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

196 MODELING FINANCIAL SCENARIOSTime Period ·q ¹ q ¾ q1913—2001 0.37 3.3% 4.0%1946—2001 0.47 4.8% 3.0%We selected the default mean reversion speed (·q) tobe0.4and the mean reversion level (¹ q ) to be 4.8% to capture the postwareconomic period. Although it might appear that the speedof mean reversion over the second half of the 20th century hasincreased, it should be noted that the standard error of the estimateof ·q is higher (which undoubtedly is due to fewer datapoints in the shorter period).Instead of being concerned with the annualized, instantaneouslevel of inflation, bond investors are more concerned with the expectedlevel of inflation over the life of their investment. Giventhe existing level of inflation (q t ) and the parameters of the assumedprocess in Formula (3.1), we can derive expectations offuture inflation over various horizons. Our process for inflationfollows the same Ornstein-Uhlenbeck process as in Vasicek [43],so we can develop a “term structure” of inflation analogous toFormula (2.3). This term structure posits an expected inflationrate over various horizons. A term structure of inflation is neededto generate nominal interest rates, since investors are concernedabout not only the time value of money, but also the erosion ofpurchasing power expected over the life of their investment.Real Interest RatesTo derive real interest rates, we selected a simple case of thetwo-factor Hull-White model (Formula (2.8)). In this model, theshort-term rate (denoted by r) reverts to a long-term rate (denotedby l) that is itself stochastic. The long rate reverts to an averagemean reversion level ¹ rdr t = ·r(l t ¡ r t )dt + ¾ r dB r ,dl t = ·l(¹ r ¡ l t )dt + ¾ l dB l :(3.8)