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

MODELING FINANCIAL SCENARIOS 197In order to estimate the parameters of the model, we look atthe discrete analog of the model:¢r t = ·r(l t ¡ r t )¢t + ¾ r " rtp¢t,¢l t = ·l(¹ r ¡ l t )¢t + ¾ l " ltp¢t:(3.9)r t+1 ¡ r t = ·r(l t ¡ r t )¢t + ¾ r " rtp¢tp=(·rl t ¡ ·rr t )¢t + ¾ r " rt ¢t,p (3.10)l t+1 ¡ l t = ·l(¹ r ¡ l t )¢t + ¾ l " lt ¢tp=(·l¹ r ¡ ·ll t )¢t + ¾ l " lt ¢t:r t+1 = r t +(·rl t ¡ ·rr t )¢t + ¾ r " rtp¢t= ·r¢t ¢ l t +(1¡ ·r¢t) p¢ r t + ¾ r " rt ¢t,p (3.11)l t+1 = l t +(·l¹ r ¡ ·ll t )¢t + ¾ l " lt ¢t= ·l¢t ¢ ¹ r +(1¡ ·l¢t) p¢ l t + ¾ l " lt ¢t:From these equations, we can see that the short rate is againa weighted average of the current levels of r t and the mean reversionfactor l t . The mean reversion factor is itself a weightedaverage of its long-term mean (¹ r ) and its current value (l t ).Hibbert, Mowbray, and Turnbull [24] (hereafter HMT) alsouse this process for real interest rates. They derive closedformsolutions for bond prices (and therefore yields), whichare slightly more complicated than the one-factor Ornstein-Uhlenbeck process for inflation:P r (t,T)=A r (t,T)e ¡r t B 1 (t,T)¡l t B 2 (t,T) (3.12)where r t and l t are the values for the short and long real interestrates and A r , B 1 ,andB 2 are functions of underlying parametersin the two-factor Hull-White specification for real interest rates.Estimating the parameters in Formula (3.11) is a difficult proceduresince real interest rates are not directly observable in the