, on average, 2.5 percent RPIX inflation. Spiegel (1998) performed an event study into the impact of central bank independence on inflation expectations. Looking at a pair of index-linked and conventional bonds with maturities in 2016, Spiegel found that the break-even inflation spot rate fell by 34 basis points on May 6, the day of the announcement, and by 60 basis points over the longer two-week event window. But credibility generally takes longer to establish than it does to lose, and as the Chancellor of the Exchequer, Gordon Brown, stated at the time, “The ultimate judgment of the success of this measure will not come next week, or indeed in the next year, but in the long term.” Since May 1997, 10-year break-even inflation forwards have fallen by around 150 basis points, and they are very close to the government’s 2.5 percent inflation target.

Summary and Conclusions

This chapter has outlined how inflation-linked securities can be used to infer market-based expectations of future inflation. As a source of information on inflation expectations, inflation-linked securities provide an alternative to surveys and econometric forecasting approaches, with the advantage that they are available for a wide range of maturities, entirely forward looking, timely, and updated every working day.

In the United Kingdom, market inflation expectations can be directly derived from inflation swaps (albeit with some difficulty) or from a comparison of conventional and index-linked gilt prices. By fitting real and nominal yield curves to conventional and index-linked gilts, it is possible to infer zero-coupon and forward break-even inflation rates. These break-even inflation rates contain information about inflation expectations, though to extract this information one has to allow for both technical complications and the possibility of institutional distortions. Technical complications arise from indexation lags, the convexity of bond prices, and inflation risk premiums. Unfortunately, convexity biases depend on changing interest-rate volatility, and inflation risk premiums depend also on time-varying investor risk preferences. Stripping out inflation risk premiums from break-even rates to extract objective inflation expectations therefore requires sophisticated modeling. Institutional distortions, on the other hand, can arise from legislative measures, such as the United Kingdom’s minimum funding requirement, which has been blamed for distorting the yields of U.K. gilt-edged securities and hence implied break-even inflation rates.

Because of the near-continuous nature of gilt trading, a great advantage is that break-even inflation rates provide policymakers with an immediate verdict on the market’s view of the impact of economic news on the anticipated path of future inflation, and on investors’ attitudes to inflation risk. To gauge what incremental, policy-relevant information can, in practice, be gained from a comparison of index-linked and conventional gilt prices, the Bank of England compared the two-year breakeven inflation rates with two-year Basix inflation surveys. The results indicate that, despite the possible influence of risk premiums and institutional distortions, two-year break-even inflation rates do provide information beyond that already contained in Basix surveys of inflation expectations. Longer-term break-even inflation rates, meanwhile, provide a barometer of inflation credibility. It is interesting, for example, to compare the immediate (negative) impact of September 1992 on U.K. monetary policy credibility in 10-year break-even rates with the gradual gains in credibility accumulated since the Bank of England was granted operational independence in 1997.

Appendix 12.1: The Real Stochastic Discount Factor/Pricing Kernel

Consider a discrete state setting. Note that all payouts are in constant-purchasing-power dollars, and therefore rates of return are real. For notation,

letbe the
s = 1,…,Sstates of the world
i = 1,…,Nassets
viinitial price of asset i in state s
v(N × 1) vector of asset prices
Yispayoff to asset i in state s (in constant-purchasing-power dollars)
X(N × S) matrix of state-contingent asset payoffs
Gis = 1 + Ris = Xis/vigross return on asset i in state s
G(N × S) matrix of state-contingent gross returns on assets
ps“state price”: today’s price for one constant-purchasing-power dollar to be paid tomorrow in state s
p(S × 1) state price vector
Zsobjective probability of state s occurring
Z(S × 1) state-probability vector
Rsrisk-free real interest rate

Each asset price vi can be represented as the sum of its state contingent payoffs times the appropriate state prices: vi = Σs Xisps. Equivalently, dividing through by vi, one obtains 1 = ΣsGisps = Σs(1 + Ris)ps, or i = Gp, where i is an (S × 1) vector of ones.

