Parameterizing Interest Rate Models
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Parameterizing Interest Rate Models
Kevin C. Ahlgrim, ASAStephen P. D’Arcy, FCASRichard W. Gorvett, FCAS
Casualty Actuarial SocietySpecial Interest Seminar onDynamic Financial Analysis
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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
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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
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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
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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
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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 ),(),(
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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 ),(),(
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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 )(
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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 )(
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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 )),(,,()),(,,(),(
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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
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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
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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
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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
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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
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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