Certain countries have markets in bonds whose coupon or final redemption payment, or both, are linked to their consumer price indexes. Generally, the most liquid markets in these inflation-indexed, or index-linked, debt instruments are the ones for government issues. Investors’ experiences with the bonds differ, since the securities were introduced at different times in different markets and so are designed differently. In some markets, for instance, only the coupon payment, and not the redemption value, is index-linked. This makes comparisons in terms of factors such as yield difficult and has in the past hindered arbitrageurs seeking to exploit real yield differences. This chapter highlights the basic concepts behind indexed bonds and how their structures may differ from market to market.

The features considered in the design of index-linked bonds are the type of index, the indexation lag, the coupon frequency and the type of indexation.

In principle, bonds can be linked to almost any variable, including various price indexes, earnings measures, GDP output, specific commodities and the exchange rate of foreign currencies against another currency. Ideally, the chosen index should reflect the hedging requirements of both parties - that is, the issuer and the investor. Their needs, however, may not coincide. For instance, retail investors overwhelmingly favour indexation to consumer prices, to hedge against inflation, which erodes bond earnings. Pension funds, on the other hand, prefer linking to earnings levels, to offset their earnings-linked pension liabilities. In practice, most bonds have been tied to inflation indexes, since these are usually widely circulated, well-understood and issued on a regular basis. US Treasury inflation-indexed securities (TIIS) or Treasury inflation-protected securities (TIPS), for instance, are linked to the US Consumer Price Index (CPI-U), the non-seasonally adjusted average of prices for urban consumers. The securities’ daily interest accrual is based on straight-line interpolation, and there is a 3-month lag. So, for example, the October 2003 index level is used to determine the adjustment for 1 January 2004. Figure 9.1 is the Bloomberg DES (‘description’) screen for the TIPS maturing in July 2013. Figure 9.2 shows the yield analysis page for this bond as at 2 December 2005.

Figure 9.3 is the DES screen for the CPI index in July 2003, which was the base month for this security when it was issued.

To provide precise protection against inflation, interest payments for a given period would need to be corrected for actual inflation over the same period. Lags, however, exist between the movements in the price index and the adjustment to the bond cash flows. According to Deacon and Derry (1998), such lags are unavoidable for two reasons. First, inflation statistics for 1 month are usually not known until well into the following month and are published some time after that. This causes a lag of at least 1 month. Second, in some markets the size of a coupon payment must be known before the start of the coupon period in order to calculate the accrued interest. There is thus a delay - between the date the coupon amount is fixed and the time the inflation rate for the period affecting that payment is known - that is equal to the length of time between coupon payments. Deacon and Derry (1998) also note that the lag can be minimised - for example, by basing the accrued interest calculation on cumulative movements in the consumer price index since the last coupon date, as is done for Canadian Real Return Bonds.

Figure 9.3 Bloomberg DES screen for the July 2003 Consumer Price Index.

© Bloomberg Finance L.P. All rights reserved. Used with permission.

Index-linked bonds often pay interest semiannually. Certain long-dated investors, such as fund managers whose liabilities include inflation-indexed annuities, may be interested in indexed bonds that pay on a quarterly or even monthly basis.

There are five basic methods of linking the cash flows from a bond to an inflation index: interest indexation, capital indexation, zero-coupon indexation, annuity indexation and current pay. Which method is chosen depends on the requirements of the issuers and of the investors they wish to attract. The principal factors considered in making this choice, according to Deacon and Derry (1998), are duration, reinvestment risk and tax treatment.

Interest indexation Interest-indexed bonds have been issued in Australia. They pay a coupon fixed rate at a real, inflation-adjusted interest rate. They also pay a principal adjustment (equal to the percentage change in the CPI from the issue date times the principal amount) every period. The inflation adjustment is thus fully paid out as it occurs, and no adjustment to the principal repayment at maturity is needed.

