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

Common Term Structure Shapes

0

2

4

6

8

10

12

1 3 5 10 30

Years

Inte

rest

Rat

e (%

)

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

Maturity

Yie

ld

Steepness Shifts

• Short rates move more (or less) than longer term interest rates

• Changes the slope of the yield curve

Maturity

Yie

ld

Curvature Shifts

• Shape of curve is altered

• Short and long rates move in one direction, intermediate rates move in the other

Maturity

Yie

ld

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 Rates

April 1953-July 1998Shape

Normal Inverted Humped Other68.8% 11.6% 13.4% 6.3%

Yield Statistics1 yr. 3 yr. 5 yr. 10 yr.

Mean 6.08 6.47 6.64 6.81S.D. 3.01 2.88 2.84 2.81Skewness 0.97 0.84 0.77 0.68Exc. Kurtosis 1.10 0.69 0.48 0.16

Percentiles1 yr. 3 yr. 5 yr. 10 yr.

1% 1.07 1.59 1.94 2.385% 2.05 2.52 2.72 2.9050% 5.61 6.20 6.44 6.6895% 12.08 12.48 12.59 12.5699% 15.17 14.69 14.59 14.29

Corr (1 yr,10 yr) = 0.944

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

Yield Curve December 2001

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

Years to Maturity

Inte

res

t R

ate

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)

dzdtrrdr )ˆ(

dzrdtrrdr )ˆ(

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

tdBTtfTtdtTtfTtTtdf )),(,,()),(,,(),(

Table 2Summary Statistics for Vasicek Model

ShapeNormal Inverted Humped Other41.6% 54.8% 3.6% 0.0%

Yield Statistics1 yr. 3 yr. 5 yr. 10 yr.

Mean 8.81 8.75 8.68 8.52S.D. 3.83 3.24 2.77 1.95Skewness -0.16 -0.16 -0.16 -0.16Exc. Kurtosis -0.19 -0.19 -0.19 -0.19

Percentiles1 yr. 3 yr. 5 yr. 10 yr.

1% -0.38 0.97 2.04 3.845% 2.33 3.27 4.00 5.2250% 8.94 8.86 8.77 8.5995% 14.69 13.73 12.94 11.5399% 17.22 15.87 14.76 12.82

Corr (1 yr,10 yr) = 1.000

Notes: Number of simulations = 10,000, = 0.1779, = 0.0866, = 0.0200r̂

Table 3Summary Statistics for CIR Model

ShapeNormal Inverted Humped Other44.7% 44.6% 4.7% 0.0%

Yield Statistics1 yr. 3 yr. 5 yr. 10 yr.

Mean 8.08 8.04 7.98 7.86S.D. 2.89 2.31 1.88 1.20Skewness 0.92 0.92 0.92 0.92Exc. Kurtosis 1.49 1.49 1.49 1.49

Percentiles1 yr. 3 yr. 5 yr. 10 yr.

1% 2.92 3.90 4.62 5.715% 3.95 4.73 5.29 6.1450% 7.71 7.73 7.73 7.7095% 13.42 12.31 11.45 10.0999% 17.19 15.33 13.90 11.66

Corr (1 yr,10 yr) = 1.000

Notes: Number of simulations = 10,000, = 0.2339, = 0.0808, = 0.0854r̂

Table 4Summary Statistics for HJM Model

Yield Statistics1 yr. 3 yr. 5 yr. 10 yr.

Mean 7.39 7.51 7.60 7.80S.D. 2.26 2.27 2.31 2.44Skewness 0.51 0.53 0.54 0.54Exc. Kurtosis -0.88 -0.85 -0.85 -0.86

Percentiles1 yr. 3 yr. 5 yr. 10 yr.

1% 4.45 4.48 4.52 4.595% 4.79 4.85 4.90 4.9950% 7.48 7.58 7.65 7.8395% 11.57 11.74 11.92 12.3899% 12.09 12.26 12.44 12.89

Corr (1 yr,10 yr) = 0.999

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