17

Inflation-linked securities

17.1 INTRODUCTION

Inflation-linked securities present some additional challenges for the attribution analyst compared to vanilla sovereign and corporate bonds, because of their dependence on inflation rates and real, rather than nominal, yields.

This chapter presents an overview of what inflation-linked securities are and how they are priced, highlights some of their major differences compared to related asset classes and presents an attribution framework that can be used for the majority of inflation-linked bonds.

Inflation-linked bonds are also referred to as index-linked bonds or just linkers, and the reader will often find these terms used interchangeably in the marketplace.

17.2 OVERVIEW OF THE INFLATION-LINKED BOND MARKET

In the twentieth century, inflation-linked bonds were exclusively issued by sovereign governments, but many are now offered by corporate and other entities. From 2011 to 2012, inflation-linked issuance made up 23% of the total gilts offered in the UK and more than 10% of issuance by the US Federal Reserve. Countries that have issued inflation-indexed debt now include the United States, the United Kingdom, France, Germany, Italy, Sweden, Canada, Australia, Israel and Japan.

Reasons for issuing inflation-linked debt include:

• the high and volatile levels of inflation seen in Latin America in the 1960s and 1970s;
• demonstrating commitment to disinflationary policies;
• capitalising on inflationary expectations;
• encouragement of long-term inflation hedges for liabilities such as pensions.

Inflation-linked bonds and related securities, such as inflation swaps, are common, and any attribution system should be able to measure the specific types of return generated by this asset class.¹

17.3 WHAT IS AN INFLATION-LINKED BOND?

An inflation-linked bond is a bond that has its interest and principal payments indexed against inflation, using a measure such as the General Index of Retail Prices (RPI) in the UK or the Consumer Price Index (CPI) in the United States. The index is typically based on a complex mixture of assets that represents the cost of living in Western societies.

As the inflation index rises, the bond’s principal rises in proportion. Since coupons are calculated as a proportion of the bond’s principal, any coupons also change in line with inflation. For instance, suppose a bond is issued with $100 face value and a 5% coupon. The CPI then rises 60% over the security’s lifetime. In this case, the principal rises to $160, so the coupon payable at maturity will also have risen by 60%, and the payment at maturity will be $160. However, the principal and coupon are still $100 and 5% in real terms, whereas the value of a non-indexed bond will have fallen to $100/1.6 = 62.5% of its initial purchasing power.

The holder of such a bond therefore expects their investment at least to be immunised against inflation, and to generate a known, predictable return relative to the current rate of inflation. The bond’s market-quoted real yield measures the bond’s guaranteed rate of return over and above that of inflation if held to maturity.

Therefore, inflation-linked bonds can be used in a number of different ways. They both allow investors to hedge their assets against the effects of inflation, and to take speculative positions on changes in inflation rates and real yields. The attribution analyst’s task is to measure these inflation-generated returns, in addition to returns from other sources of risk such as carry, credit and changes in the market’s term structure.

17.4 THE CANADIAN MODEL FOR INFLATION-LINKED DEBT

Inflation-linked bonds pay periodic coupons that are set to the product of a nominal coupon rate and the inflation index. The majority of such bonds follow the Canadian model (Deacon et al., 2004) where daily inflation is measured as the ratio of the inflation index over two successive months, lagged by three months. I follow this convention in the remainder of this chapter, but there are some exceptions of which the reader should be aware:

• UK inflation-linked gilts issued before 2005 used an inflation ratio that was constant over each calendar month.
• Australian indexed annuity bonds are a stream of cash flows indexed to inflation, without any principal payment at maturity. These can be modelled as an inflation-linked bond minus a zero coupon inflation-linked bond.

17.5 INFLATION RATIOS

Inflation-linked securities use various forms of inflation ratio to measure the day-to-day effects of inflation.

The current rate of inflation will not be known on the date that the security is traded, because of the time and effort required to collect and collate the index. Instead, recent inflation measurements are used instead to calculate an indicative daily inflation rate.

For example, consider a UK inflation-linked gilt with a three-month indexation lag. The daily index ratio is defined as follows:

where

• IR_t is the (calculated) index ratio at time t;
• RPI_T and RPI_{T + 1} are the (published) applicable RPI values for the first day of the month three months ago, and the RPI value for the first day of the next month after that date, respectively;
• t is the calendar date at which the ratio is to be calculated;
• D is the number of days in the current month.