Capital indexation Capital-indexed bonds have been issued in the US, Australia, Canada, New Zealand and the UK. Their coupon rates are specified in real terms, meaning that the coupon paid guarantees the real amount. For example, if the coupon is stated as 2%, what the buyer really gets is 2% after adjustment for inflation. Each period this rate is applied to the inflation-adjusted principal amount to produce the coupon payment amount. At maturity the principal repayment is the product of the bond’s nominal value times the cumulative change in the index since issuance. Compared with interest-indexed bonds of similar maturity, these bonds have longer durations and lower reinvestment risk.

Zero-coupon indexation Zero-coupon indexed bonds have been issued in Sweden. As their name implies, they pay no coupons; the entire inflation adjustment occurs at maturity, applied to their redemption value. These bonds have the longest duration of all indexed securities and no reinvestment risk.

In the US, Canada and New Zealand, indexed bonds can be stripped, allowing coupon and principal cash flows to be traded separately. This obviates the need for specific issues of zero-coupon indexed securities, since the market can create products such as deferred-payment indexed bonds in response to specific investor demand. In markets allowing stripping of indexed government bonds, a strip is simply a single cash flow with an inflation adjustment. An exception to this is in New Zealand, where the cash flows are separated into three components: the principal, the principal inflation adjustment and the inflation-linked coupons - the latter being an indexed annuity.

Annuity indexation Indexed-annuity bonds have been issued in Australia, although not by the central government. They pay a fixed annuity payment plus a varying element that compensates for inflation. These bonds have the shortest duration and highest reinvestment risk of all index-linked debt securities.

Current pay Current-pay bonds have been issued in Turkey. They are similar to interest-indexed bonds in that their redemption payments at maturity are not adjusted for inflation. They differ, however, in their term cash flows. Current-pay bonds pay an inflation-adjusted coupon plus an indexed amount that is related to the principal. In effect, they are inflation-indexed floating-rate notes.

Duration Duration measures something slightly different for an indexed bond than it does for a conventional bond, indicating price sensitivity to changes in real, inflation-adjusted interest rates, instead of in nominal, unadjusted ones. As with conventional bonds, however, the duration of zero-coupon indexed bonds is longer than that of equivalent coupon bonds. As noted above, indexed annuities will have the shortest duration of the inflation-linked securities. Investors with long-dated liabilities should theoretically prefer hedging instruments with long durations.

Reinvestment risk Like holders of a conventional bond, investors in a coupon indexed bond are exposed to reinvestment risk: because they cannot know in advance what rates will be in effect when the bond’s coupon payments are made, investors cannot be sure when they purchase their bond what yield they will earn by holding it to maturity. Bonds, such as indexed annuities, that pay more of their return in the form of coupons carry more reinvestment risk. Indexed zero-coupon bonds, like their conventional counterparts, carry none.

Tax treatment Tax treatment differs from market to market and from product to product. Some jurisdictions, for example, treat the yearly capital gain on zero-coupon bonds as current income for tax purposes. This is a serious drawback, since the actual gain is not available until maturity, and it reduces institutional demand for these instruments.

As noted above, index bonds differ in whether their principal payments or their coupons, or both, are linked to the index. When the principal alone is linked, each coupon and the final principal payment are determined by the ratio of two values of the relevant index. US TIPS’ coupon payments, for instance, are calculated using an accretion factor based on the ratio between two CPI-U levels, defined by (9.1): where

For a settlement or issue date of 1 May, for instance, the relevant CPI level would be the one recorded on 1 February. For a settlement or issue occurring on any day besides the first of the month, linear interpolation is used to calculate the appropriate CPI level. This is done by subtracting the reference month’s CPI-U level from the following month’s level, then dividing the difference by the number of days between the readings and multiplying the result by the number of days in the month leading up to the reference date. As an illustration, consider an issue date of 7 April. The relevant index level would be the one for 7 January. Say the 1 January CPI-U level is 160.5 and the 1 February level 160.6. The difference between these two values is:

Dividing this difference by the number of days between 1 January and 1 February gives: and multiplying the result by the number of days in January before the reference date gives:

So, the CPI-U for January 7 is 160.5 + 0.19, or 160.69.

TIPS periodic coupon payments and their final redemption payments are both calculated using an inflation adjustment. Known as the inflation compensation, or IC, this is defined as in (9.2): where P = Bond’s principal.

