The Cox-Ingersoll-Ross (CIR) Interest Rate Model: Practice Problems and Solutions

In the Cox-Ingersoll-Ross (CIR) interest rate model, the short-term interest rate follows this Brownian process:

dr = a(b - r)dt + σ√(r)dZ

Meaning of some variables:

σ = volatility factor.

a = drift factor.

b = the mean around which mean reversion occurs.

r = the short-term interest rate.

The way to solve this equation for the price P[t, T, r(t)] at time t of $1 paid with certainty at time T, where t ≤ T and P(T, T, r) = 1, is

P[t, T, r(t)] = A(t, T)e^{-B(t, T)r(t)}, as in the Vasicek Model. The values of A(t, T) and B(t, T) are calculated differently, however, and the formulas for doing so are much more complicated than those in the Vasicek Model. The chance you will need to use these formulas on the exam will be minuscule.

It is more instructive to compare the CIR model to the Vasicek model. The following matters stand out in particular.

1. In the Vasicek model, interest rates can be negative. In the CIR model, negative interest rates are impossible. If r = 0 in the CIR model, the drift factor a(b - r) becomes ab > 0, while the volatility factor is σ√(0) = 0, so the interest rate will increase. As one's time horizon increases, the likelihood of interest rates becoming negative in the Vasicek model greatly increases as well.

2. In the Vasicek model, the volatility of the short-term interest rate is constant. In the CIR model, the volatility of the short-term interest rate increases as the short-term interest rate increases.

3. As in the Vasicek model, the short-term interest rate in the CIR model exhibits mean reversion, where b is the mean.

4. In both the Vasicek and the CIR model, the delta and gamma Greeks for a zero-coupon bond are based on the change in the short-term interest rate.

Learning Objective A2 on the syllabus for Exam 3F / Exam MFE asks students to "explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed."

What does this mean, and why is it true?

An exogenous prescription of the time-zero yield curve would mean that you know-empirically, or from some source external to the models - the data regarding the yields to maturity for many different time horizons, where currently, t = 0. Then you would be able to put that data into the Vasicek or CIR model and get consistent results as well as the ability to predict yields to maturity for other time horizons. Alas, this is not the case.

For instance, you might know empirically that for

t = 0, T = 1: r = 0.02

t = 0, T = 2: r = 0.03

t = 0, T = 3: r = 0.04

t = 0, T = 4: r = 0.05

t = 0, T = 5: r = 0.08

Why can this data not be used consistently with a Vasicek or CIR model? Both models are based on four parameters: a, b, σ, and r. But if you have four or more empirical data points, it is possible (indeed, likely) for a 4-parameter model to give conclusions inconsistent with one or more of those data points.

We can think of this via an analogy.

Let us say we have the following model: x + 2y = 5.

If we are just given x, we can solve for y consistently with the model.

If we are given both x and y, however, we will not always be able to do so. For instance, if x = 3 and y = 4, there is no way for x + 2y to equal 5. If you have even more externally prescribed data, the likelihood of the model working decreases even further.

So a model based on m parameters can only be used consistently with exogenously prescribed data consisting of m-1 or fewer data points.

Sources: Cross, Bill. Post on Actuarial Outpost Discussion Forum. April 9, 2007.

McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 24, pp. 787-789.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem CIRIRM1.

Assume the CIR model holds. A particular interest rate follows this Brownian motion:

dr = 0.22(0.06 - r)dt + 0.443√(r)dZ

At some particular time t, r = 0.11. Then, r suddenly becomes 0.02. What is the resulting change in the volatility?

Solution CIRIRM1. The volatility in the CIR model is σ√(r). For r = 0.11, and σ = 0.443, the volatility is 0.443√(0.11). For r = 0.02, and σ = 0.443, the volatility is 0.443√(0.02). The change in volatility is thus 0.443√(0.02) - 0.443√(0.11) = -0.08427668174

Problem CIRIRM2. Assume the CIR model holds. The following values are true with regard to a particular interest rate at some specific time t: r = 0.04, a = 0.88, b = 0.09, σ = 0.23, dt = 1, dZ = 0.4. Find dr in the CIR model.

Solution CIRIRM2.

We use the formula dr = a(b - r)dt + σ√(r)dZ = 0.88(0.09 - 0.04)1 + 0.23√(0.04)0.4 =

dr = 0.0624

Problem CIRIRM3. Which of the following statements are true? More than one answer may be correct.

(a) The Vasicek model incorporates mean reversion.
(b) The CIR model incorporates mean reversion.
(c) In the Vasicek model, interest rates can be negative.
(d) In the CIR model, interest rates can be negative.
(e) In the Vasicek model, volatility is a function of the interest rate.

(f) In the CIR model, volatility is a function of the interest rate.

Solution CIRIRM3. Both the Vasicek and CIR models incorporate mean reversion, so (a) and (b) are correct. Interest rates can be negative in the Vasicek model, but not in the CIR model, so (c) is true and (d) is false. In the Vasicek model, volatility is constant, so (e) is false But in the CIR model, volatility = σ√(r), so (f) is true. Thus, (a), (b), (c), and (f) are the correct answers.

Problem CIRIRM4. Assume that the CIR model holds. When a particular interest rate is 0, the Brownian motion it follows is dr = 0.55dt.

You know that a + b = 0.99 and a > 1. What is the drift factor of the Brownian motion in this model when r = 0.05?

Solution CIRIRM4. The drift factor in the CIR model is a(b - r). When r = 0, this is equal to ab. We are thus given that ab = 0.55 and a + b = 1.7. Thus,

b = 0.55/a and so a + 0.55/a = 1.7 and a² + 0.55 = 1.7a. Thus,

a² - 1.7a + 0.55 = 0. By the quadratic formula, a = 1.265331193 or a = 0.4346688069. Since a > 1, a = 1.265331193.

Thus, b = 1.7 - a = 1.7 - 1.265331193 = b = 0.4346688069.

When r = 0.05, a(b - r) = 1.265331193(0.4346688069 - 0.05) =

The drift factor = 0.4867334405

Problem CIRIRM5. For which of these exogenously prescribed data sets regarding yields to maturity for various time horizons can the Vasicek and CIR models be used? More than one answer may be correct.

Set A:

t = 0, T = 1: r = 0.22

t = 0, T = 2: r = 0.26

t = 0, T = 3: r = 0.28

Set B:

t = 0, T = 0.5: r = 0.04

t = 0, T = 5: r = 0.05

Set C:

t = 0, T = 1: r = 0.12

t = 0, T = 2: r = 0.09

t = 0, T = 3: r = 0.18

t = 0, T = 6: r = 0.21

t = 0, T = 7: r = 0.03

t = 0, T = 8: r = 0.44

Set D:

t = 0, T = 1: r = 0.11

t = 0, T = 2: r = 0.31

t = 0, T = 7: r = 0.34

t = 0, T = 8: r = 0.54

Solution CIRIRM5. The time-zero yield curve for an interest rate model can only be interpreted consistently with the model (in most cases) when the number of data points in the time-zero yield curve is less than the number of parameters in the model. The Vasicek and CIR models each have 4 parameters: a, b, σ, and r. So only data sets with 3 points or fewer can have the models consistently applied to them. Thus, only Sets A and B, with 3 and 2 data points respectively, can have the models applied to them.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.