The applicable starting RPI used in this formula is the published RPI for the calendar month falling three months earlier. For instance, to calculate an index ratio during January 2012, the values used should be for October and November 2011. By convention, the result is rounded to the fifth decimal place.

Example

The RPI index ratio for 5 January 2012 uses the RPI factors for October (238.0) and November (238.5). Its value at this date is

17.6 REAL YIELDS AND NOMINAL YIELDS

A common point of confusion arising when discussing inflation-linked bonds is the meaning of yield. In the following sections I present the terms real yield, nominal yield and break-even yield.

17.6.2 Nominal yield

Fixed-coupon bonds and inflation-indexed bonds both have a nominal yield.

For the fixed-coupon bond, the nominal yield is the same as its yield to maturity.

For the inflation-indexed bond, the nominal yield is given by the Fisher equation, which states that the nominal interest rate y is the sum of the real interest rate r and the inflation rate i:

17.6.3 Break-even yield

The break-even inflation rate (BEIR) is the rate of inflation for which the nominal yield on an index-linked bond is the same as the nominal yield of a conventional bond having the same maturity. In other words, it is the rate of inflation above which an inflation-linked investment will outperform a fixed-coupon investment. Therefore, the break-even rate represents the market’s consensus on future inflation. Break-even rates are published for a range of countries and maturities on a daily basis in the Financial Times and on Bloomberg.

For instance, at the time of writing the yield on 10-year US notes was 1.6%, and the yield on 10-year TIPS was −0.8%. The 10-year US breakeven rate was therefore 1.6% −(−0.8%) = 2.4%, indicating that the market expects average annual US inflation of 2.4% over the next 10 years.

Similar measures can be calculated for other maturities to generate an inflation term structure, showing the market’s inflationary expectations over a range of timescales.

At a given maturity, the break-even yield y_b is the difference between the nominal yield y_n and the real yield y_r:

If future inflation is higher than the level implied by the current break-even level, then inflation-linked securities will give a better rate of return than ordinary gilts, and vice versa. Break-even rates therefore present an additional source of return for portfolio managers, and the attribution analysis framework in this chapter allows the returns of an inflation-linked bond to be decomposed into returns made by break-even yields and real yields.

Depending on the investment strategy, managers of inflation-linked funds may prefer to see returns either on a total return basis or on a nominal plus break-even return basis.

17.7 PRICING AN INFLATION-LINKED BOND

A fixed-coupon bond requires only a nominal yield for its price to be calculated. In contrast, an inflation-linked bond requires two sources of information:

• its real yield;
• the inflation rate.

If the inflation-linked bond’s real yield is positive, it is outperforming inflation. At the time of writing (January 2013) many real yields on inflation-linked securities were negative, implying that expected inflation is higher than Treasury bond yields. The implication is that holders of these inflation-linked bonds are accepting a return on their investment that is lower than inflation. This is not the same as a negative absolute return; the bond’s return is still positive, but its return is not keeping pace with inflation.

One reason investors are willing to accept this poor performance is that this negative real return is known and certain. The bond is government guaranteed, so the risks of default or other credit events are minimal.

17.8 REAL YIELD TERM STRUCTURE

Some central banks publish real yield curves, showing expected real yields as a function of maturity date. Real yield curves should always be used for attribution of inflation-linked bonds, as they may have a different shape and behaviour to the nominal term structure.

The Bank of England also publishes an implied inflation curve, showing the term structure of inflation implied from relative yields of gilts and inflation-linked securities. This indicates expected levels of inflation at various dates in the future, as implied by the bond markets.

If the effects of changes in the shape of the real curve need to be measured, real yield curves will be required for attribution analysis.

17.9 PRICING AN INFLATION-LINKED BOND

The pricing formula for an inflation-linked bond is identical to that of a vanilla bond, save that real yields are used rather than nominal yields. The price of a UK index-linked gilt, paying two coupons a year, is

where

• P is the real dirty price per £100 nominal;
• c is the coupon per £100 nominal;
• r is the number of calendar days from the settlement date to the next coupon date;
• s is the number of calendar days in the full coupon period in which settlement occurs;
• n is the number of quasi-coupon periods from the next coupon period to maturity;
• where y is the real yield.

Note that the market price must be indexed using the inflation ratio, shown in the previous section.

Alternatively, an inflation-linked bond can be priced off the real yield curve using the expression

where c has the same meaning as above, y_n,t is the real yield for the n th cash flow at maturity t and N is the number of outstanding coupons.

