Real yields

This expression establishes a variation of the real yield factor as a function of specific growth rate (Equation 27). 

Explosions are rated in terms of the amount of energy released, commonly expressed as an equivalent quantity of trinitrotoluene (TNT). The theoretical maximum values of explosive potentials for fuels (TNT equivalents) are recorded in Table 4.7. Experimental data indicate, that real yield factors of 10 % are considered reasonable. Note that hydrogen is most potent on a mass basis and least potent on a volumetric basis. It also has the least theoretical explosive potential, when equivalent energy storage is taken into account. 

From market observation we know that index-linked bonds can experience considerable volatility in prices, similar to conventional bonds, and therefore, there is an element of volatility in the real yield return of these bonds. Traditional economic theory states that the level of real interest rates is cmistant however, in practice they do vary over time. In addition, there are liquidity and supply and demand factors that affect the market prices of index-linked bonds. In this chapter, we present analytical techniques that can be applied to index-linked bonds, the duration and volatility of index-linked bonds and the concept of the real interest rate term structure. Moreover, we show the valuation of inflation-linked bonds with different cash flow structures and embedded options. 

The effect of general economic conditions and the change in these over time results in real yields rai index-linked braids fluctuating over time, in the same... 

Also the simple difference between the yield of conventional and inflation-linked bonds or yield spread is an indicator for expected inflation. For example, on 18 September 2014 the 10-year UK 0 1/8% inflation-linked 2024 had a money yield of 2.125% and a real yield of -0.410%, assuming an inflation of 2.571%. Differently, the 10-year benchmark, the UK T A% 2023 had a gross... 

FIG U RE 6.3 Example of index-linked yield analysis, UK 01/8% Treasury 2024 (assumed annual inflation rate 2.57%, base inflation index 242.4, reference RPI value 256.1), showing real yield and money yield, 18 September 2014. ( Bloomberg L.P. Reproduced and used with permission.)... 

Figure 6.4 shows the inflation beta calculated as the correlation between nominal and real yield of the United Kingdom gilts, based to 30 days. How can be seen the inflation beta is unstable over time. 

In the traditional approach the duration value is calculated using nominal cash flows, discounted at the nominal yield. A more common approach is to assume a constant average rate of inflation, and adjust cash flows using this inflation rate. The real yield is then used to discount the assumed future cash... 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

A repeat of the above procedure, with the additional step, after the shift in the yield curve, of recalculating the bond cash flows based on a new inflation forecast. This produces a duration measure that is a function of the level of nominal yields. This measure is in effect an inflation duration, or the sensitivity to changes in market inflation expectations, which is a different measure to the real yield duration ... 

One final point regarding duration is that it is possible to calculate a tax-adjusted duration for an index-linked bond in markets where there is a different tax treatment to indexed bonds compared to conventional bonds. In the United States market, the returns on indexed and conventional bonds are taxed in essentially the same manner, so that in similar fashion to Treasury strips, the inflation adjustment to the indexed bond s principal is taxable as it occurs, and not only on the maturity date. Therefore, in the US-indexed bonds do not offer protection against any impact of after-tax effects of high inflation. That is, Tips real yields reflect a premium for only pretax inflation risk. In the United Kingdom market however, index-linked gilts receive preferential tax treatment, so their yields... 

The simple approach, where the average expected inflation rate is calculated using the Fisher identity, so that the inflation estimate is regarded as the straight difference between the real yield on an index-linked bond, at an assumed average rate of inflation, and the yield on a conventional bond of similar maturity ... 

The break-even inflation expectation, where average inflation expectations are estimated by comparing the return oti a conventional bond against that on an indexed bond of similar maturity, but including an appUcatimi of the compound form of the Fisher identity. This has the effect of decomposing the nominal rate of return on the bond into comprments of real yield and inflation ... 

The drawbacks of each of these approaches are apparent. A rather more valid and sound approach is to constmct a term structure of the real interest rates, which would indicate, in exactly the same way that the conventional forward rate curve does for nominal rates, the market s expectatimis rat future inflation rates. In countries where there are liquid markets in both conventional and inflation-indexed bmids, we can observe a nominal and a real yield curve. It then becomes possible to estimate both a conventional and a real term structure using these allows us to create pairs of hypothetical conventional and indexed bonds that have identical maturity dates, for any point on the term structure. We could then apply the break-even approach to any pair of bonds... 

