Parameterizing Interest Rate Models

Kevin C. Ahlgrim, ASAStephen P. D’Arcy, FCASRichard W. Gorvett, FCAS

Casualty Actuarial SocietySpecial Interest Seminar onDynamic Financial Analysis

Overview

Objective of PresentationTo 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

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

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

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

Understanding a general interest rate model

• Change in short-term interest rate

• a(rt,t) is the expected change over the next instant– Also called the drift

• dBt is a random draw from a standard normal distribution

tttt dBtrdttradr ),(),(

Understanding a general interest rate model (p.2)

• (rt,t) is the magnitude of the randomness

– Also called volatility or diffusion

• Alternative models depend on the definition of a(rt,t) and (rt,t)

tttt dBtrdttradr ),(),(

Vasicek model

• Mean reversion affected by size of • Short-rate tends toward • Volatility is constant

• Negative interest rates are possible

• Yield curve driven by short-term rate– Perfect correlation of yields for all maturities

ttt dBdtrdr )(

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

tttt dBrdtrdr )(

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 1Summary Statistics for Historical Rates

ShapeNormal 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

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, = 0.0200

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, = 0.0854

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

• A variety of models exist to help value contingent claims

• Pick parameters that reflect current environment or view

• Analogy to a rabbit

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