Interest Rate Models: Vasicek, CIR, and Hull-White
An overview of the major interest rate models used in actuarial valuation and risk management.
Short-Rate Models
Interest rate models are essential tools for actuaries valuing insurance liabilities with interest-sensitive cash flows. The Vasicek model describes the short rate as a mean-reverting Ornstein-Uhlenbeck process: dr = a(b - r)dt + sigma dW, where a is the speed of mean reversion, b is the long-run mean, and sigma is the volatility. This model produces normally distributed rates, which allows for negative rates (sometimes considered a drawback). The Cox-Ingersoll-Ross (CIR) model modifies the volatility term to sigma times the square root of r, ensuring rates remain non-negative and producing a chi-squared distribution for the short rate.
Hull-White and Practical Use
The Hull-White model extends Vasicek by allowing the long-run mean to be a time-dependent function, enabling exact calibration to the observed yield curve. This feature makes it particularly practical for actuarial applications where consistency with market prices is important. Actuaries use these models in economic scenario generators (ESGs) for cash flow testing, principle-based reserving, and asset-liability management. The choice among models depends on the application: Vasicek for simplicity and analytical tractability, CIR for non-negative rates, and Hull-White for term structure fitting. Parameter estimation uses historical data, market prices, or a combination of both.