Term Premium Comments

I have been looking at term premium models as part of a non-writing project. I decided to take a look at the Kim & Wright model output (available via the Federal Reserve, paper link: http://www.federalreserve.gov/pubs/feds/2005/200533/200533abs.html). By way of background, there are two very popular term premium models, the Kim & Wright paper, and the one by Adrian, Crump, and Moench (ACM).

Using Fed Projections To Infer The Term Premium?

I was passed along the article “Views from the Floor — Tighter and Tighter” by the Man Institute published last month. It discusses using the FOMC long-term projections to infer the term premium in the 10-year Treasury yield.

The methodology is straightforward (I have a busy week, so I have not gathered the data to replicate it myself). They describe it as follows:

A better approach is to incorporate the FOMC’s projections of the Fed Funds Rate into the expected path of short rates. This will make term premia estimates more consistent with sub-2% growth. Figure 1 shows that model applied, with an expectation that the short rate matches the Fed Funds Rate over the next year, then it linearly converges to the long-run projection over the next three years, and then remains constant.

Measuring Term Premia (And Other Comments)

I have been slowly getting back into things, and been thinking about what to write next. Based on a conversation with Gabriel Mathy on Twitter, I just want to quickly discuss how term premia ought to be measured. I think I have mentioned this before, but I felt like running through the arguments briefly again. I then have a few other unrelated comments to fill out this article.

The Yield Curve Provides Limited Economic Information

The relentless flattening of the Treasury yield curve has been a topic of ongoing debate -- is this a signal that a recession is near? The key to interpreting the flattening is that bond market participants are not paid to to anticipate economic outcomes (outside the corner case of the inflation-linked market), rather to anticipate the path of short-term rates (and the term premium). The flattening yield curve tells us that market participants (on average) believe that we are near the end of the rate hike cycle, but that does not necessarily mean that a recession is imminent.

Is Breakeven Inflation The Same Thing As A Forecast?

One of the difficulties in discussing the index-linked market is pinning down what we mean when say a certain rate of inflation is priced into the market. The breakeven inflation rate is (roughly) the rate of future inflation for which the nominal bond and the inflation-linked bond have the same total return. Alternatively, we might refer to the breakeven inflation rate as the market expectations for inflation. Unfortunately, we could have a situation in which the breakeven rate does not match what market participants forecast for inflation.

Weaknesses Of Term Premium Estimates Derived From Yield Curve Models

Term structure models have been a growth industry for researchers in academia and at central banks. These models can be structured in many different ways, which makes generalisations about them difficult. For the purposes of this article, I am only concerned about the use of these models to estimate a term premium that is embedded in nominal yields (although my comments can be extended to cover related exercises, such as calculating an inflation risk premium). When I examine individual models, the term premium estimates appear unsatisfactory, but the issues are different for each model. I believe that the root problems for this exercise are fundamental, and we need to understand these fundamental problems before looking at individual models.

How To Approach The Term Premium

The term premium is an important concept in fixed income analysis. For our own analysis, there are a few ways of using the term premium. Unfortunately, there is no way of extending the analysis for an individual to the market in general, as there is no need for market participants to agree on the term premium before undertaking a transaction. As a result, we should not expect to be able to infer an average term premium implied by market pricing using any algorithm.

The Term Premium Problem

Discussions of the behaviour of term premia have come up recently in online discussion. (For example, among people I follow on Twitter.) When we discuss the term premium, we are usually discussing the estimates derived from arbitrage-free term structure models (such as affine term structure models). I am not a fan of these term premium estimates, but explaining my views has always been difficult. I have come to the conclusion that the mathematics behind these models is part of the problem, not part of the solution.

Higher Debt-To-GDP Ratio And Lower Bond Yields: Japanese Experience

I recently referred to an older article in which I showed how higher debt-to-GDP ratios for the U.S. Federal Government was correlated with lower bond yields. The post-war U.S. experience was not an isolated case; similar relationships hold for the countries with free-floating currencies. The Japanese Government Bond (JGB) market is the poster child for this correlation.

