1. Financial Risk Management of Insurance Enterprises
Interest Rate Models

2. Interest Rate Models Classifications of Interest Rate Models
Term Structure of Interest Rate Shapes
Historical Interest Rate Movements
Parameterizing Interest Rate Models

3. Classifications of Interest Rate Models
Discrete vs. Continuous
Single Factor vs. Multiple Factors
General Equilbrium vs. Arbitrage Free

4. Discrete Models
Discrete models have interest rates change only at specified intervals
Typical interval is monthly
Daily, quarterly or annually also feasible
Discrete models can be illustrated by a lattice approach

5. Continuous Models
Interest rates change continuously and smoothly (no jumps or discontinuities)
Mathematically tractable
Accumulated value = ert
Example
$1 million invested for 1 year at r = 5%
Accumulated value = 1 million x e.05 = 1,051,271

6. Single Factor Models
Single factor is the short term interest rate for discrete models
Single factor is the instantaneous short term rate for continuous time models
Entire term structure is based on the short term rate
For every short term interest rate there is one, and only one, corresponding term structure

7. Multiple Factor Models
Variety of alternative choices for additional factors
Short term real interest rate and inflation (CIR)
Short term rate and long term rate (Brennan-Schwartz)
Short term rate and volatility parameter (Longstaff-Schwartz)
Short term rate and mean reverting drift (Hull-White)

8. General Equilibrium Models
Start with assumptions about economic variables
Derive a process for the short term interest rate
Based on expectations of investors in the economy
Term structure of interest rates is an output of model
Does not generate the current term structure
Limited usefulness for pricing interest rate contingent securities
More useful for capturing time series variation in interest rates
Often provides closed form solutions for interest rate movements and prices of securities

9. Arbitrage Free Models
Designed to be exactly consistent with current term structure of interest rates
Current term structure is an input
Useful for valuing interest rate contingent securities
Requires frequent recalibration to use model over any length of time
Difficult to use for time series modeling

10. Which Type of Model is Best?
There is no single ideal term structure model useful for all purposes
Single factor models are simpler to use, but may not be as accurate as multiple factor models
General equilibrium models are useful for modeling term structure behavior over time
Arbitrage free models are useful for pricing interest rate contingent securities
How the model will be used determines which interest rate model would be most appropriate

11. Term Structure Shapes
Normal upward sloping
Inverted
Level
Humped

12. How Do Curves Shift?
Litterman and Scheinkmann (1991) investigated the factors that affect yield movements
Over 95% of yield changes are explained by a combination of three different factors
Level
Steepness
Curvature

13. Level Shifts
Rates of maturities shift by approximately the same amount
Also called a parallel shift

14. Steepness Shifts
Short rates move more (or less) than longer term interest rates
Changes the slope of the yield curve

15. Curvature Shifts Shape of curve is altered
Short and long rates move in one direction, intermediate rates move in the other

16. Characteristics of Historical Interest Rate Movements
Rule out negative interest rates
Higher volatility in short-term rates, lower volatility in long-term rates
Mean reversion (weak)
Correlation between rates closer together is higher than between rates far apart
Volatility of rates is related to level of the rate

17. Table 1 Summary Statistics for Historical Rates April 1953-July 1998

18. Run Graph Show of Interest Rates
Go to:
Download Graph Show
Click on Historical (4/53-5/99)
Click on Start Graph Show
You may want to shorten the time interval to speed up the process
Note how interest rates have moved over the last 46 years
Pay attention to the level of interest rates, the shape of the yield curve and the volatility over time

19. Current Interest Rates
Yields
Spot rates
Implied forward rates

21. Distortions
U. S. Government stopped issuing 30 year bonds in October, 2001
Reduced supply of long term bonds has increased their price, and reduced their yields
Effect has distorted the yield curve

22. Parameterizing Interest Rate Models
Vasicek
Cox-Ingersoll-Ross (CIR)
Heath-Jarrow-Morton (HJM)

23. 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

24. Table 2 Summary Statistics for Vasicek Model
Notes: Number of simulations = 10,000,  = , = ,  =

25. Table 3 Summary Statistics for CIR Model
Notes: Number of simulations = 10,000, = , = ,  =

26. Table 4 Summary Statistics for HJM Model
Notes: Number of simulations = 100,  = ,  = 0.5

27. Concluding remarks Interest rates are not constant
Interest rate models are used to predict interest rate movements
Historical information useful to determine type of fluctuations
Shapes of term structure
Volatility
Mean reversion speed
Long run mean levels
Don’t assume best model is the one that best fits past movements
Pick parameters that reflect current environment or view
Recognize parameter error
Analogy to a rabbit