# Inflation linked bond definition

### Introduction

This security type refers to a generic inflation-linked bond. Such securities are also referred to as inflation-indexed securities.

### Security description

An inflation linked bond is similar to a standard bond in that it has a known maturity date and pays regular coupons. Unlike a regular bond, however, its principal and coupons are linked to an underlying inflation index using a measure such as the General Index of Retail Prices (RPI) in the UK, or the Consumer Price Index (CPI) in the US. The owner of an inflation-linked bond is offered some degree of protection against the value of the asset being eroded by inflation.

From an attribution analyst's perspective, an inflation-linked bond differs from a vanilla bond is different in two main ways:

- The bond is priced using a real yield curve instead of a nominal yield curve
- The bond's return includes an inflation term, measuring the return generated by the underlying rate of inflation in addition to the return generated by carry and curve movements. This term is sometimes called inflation carry.

#### Inflation ratios

Most inflation-linked securities use some form of *inflation ratio* to measure inflation.

In most cases the current rate of inflation will not be known at the date the security is traded, so recent inflation measurements are used instead to calculate an indicative daily rate of inflation. This allows the return due to inflation to be estimated.

For example, consider a UK inflation-linked gilt with a three month indexation lag, which is typical of most bonds in this class. The daily index ratio <math>IR_t</math> is defined as follows:

<math>IR_t = RPI_t + \frac{RPI_{t+1} - RPI_{t}}{D}\,</math>

where

- <math>RPI_t</math> and <math>RPI_{t+1}</math> are the published values of the RPI (Retail Price Index) for the first day of the month three months ago, and the first day of the month after that date, respectively;

- <math>t</math> is the calendar date at which the ratio is to be calculated;

- <math>D</math> is the number of days in the current month.

Note that the applicable RPI used in this formula is the published RPI for the calendar month falling three months earlier than the calculation date. 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 5th January 2012 uses the RPI factors for October (238.0) and November (238.5). The inflation ratio at this date is therefore

IR(t) = 238.0 + (4/31) * (238.5 - 238.0) = 238.06452.

### Real yields, nominal yields and breakeven rates

A common source of confusion in discussing inflation-linked bonds is the meaning of yield. Here we describe and clarify the meanings of real yield, nominal yield and breakeven yield.

#### Real yield

For inflation-linked bonds, the effect of inflation is explicitly separated from the actual yield of the bond. Trades and revaluations are made on the basis of the real yield.

Real yield measures the bond's return over its lifetime assuming zero inflation. The term is used because it is a measure of the true, or real, growth in the bond's purchasing power over its lifetime.

The real yield of an inflation-linked bond is known at all points during its lifetime. In comparison, the real yield of a vanilla (non-indexed) bond can only be measured after the bond matures, since this is the first point at which the effects of inflation on principal and interest can be measured.

If an inflation-linked bond's real yield is positive, its return will outperform inflation. However, at some points in recent years many real yields have traded at negative values, implying that expected inflation was higher than Treasury yields. The implication is that holders of such bonds are accepting a negative real return on their investment.

The reason they are willing to do so is that this negative return is known and certain, and that other investments may well show larger losses over the same interval. In other words, the owner of an inflation linked security is paying the issuer to keep the money safe.

#### Nominal yield

Non-indexed bonds are often compared using their nominal yield. This is the yield to maturity of the bond, which measures the return the owner can expect if holding the bond to maturity, without adjusting for inflation. Inflation may have adjusted the value of the bond, but this is not accounted for in the bond's return.

Inflation-linked bonds also have a nominal yield. This is given by the Fisher equation, which states that the nominal interest rate <math>y</math> is approximated by the sum of the real interest rate <math>r</math> and the inflation rate <math>i</math>:

<math>y = i + r</math>

#### Break-even yield

The break-even yield is the rate of inflation for which the nominal yield on an inflation-linked bond is the same as the nominal yield of a conventional bond with the same maturity. In other words, it measures whether a fixed-rate investment is outperforming an inflation-linked investment. The break-even rate is published on a daily basis in the financial press.

The break-even yield <math>Y_b</math> is the difference between the nominal yield <math>Y_n</math> and the real yield <math>Y_r</math> at a given maturity:

<math>Y_b = Y_n - Y_r</math>

If 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 FIA's attribution analysis framework allows the return of an inflation-linked bond to be decomposed into returns made by break-even yields and real yields.

### Attribution on an inflation-linked bond

The return of an inflation linked bond is decomposed as follows:

<math> r = y \times \delta t - MD \times \delta y + IR \times \delta t </math>

where

- <math>r</math> is the inflation-linked bond's return
- <math>y</math> is the real yield to maturity
- <math> dt </math> is the elapsed time
- <math>MD</math> is the modified duration with respect to real yield
- <math>\delta y</math> is the change in real yield
- <math>IR</math> is the inflation return for the bond in the current month.

The third term is required because the return due to inflation has been removed from the returns due to market movement, since this expression use real yields rather than nominal yields. For conventional bonds, inflation and real returns are already combined in the form of nominal yield.

The derivation of the inflation return is described below. This term is sometimes called inflation carry, but in our view it is a separate source of return distinct from other carry terms, and should be displayed separately.

### Security code

Inflation-linked bonds have security type `INFLATION_LINKED`.

### Calculation of returns

Bonds are priced as the sum of the various discounted cash flows generated by the security. Because coupon-paying bonds are ubiquitous within the fixed income markets, FIA provides specialised internal routines to price these securities to a very high degree of accuracy, using the relevant yield curve.

### Descriptions

All quantities follow the pattern described for vanilla bonds. The exception is the daily inflation return, which is calculated as follows:

<math>I_R = \frac{\left[\frac{RPI_2}{RPI_3} - 1\right]}{d}</math>

where

- <math>RPI_2</math> and <math>RPI_3</math> are the values of the retail price index two and three months ago, respectively;
- <math>d</math> is the number of days in the current month.

For example, to calculate the daily inflation return for UK inflation-linked gilts during April 2010, the value of the UK Retail Price Index for January and February are required; these are 217.90 and 219.20, respectively. There are 30 days in April. Therefore, the daily inflation return is

<math>I_R = \left[\frac{219.20}{217.90}-1\right] \div 30 = 0.019887\%</math>

This quantity is calculated automatically within FIA.

RPI data for the UK is available from the Debt Management Office website at http://www.dmo.gov.uk.

### Notes

1. The calculation of the inflation return is an approximation, but will be very close to the correct value. The exact value is given by the ratio of the RPI index over two days, and this index is subject to rounding on each day. The cumulative effect of the approximation will be less than 0.5 basis points per year.