If there are no “type 1” arbitrage opportunities—that is, there are no available assets or combinations of assets with nonpositive cost today and nonnegative payoffs tomorrow, with a strictly positive payoff in at least one state—then the state price vector is positive. If one has a positive state price vector and defines the random variable Ms = ps/zs, where zs is the objective probability of state s, for any asset i the relationship i = Gp implies:

where Ms is the ratio of the state price of state s to the objective probability of state s (Ms = ps/zs). In the absence of arbitrage, Ms is always positive because both state prices and probabilities are positive. Ms is the real stochastic discount factor for state s. If Ms is small, then state s is “cheap,” in the sense that investors are not willing to pay a high price for an asset that will deliver one unit of purchasing in that state. An asset that tends to deliver wealth in cheap states thus has a real return that covaries negatively with the pricing kernel.

Attaching time subscripts and taking expectations conditional on information available at time t allows one to express equation (12.15) as:

This relationship can be rearranged to explicitly determine expected asset returns.


Because E[M] = ΣszsMs = Σszs(ps/zs) = Σsps = 1/(1 + Rf), where Rf is the risk-free real rate of interest:

This shows that an asset’s expected return is greater the smaller its covariance with the pricing kernel.

Appendix 12.2: The Inflation Risk Premium

This appendix develops expressions for the inflation risk premium, using the pricing-kernel framework adapted for analysis of conventional bonds outlined in Campbell, Lo, and MacKinlay (1997). For notation,

letbe the
PntRprice of n-period maturity perfectly indexed (real) bond at time t
PntNprice of n-period maturity conventional bond at time t
RntR(h)real net return on n-period perfectly indexed bond at time t, when held over h periods
RntN(h)real net return on n-period conventional bond at time t, when held over h periods
Mt+1real stochastic discount factor (RSDF) for real payouts due in period t + 1
Пtprice level index at time t
πt = ln(Пt+1t)inflation rate over period t to t + 1

Consider the expected real rates of return over one period on both nominal and perfectly indexed government bonds, both with one period to maturity. The real payoff and therefore real one-period return on the indexed bond is riskless. The (expected) real return over one period on a one-period maturity real bond is guaranteed to equal the inverse of the current real price of the bond:

Conversely, in real terms, the return on the conventional bond is risky because of inflation uncertainty. If one denotes the price level at time t by Пt, and allows Пt to equal 1, then the real price of the nominal bond by P1,tN, then the expected real return on the conventional bond is given by:

Now consider the RSDF, Mt+1, which values one unit of real wealth realized in every future state at time t + 1. The real price of the real bond with one period to maturity is denoted by PI1,t. P0,t = 1 so P1,t equals the expected value of Mt+1:

Similarly, the real price of the matching maturity nominal bond equals the expected value of the discounted real payoff:

If one assumes that the RSDF and future price level are individually and jointly lognormal, then a comparison of expected log real returns on the two bond types (where lowercase denotes logs) gives:

From equations (12.15), (12.23), and (12.24), and after some rearranging, one can thus derive an expression for the one-period inflation risk premium on a one-period conventional bond (h = n = 1):

For bonds with maturities greater than one-period, the inflation risk premium for a bond held to maturity (h = n > 1) is:

Extending the analysis to holding period returns where n > h > 1, the representation is less straightforward. When the holding period is less than the time to maturity of the bond, both conventional and index-linked bonds are subject to real return risk, because real values at the end of the holding period are unknown. Both are subject to revisions in the future value of the RSDF during the holding period.

The holding period inflation risk premium for h = 1<n can be expressed as:

In this situation, investors are concerned about the real value of the bond at time t + 1, which will be affected by any revisions to inflation expectations during the holding period. To the degree that investors might anticipate this to covary with fluctuations in the pricing kernel over the holding period, investors will require compensation.


The views expressed are those of the author and do not necessarily reflect those of the Bank of England. The author would like to thank Roger Clews, Mark Salmon, and Anne Vila Wetherilt (Monetary Analysis), and Martin Brooke (Financial Market Operations) at the Bank of England, as well as Kevin Cummings of Allen & Overy. All remaining errors are, of course, my own. A short version of his paper for this seminar appeared in Bank of England Quarterly Bulletin, Spring 2002, pp. 67–77.


