1. Parameterizing Interest Rate Models Kevin C. Ahlgrim, ASA Stephen P. D’Arcy, FCAS Richard W. Gorvett, FCAS Casualty Actuarial Society Special Interest Seminar on Dynamic Financial Analysis

2. Overview Objective of Presentation To help you understand models that attempt to mimic interest rate movements Sections of Presentation Provide background of interest rate models Introduce popular interest rate models Review statistics - models and historical data Provide advice for use of interest rate models

3. What are we trying to do? Historical interest rates from April 1953 through May 1999 provide some evidence on interest rate movements We want a model that helps to fully understand interest rate risk Use model for valuing interest rate contingent claims

4. Characteristics of interest rate movements Higher volatility in short-term rates, lower volatility in long-term rates Mean reversion Correlation between rates closer together is higher than between rates far apart Rule out negative interest rates Volatility of rates is related to level of the rate

5. General equilibrium vs. Arbitrage free GE models are developed by assuming that investors are expected utility maximizers –Interest rate dynamics evolve from the equilibrium of supply and demand of bonds Arbitrage free models assume that the dynamics of interest rates must be consistent with securities’ prices

6. Understanding a general interest rate model Change in short-term interest rate a(r t,t) is the expected change over the next instant –Also called the drift dB t is a random draw from a standard normal distribution

7. Understanding a general interest rate model (p.2) F (r t,t) is the magnitude of the randomness –Also called volatility or diffusion Alternative models depend on the definition of a(r t,t) and F (r t,t)

8. Vasicek model Mean reversion affected by size of 6 Short-rate tends toward 2 Volatility is constant Negative interest rates are possible Yield curve driven by short-term rate –Perfect correlation of yields for all maturities

9. Cox, Ingersoll, Ross model Mean reversion toward a long-term rate Volatility is (weakly) related to the level of the interest rate Negative interest rates are ruled out Again, perfect correlation among yields of all maturities

10. Heath, Jarrow, Morton model Specifies process for entire term structure by including an equation for each forward rate Fewer restrictions on term structure movements Drift and volatility can have many forms Simplest case is where volatility is constant –Ho-Lee model

11. Table 1 Summary Statistics for Historical Rates

12. Table 2 Summary Statistics for Vasicek Model Notes: Number of simulations = 10,000, 6 = 0.1779, 2 = 0.0866, F = 0.0200

13. Table 3 Summary Statistics for CIR Model Notes: Number of simulations = 10,000, 6 = 0.2339, 2 = 0.0808, F = 0.0854

14. Table 4 Summary Statistics for HJM Model Notes: Number of simulations = 100, F = 0.0485, ( = 0.5

15. Concluding remarks Interest rates are not constant A variety of models exist to help value contingent claims Pick parameters that reflect current environment or view Analogy to a rabbit

16. How to Access Interest Rate Programs Go to website: http://www.cba.uiuc.edu/~s-darcy/index.html July 1999 CAS DFA Presentation Parameterizing Interest Rate Models Call Paper Interest Rate Graphing Models PowerPoint Presentation