Recently the French government issued a 30-year OATi, an inflation-linked bond (ILB), boosting this nascent French debt sector. In other countries, especially in the U.K., the market for government-issued inflation-linked bonds is already well developed. An inflation index-linked bond is a bond with both coupons and redemption payments pegged to an inflation indicator. Typically the future cashflows are adjusted to the performance of a consumer price index (CPI). The latter is based on the price evolution of a broadly diversified basket of consumer goods and is published by government authorities, like the Office for National Statistics for the U.K. Recently, OTC derivatives, mainly interest rate swaps, which are linked to an inflation index, have also started to gain in prominence.
In most cases the inflation adjustment for a coupon-bearing bond works as follows:
Suppose a bond pays a coupon C and Xt is the inflation index fixed at time t. If the index was X0 at issue date of the bond, then the index-linked bond pays
at the coupon date t.
Analogously to the coupon payments, the redemption amount is also inflation-adjusted. In some markets (e.g. in France) the redemption payment is even protected against deflation, i.e. the final payment can never fall short of the bond notional.
For an inflation-linked swap the fixed swap leg is adjusted to the performance of the corresponding CPI in exactly the same way as for an ILB. Consequently, from the pricing point of view swaps can be treated analogously to bonds. One can often observe that the payment schedule of such a swap closely resembles one of the cornerstone ILBs in the respective market, as this provides a strategy to immunize the swap position against the inherent inflation risk.
PRICING
The fair price of an ILB is simply the sum of the discounted inflation-adjusted coupon payments and the redemption amount. Obviously, the future indices are unknown. Expressing these indices in terms of an inflation rate, one can derive--analogously to the implied volatility of an equity option--an implied inflation rate, which can be seen as the market anticipation for the average future inflation.
American economistIrving Fisher (1867-1967) postulated a relationship for interest rates in terms of so-called real and nominal yields, which are the key elements in order to understand the pricing of inflation-linked instruments.
Let us first clarify the notion of real and nominal yields. What would a rational market participant charge as compensation for granting a certain amount of money only to be repaid after a pre-fixed period of time?
Firstly, he would certainly insist on some compensation, solely covering the fact that the money will only be repaid at some later point in time. This translates to what goes by the name of real yield. Secondly, he would demand an allowance to capture the expected average future inflation until redemption, the expected inflation rate. Finally, the investor would typically, in addition to the above, require a risk premium on the inflation, covering unexpected adverse moves in the inflation rate. Taken all together, this constitutes the nominal rate.
Thus the nominal yield ynom corresponds to the yield of conventional bonds. Note that the real rate can be seen as a yield, which would apply in a world without inflation. The relationship between the real yield yreal, the expected inflation rate x and an inflation risk premium p is now given by Fisher's identity:
As the inflation risk premium cannot be extracted from the market, one assumes for all practical purposes p = 0. Under this assumption x is known as break-even inflation rate. Applying this relation to the pricing equation of an ILB we obtain in general the usual bond price-yield relationship applied to the fixed coupon cash flows of the ILB (not inflation adjusted!). The only difference is that we discount with the real yield instead of the nominal yield. As a consequence, in the world of real rates, an inflation-linked bond does not bear any inflation risk.
THE MARKET
In July 1981 the British government issued an ILB for the first time. More than 10 have followed since then. The reason for linking the coupon payments to the observed inflation arose from the high inflation in U.K. during the 1970s. Most of the issues are held by pension funds, which use them as a shield against rising inflation. For them it makes sense to hold them up to maturity, in order to inflation-protect the redemption amount. As a consequence the index-linked Gilt market is not as liquid as the one for conventional bonds.
In Europe, France and Sweden are further markets for index-linked stocks. The Swedish National Debt Office has been issuing ILBs since April 1994. In France there are two government issues on the market, which mature in about nine and 30 years, respectively. In both countries the first issues were initially not too well accepted by the market, mainly due to the low inflation level in Europe during the 1990s. Obviously, during periods of low inflation index-linked securities are not very attractive to investors because of their lower coupon rate in comparison to conventional bonds.
In North America, Canada was the first to issue index-linked stocks, which was in October 1991. About two years ago the U.S. government also pushed into this market. But similarly to Sweden and France this emission was not as successful as expected.
As already mentioned there is an increasing demand from the market for inflation-linked swaps, particularly in those countries where a government market does not exist. In Germany, for instance, due to the experience of hyperinflation from 1921-1923 the post-war government did not admit issuing any index-linked securities. Another perspective market in the previous catogory constitute swaps linked to the CPI evolution in Euroland. This raises the problem that due to the lack of ILBs one cannot extract implied inflation rates, which are, however, necessary for pricing and hedging index-linked swaps. Therefore the challenge is to transfer the implied inflation expectations derived implicitly from the quoted ILB prices of those markets, that are in economic terms closest to the considered one. Thus, index-linked swaps are typically structured in congruence to the available ILBs.
This above-mentioned transformation can be accomplished as follows: Let us recall the example of a swap linked to the German CPI, whose payment schedule resembles that of a French OATi. We extract the implied inflation rate from the French ILB and transfer it via a regression analysis (on the historical CPI changes between the two countries) to the German market. In light of the common European currency, a further convergence of the European economies is anticipated, which will make this procedure even more justifiable in the future.
This week's Learning Curve was written by Drs. Hermann Haaf andHannes Wilhelm, group risk control at Dresdner Bankin Frankfurt.
