PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

MODELING FINANCIAL SCENARIOS 183the slope of the term structure improves explanatory power toover 95%. Finally, including U-shaped shifts (called curvature)explains over 99% of the variation observed in historical termstructure movements. Chapman and Pearson [12] confirm thatthese three factors are persistent over different time periods.² Volatility of interest rates is related to the level of the shortterminterest rate. Chapman and Pearson [12] further point outthat the appropriate measure for volatility depends on whetherthe period from 1979 to 1982–when the Federal Reserveshifted policy from focusing on interest rates to controlling inflation,resulting in a rapid increase in interest rates–is treatedas an aberration or included in the sample period.Equilibrium and Arbitrage Free ModelsSeveral popular models have been proposed to incorporatesome of the characteristics of historical interest rate movements.Often these continuous time models are based on only onestochastic factor, movements (changes) in the short-term interestrate (the instantaneous rate). A generic form of a one-factor termstructure model isdr t = ·( ¡ r t )dt + ¾rt ° dB t : (2.1)Formula (2.1) incorporates mean reversion. To see this, considerthe case where the current level of the short-term rate (r t )isabove the mean reversion level . The change in the interest rateis then expected to be negative–interest rates are expected tofall. The speed of the reversion is determined by the parameter·. The last term in (2.1) incorporates the unknown, volatilecomponent of interest rate changes over the next instant. Thelast term, dB t , is the change in a Brownian motion–it has meanzero and variance equal to dt. This uncertainty is scaled by thevolatility parameter ¾. If°>0, then interest rate volatility is relatedto the level of the interest rate. When ° = 0, this modelis equivalent to the formulation of Vasicek [43]; when ° =0:5,the model is the process proposed by Cox, Ingersoll, Ross [13](hereafter CIR). Chan et al. [10] estimate this class of interest