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Wednesday, February 7, 2018

Primer: U.K. Index-Linked Gilts

Her Majesty's Treasury of the United Kingdom was one of the earliest modern issuers of inflation-linked bonds, with issuance starting in 1981. (There were earlier bonds that featured some form of indexation, but those were less systematic in design.) Unfortunately, the original design used is much harder to work with than the structure used by Canada ("Canadian model").

The U.K. has switched over to the Canadian model in 2005, although some of the older structures remain. However, even once the old structure bonds all mature, analysts will still run into pricing data based on them in the back history. In particular, if time series are properly labelled, they will have a caption to the effect that the real yields are based on an inflation assumption of 3% (or possibly 5%), which sounds counter-intuitive if one is familiar with the Canadian model. (How can we assume a level of inflation? Isn't that determined in the market?) This section just presents a high level overview of the old structure in case the reader encounters that back history.

(This article is an unedited draft of a section that will be incorporated into my inflation breakeven report. I am nearing the end of adding material for the first draft version.)

The U.K. Treasury has a page describing inflation-linked gilts at There is also a document describing yield calculations available at:

There are two areas that distinguish the old design and the Canadian model.
  1. Cash flow calculations.
  2. Yield calculations.

Cash Flows

The cash flow calculations in the old structure have a calculation lag of 8 months, instead of 3 months under the Canadian model. (The U.K. Treasury documents will distinguish the issues based on this lag, but as discussed below, this is not the only difference between the issues in practice.) The index used in indexation calculations is the Retail Price Index (RPI), and not the Consumer Price Index (CPI), which is the Bank of England's target inflation index. (This creates basis risk between the inflation target and the inflation-linked bond index. That is, if one is comparing inflation breakevens to the inflation target, one needs to account for the statistical bias between the RPI relative to the CPI.)

The longer lag used means that the next coupon payment will always be covered by already-released CPI data. Nominal cash flows can then be multiplied by the indexation factor for the bond. This means that prices can be safely quoted in nominal terms. This means that quoted prices move well away from the 100 par price, unlike the Canadian model, where quoted prices are real and have a pull-to-par as they evolve towards maturity.

I am unsure why they chose this structure, but it appears it has one advantage: since everything was quoted in nominal terms, the existing trading technical infrastructure could handle these bonds. 

The added calculation lag is slightly unattractive, but in a contained inflation environment, the differences from the Canadian model for cash flows is not that significant. You just need to remember that quoted prices are in nominal terms, and so increase along the RPI. 

Yield Calculation Differences

From an analytical point of view, the big difference between the two bond structures lies in how yields are calculated.

Since calculations were being done in nominal terms, and not in indexed terms, there needed to be a way to project future nominal cash flows. The way this was done was to assume a forward rate of inflation. The current assumption is 3%. In other words, we project the RPI forward at an annual rate of 3%, then use that to generate the index ratios for all cash flows. 

We can then calculate the bond price as a function of the real yield r by using a nominal discount rate (1+r)(1+0.03), or the Fisher relationship, with the assumed inflation rate of 3%. We can invert the price function (probably numerically) to get the real yield as a function of price.

As can be seen, the 3% inflation assumption is arbitrary. If some simplifying assumptions were true, it would not really matter. For example, assume that the nominal yield implied by a 3% inflation assumption was 4%. This would mean that the real yield is approximately equal to 1% (=4% minus the inflation assumption). If we then switched to a 4% inflation assumption, the implied nominal return would be approximately 5%. Once we subtract the 4% assumed inflation, we are back to a 1% real yield.

To get the simple breakeven inflation rate, we subtract the calculated real yield from the yield on a conventional gilt. Since there is no reason that the conventional yield must equal the assumed nominal return of the inflation-linked gilt under the 3% inflation assumption, the breakeven inflation rate will vary from the 3% inflation (as would be expected). If we could assume that the real yield is independent of the inflation assumption of the calculation, the simple breakeven will be as well.

The problem is that those simplifying assumptions do not actually apply. The real yield is somewhat sensitive to the inflation rate assumption, with the deviations growing as the economic breakeven diverges from the assumed inflation rate in calculations. In other words, if the simple inflation breakeven is near 3%, it is a good approximation of the true economic breakeven. However, if the simple inflation breakeven is not near 3%, it will be biased relative to the true economic breakeven.

I am not in a position to judge how significant this effect is in practice. It might only matter if one wants to dig very deeply into the market history of the inflation-linked gilt market. Although that may sound like an obscure topic, it is of some interest. This U.K. history is the only span of inflation-linked bond market data that hails from a period without stable inflation.

Concluding Remarks

If you look at time series data for U.K. linkers, you can treat the real yields as being similar to the indexed yields for Canadian model linkers. We may only need to be concerned about the series being biased if the (simple) inflation breakeven considerably departs from the inflation assumption associated with the series.

(c) Brian Romanchuk 2018

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