Financial Risk Management of Insurance Enterprises

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

Parameterizing the Yield Curve Level = 6 month yield Steepness (or slope) = 10 year yield – 6 month yield Curvature = 6 month yield + 10 year yield – 2 x 2 year yield Based on Brandt and Chapman (2002)

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 1Summary Statistics for Historical RatesApril 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 • Alternative source for the yield curve movements: http://www.smartmoney.com/onebond/index.cfm?story=yieldcurve

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

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

Table 4Summary 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

Financial Risk Management of Insurance Enterprises