*(NOTE: This article is an unedited first draft of a section of my upcoming book on inflation breakeven analysis.)*

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*(Comments: I have largely rewritten an earlier article that was meant to be used as a section of the breakeven inflation book. As a result, some text near the end may be familiar to regular readers. I am not hugely satisfied with my discussion of affine term structure models herein. Once again, I am avoiding discussing the mathematics. The book is meant to be more advanced, and it would be appropriate to delve into the mathematics. However, I would largely end up where I am here: the inclusion of consensus forecasts into the curve fitting algorithm is dubious. I do not see any point in paying thousands of dollars for consensus forecast data just so that I can replicate an algorithm that I have zero confidence in. I am currently going through my first draft, and getting it ready for publication. This section was the only section of content that I was unsure about. I have just started this pass of work; I hope to wrap it up within a couple of weeks. I will see then whether it is ready for publication.)*

For the affine term structure approach, one of the issues is that academic and central bank researchers are continuously churning out papers. The class of models is wide, and offers plenty of opportunities for adding to the publication count. I will briefly discuss the paper “Estimating Inflation Expectations with a Limited Number of Inflation-Linked Bonds” by Richard Finlay and Sebastian Wende. (URL: https://www.rba.gov.au/publications/rdp/2011/pdf/rdp2011-01.pdf) I believe that this paper is indicative of the relevant portion of the literature for this discussion, and is interesting in that it discusses the issues of fitting against a limited number of bonds.

In an affine term structure model, we are attempting to characterise a yield (or inflation) curve as a the evolution of a stochastic discount factor (SDF). In this case, SDF is approximated by a function of the instantaneous inflation rate and a market price of risk (term premium) terms. Both of these terms are assumed to be modelled by three latent factors. They then calibrate observed inflation-linked bond prices against what is implied by those stochastic factors, and a nominal discount curve. (The nominal discount curve is a separate fitting exercise.)

The fundamental problem for this fitting exercise is that there is only one observed breakeven inflation curve, yet there are two variables explaining it – the instantaneous inflation rate, and the instantaneous market price of risk (term premium). Without any additional information, any decomposition of the observed inflation curve is legitimate. For example, if the observed inflation breakeven rate is 2%, one could create an affine term structure model that decomposes that 2% into an inflation expectation of 1,000,0002% and an inflation risk premium of -1,000,000%. This is not particularly helpful.

Finlay and Wende inject information by taking an economist consensus inflation forecasts from Consensus Economics. These consensus forecasts pin down the inflation forecast.

This step explains why I personally stop paying to the description of the model. If I wanted to use a consensus economist forecast as an estimate of future inflation, I would just subscribe to the information service, and read off the forecast from the table. I certainly would not grind the data through a nonlinear variant of the Kalman Filter as a first step.

This calibration step explains the results of the paper: “… long-term inflation expectations are well anchored within the 2 to 3 per cent inflation target range, while short-run inflation expectations are more volatile and more closely follow contemporaneous inflation.” This is exactly what you would expect would happen if economist long-term inflation forecasts are anchored by the inflation target. Meanwhile, market inflation expectations are tearing around all over the place. Well, that is just the risk premium being its volatile self.

Consensus economist forecasts are sticky. In most institutions, they are being set by some type of committee that oversees the consistency of the firm’s forecast. The members of these committees rarely always agree with each other, and so an internal consensus needs to be hammered out. Once the compromise position is reached, it takes a lot to change the view, as it implies that someone was wrong. Therefore, even if one updates a consensus view daily, you are just sampling at a daily frequency an underlying series that might change values once very few months. Meanwhile, there is a tendency for economists to stick to their views, and then only capitulating with a lag after the market has already moved.

Conversely, the bulk of information in price setting in fixed income markets is coming from market makers. They have to react very quickly to new data, on the basis that they need to remain in the middle of potential flows. As a result, the market data that you are calibrating against is moving at a high frequency and at real time.

A Kalman filter cannot do magic, all it does is infer the central tendency of data that you feed into it. If you want to decompose a high frequency signal into two components, and you use a low frequency signal as the unbiased estimator of one component, the filter has no choice but to load all of the high frequency dynamics of the signal onto the other. It is no surprise that the term premia estimates are highly volatile in affine term structure models that use low frequency data like fundamental economic data or economist surveys. That lack of volatility just reflects the model construction.