New York Fed - Beating The Bond Market With Momentum

J. Benson Durham of the New York Fed has recently published a staff report "Momentum and the Term Structure of Interest Rates". The report describes a portfolio construction technique that is long-only, duration neutral versus the index, yet it generates up to 120 basis points of annual excess return. I explain what I think explains this strong performance, although it would require detailed testing to validate this. I conclude with a less technical discussion of how this relates to the concept of market efficiency.

Did Fed Purchases Reduce 10-Year Yields By 140 Basis Points? Probably Not.

In this article, I look at the paper "Official Demand for U.S. Debt: Implications for U.S. Real Interest Rates" by Iryna Kaminska and Gabriele Zinna (of the IMF, but the paper does not represent the official views of the IMF). Their conclusion was that Fed purchases of Treasurys (Quantitative Easing, or "QE") lowered 10-year real rates by 140 basis points. I explain why I question their methodology.

What Is Ricardian Equivalence, And Why It Does Not Hold

Ricardian Equivalence is a theoretical concept that has been used to argue that fiscal policy is not effective. The argument is that increased government spending implies higher future taxes, so households will increase savings to cancel out the increase in government spending. This concept has been heavily disputed; see this blog entry by Bill Mitchell for an example. He argues that various assumptions are too restrictive in the models that show the “Ricardian Equivalence” effect.

In this article, I will introduce the “inter-temporal governmental budget constraint”, which is the equation which provides the justification for Ricardian Equivalence. I also show why this equation does not hold if term premia are non-zero. The chart above gives an example that shows the error in its prediction about the Net Present Value of future primary surpluses can become arbitrarily large. This has obvious implications for models that incorporate this constraint, as well as the usage of the concept in the analysis of fiscal sustainability.

Primer: Should You Use Affine Term Structure Models? (Wonkish)

In this article I describe a very popular class of models in academia and central bank research circles: affine term structure models. These models attempt to provide an answer to the important question: what is the term premium in the yield curve? (See here and here for previous articles on this topic.) I will not even attempt to cover the mathematics involved here.

I will begin with a personal anecdote which explains how my philosophy of term structure modelling developed. At the beginning of my career in finance, I inherited the task of maintaining a model which calculated the unbiased expected forward level of short rates; i.e., a modelled term premium was subtracted from observed forward rates. Everybody loved the concept, but snags developed when it was used.

A typical problem: the forward curve rose by 5 basis points in response to some data, and a strategist at the firm wanted to say that the expected fed funds rate rose by 5 basis points, right? Nope. The model decided that the term premium rose by 7 basis points that day, and so the expected rate fell by 2 basis points. I would then be told to inspect the model, because this made no sense. After this happened a dozen times or so, it became the first mathematical model in my life that I truly loathed.

I luckily had an excuse to “recalibrate” the model, and I clamped down on the volatility of the term premium hard. It was not constant, as I needed to have some “quant-y” black box stuff in there to justify my salary, but it was stable enough so that I did not have to re-examine the model every 20 days. And my advice to anyone out there who have to build a model like this: stabilise the volatility of the term premium in your model output by any means possible.

Returning to the affine term structure models, I recommend this working paper by David Jamieson Bolder at the Bank of Canada. The paper is dated, but it covers the mathematical basics which are not put into other papers for reasons of space. The state of the art has moved on, but it will be easier to follow the other papers once the basic concepts are covered. If I had a working copy of an affine term structure model, I would work backward: start with the final algorithm, and then see what mathematical model is implied later.

The basic idea of an affine term structure model is very similar to factor analysis used in other parts of finance. The expected path of short rates is modelled by some kind of a random walk influenced by some fundamental factors ("unobserved latent factors"), and then a time-varying random term premium is added to reproduce observed bond prices. (The models are called affine because there is an assumption that the term structure is an affine function of the unobserved latent factors, which is function of the form f(x) = a + bx; in other words a “linear function plus a constant”.) [UPDATE: added italicised words, as the original sentence did not have the intended meaning.]

Central banks are natural consumers of these models. They are not interested in forecasting bond portfolio returns, so they want to strip out the term premium. Market practitioners, on the other hand, should really only be interested in expected returns, and it does not really matter whether the returns come from the path of short rates or the term premium.

