Monetary Review - 2nd Quarter 2004

Bo William Hansen, Financial Markets

Introduction

Index-linked bonds are bonds where the principal, outstanding debt, instalments and/or repayments are regulated using indices. This article considers index-linked bonds from an investment point of view, i.e. how the return/risk is perceived and how index-linked bonds are managed. The starting point is an investor presenting nominal accounts. An interesting aspect in this connection is that index-linked bonds in some respects resemble bonds denominated in a foreign currency. In that case the foreign currency is the index factor.

The motive for introducing index-linked bonds was the wish to compensate investors for inflationary erosion of the future nominal payments. Payments are therefore written up by a factor corresponding to the price development so that payments are in real amounts. In principle, payments on index-linked bonds can be linked to any index – e.g. gold, dollars or stocks. In practice, they are linked to an index of consumer prices.

Index-linked bonds should be seen as an independent class of assets on a par with nominal bonds and shares. The yield on index-linked bonds is a real yield (i.e. a yield with no built-in compensation for expected inflation).^[1] In addition, the investor is compensated for the erosion of purchasing power. The real repayments are thus known in advance, while the nominal repayments are not known until later. The opposite applies to nominal bonds.

In connection with investments in index-linked bonds, new measures must be calculated for assessing and managing the portfolio. As far as calculation goes, these measures are not complicated. They are break-even inflation, inflation sensitivity and real interest-rate sensitivity.^[2] Break-even inflation is used to assess the relative pricing of index-linked and nominal bonds, while the other two measures indicate the exposure to the development in inflation and the real yield, respectively.

Index-linked bonds have long been a topic in academic literature, and over time a number of leading economists (including Keynes, Musgrave, Friedman and Barro) have argued in their favour – from the borrower's as well as the investor's point of view.^[3] In Denmark the issue was intensely analysed and discussed in the 1970s.^[4] Previously index-linked bonds were mainly introduced in countries with high inflation. Today index-linked bonds are issued for other reasons, primarily diversification, and several countries now include index-linked bonds in their government-debt programmes. The most liquid markets for index-linked bonds are Australia, Canada, France, Italy, the UK, South Africa, Sweden and the USA.^[5] Chart 1 shows the market value in the individual markets.

Market value of index-linked bonds issued, 31 december 2003	Chart 1

Source: Barclays Capital.

A Danish market for index-linked bonds was introduced in 1982, but is now languishing, inter alia due to the Danish rules governing the issue and taxation of index-linked bonds.^[6] The market for Danish index-linked bonds has also been characterised by various borrower considerations. This has resulted in the issue of many bonds with different structures and contributed to reducing liquidity.

Recent years have seen renewed interest in index-linked bonds. One reason is that investors and borrowers are often interested in repayments linked to inflation, so they can achieve a diversification gain by holding index-linked bonds in their portfolios. Another – perhaps more important – reason is that they are beginning to comprehend the special features of index-linked bonds. The increasing interest in index-linked bonds has led to the creation of a rapidly growing market for inflation-linked financial derivatives – e.g. inflation swaps.

Nominal and real yields – fisher's equation

Chart 2 shows the (nominal) yield to maturity on a French nominal government bond and the (real) yield to maturity on a French index-linked government bond. Both bonds mature in 2009.^[7]

Yield to maturity on french nominal and index-linked bonds	Chart 2

Source: Barclays Capital.

To illustrate the difference between index-linked and nominal bonds the classical Fisher identity is a good starting point. Fisher sees the nominal yield (i) as the sum of the real yield (r) and expected inflation (p^e):

(1)

Fisher's equation is often extended with an "inflation risk premium" (n):

(2)

where n is, inter alia, a function of the variability in inflation and the maturity of the bond, cf. below.^[8]

The inflation risk premium reflects a wish on the part of investors for further compensation for the inflation risk associated with buying nominal bonds, e.g. for incurring a risk that actual inflation exceeds expected inflation.^[9] The existence of the inflation risk premium is an empirical question. Estimation of the individual elements is complicated by the fact that neither p^enor n can be observed in practice. At the same time, the value of the risk premium can be expected to vary over time and between markets.^[10] Most studies are based on data from the USA and the UK and estimate a significantly positive inflation risk premium, i.e. a higher expected return on nominal bonds.