The Massachusetts notes had the following terms: “Both principal and interest to be paid in the then current Money of said State, in a greater or less sum, according as five bushels of com, sixty-eight pounds and four-seventh parts of a pound of beef, ten pounds of sheep’s wool, and sixteen pounds of sole leather shall then cost more or less than one hundred and thirteen pounds current money, at the then current prices of the said articles.”


In October 2001, the French Trésor issued a new bond (OATi€ 3% 25/07/2012) indexed to the Eurozone harmonized index of consumer prices minus tobacco. The French government already issues (OATi) bonds linked to the French consumer price index excluding tobacco.


Source: U.K. Debt Management Office.


Typical bid-ask spreads are around 25 pence for a £5m trade, as against 5 pence for conventionals.


In 1985 the New York Coffee, Sugar, and Cocoa Exchange introduced a futures contract on the U.S. CPI-W, the consumer price index for urban wage earners and clerical workers. The experiment failed, and the contracts were withdrawn 18 months later.


In the United Kingdom, many utility companies’ pricing schedules are regulated by government and are explicitly linked to RPI-based formulas.


Aggregated inflation swap turnover figures are not yet available.


Note that the Fisher equation is an identity, and quite distinct from the Fisher hypothesis. The Fisher hypothesis is a behavioral assumption about interest rate determination that states that developments in expected inflation (Etn]) should be associated with equal changes to nominal interest rates (Yn,t), and that real interest rates (Rn-t) will remain approximately constant. (With non-constant real rates, the Fisher hypothesis can be modified to require statistical independence between real rates and expected inflation.)


A market is complete when there exists a full set of Arrow-Debreu securities. That is, for any possible future state of the world, one can purchase a security that will generate a known payoff in that state and nothing in all other states.


This assumes the multiplicative functional form. Note also that the risk premium will be a function of the perceived future distribution of inflation, not necessarily the historical or “true” distribution.


The report by Boskin and others (1996) found evidence to suggest that changes in the U.S. CPI overestimate changes in the true cost of living. Cunningham (1996) finds that measures of U.K. inflation, calculated on the basis of the RPI, are subject to similar biases. Cunningham suggests that growth in the RPI may overestimate the true increase in the cost of living by 0.35 percent to 0.8 percent a year. The U.K. Office for National Statistics is currently concluding research into the methodology used to construct the RPI (Office for National Statistics, 1999).


For issues before March 1982, the prospectuses for U.K. ILGs include an early redemption clause stating that “if any change should be made to the coverage or basic calculation of the index which, in the opinion of the Bank of England, constitutes a fundamental change in the index which would be materially detrimental to the interests of the stockholders, HM Treasury will … offer [investors] the right to require Hit Majesty’s Treasury to redeem their Stock [at index-adjusted par].” For new issues from March 1982, the prospectuses allowed for the possibility of switching to a substitute index (so long as it did not result in material detriment to the holders). However, in January 2002, the Debt Management Office (DMO) announced that the early redemption clause for future issues would be removed, and that the Chancellor’s choice of index, after consultation with an independent and expert body, would be conclusive and binding on all stockholders (Debt Management Office, 2001, 2002).


The minimum indexation lag is determined by two factors; (1) (unavoidable) reporting delays and (2) the method used for calculating accrued interest payment (essential for trading in the secondary market). The indexation lag on Canadian Real Return Bonds is three months, and accrued interest is calculated by interpolating between the three-month lagged CPI and the two-month lagged CPI value. U.S. Treasury Inflation Protected Securities (TIPS) also have a lag of three months. In the United Kingdom, on the other hand, accrued interest is calculated as a linear interpolation to the next coupon payment (which must therefore be known in advance). Consequently, an eight-month lag is required: two months for reporting delays and six months to calculate the next semiannual coupon. By comparing the two methods of calculating accrued interest against a perfect indexation scenario, the DMO discovered (unsurprisingly) that for the United Kingdom, in almost all cases the Canadian design would have offered inflation protection superior to that of the U.K. design. Note that, in response to a consultation exercise with market participants, the DMO decided not to alter the design of future index-linked gilt issues (Debt Management Office, 2001, 2002).