If the reader takes consensus economist surveys seriously, feel free to go ahead and use such models. However, one should not believe that the term premium estimate is truly an estimate of future excess returns based on market movements. In the post-1990 era, the estimates may have worked – since inflation did end up near target, and so the consensus inflation estimate may have been a better forecast of 10-year average inflation than market pricing. However, as all the investment research disclaimers note – historical performance may not be indicative of future results. Those consensus forecasts could end up being delusional, and the markets correct.

For those of us who are skeptical about black box models and consensus surveys, we need another method to determine whether there is an inflation risk premium.

Going back to first principles, an inflation risk premium implies that there is a systematic bias in economic breakeven inflation rates. On average, either buying inflation or selling inflation is profitable.

In conventional bonds, we have an analogy: the term premium. (As noted earlier, the inflation risk premium is equivalent to the term premium for inflation-linked bonds not equalling the term premium for conventional bonds.) Once again, one could try to use affine term structure models.

Alternatively, one could just crunch historical return data. I will first discuss the situation for the term premium for conventional bonds, then return to the inflation risk premium.

We have two options.

- Assume that the term premium for a tenor is constant.
- Assume that the term premium for a tenor is a function of some observed variables.

A constant term premium is straightforward. We just look at historical outperformance of bonds of a certain tenor versus (risk-free) cash. The estimate of the term premium is the average of this historical experience. (Given autocorrelation, one should probably take into account that outperformance is auto-correlated – the outperformance of a 10-year bond today is likely going to be very similar to the outperformance of a 10-year bond tomorrow. (If we do not care about the statistical analysis of errors, we can largely gloss over this concern.) In any event, it seems safe to argue that longer maturity bonds outperform shorter maturities (particularly cash); the outperformance is sensitive to the data set chosen.

There are reasons to be suspicious of this analysis for long maturity bonds. We do not have that many independent observations of 30-year government bonds for any particular currency, and one would question the applicability of data from the Gold Standard era to a sovereign borrowing in a free-floating currency it controls. The fact that interest rates were regulated in the developed countries (with regulations dismantled at different times) cuts down the range of useful data even more. Furthermore, adding currencies to the data set does not add too much information: interest rate and inflation trends were highly correlated in the developed countries in the period of deregulated interest rates.

However, this does not apply to short-term debt. A government will issue 12 completely independent 1-month Treasury bills in a year, and we have decades of relevant data to work with. (Starting in 1990 or so is relatively safe; if we go back to the 1970s, we run into the problem that interest rates were regulated in some countries. This makes the data non-comparable to our highly non-regulated present institutions.) So we can tune estimates of short-term (under 1-year tenor) as much as we would wish.

Unfortunately, a constant term premium estimate is arguably not entirely satisfactory. It is possible that it is being biased by some factor. We could then compare this factor to the realised outperformance of bonds. One article in this vein is “Bond Supply and Excess Bond Returns,” by Robin Greenwood and Dimitri Vayanos (URL: https://www.bankofcanada.ca/wp-content/uploads/2010/09/greenwood_robin.pdf). I had looked at this paper because of some other research; it attempts to relate relative performance to the maturity structure of bond supply.

If we step back from the problem, we realise that finding a term premium estimate based on some variables is almost mathematically equivalent to finding a bond fair value model. Therefore, one’s belief that one can develop an accurate term premium estimate should align with one’s belief in the ability to develop an accurate bond valuation model. It is safe to this is the subject of a massive amount of proprietary research in the fixed income management business, but since the research is proprietary, it is not clear how strong the conclusions are.

We can now turn to the question of the inflation risk premium. Do we have a reason to believe that inflation breakevens are biased, or equivalently – are inflation-linked bonds inherently rich or cheap versus the nominal curve?

We quickly realise we have much less data to work with. In particular, there are no inflation-linked bonds with money market maturities. We do not have a dozen one-month inflation-linked bonds maturing each year to hone our estimates of realised outperformance to maturity.