In order to gain a better understanding of the underlying risks associated with investing in index-linked bonds, it is useful to compare with an investment in e.g. US nominal dollar-denominated bonds. The US investor who solely cares about the return in dollars naturally incurs no exchange-rate risk when investing in dollar-denominated bonds, but solely exposure to the development in US interest rates. A foreign investor in dollar-denominated bonds, on the other hand, is typically interested in the return measured in his own currency and therefore incurs an exchange-rate risk.

For the foreign investor, the dollar rate may in principle be interpreted as the bond's "index", and the development in the "index" constitutes a risk in terms of the return. On the basis of equation (2), the foreign investor's total expected return comprises an expected return in dollars, as well as a return in the form of the expected development in the dollar rate, and finally an exchange-rate-risk premium.

In the same way, an investor in index-linked bonds who is only interested in the real return naturally runs no inflation risk. However, he still incurs a risk in terms of the development in real interest rates. This corresponds to the dollar-based investor only incurring a risk in terms of the dollar interest rate.

An investor in index-linked bonds who is interested in the nominal return, however, does incur an inflation risk, i.e. a risk connected with the development in the price index applied, in the same way that the foreign investor runs an exchange-rate risk. It should be borne in mind that although an investor has nominal accounts he may well have real liabilities, which naturally means that investments in index-linked bonds will reduce his sensitivity to inflationary changes.

Break-even inflation

When assessing index-linked bonds in relation to nominal bonds, break-even inflation is the basic key indicator. This is the rate of inflation at which the return on an index-linked bond is equal to the return on an equivalent nominal bond.

Break-even inflation is comparable to e.g. forward rates in the foreign-exchange market balancing the return on bonds denominated in different currencies.

In practice, break-even inflation is often approximated by subtracting the yield to maturity on an index-linked bond from the yield to maturity for a (comparative) nominal bond. Thus break-even inflation (bei) is equal to

(3)

A more advanced method consists in deriving break-even inflation using the market price of the index-linked bond. Break-even inflation can be applied as a measure of the relative pricing of index-linked bonds and nominal bonds. If break-even inflation is higher than the investor's own inflation expectations, the expected return on index-linked bonds is lower than for nominal bonds.

Break-even inflation is typically volatile since the ratio of nominal to real yields is not constant. In periods with relatively strong demand for index-linked bonds, break-even inflation rises. At the same time, break-even inflation typically rises with the maturity of the bond. The reason is that the inflation risk premium increases with the maturity, cf. Chart 3.

Break-even inflation for selected us index-linked bonds	Chart 3

Source: Barclays Capital.

Pricing

For index-linked bonds a distinction should be made between the real price, i.e. the price before indexation, and the indexed value, the market value. The following relationship applies

(4)

where Vis the indexed value, P is the real price^[11] and I₀ is the current indexing factor. The indexing factor is a function of inflation (p). The indexed value is the amount which an investor must pay to purchase the bonds.

In the previous example, where the index is the dollar rate, P corresponds to the dollar price, I to the exchange rate and V to the price in the domestic currency.

In the markets the real price is typically quoted. Provided that the real price is known, the real yield to maturity can be calculated in a standard manner using the following equation

(5)

where C_t is the bond's known real payment at the time t, r is the real yield to maturity and T is the expiry date. All other things being equal, a higher real price lead to a lower real yield to maturity. Equation (5) can also be used to calculate the real theoretical price on the basis of the real yield to maturity or real-yield curve and a known series of real payments.