These models are also very popular in academia, paradoxically because they do not work too well. There is always room to tweak the models, hence publish a new paper. (By contrast, look at principal component analysis. Once you fix the estimation period, your estimates for hedging ratios will not change much even if you make (sensible) changes to the algorithm. This means that the model is useful for practitioners, but there is no capacity to keep publishing papers on the subject.)

As a typical example of how they have been used recently: imagine that we have calibrated an affine term structure model on data pre-2008. We then freeze the model structure, and see how the model develops. And imagine then that the 10-year model term premium falls 150 basis points in recent years, once Quantitative Easing (QE) started. If you are an academic or central bank researcher, you publish a paper explaining that this means that QE has lowered the 10-year bond yield by 150 basis points. However, a cynic might suggest that your model just blew up when you went “out-of-sample”. There is no way of distinguishing these explanations with the data available.

As a final example, take the recent hammering of the 10-year Treasury: going from 1.65% to about 3% in a few months (with a small retracement going on at the time of writing). The answer you get depends on how you structure your model, but for a lot of the term structure models, it is possible that almost all the move could be in the term premium. In other words, the model expectations for short rates did not move much. (Since the expectations in a lot of these models are based on macro data, they are slow-moving.) This is very unsatisfying to me. I could hope to forecast where the Fed might be going, but I see no way of forecasting such violent moves in an unobservable model variable.

 

(c) Brian Romanchuk 2013

Historical Treasury Term Premia: Huge!

This post is an illustration of the concepts discussed in my primer on the term premium. Although we do not really know what the term premium is at any particular time, historical excess returns over a long period of time should average out near the average term premium. However, those excess returns have been implausibly high. Why this matters: if we do not know what the term premium is, we cannot know what the Treasury bond curve is pricing for the Fed outlook.

In the chart below, I show the behaviour of the realised (historical) excess returns for the 5-year Treasury.

To explain the chart, in the top panel we see the 5-year Treasury yield versus the 5-year average of the effective Fed Funds rate for the following 5 years. Since FRED does not yet have a time machine option, the data for the average ends in 2008 (i.e., 5 years ago).

In the bottom panel, the “Realised Excess Return” is the 5-year bond yield minus the average fed funds rate depicted above. This is a fairly good approximation of the excess return of a buy-and-hold position in a 5-year bond entered into a particular date versus a cash investment.

For example, in October 2008 (the end point of my sample), the 5-year yield was 2.73%, while the realised average effective fed funds rate since then was 0.16%, generating an excess return of  2.57%.

The table below shows the average realised excess returns for various periods, for the 2-, 5-, and 10-year points on the Treasury Curve. (Charts for the 2-year and 10-year are at the bottom of this post.)

Maturity	Start Date of Data	Mean Excess Return
		Entire Dataset	Since 1980-01-01	Since 1990-01-01
2-year	1976	0.51%	0.80%	0.81%
5-year	1962	0.77%	1.95%	1.74%
10-year	1962	1.01%	3.02%	2.56%

The experience for the 5-year maturity is particularly interesting. The negative premia pre-1980 could be explained by the various regulations that led to “financial repression”: yields held below what market forces would suggest. Once the deregulation of interest rates was completed, the premium has not been significantly been negative. Although the disinflation post-1980 was a surprise, the disinflation was largely finished by 1990. The market did not catch on to this, and the 5-year yield was on average 1.74% above the realised fed funds rate since 1990. In my opinion, the historical premium appears outsized for the amount of price risk associated with a 5-year bond; for example it is about double 5-year investment grade spreads right now (using the CDX index). Such a persistent miss by the market is hard to explain if it is in fact efficient.

This makes it hard to calibrate a model for calculating the “true” expected path of interest rates after adjusting for a term premium. If we blindly applied the post-1990 premium to the current curve, the implied expected average fed funds rate over the next 5 years is around -0.25%. This does not appear very plausible.

This also messes up any models for the fair value of bond yields which are based on historical data. In the post-1990s sample, the whole bond curve exhibited a very large term premium (or else market forecasts were consistently terrible). Therefore you end up basing your model target upon a too-high yield. I believe that this was a common analysis error made for the past 30 years, which explains the persistence of the bull market (whereas excess returns should theoretically be a random walk).

In an upcoming post, I will turn to model-based approaches to calculating the term premium – affine curve models.

(c) Brian Romanchuk 2013