Forward inflation rates

The Term Structure of Implied Forward Inflation Rates... 

Using any of the methods described in Chapter 5 or the discount function approach summarised above, we can construct curves for both the nominal and the real implied forward rates. These two curves can then be used to infer market expectations of future inflation rates. The term stmcture of forward inflation rates is obtained from both these curves by applying the Fisher identity ... 

This result has been derived, in different forms, in all three references noted previously. As with other options pricing models, it needs to be calibrated to the market before it can be used. Generally, this will involve using actual and project forward inflation rates to fit the model to market prices and volatilities. 

The drawbacks of each of these approaches are apparent. A rather more valid and sound approach is to constmct a term structure of the real interest rates, which would indicate, in exactly the same way that the conventional forward rate curve does for nominal rates, the market s expectatimis rat future inflation rates. In countries where there are liquid markets in both conventional and inflation-indexed bmids, we can observe a nominal and a real yield curve. It then becomes possible to estimate both a conventional and a real term structure using these allows us to create pairs of hypothetical conventional and indexed bonds that have identical maturity dates, for any point on the term structure. We could then apply the break-even approach to any pair of bonds... 

From this average inflation curve, we can select specific inflation rates for each index-linked bond in our sample. The real yields on each indexed bond are then recalculated using these new inflation assumptions. From these yields the real forward curve is calculated, enabling us to produce a new estimate of the inflation term stmcture. This process is repeated until there is consistency between the inflation term stmcture used to estimate the real yields and that produced by Equation (6.5). 

Where a liquid market in indexed bonds exists across a reasonable maturity term structure, it is possible to construct a term structure of inflation rates. In essence, the process involves constructing the nominal and real interest rate term structures, then using them to infer an inflation term structure. This, in turn, can be used to calculate a forward expected inflation rate for any term or a forward inflation curve in the same way that a forward interest rate curve is constructed. 

The U.S. Federal Reserve uses an iterative technique to construct a term structure of expected inflation rates. First the nominal interest rate term structure is constructed using a version of the model described in Waggoner (1997) and discussed in James and Webber (2000). An initial assumed inflation term structure is then used to infer a term structure of real interest rates. This assumed inflation curve is usually set at a flat 3 or 5 percent. The real interest rate curve is then used to calculate an implied real interest rate forward curve. Next, the Fisher identity is applied at each point along the nominal and real interest rate forward... 

Using equation 14.16, we can build a forward inflation curve provided we have the values of the index at present, as well as a set of zero-coupon bond prices of required credit quality. Following standard yield curve analysis, we may build the term structure from forward rates and therefore imply the real yield curve, or alternatively we may construct the real curve and project the forward rates. However, if we are using inflation swaps for the market price inputs, the former method is preferred because IL swaps are usually quoted in terms of a forward index value. 

Using the prices of index-linked bonds, it is possible to estimate a term structure of real interest rates. The estimation of such a curve provides a real interest counterpart to the nominal term structure that was discussed in the previous chapters. More important it enables us to derive a real forward rate curve. This enables the real yield curve to be used as a somce of information on the market s view of expected future inflation. In the United Kingdom market, there are two factors that present problems for the estimation of the real term structure the first is the 8-month lag between the indexation uplift and the cash flow date, and the second is the fact that there are fewer index-linked bonds in issue, compared to the number of conventional bonds. The indexation lag means that in the absence of a measure of expected inflation, real bond yields are dependent to some extent on the assumed rate of future inflatiOTi. The second factor presents practical problems in curve estimation in December 1999 there were only 11 index-linked gilts in existence, and this is not sufficient for most models. Neither of these factors presents an insurmountable problem however, and it is stiU possible to estimate a real term structure. 

So, we have argued, the economics of supply and demand make the risk premium a slippery concept. Bond mathematics now makes matters worse. This new aspect centres on the issue of convexity. We know that a forward curve of implied future short-term nominal rates can be derived from the nominal government bond curve. In principle, a forward curve of implied future short-term real rates can be similarly derived from the inflation-linked bond real yield curve. These two curves, taken together, should imply a future path of inflation, if we can set aside the risk premium for the moment. Unfortunately, that is not the case. 

The presence of convexity means that both our forward bond curves understate true expectations of the future paths of nominal and real short-term rates. However, the value placed on convexity is a function of volatility, which is much greater in nominals than in linkers. Therefore, an implied path of future inflation derived in this way will understate true inflationary expectations because of convexity, if there are no other influences, such as the risk premium. 

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