The Bank of England’s nominal and real yield-curve fitting approaches are comprehensively described in Anderson and Sleath (2001). In summary, the Bank’s variable roughness penalty (VRP) approach fits a nominal term structure using a cubic smoothing spline, where the degree of smoothing is a function of maturity. Combining the VRP technique with an enhanced version of Evans’s (1998) framework allows estimation of real and inflation term structures from index-linked gilts. This is discussed more fully in the section that follows on fitting break-even inflation rates.


A “zero-coupon” or “pure discount” bond is a bond with only one cash flow—the principal payment (by convention £100)—which is paid at maturity. There are no intermediate cash flows (coupons). Before maturity, zero-coupon bonds trade at a discount to face value. With annualized compounding, the zero-coupon or “spot” rate, z(t,T), at time t for a bond maturing at time T, is related to its price, P(t,T) as follows:


In fact, when comparing index-linked and conventional gilts with similar coupon rates and maturities, this crude approach usually generates break-even inflation yields that are very close to estimates derived from the difference between fitted real and nominal yields.


Forward rates are implied by zero-coupon rates or, equivalently, zero-coupon bond prices. A forward rate represents today’s terms for the lending of funds between two dates in the future. If P(0,t) and P(0,T) are the zero-coupon bond prices for t and T years (where t < T), then the annualized forward rate f(0,t,T) at time 0 for lending between t and T is given by:


To see why bonds are convex instruments, consider the price of an n-period zero-coupon bond with annually compounded yield, y: Pn,t = 1/(1 + y)n. Because the first derivative with respect to y is negative and the second derivative positive, the price function is convex. The practical implications can be illustrated with an example. Consider a 10-year zero-coupon bond with a face value of £100 and initial yield of 5 percent. Its current price is £61.39. Now consider the effects of a 1 percent change in the yield on price. If yield rose to 6 percent, price would fall to £55.84 (down £5.55). If yield fell to 4 percent, the bond price would rise to £67.55 (up £6.17). In other words, bond convexity means that bond prices are more sensitive to falls in yield than to increases in yield.


Formally, Jensen’s Inequality states that for a random variable X and strictly convex function g(.), then E[g(X)] > g(E[X]).


Equation (12.6) is usually referred to as the local expectations hypothesis. Cox, Ingersoll, and Ross (1981) demonstrate that this is the only form of the interest rate expectations hypothesis consistent with rational expectations.


Typically the sterling break-even inflation forward curve is observed to be broadly flat or slightly downward-sloping at the long end. This is consistent with the empirical evidence that sterling nominal interest rates are generally more volatile than real interest rates, whatever the maturity. Since 1990, for example, the standard deviation of end-of-month five-year (fitted) spot nominal rates has been 0.33 percentage points against 0.22 percentage points for real spot rates. For end-of-month 20-year spots since mid-1992, the figures are 0.24 and 0.14 percentage points, respectively.


An outline of the derivation of the pricing kernel is given in Appendix 12.1. For a comprehensive discussion of the theory, see Cochrane (2001).


In contrast to more usual definitions of risk premiums, where the emphasis is on the excess return on an asset over the one-period risk-free nominal rate of interest, the inflation risk premium is defined here by referencing two bonds with the same maturity against each other. Absence of arbitrage opportunities then requires that any two maturity-matched bonds should incorporate an equal amount of real interest rate risk premium. A comparison of conventional and perfectly indexed bonds of the same maturity should therefore provide a measure of the risk premium owing only to inflation uncertainty.


See Appendix 12.2 for derivations of the inflation risk premium when h = n = 1, h = n > 1 and n > h > 1.


Pensions Act (1995), which included the minimum funding requirement.


