*real interest rate*is relatively old, and often associated with the work of Irving Fisher (the

*Fisher Equation*). We decompose observed nominal interest rates into two components: inflation compensation, and a real rate of interest. I prefer to avoid using the term real yield, as the standard definitions cannot be easily related to real world data, and there are at least five definitions that have been used in practice. In the context of index-linked bonds ("linkers") using the Canadian pricing model, I prefer to call the quoted yield the "indexed yield," and not the "real yield" (which appears to be the market convention).

*(This article is an unedited draft of a section that is hoped to make it into a book on inflation-linked bonds. Since I am in the process of trying to finish off the formatting of my book on Python and SFC models, the writing process for this article was rushed.)*

## Possible Definitions

If we denote the nominal interest rate as*i*, inflation compensation as

*p*, and the real yield as

*r*(putting aside the definitions of those terms until later), the preferred mathematical relationship between these variables is given by:

*i = (1+r)(1+p),*

(the Fisher Equation).

If the inflation compensation and real yield variables are much smaller than 1, then we end up with the approximate relationship:

*i = r + p.*

In my writing, I will often make statements like "subtract inflation from the nominal rate to get the real rate." This is a shorthand for the slightly more complex relationship implied by the Fisher equation. If I write out the equations, the relationship I am using in that context is given exactly.

The problem with this definition is that there is no clear way to define

*inflation compensation*, and so we can get (at least) five very distinct types of definitions used that match up with the common usage of real yield. (Since nominal interest rates are very well defined in legal terms, once we come up with a fixed definition of inflation compensation, the real yield definition if fully determined. One could theoretically define the real yield and inflation compensation first, and then the nominal interest rate would be determined by those two definitions. However, it would not necessarily match up to the legal definition of interest rates, and so we would need to translate quoted nominal yields to match this theoretical definition. This is a step that most analysts would want to avoid.)

Five competing definitions for real yield/inflation compensation are given next. They are definitions that apply to a particular debt instrument (with a fixed maturity) at a particular date. Note that some of the terms are ambiguous, as will be discussed later. For the first three definitions, the instrument in question is a conventional bond (with a nominal quoted yield), the final two definitions apply only to index-linked bonds.

- Inflation compensation is equal to the latest available value of the inflation rate, and we then back out the yield yield.
- Inflation compensation is equal to the annualised inflation rate over the life of the debt instrument for which we extracting the nominal interest rate; we then back out the real yield.
*Note that we can only arrive at this measure after the maturity of the debt instrument, and so we cannot use it in real-time. However, it could be used in historical analysis.* - Inflation compensation is equal to the expected value of inflation at the valuation date; we then back out the real yield.
- The instrument in question is a Canadian model inflation-linked bond, and the real yield is the quoted yield on the bond. I
*n this definition, there is no strict definition of the nominal interest rate, nor inflation compensation.* - If the debt instrument in question follows the old model of U.K. inflation-linked gilts, the real and nominal yields are the result of a fairly complicated calculation procedure. This methodology requires the analyst to set an expected rate of future inflation, and then the implied internal rate of return in nominal terms is calculated. Since these bonds are now a curiosity, I will not attempt to discuss this in detail.

(There is an additional complication regarding nominal interest rate quote conventions, which I am avoiding. In order to fit into the equation, all nominal rates would need to be converted to a clean mathematical convention, and not the various silly quote conventions that are used by market participants.)

Historically, inflation-linked bond yields were not available, and economists mainly focused on the first and third definitions (where inflation compensation is either equal to spot historical inflation, or spot expected inflation). For a variety of reasons, the two usages ended up being commingled, and so unless the economist in question specified which definition they are using, the usage was ambiguous. However, even if they specified whether they were using spot inflation or expected inflation, those definitions themselves are ambiguous. I will discuss these in turn.

## Historical Inflation Ambiguous

For someone working with a time series database and analytical software, specifying the real yield using historical data seems like a straightforward operation. You load the time series of the bond yield (as a monthly series), get the time series of inflation, and subtract the inflation rate from the bond yield to get the real rate. (Or use the slightly more complex Fisher Equation to back out the real rate.)

Although this might satisfy market economists, it is not good enough for investors where every basis point counts.

I will take as an example the data that are publicly available as of the time of writing (Saturday, November 11, 2017) for the United States. The interest rate data comes from the Federal Reserve H.15 release. As a result of the Veteran Day's holiday, the last H.15 release was on the Thursday. covering the data for Wednesday, November 8.

- The effective Federal funds rate (an overnight bank rate) was 1.16%.
- The (fitted) constant maturity 10-year Treasury rate was 2.35%.
- The last Consumer Price Index (CPI) data was released by the Bureau of Labor Statistics on October 13, and corresponds to the month of September 2017.

Let us step back to the situation on November 8. If you are a market participant, you would have had the yield data available in real time (for traded instruments, not the fitted/averaged rates produced by the Federal Reserve).

We then want to know: what is the real effective Federal funds rate on that day? A Fed funds agreement is a lending agreement on an overnight basis, that is, from November 8 to November 9.

