## Friday, April 7, 2017

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.

To be clear, this article is not attempting to explain why I believe this. However, I just wanted to give a heads up that outlines the logic behind some upcoming articles. Each article is meant to be stand alone, and so it may be unclear how these articles relate to each other. For example, I will be publishing an article on fixed income arbitrage. Although primarily aimed at people new to the concept, there are some ideas that I will reference when I get around to discussing term structure models.

The term structure modelling literature was aimed to answer what I call the Term Premium Question. I would phrase the question as follows.

Term Premium Question. We want to create a mathematical model that takes fixed income market data, and decomposes nominal interest rates into a rate expectations component, and a risk premium. Furthermore, this model has to obey no-arbitrage conditions.

Once we assume that the question has to be phrased this way, researchers are able to dust off their stochastic calculus and attack the problem. The chain of events looked something like this.
1. There is an infinite number of decompositions of observed rates into an expected short rate and a term premium. (The no-arbitrage condition eliminates some decompositions, but it is not enough to force uniqueness.)
2. Since most potential models are not analytically tractable, researchers first picked a tractable decomposition, and then reported the results.
3. These initial decompositions had unwelcome properties, and so later researchers (possibly the same people) proposed various fixes. These new models had new drawbacks.
4. The number of potential papers exploded. Not only was it possible to create new models, one could compare existing models, or even try to relate model outputs (term premium, inflation risk premium, etc.) to other economic series.
The unbounded nature of the number of papers in this area has made it extremely attractive to academics and central bank researchers. This is similar to DSGE models. And just like DSGE models, these term structure models are largely ignored by financial market practitioners, other than those who want to emulate central bank thinking.

## What Is The Problem?

Every time I looked at these models (for over a decade), I concluded that the model outputs were obviously pathological. It was never worthwhile to reconstruct the models to see whether they could be fixed; I was being paid to work on models that were useful for making money.

Now that I am a writer, I have greater freedom of research. However, it was still largely unclear what to say about these models. The problem is that the field is highly mathematical, and it would be hard to explain the mathematics to a wider audience.

I now believe that the mathematics is a trap. If your starting point is the "Term Premium Question," and then dive into stochastic calculus, you will almost inevitably follow a similar path to earlier researchers. You might be able to tweak the models, but you would end up with a model that is qualitatively similar to previous models.

The trick is to start with mathematics, and instead ask yourself: what is (are) the right question(s) to ask? I will write out a long-winded version of the question in the upcoming articles, but one summary question is: why do you expect to be able to create a mathematical model that decomposes observed rates into a "market expectation" and a "market term premium" in the first place?

I hope to explain myself further in a few articles. The projected articles are:
• background material on fixed income arbitrage;
• what are the correct questions to ask about term premia; and
• what is the problem with the "Term Premium Question," as it is traditionally posed?
The first article is nearly complete; not sure what order the latter two will be completed in (and whether I will need to break those latter two strands into multiple article).

(c) Brian Romanchuk 2017