Example

A 1.875% coupon bond, maturing on 22 November 2022, is purchased on 5 January 2012 at a real yield of –0.60698%. The settlement date is 6 January and the real dirty price is £128.18180 per £100 nominal.

The RPI factor on the date the bond was issued was 205.658. Therefore, the price must be indexed by multiplying by the ratio of the inflation index on this date (238.06452) to the RPI factor on the date the bond was issued (205.658), giving an actual price of

17.10 MODIFIED DURATION AND RETURN OF INFLATION-LINKED GILTS

In the current context, modified duration has a slightly different meaning to that for bonds without inflation indexation. Here, modified duration is a measure of the bond’s price sensitivity to changes in its real yield. However, the Fisher equation states that the nominal yield of a bond is given by the sum of the real yield and the inflation return, so a modified duration calculated from the above pricing equation will not take inflation returns into account.

Running attribution on an inflation-linked bond therefore requires a slight modification to the usual expression. In addition to the usual returns generated by the passage of time and by movements in real interest rates, we also have a known return generated by the index ratio:

where

• r is the inflation-linked bond’s return;
• y is the real yield to maturity;
• δt is the elapsed time;
• MD is the modified duration with respect to real yields;
• δy is the change in real yield;
• IR_t−1 and IR_t are the applicable inflation ratios at the start and end of the current interval, respectively.

The third term is required because the return due to inflation has been removed from the returns due to market movements. In a conventional bond, inflation and investment returns are already combined in the form of the nominal yield.

This term is sometimes called inflation carry, but it is more properly a separate source of return unrelated to carry, and should always be displayed separately.

Equation (17.6) is the appropriate perturbational attribution equation to use for inflation-linked bonds.

Example

Table 17.1 Attribution on a 1.875% 2022 UK Index-linked gilt

Source: Compiled with data from the UK Debt Management Office, http://www.dmo.gov.uk

Table 17.1 provides an example of this expression’s use in practice. Here

• P is market price;
• Y is yield to maturity;
• MD is modified duration;
• IR is inflation ratio;
• r_M is market return, calculated from market price and coupon payments;
• I is inflation carry;
• r_C is calculated return, using (17.6).

17.11 BREAK-EVEN YIELDS IN ATTRIBUTION

This equation can be extended further to measure the effects of changes in the break-even rate. Denoting y_n as the nominal yield and y_b = y_n − y_r as the break-even yield and substituting into equation (17.6), the attribution equation becomes

In addition to the carry and inflation return terms, the market return term is now split into two new terms, one driven by changes in the nominal yield (the market return) and one driven by changes in the break-even yield (the inflation expected return).

Whether it is useful to calculate break-even returns in this way depends on the investment approach taken by the portfolio manager.

For instance, an index-linked fund with an index-linked benchmark is managed by forecasting movements in the real yield curve, so an attribution report should present returns in terms of curve changes. Break-even and nominal risk do not form part of the investment strategy, so there is no need to display returns at this level of detail.

Conversely, a trader who wants to minimise duration risk but take exposure to break-even spreads might buy a long nominal bond and sell an inflation-linked bond with the same duration. Here, the attribution report should show real yield returns broken down into nominal and break-even returns, as the trader is taking explicit inflation bets in their portfolio.

17.12 INFLATION SWAPS

An inflation swap is an elegant way of taking exposure to inflation risk.

The simplest (and most common) inflation swap is an exchange of cash flows at maturity. The holder of such a security pays a fixed amount at maturity but receives an amount indexed to inflation over the life of the swap, and the value of the swap at times before maturity is determined by movements in inflation since inception. For more details on how to treat swaps in an attribution framework, refer to Chapter 20.

17.13 PRACTICAL CONSIDERATIONS

• Pricing – If attribution is run using an explicit pricing scheme, inflation-linked bonds can use the same pricing formula as vanilla bonds, but their price and coupons must be indexed to the appropriate value. If a perturbational scheme is used, an inflation term must be added to the fundamental attribution equation.
• Yield data – Inflation-linked bonds require real yields rather than nominal yields.
• Inflation data – Depending on the approach used, price indices and inflation ratio time series must be provided.
• Risk data – Typically the duration measure on an inflation-linked bond is the real duration, or the sensitivity of the bond’s price to parallel shifts in the real yield curve.
• Presentation – Since conventional bonds and inflation-linked bonds show different risks, their returns should be shown in separate categories on an attribution report. This applies to return from both carry and curve movements, since the shape and movements of the curves driving the two markets may be quite different, even though they refer to the same currency.