Equilibrium Term Structure Models

Equilibrium term structure models use fundamental variables to explain the term structure of interest rates. These simple models aren't the most accurate, but they do provide some insights. Two of the most popular models for equilibrium term structure are both single-factor models, and that factor is the short-term interest rate, _r_. Since parallel shifts in the yield curve cause the vast majority of yield changes, it's not such a stretch.

The first of the two models is the __Cox–Ingersoll–Ross (CIR) model__, and it looks like this: $$\displaystyle dr_t = k(\theta - r_t)dt + \sigma \sqrt{r_t}dZ $$ The _d_ in front of $$r_t$$ for interest rate, _t_ for time, and $$Z$$ for random noise (or "noise" if that helps) is the derivative notation you would find in calculus. It means very short change, so that you can think of this as a continuous time model. So the whole thing is an expression that represents a change in the short-term interest rate and it captures the fact that individuals must determine their own optimal trade off between consumption and investment decisions.

Now look carefully at the first part with $$(\theta - r_t)$$ in it, and imagine that interest rates start moving in one direction. What does the model suggest will happen?

Right!

No. It actually suggests that it will turn around and come back.

The "$$-r_t$$" in that first term suggests that the change will move against the level. That is, the interest rate is mean reverting. That's fairly important, since it really is in practice. Interest rates can't run off in either direction forever. This deterministic term in the model is called the "drift term," since it causes the rate to drift back toward equilibrium.

Now consider that the first term has no effect on the change of interest rates. It's at equilibrium. What would the interest rate have to be?

No. This would still allow a deterministic change to the interest rate.

Yes! If $$\theta = r_t$$, then that term goes to zero, and the interest rate is home. So $$\theta$$ is the long-term interest rate. Of course it won't just sit there according to the model since the stochastic component, that messy risk term on the right, will still provide some turbulence.

No. That's not a value. It's just that very short-term change to the interest rate.

And just to clarify that volatility term on the right: you can see that sigma is there to represent standard deviation, with $$dZ$$ kind of like a random walk. Since _r_ is included there, it looks like higher interest rates are associated with higher volatility of the stochastic term. So what would it mean if that "square root of _r_" term were removed?

No. That would be the case if the _r_ were replaced by zero.

Not necessarily. The sigma term is a parameter, so it could be estimated at any level with or without the _r_ term there.

Exactly! No $$r_t$$, then no change to volatility. It's just estimated by the parameter sigma and $$dZ$$. The variables $$k$$, $$\theta$$, and $$\sigma$$ are all parameters that would need to be estimated somehow.

And just like that, you have the second model. $$\displaystyle dr_t = k(\theta - r_t)dt + \sigma dZ $$ This is called the __Vasicek model__, and you can see a startling similarity to the CIR model. The $$\sqrt{r_t}$$ is now gone. That's the only difference. But for a main weakness here, consider what happens when interest rates approach zero. In the CIR model, the volatility goes to zero, and the drift term takes over, pushing rates back up. What would happen with the Vasicek model?

Not necessarily, no. Interest rates could actually go negative.

Absolutely.

The drift term would still push rates upward, but rates could still randomly walk further downward, taking them to negative territory. With only rare exceptions, nominal interest rates are historically positive.

To summarize: [[summary]]

They will continue

They will turn around and come back

$$k$$

$$\theta$$

$$dr_t$$

Volatility would be lower

There would be no volatility

Volatility would be constant

Same thing

Interest rates could wander negative

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