Rates That Cannot Go Below Zero: the Cox-Ingersoll-Ross Term Structure

Vasicek's 1977 model was elegant, tractable and slightly embarrassing: the interest rate it described could wander below zero, which in 1977 seemed absurd. It also came from nowhere in particular. It assumed a process for the short rate and then leaned on a no-arbitrage argument, but it never said why the rate should behave that way or where the compensation for risk came from.

Cox, Ingersoll and Ross set out to fix both problems at once, and the result became, alongside Vasicek, one of the two pillars of interest rate modelling.

The problem: a model with no economy inside it

If you write down "the short rate wanders like this" and stop, you have made an assumption, not a theory. Two awkward questions follow immediately.

First, where does the risk premium come from? Vasicek's model has a "market price of risk" parameter that determines how much extra return long bonds pay. You have to feed it in by hand. Nothing in the model explains why it is the size it is, or whether it should be positive at all.

Second, why should the rate behave that way in the first place? Interest rates are not a physical constant. They emerge from an economy: from how productive investment is, from how much people want to consume now versus later, from how risk-averse they are. A model that ignores all that is a description, not an explanation.

Cox, Ingersoll and Ross wanted a term structure that came out of an economy, so that the risk premium, the shape of the curve and the behaviour of the short rate were all consequences rather than inputs.

The key idea via analogy: build the farm, then read the interest rate off it

Their approach is best understood in reverse. Instead of starting with the interest rate, they start with the world.

Imagine a simple economy with a single productive technology, call it a farm. The farm's output is uncertain: sometimes the harvest is good, sometimes bad. People in this economy have to decide how much to consume today and how much to plough back into the farm. They dislike risk, and they prefer consumption sooner rather than later.

Solve that whole problem, and out of it pops an interest rate. It is not assumed; it is the price at which people in this economy are indifferent between eating now and eating later, given how productive the farm is and how nervous they are. The risk premium on long bonds is not a free parameter either: it is exactly the compensation these people demand for holding an asset whose value swings around with their own consumption. If a bond tends to do badly when the harvest is bad, they will pay less for it. The model tells you so.

Once you turn the crank, the short rate that emerges from this economy has a specific character, and it is here that the paper's most famous technical feature lives.

The CIR short rate mean-reverts, like Vasicek's dog on an elastic lead. But its jitters are not constant: the closer the rate gets to zero, the calmer it becomes. Volatility scales with the square root of the level of rates. Picture a bouncing ball whose bounces get smaller and smaller as it settles: when rates are high, they are volatile and jumpy; when rates are near zero, the randomness shrinks toward nothing, and the upward pull of mean reversion dominates. The rate is pushed back up before it can cross into negative territory. Rates stay non-negative not by a hard floor bolted on afterwards, but because of how the process is built.

The reward for all this is that the model stays workable. Bond prices still come out in closed form, ugly but explicit. You can price a bond of any maturity with a formula rather than a simulation, which is exactly the property that let CIR become an industry standard rather than a theoretical curiosity.

Why it mattered

• It made risk premia endogenous. In CIR, the extra return on long bonds is not a plug. It is the answer to a question about preferences and technology. That is a genuinely deeper kind of model, and it is why the paper appeared in Econometrica and not a finance journal.
• It gave everyone a positive-rate model. For twenty-five years, "use CIR if you need rates to stay positive" was standard practice, and it is still the default when you are modelling something that genuinely cannot go below zero, like a default intensity or a variance process.
• The square-root process escaped fixed income entirely. The same mathematical object turns up as the variance process in the Heston stochastic volatility model and as the default intensity in reduced-form credit models. When quants say "CIR process," they very often mean nothing about bonds at all.
• It anchored the affine family. Vasicek and CIR turned out to be the two extreme cases of a much larger class of tractable models, later mapped out by Duffie and Kan and classified by Dai and Singleton. CIR is the corner of that class where the volatility is driven by the factors themselves.

The honest limitations

• It still does not fit the observed curve. Like Vasicek, CIR produces a yield curve rather than matching the one on your screen. With a handful of parameters you will never nail every point on the real curve, which means a trader pricing a bond option will start from prices that disagree with the market. Hull and White's extensions exist to fix precisely this.
• One factor, again. A single short rate drives every yield, so the curve can shift but it cannot really twist. The model cannot produce the rich variety of shapes real curves take, and it cannot generate the imperfect correlations between short and long rates that show up in any dataset.
• The economy is a caricature. A single representative agent, a single production technology, one source of uncertainty. This is the price of a closed-form answer, and it means the "general equilibrium" foundations are more of a coherence check than a realistic description of a modern economy.
• The zero lower bound became optional. CIR's headline selling point was that rates cannot go negative. Then Europe, Japan and Switzerland ran negative policy rates for years. The property that made the model famous became, in some markets, an active liability, and desks quietly went back to Gaussian models that permit negative rates.
• Empirically, it forecasts poorly. Later work, notably Duffee, showed that the standard affine models in this family (CIR very much included) produce worse forecasts of future yields than simply assuming yields follow a random walk. The tight link the model imposes between volatility and risk compensation turns out to be the culprit.

The one-line takeaway

Cox, Ingersoll and Ross derived the yield curve from an actual economy rather than an assumption, producing a short rate that grows calmer as it approaches zero and therefore never crosses it, a construction so useful that it now underpins stochastic volatility and credit modelling as much as it does bonds.