The semiannual coupon payment, or interest, on a particular dividend date is calculated using (9.3): where C = Annual coupon rate.

The principal repayment is computed as in expression (9.4). Note that the redemption value of a TIPS is guaranteed by the Treasury to be a minimum of $100 - that is, 100% of the face value.

(9.4)

where CPI₀ = Base CPI level - that is, the level three months before the bond’s issue date.

The price of a TIPS comprises its real price plus any accrued interest, both of which are adjusted for inflation by multiplying them times the index ratio for the settlement date. The bond’s unadjusted accrued interest, as explained in Chapter 1, is calculated using (9.5): where

The TIPS real price is given by (9.6): where

The markets use two main yield measures for all index-linked bonds: the money, or nominal yield, and the real yield. Both are varieties of yield to maturity.

To calculate a money yield for an indexed bond, it is necessary to forecast all its future cash flows. This requires forecasting all the relevant future CPI-U levels. The market convention is to take the latest available CPI reading and assume a constant future inflation rate, usually 2.5% or 5%. The first relevant future CPI level is computed using equation (9.7): where

Consider an indexed bond that pays coupons every June and December. To compute its yield, it is necessary to forecast the CPI levels registered 3 months before June and 8 months before December - that is, the October and April levels. Say this computation takes place in February. The first CPI level that must be forecast is thus next April’s. This means that in equation (9.7), m = 2. Say the February CPI is 163.7. Assuming an annual inflation rate of 2.5 %, the CPI for the following April is computed as follows:

Equation (9.8) is used to forecast the subsequent relevant CPI levels: where j = Number of semiannual forecasts after CPI₁.

The forecast CPI level for the following October is calculated as follows:

Once the CPIs have been forecast, the bond’s yield can be calculated. Assuming that the analysis is carried out on a coupon date so that accrued interest is 0, the money yield of a bond paying semiannual coupons is calculated by solving equation (9.9) for r_i: where

The equation for indexed bonds paying annual coupons is (9.10):

The real yield, ry, first described by Fisher in Theory of Interest (1930), is related to the money yield through equation (9.11) - for bonds paying semiannual coupons:

To illustrate this relationship, say the money yield is 5.5 % and the forecast inflation rate is 2.5%. The real yield would then be:

Re-arranging equation (9.11) to express ri in terms of ry and substituting the resulting expression for ri in equation (9.9) gives equation (9.12), which can be solved to give the real yield, calculated on a coupon date, of index bonds paying semiannual coupons: where

The equations for money yield and real yield can be interpreted as indicating what redemption yield to employ as the discount rate in calculating the present value of an index bond’s future cash flows. From this perspective, equation (9.9) shows that the money yield is the appropriate rate for discounting money or nominal cash flows. Equation (9.12) shows that the real yield is the appropriate rate for discounting real cash flows.

Index-linked bonds do not offer complete protection against a fall in the real value of an investment. These bonds, including TIPS, do not have guaranteed real returns, despite having their cash flows linked to a price index such as the CPI. The reason for this is the lag in indexation, which for TIPS is 3 months. The time lag means that an indexed bond is not protected against inflation for the last interest period of its life. Any inflation occurring during the final interest period will not be reflected in the bond’s cash flows and will reduce the real value of the redemption payment and hence the bond’s real yield. This may not be a major consideration when the inflation rate is low, but it can be a worry for investors when the rate is high. The only way to effectively eliminate inflation risk is to reduce the time lag in indexation of payments to 1 or 2 months.

Bond analysts frequently compare the yields on index-linked bonds with those on conventional bonds of the same maturity to determine the market’s expectation with regard to inflation rates. Of course, many factors can influence the gap between conventional and indexed bond yields, including supply and demand, and liquidity (conventional bonds are generally more liquid than indexed ones). A large part of the difference, however, is the inflation premium, which reflects the market’s expectations about inflation during the life of the bond. To determine the implied expectation, analysts calculate the break-even inflation rate, the rate for which the money yield on an index-linked bond equals the redemption yield on a conventional bond of the same maturity.