We can immediately see problems with the definition using historical inflation. Our last data point is for the month of September; we will not get November CPI data until several weeks have passed. Furthermore, the CPI data for September is a level for September, not a rate of change. The usual technique is to look at the annual rate of change in the index; in this case, from September 2016 to September 2017. In other words, we deflating the returns on a lending operation between November 8, 2017 and November 9, 2017 by the percentage change in the CPI index between September 2016 and September 2017. What sort of information is supposed to be conveyed by that comparison?

We can attempt to make the inflation data more comparable in time by looking at a shorter-term rate of change. The closest we can get to the valuation date is to look at the 1-month annualised inflation rate between August 2017 and September 2017. However, that operation runs into the problem that monthly rates of change are extremely erratic, especially if we use the non-seasonally adjusted index data. (The use of the seasonally adjusted CPI poses other challenges; it is generally not used in legal contracts -- like in inflation-linked bond calculations -- for these reasons.)

The situation for the 10-year bond is even more challenging. The hypothetical constant maturity 10-year bond has cash flows every six months from November 8, 2017 to November 8, 2027 (with holiday and weekend dates moved according to predetermined rules). What relevance does the change of the CPI index from September 2016 to September 2017 have to those cash flows?

## Expected Inflation: Also a Mess

Many economists would have looked at the previous discussion of the definition using historical inflation and tut-tutted. The correct answer -- allegedly -- is to use expected inflation to stand in for inflation compensation in the Fisher equation.

Unfortunately, this alternative is largely as problematic as the use of historical inflation.

The obvious problem is that we have no idea whose expectations we should use. (I am using expected inflation here as a synonym for

*forecast*inflation, and not the mathematical expectations operator. The problem with the use of the mathematical definition of expectations is that it is defined in terms to a probability distribution for future inflation outcomes, and so we still have the problem of deciding whose probability distribution to use.)
If you wish, you can substitute your own forecast for inflation into the expression. Although I think it is a silly way to approach fixed income investing, using your own forecasts is an entirely reasonable internally coherent methodology. The problem with the approach is straightforward: the results are only useful to yourself; different individuals/entities will have different inflation forecasts, and so different implied real rates. Furthermore, there is no way of doing historical analysis, unless you have written down your inflation forecasts every day -- and what happens for dates before you started writing down those forecasts?

An alternative is to use some surveys of expected inflation. This eliminates much of the subjectivity, although it still raises the question of how to analyse data from before the inception of the survey. However, it raises the problem that the people being surveyed may have entirely useless opinions about the path of inflation. Let us take an extreme example: imagine a survey of inflation expectations taken from a sample of kindergarten students. Under what circumstances would we expect such data to provide useful information?

However, even if we accept the surveys as being slightly useful, we still have issues with timing. Firstly, we generally do not have inflation survey data available on the same day as the survey was taken. Therefore, we would be comparing the yields on November 8 with a survey taken at some earlier date. Secondly, in order for the data to be useful, we need to match up the maturity of the debt instrument with the period for which we are doing a survey.

Even if we can get a survey that matches the monthly frequency of the CPI data, we have no way of determining the inflation rate between November 8 and November 9 using the monthly CPI. We need a daily frequency price index in order to compare properly to debt instruments. (In hyperinflations, we get this level of granularity by estimating the price level by the use of inferring the price level from the level of the currency.) There are no surveys that cover inflation at a daily frequency.

We end up in a curious situation. According to conventional economic models, we need to use inflation expectations to determine the real yield. However, in practice, there are no data available that meets the requirements. We have economic theory that literally cannot be applied to real world data.

## Quoted Inflation-Linked Bond Yields has a Hidden Twist

Using the quoted yields on Canadian-style inflation-linked bonds appears to be the only rigorous way to approach the definition. This makes sense: we need to be able to determine the invoice price on a bond trade to the penny, based on an agreed yield quote. Like nominal interest rates, the contract law ensures clarity of definition.

There is a hidden trick to the story: the reason why it works as we have dropped the other two components of the Fisher Equation.

*There is no nominal yield quoted on the instrument, and so there is no inflation rate associated with it.*If we want to go from the quoted yields on inflation-linked bonds to nominal interest rates and/or inflation compensation, we need to find a nominal interest rate to compare the yield on the inflation-linked bond. This is normally done via breakeven inflation analysis, but there are multiple potential definitions of breakeven inflation.
(The old U.K. style inflation-linked bonds did tie together inflation compensation, real yields, and nominal yields. However, they did so in a fairly horrific fashion, using an inflation assumption that causes other problems.)

*[My book will be covering the gory details of rigorous definitions of inflation breakevens. A draft may be published later.]*

## Concluding Remarks

The concept of a real yield, although ubiquitous in economics, is extremely poorly defined. Furthermore, there is considerable mythology regarding the concept. The safest way to approach index-linked bonds is to drop the use of the term completely. The best replacement I am aware of indexed yields. One could rightfully complain that this term is problematic: the yield is not the subject of some indexation procedure. Instead, it is a contraction of "quoted yield on index-linked bonds."

(As an aside, my awareness of the linguistic minefields associated with the term real yield probably came from some observation by my boss at the time, Gerard MacDonell.)

(c) Brian Romanchuk 2017

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