Sensitivity measures

For an investor investing in index-linked bonds the development in V – the market value of the bond – is interesting from a nominal point of view. The development in the market value depends on the parameters affecting I and P, cf. equation (4). V is thus a function of inflation and real interest rates, which are the primary first-order derivative. The return in kroner over a period can be written as

(6)

where

∂V/∂r is the sensitivity to the real interest rate (real krone duration).
∂V/∂p is the sensitivity to inflation (which is close to 1).
∂V/∂t is the direct return (carry) in the form of coupon and reduction of maturity.

For investments in foreign index-linked bonds, the return is furthermore exposed to the development in exchange rates.

In practice many investors are interested in the exposure to the nominal interest-rate, e.g. if their liabilities are calculated on the basis of a nominal yield. An investor with a portfolio comprising both index-linked and nominal bonds thus wants to know the portfolio's aggregate nominal interest-rate sensitivity.

For index-linked bonds it only makes sense to calculate the nominal sensitivity if the co-variation between the nominal and real yield is 1. This is seldom the case since inflation expectations and premiums change, cf. equation (2). The nominal and real yields are thus not fully correlated. Typically, the nominal interest-rate sensitivity is lower for index-linked bonds than for otherwise comparable nominal bonds. For an investor with a nominal portfolio the introduction of index-linked bonds may therefore give a lower expected risk due to diversification, a so-called diversification gain.

Why invest in index-linked bonds?

Index-linked bonds are an obvious investment object for investors with commitments that are sensitive to inflation. This could be pension funds with real commitments. However, if assets and liabilities are to be matched, the index-linked bonds must have the same maturity as the liabilities. If the maturities differ, a real-interest-rate risk will persist. Pension funds with nominal commitments may also find it expedient to hedge the real value of the pensions to a certain degree.

As a result of the replacement need in connection with the regular interest payments and instalments, a strategy comprising index-linked bonds may not always ensure a given real return, just as a strategy comprising nominal bonds may not always ensure a given nominal return.

Even though an institution does not have index-linked liabilities, it may make sense to invest in index-linked bonds due to diversification gains. Empirically there have been small or negative yield correlations between index-linked bonds and other assets, cf. e.g. Barclays Capital (2004).

Literature

Andersen, Jens Verner and Jacob Gyntelberg (1999). Index-Linked Mortgage Bonds, Danmarks Nationalbank, Monetary Review, 1st Quarter.

Barclays Capital (2004). Global Inflation-Linked Products – A User's Guide, New York: Barclays, January.

Report No. 732 (1975). Statement from the working group on index-linking of monetary claims (in Danish).

Price, R. (1997). The Rationale and Design of Inflation-Indexed Bonds, IMF Working Paper 97/12.

Ølgaard, Anders (1975). Financing during inflation – index-linked loans and other options (in Danish), Nationaløkonomisk Tidsskrift, volume 113, No. 1.

[1] It should be noted that these are real yields in relation to the index applied, and thus only "pure" real yields to the extent that the index in question reflects the actual price development in society.

[2] Inflation sensitivity is relevant from an accounting point of view where the investor takes the nominal return into account.

[3] For a review of the literature and the various rationales behind index-linked bonds, see Price (1997).

[4] Cf. e.g. Report No. 732 (1975) and Ølgaard (1975).

[5] Index-linked bonds are also issued in Greece, Iceland, Israel, Mexico and New Zealand, among other countries. The Japanese government-debt office opened a new 10-year index-linked bond series in March 2004. For a detailed description of the individual markets, see Barclays Capital (2004).

[6] For a detailed description, see Andersen and Gyntelberg (1999).

[7] Here we will consider the choice between two placements of equal horizon. In practice the decision involves a horizon choice between short-term and long-term placements.

[8] In principle liquidity premiums and any other premiums can be added to Fisher's equation.

[9] A high inflation risk premium has been used as an argument for issuing index-linked bonds and thus reducing issuing costs.

[10] The inflation risk premium must therefore be expected to have been larger in the 1970s and 1980s when inflation was far higher and more volatile than now.

[11] The real clean price plus real accrued interest.