- In Canada, the shortest maturity linker matures in 2021, and so at the time of writing, no Government of Canada Real Return Bond was below 1 year maturity. (There may have been provincial issues that matured.)
- The old U.K. index-linked gilt design was horror show, and coupon payment was fixed six months in advance. If you have the price data and associated pricing algorithms, this market gives the longest back history. That said, it appears that such bonds would have been completely illiquid, and so the reliability of pricing data would be open to question.
- Since 2008, euro area linker pricing is very sensitive to default risk. There is a decent sample size, but you would need to be very careful with the data.
- There were a few U.S. TIPS that matured to provide a sample.

We cannot use a fitted curve to create hypothetical bonds that are near maturity. Your statistical tests of market efficiency would be purely an analysis of how well the yield curve fitting algorithm extrapolates the short end of the curve. Based on my experience, I have limited confidence in any algorithms ability to extrapolate a linker curve. (For conventional bonds, you can start pulling in other instruments to pin down short maturities.) This is why I am not pursuing measuring a term premium based on the fitted U.K. gilt curve data (which is what I have access to); I would need the underlying bond data (and associated pricing algorithms) to do a proper estimate.

The next problem with looking at breakeven inflation pricing is that this is a test whether investors are clairvoyants. When we are discussing the term premium estimates in short-term nominal instruments, we are implicitly assuming that investors can on average forecast the path of the short rate. Outside of the onset of recessions, modern central bankers are transparent and respond with a lag to economic data. It is somewhat reasonable to believe that investors’ policy rate forecasts would be correct most of the time. This is less clear for short-term inflation forecasts.

The main driver of CPI inflation in the short term is oil price (technically, gasoline) movements. All useful CPI forecasts are effectively conditional upon oil price movements. This means that short-dated index-linked positions are extremely interesting to fixed income macro investors: this is one of the few ways within fixed income to take a position on anything other than interest rates (or credit spreads).

The side effect of this oil dependence is that the uncertainty of the effect of oil prices on the oil forecast is probably an order of magnitude larger than any term premium that might exist in the instrument. If we try to see whether there is a bias in realised inflation versus the economic breakeven, all you are doing is testing whether these fixed income investors were correct in their oil forecasts.

You just need to look at an oil price chart from 2007-2008 to see that many people had to be wrong about oil prices in both directions. There is no particular reason to believe that fixed income investors did a better job forecasting oil than investors in other markets did. Since oil prices generally follow trends, it would not be a surprise to see autocorrelation in forecast errors.

Once the Financial Crisis hit, pricing in the index-linked market bore no resemblance to serious inflation forecasts. The reasoning was simple: levered fixed income investors had been bullish on oil, and were trapped in long index-linked positions that everyone knew that they could not finance. There was a large “squeeze discount” in inflation-linked yields.

If we put aside the period of oil market volatility and the aftermath of the Financial Crisis, the result is that we have an even smaller data set to work with for judging whether there is a persistent term premium. We will be looking at the pricing of just a couple of dozen bonds across the developed markets. This is too small a sample size to be interesting, but that would change by the mid-2020s.

If we want to examine long maturity linkers, we have even less matured bonds to work with (outside the United Kingdom). Furthermore, it is clear that the post-1990 era of low and stable inflation was not forecast by investors, so that bonds issued in the 1980s will have had abnormally high returns.

One way to increase the apparent sample size for long-dated bonds is to look at excess returns over fixed horizons, such as one to five years. That is, what are the excess returns of a hypothetical breakeven trade versus realised inflation over the horizon period for bonds of various tenors? This is a typical financial market analysis exercise. Unfortunately, there is no obviously correct way to approach the analysis. The return for a 10-year bond on a 3-year horizon depends almost entirely on the pricing of the 7-year stub at the end of the analysis horizon. If the analysis period chosen is a secular bull market, it is almost certain that this technique will suggest that there is an increasing term premium as a function of the tenor. This is a greater problem for conventional bonds, as breakeven inflation has been largely bound in a range for the period of data availability. However, this range-bound nature appears to imply that the realised term premium will just equal the average slope of the breakeven inflation curve. That result will be only robust if we remain in an era of stable realised inflation.

In summary, I do not think there are yet enough data to prove that there is a significant inflation risk premium (or even the sign of the premium). If you are an investor, you should look at the raw economic inflation breakeven, and decide whether that is an attractive proposition. If you are a market observer, I see little value in destroying the information embedded in the inflation-linked bond market by passing it through a complicated statistical procedure that has very little empirical support.

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