Financial Risk Management of Insurance Enterprises Interest Rate Models

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

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

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

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

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

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)

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

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

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

Term Structure Shapes Normal upward sloping Inverted Level Humped

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

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

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

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

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

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

Run Graph Show of Interest Rates Go to: http://www.cba.uiuc.edu/~s-darcy/present/casdfa3/intmodels.html 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

Current Interest Rates Yields Spot rates Implied forward rates

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

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

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

Table 2 Summary Statistics for Vasicek Model Notes: Number of simulations = 10,000,  = 0.1779, = 0.0866,  = 0.0200

Table 3 Summary Statistics for CIR Model Notes: Number of simulations = 10,000, = 0.2339, = 0.0808,  = 0.0854

Table 4 Summary Statistics for HJM Model Notes: Number of simulations = 100,  = 0.0485,  = 0.5

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