Specifically, a scheme’s liabilities were divided between pensioners and those who had not yet retired. The value of liabilities for pensioners was obtained by discounting expected future payments using the yield on a given specified gilt index. For pensioners entitled to level pensions or pensions with fixed annual percentage increases, the 15-year fixed interest gilt index applied. For inflation-proofed pensions, the 5-year index-linked gilt index applied. For scheme members not yet retired, the discount rate was broadly the assumed long-term total rate of return on U.K. equities before retirement and the appropriate gilt index after.


Looking at developments in the distribution of gilt ownership, one sees that since 1993 insurance firms and LAPFs have greatly increased their holdings of gilts covered by the FTSE 15-year-plus conventional and 5-year-plus index-linked gilt indices. In part, this has been a response to an aging of the U.K. population. But the prominent increase in ownership share during 1997–98 is probably attributable to MFR regulations.


The author is most grateful to Martin Brooke for advice on this issue. Price sensitivities to index compositional changes effects can be measured by looking at developments in the butterfly spread of the bond exiting the index against two other bonds (one shorter, one longer) around the date of exit from the index. Price effects appear to have become larger in recent years, probably as a result of increasing use of FTSE gilt indices for benchmarking and tracking purposes. Price effects also seem to have been greatest for bonds falling out of the FTSE five-year-plus IL gilt index, probably because of (1) the low number of IL gilts in this index, leading to significant changes in index duration action by fund managers, and (2) relatively lower liquidity in the index-linked gilt market. Note, however, that the window used in the event study is quite short, and compositional changes may have only short-term effects on prices.


It can be difficult to identify the impact of compositional changes on gilt prices. First, investors may anticipate changes to the index well in advance. Second, the recent practice by the DMO to hold switch auctions has allowed investors to exchange gilts falling out of the FTSE indices into new benchmark issues, removing the need to sell and reinvest to maintain an index-tracking portfolio, and limiting the impact of index composition changes on yields.


A revealing measure of the impact of institutional factors on gilts prices is provided by comparing the common currency cost of borrowing gilts relative to other fixed-income instruments of similar maturity and credit quality. For example, on December 1, 1999, the U.K. Treasury 9% 06/08/2012 could be swapped into British pound (GBP) six-month LIBOR minus 102.7 basis points. The French government OAT 8.5% 26/12/2012 bond, on the other hand, could be swapped into GBP six-month LIBOR less 47.9 basis points. In effect this meant that demand for gilts at the time was such that the U.K. Treasury was able to borrow some 55 basis points more cheaply than the French Trésor. Because the two issuers are almost identical in terms of credit quality, this difference must have reflected institutional distortions, such as MFR legislation.


Break-even inflation rates for single index-linked government issues are calculated as the difference in redemption yields between the index-linked bond and a conventional bond of similar maturity and coupon. The indexed bonds were selected to have sufficient history and remaining time to maturity.


The impact of MFR legislation on the yield curve is difficult to isolate because (1) the MFR came into force in 1997, approximately coinciding with government reforms giving the Bank of England operational independence, and (2) the effects of the incentive structure encouraging LAPFs to purchase gilts created by the MFR would have been cumulative.


The forward curves on December 20 are not atypical of the forward curve profiles during the second half of 1999.


Note also that, by construction, forward rates exaggerate much smaller differences in spot rates.


Note that the technical and institutional distortions to break-even inflation forwards may be relatively small at short maturities. Most surveys are also limited to expectations of inflation one or two years ahead.


The general public survey figures are excluded because of their consistent positive bias.


The authors also found, however, that the real term structure tended to underpredict the level of future real interest rates, possibly because of an inflation risk premium that drives up index-linked bond prices and reduces real yields.


The break-even inflation rate is calculated at end-of-month, and the September 1992 data point therefore falls after the United Kingdom’s exit from the ERM.


If an inflation forecast is rational, then in the long run the forecast and actual inflation series should also be cointegrated. Cointegration is a necessary but not sufficient condition for the rational expectations hypothesis to hold, and this chapter does not consider long-run cointegration properties.


RPIX is the retail price index excluding mortgage interest payments.

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