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