Accepting that developed, liquid markets - such as that for Treasuries - are efficient, with near-perfect information available to most if not all participants, then the inflation expectation is built into the conventional treasury yield. If the inflation premium understates what certain market participants expect, investors will start buying more of the index-linked bond in preference to the conventional bond. This activity will force the indexed yield down (or the conventional yield up). If, on the other hand, investors think that the implied inflation rate overstates expectations, they will buy more of the conventional bond.

The higher yields of the conventional bonds compared with those of the index-linked bonds represent compensation for the effects of inflation. Bondholders will choose to hold index-linked bonds instead of conventional ones if they are worried about unexpected inflation. An individual’s view on future inflation will depend on several factors, including the current macroeconomic environment and the credibility of the monetary authorities, be they the central bank or the government. Fund managers take their views of inflation, among other factors, into account in deciding how much of the TIPS and how much of the conventional Treasury to hold. Investment managers often hold indexed bonds in a portfolio against specific index-linked liabilities, such as pension contracts that increase their payouts in line with inflation each year.

In certain countries, such as the UK and New Zealand, the central bank has explicit inflation targets, and investors may believe that over the long term those targets will be met. If the monetary authorities have good track records, investors may further believe that inflation is not a significant issue. In such situations, the case for holding index-linked bonds is weakened.

Indexed bonds’ real yields in other markets are also a factor in investors’ decisions. The integration of markets around the world in the past 20 years has increased global capital mobility, enabling investors to shun markets where inflation is high. Over time, therefore, expected returns should be roughly equal around the world, at least in developed and liquid markets, and so should real yields. Accordingly, index-linked bonds should have roughly similar real yields, whatever market they are traded in.

The yields on indexed bonds in the US, for example, should be close to those in the UK indexed market. In May 1999, however, long-dated indexed bonds in the US were trading at a real yield of 3.8%, compared with just 2% for long-dated index-linked gilts. Analysts interpreted this difference as a reflection of the fact that international capital was not as mobile as had been thought and that productivity gains and technological progress in the US had boosted demand for capital there to such an extent that the real yield had had to rise. In the first quarter of 2010, the difference in yield between 10-year Treasuries and 10-year TIPS was at 1.8%, suggesting that investors were not concerned with inflation being a problem for the time being.

Inflation-indexed derivatives, also known as inflation-linked derivatives or inflation derivatives, have become widely traded instruments in the capital markets. They are traded generally by the same desks in investment banks that trade inflation-linked sovereign bonds, who use these instruments for hedging as well as to meet the requirements of clients such as hedge funds, pension funds and corporates. They are a natural development of the inflation-linked bond market.

Inflation derivatives are an additional means by which market participants can have an exposure to inflation-linked cash flows. They can also improve market liquidity in inflation-linked products, as an earlier generation of derivatives did for interest-rate and credit risk. As flexible OTS products, inflation derivatives offer advantages over cash products in certain circumstances. They provide

The inflation derivatives market in the UK was introduced after the introduction of the gilt repo market in 1996. In most gilt repo trades, index-linked (IL) gilts could be used as collateral; this mean that both IL and conventional gilts could be used as hedging tools against positions in inflation derivatives. In the euro area, IL derivatives were introduced later but experienced significant growth during 2002-2003. The existence of a sovereign IL bond market can be thought of as a necessary precursor to the development of IL derivatives, and although there is no reason why this should be the case, up to now this has been the case. The reason for this is probably because such a cash market suggests that investors are aware of the attraction of IL products, and wish to invest in them. From a market in IL bonds there developed a market in IL swaps, which are the most common IL derivatives. The IL bond market also provides a ready reference point from which IL derivatives can be priced.

We describe first some common inflation derivatives, before considering some uses for hedging and other purposes. We then consider IL derivatives pricing.

This is also known as a synthetic index-linked bond. It is a swap with the following two cash flow legs:

This converts existing conventional fixed- or floating-rate investments into inflation-linked investments. An example of such a swap is given below.

The symbols in the formulae above are

The ‘minus 3’ in the formula for HCIP refers to a 3-month lag for indexation, common in euro sovereign IL bond markets.

The swap is illustrated at Figure 9.4.