Using the prices of index-linked bonds, it is possible to estimate a term structure of real interest rates. The estimation of such a curve provides a real interest counterpart to the nominal term structure that was discussed in the previous chapters. More important it enables us to derive a real forward rate curve. This enables the real yield curve to be used as a somce of information on the market s view of expected future inflation. In the United Kingdom market, there are two factors that present problems for the estimation of the real term structure the first is the 8-month lag between the indexation uplift and the cash flow date, and the second is the fact that there are fewer index-linked bonds in issue, compared to the number of conventional bonds. The indexation lag means that in the absence of a measure of expected inflation, real bond yields are dependent to some extent on the assumed rate of future inflatiOTi. The second factor presents practical problems in curve estimation in December 1999 there were only 11 index-linked gilts in existence, and this is not sufficient for most models. Neither of these factors presents an insurmountable problem however, and it is stiU possible to estimate a real term structure. 

From this average inflation curve, we can select specific inflation rates for each index-linked bond in our sample. The real yields on each indexed bond are then recalculated using these new inflation assumptions. From these yields the real forward curve is calculated, enabling us to produce a new estimate of the inflation term stmcture. This process is repeated until there is consistency between the inflation term stmcture used to estimate the real yields and that produced by Equation (6.5). 

Real yield curves are of some use to investors, for a number of reasons. These include applications that arise in insurance investment management and corporate finance, such as the following ... 

In essence, the real yield curve can and should be used for all the purposes for which the nominal yield curve is used. Provided that there are enough liquid index-linked bonds in the market, the real term stmcture can be estimated using standard models, and the result is more valid as a measure of market inflation expectations than any of the other methods that have been used in the past. 

Bond valuation with real cash flows and yields We currently illustrate the pricing of inflation-linked bonds adopting real cash flows. As noted earlier, according to Fisher s theory, the real yield is given by the following equation ... 

Therefore, in Equations (6.12) and (6.13) we use both nominal cash flows and discount rates. Conversely, in Equation (6.15) the real yield is the appropriate discount rate for discounting real cash flows. This relationship is tme only in the perfect indexation scenario without indexation lag. 

So the prime obstacle to the development of a Euro-zone real yield curve was gone. And, as we have said, Greece has now joined the sector and other governments are understood to be considering entry, potentially adding points to the real yield curve and investor choice. 

The first linker to be issued was a 20-year bond with a zero-coupon structure (No. 3001, 0% 2014). A selection of the eight Primary Dealers in the nominal market took responsibility to quote two-way prices for the new bond. The Debt Office held five common price auctions from April to June, which saw a face value of SEK16 billion being offered to the market. But demand for much higher real yields from the market meant that only SEK6.7 billion was allotted. 

Here, and in the complexity of the real yield calculation, the United Kingdom suffers perhaps for being the prototype linker market. An eight-month lag is needed in order to always know with certainty the future money, or cash, value of the next coupon, so that the money value of that coupon can be accrued. The major innovation of the Canadian model is the use of a formula that accrues coupons on a real basis, removing the need to know precisely what will be paid in cash terms on the next coupon day. 

These matching errors are less alarming than they might appear, because all known price information is built into the real yield formula as soon as it is published. Two things really matter, in terms of the achieved real yield relative to the quoted real yield at purchase firstly, the difference between future inflation over the life of the bond and the 3% inflation assumption used in the market convention for calculating yields, and secondly (as with any coupon bond) reinvestment risk. However, we are getting ahead of ourselves—the next section handles the market s yield conventions. 

The real yield formula below has been taken from the Debt Management Office s Formulae for Calculating Gilt Prices from Yields, 15 January 2002 update, and it covers bonds with two or more remaining cash flows. The term quasi-coupon date, in the notes that follow the formula, means the theoretical cash flow dates determined by the redemption date—they are quasi dates because weekends and holidays may mean the true payment dates differ. 

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