You Cannot Have Everything: Dai and Singleton Audit the Affine Models

Duffie and Kan had shown that a whole family of interest rate models, the affine ones, come with free closed-form bond prices. Within a few years, researchers had written down dozens of members of that family, and the literature had turned into a mess. Different papers used different parameterisations, some models were secretly the same model wearing a different hat, some proposed models were not actually valid at all, and nobody could say which specification the data preferred.

Qiang Dai and Kenneth Singleton did the unglamorous, essential job: they went in and organised the whole thing. The result is one of the most-cited papers in fixed income, and its central finding is a genuinely uncomfortable one.

The problem: a family with no family tree

Three problems had piled up.

Nobody knew what the family contained. With several factors and lots of parameters, there was no systematic account of which affine models were distinct and which were re-labellings of each other. Papers routinely proposed "new" models that were equivalent to existing ones.

Some models were secretly broken. In an affine model, if a factor is driving volatility, that factor must stay positive, a variance cannot be negative. Whether the parameters you have chosen actually guarantee positivity is not obvious, and it is easy to write down a model that looks fine on paper and is internally inconsistent. Dai and Singleton call the valid ones admissible, and establishing admissibility conditions was a real service.

Nobody knew what the data wanted. Should you use a Gaussian model (like multi-factor Vasicek), a square-root model (like multi-factor CIR), or a blend? Each was defended by somebody. There was no head-to-head comparison on equal footing.

The key idea via analogy: sort the models by how many volatility drivers they have

Dai and Singleton's organising move is beautifully simple. Take an affine model with N factors. Ask: how many of those factors are involved in driving the volatility?

That number, call it m, can be anything from 0 to N. And it slices the family into N+1 distinct, non-overlapping sub-families. With three factors you get exactly four possible model types:

• Zero volatility drivers. All three factors are Gaussian, volatility is constant. This is multi-factor Vasicek. Rates jiggle by the same amount whether they are at 1 percent or 15.
• One volatility driver. One factor is a square-root process controlling how volatile things are; the other two are Gaussian.
• Two volatility drivers. Two square-root factors driving volatility, one Gaussian.
• Three volatility drivers. All three factors are square-root processes. This is multi-factor CIR.

Within each of these sub-families, Dai and Singleton construct a maximal model: the most general version, from which every other model in that box can be obtained by switching parameters off. So instead of a swamp of ad hoc specifications, you get a clean ladder with a canonical representative on each rung. Any model anyone proposes can now be located on the ladder.

Then they run the horse race on US swap yields, and the answer is the paper's real punch.

The uncomfortable trade-off

Here is what they found, and it is the reason people still cite this paper.

There is a fundamental tension inside the affine class between stochastic volatility and factor correlation, and you cannot escape it.

The logic runs like this. If a factor is going to drive volatility, it has to stay positive forever, because volatility cannot be negative. To guarantee that a random process never goes below zero, you must restrict how it interacts with the other factors. In particular, the admissibility conditions in the affine world force those volatility-driving factors to have non-negative relationships with the others. Negative correlations get ruled out.

But real yield curve factors are negatively correlated. That is an empirical fact: level and slope move against each other in the data, and any model that forbids that relationship will fail to reproduce how the curve actually behaves.

So you face a straight choice:

• Load up on square-root factors, so you get rich, realistic stochastic volatility (rates really are more volatile in some regimes than others), and pay for it by being unable to represent the negative correlations the data shows.
• Go Gaussian, so you get whatever correlations you want, including negative ones, and pay for it by having constant volatility, which is flatly counterfactual.

Dai and Singleton's empirical work indicates that the specifications fitting the swap data best are ones that keep some stochastic volatility but relax the model as far as admissibility allows in order to permit negative correlations among the factors. In other words: the data wants both, the affine class will not give you both, and the best you can do is a grudging compromise.

The analogy is a duvet that is too small. Pull it up to cover your shoulders and your feet stick out. Cover your feet and your shoulders freeze. You can shuffle it around, and Dai and Singleton tell you exactly where the edges are, but the duvet is the size it is.

Why it mattered

• It imposed order on chaos. The N+1 classification is now the standard vocabulary. When a modern paper says it is using an "A_1(3) model," it is speaking Dai and Singleton's language: three factors, one of them driving volatility.
• It made admissibility a first-class concern. After this paper, you cannot publish an affine model without demonstrating it is internally consistent. That killed off a category of quietly broken models.
• It identified a structural limit, not just a bad fit. The volatility-versus-correlation trade-off is not a calibration problem you can solve with better software. It is baked into the affine assumption. That is a much deeper and more useful kind of criticism, and it directly motivated the search for non-affine alternatives: quadratic term structure models, regime-switching models, and eventually shadow-rate models.
• It set the standard for how to compare models. Building a maximal model per class and nesting everything else inside it is now the template for any specification search.

The honest limitations

• It only audits the affine class. The paper's whole frame is "given that you are going to use an affine model, which one?" If the right answer is "do not use an affine model," this paper cannot tell you. The trade-off it identifies is an argument for leaving the class entirely, and the paper does not follow that argument out.
• The empirical work is one market, one sample. US dollar swap yields over a particular period. The conclusions about which specification wins are conditional on that data, and the interest rate world of the 1990s (positive rates, an active Fed, no quantitative easing) looks very little like what came after.
• The models still forecast poorly. Dai and Singleton are asking which affine model fits the cross-section of yields and their volatilities. Duffee, two years later, asked a different question, which one forecasts future yields, and found the answer was "none of them, they lose to a random walk." Fitting today's curve well and predicting tomorrow's are different skills, and this paper only tests the first.
• The zero lower bound breaks the frame. Square-root factors go to zero and stay well behaved. Gaussian factors go negative. Neither captures the very particular behaviour of interest rates pinned near zero for years, where volatility collapses and the distribution becomes sharply asymmetric. The post-2009 world required tools the affine class does not naturally contain.
• Estimation remains brutal. The paper does not solve the notorious difficulty of actually fitting these models: multiple local optima, flat likelihood surfaces, sensitivity to starting values. That problem persisted for another decade until Joslin, Singleton and Zhu attacked it directly.

The one-line takeaway

Dai and Singleton sorted every affine term structure model onto a ladder by how many of its factors drive volatility, and discovered a trap built into the whole class: the more realistically you model volatility, the less realistically you can model how the factors move together.