This swap is commonly used to hedge issues of IL bonds. The swap is comprised of

With these swaps, the IL leg is usually set at a floor of 0% for the annual change in the underlying index. This guarantees the investor a minimum return of the fixed-rate coupon.

This swap is also known as a pay-as-you-go swap. It is shown at Figure 9.5.

The TIPS swap is based on the structure of US TIPS securities. It pays a periodic fixed rate on an accreting notional amount, together with an additional one-off payment on maturity. This payout profile is identical to many government IL bonds. They are similar to the synthetic IL bonds described above.

TIPS swaps are commonly purchased by pension funds and other long-dated investors. They may prefer the added flexibility of the IL swap market compared with the cash IL bond market. Figure 9.6 shows the TIPS swap.

This is also known as a zero-coupon inflation swap or zero-coupon swap. It allows the investor to hedge away a breakeven exposure. Compared with IL swaps such as the synthetic bond swap, which hedge a real yield exposure, the breakeven swap has both cash flow legs paying out on maturity. The legs are

This structure enables IL derivative market makers to hedge their books. It is illustrated at Figure 9.7.

A real-annuity swap is used to hedge inflation-linked cash flows where this applies for payments such as rental streams, lease payments, project finance cash flows and so on. It enables market participants who pay or receive such payments to replace the uncertainty of the future level of these cash flows with a fixed rate of growth. The swap is written on the same notional amount for both legs, but payout profiles differ as follows:

These swaps are one of the most commonly traded. The fixed rate quoted for the swap provides a ready reference point against which to compare expected future rates of inflation. So, for instance, if a bank is quoting for a swap with a fixed rate of 3.00%, and an investor believes that inflation rates will not rise above 3.00% for the life of the swap, then it will receive ‘fixed’ (here meaning a fixed rate of growth) and pay inflation-linked on the swap.

We now describe some common applications of IL derivatives.

This is perhaps the most obvious application. Assume a life assurance company or corporate pension fund wishes to hedge its long-dated pension liabilities, which are linked to the rate of inflation. It may invest in sovereign IL bonds such as IL gilts, or in IL corporate bonds that are hedged (for credit risk purposes) with credit derivatives. However the market in IL bonds is not always liquid, especially in IL corporate bonds. The alternative is to buy a synthetic IL bond. This is structured as a combination of a conventional government bond and an IL swap, in which the pension fund pays away the bond coupon and receives inflation-linked payments.

The net cash flow leaves the pension fund receiving a stream of cash flows that are linked to inflation. The fund is therefore hedged against its liabilities. In addition, because the swap structure can be tailor-made to the pension fund’s requirements, the dates of cash flows can be set up exactly as needed. This is an added advantage over investing in the IL bonds directly.

Assume that a bank or corporate has an income stream that is linked to inflation. Up to now, it has been funded by a mix of fixed- and floating-rate debt. Say these are floating-rate bank loans and fixed-rate bonds. However, from an asset-liability management (ALM) point of view this is not optimal, because of the nature of a proportion of its income. It makes sense, therefore, to switch a part of its funding into an inflation-linked segment. This can be done using either of the following approaches:

The choice will depend on which approach provides cheapest funding and most flexibility.

Assume that a bank or corporate intends to issue an IL bond, and wishes to hedge against a possible fall in government IL bond prices, against which its issue will be priced. It can achieve this hedge using an IL gilt-linked derivative contract.

The bank or corporate enters into a cash-settled contract for difference (CFD), which pays out in the event of a rise in government IL bond yields. The CFD has a term to maturity that ties in with the issue date of the IL bond. The CFD market maker has effectively shorted the government bond, from the CFD trade date until maturity. On the issue date, the market maker will provide a cash settlement if yields have risen. If yields have fallen, the IL bond issuer will pay the difference. However, this cost is netted out by the expected ‘profit’ from the cheaper funding when the bond is issued. Meanwhile, if yields have risen and the bank or corporate issuer does have to fund at a higher rate, it will be compensated by the funds received by the CFD market maker.

Deacon, J. and Derry, M. (1998). Inflation-linked Securities. FT Prentice Hall, London.

Fisher, I. (1930). Theory of Interest. Oxford University Press, Oxford, UK.