Cox-Ingersoll-Ross Adjustment

When defining a stress test, we often start with a base scenario. Our base scenario might be an actual historical change in interest rates, or it might be a hypothetical scenario. Either way, we face a challenge when translating our base scenario into higher or lower interest rates regimes.

For example, suppose we chose as our base scenario a date in the past, when an interest rate changed from 2% to 3%, we could say that the rate increased by 1% or 50%,

dR = 3% − 2% = 1%

dR/R = (3% − 2%)/2% = 50%

Now, suppose we want to use this base scenario to create a stress scenario, but the current interest rate is 10%, and we want to apply the same shock, we could argue that the interest rate should go to 11% or 15%,

10% + dR = 11%

10% + 10% x dR/R = 15%

There is a third option, though. The Cox-Ingersoll-Ross (CIR) interest rate model has a stochastic term that is proportional to the square root of the interest rate. We can use this square root rule when translating our base scenario shock,

10% + dR x SQRT(10%/R) = 12.24%

Of the standard single-factor interest rate models, the CIR model is often the closest fit to historical data. For this reason, there is a good argument to be made for translating historical scenarios using the CIR adjustment.

To summarize, we would say that in the base scenario, the interest rate went from 2% to 3%, but in the stress scenario, based on the CIR adjustment, the interest rate went from 10% to 12.24%.