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

198 MODELING FINANCIAL SCENARIOSmarket. We compute ex post real interest rates based on the differencebetween nominal rates observed in the market less themonthly (annualized) inflation rate. We use the three-month ConstantMaturity Treasury (CMT) as a proxy for the instantaneousshort rate and the 10-year CMT yield as a proxy for the longrate. (We also looked at longer Treasury yields as a proxy for thelong rate. Results were not sensitive to the choice of maturity.)Nominal interest rates are from the Federal Reserve’s historicaldatabase. (See http://www.federalreserve.gov/releases/.)There are several issues related to the Federal Reserve’s interestrate data. First, at the long end of the yield curve, thereare significant gaps in many of the time series. For example,the 20-year CMT was discontinued in 1987; yields on 20-yearsecurities after 1987 would have to be interpolated from otheryields. Also, the future of 30-year rate data is uncertain, giventhe decision of the Treasury to stop issuing 30-year bonds (infact, the Fed stops reporting 30-year CMT data in early 2002).At the short end of the yield curve, there are several choicesfor a proxy of the short rate. Ideally, one would want an interestrate that most closely resembles a default-free instantaneousrate. While the one-month CMT is reported back only to 2001,the three-month rate is available beginning in 1982. While wecould have reverted to a private, proprietary source of data tocreate a longer time series, we restricted ourselves to only publiclyavailable data sources that would be available to any userof the model.Based on Formula (3.11), we use the following regressionson monthly data from 1982 to 2001:r t+1 = ® 1 l t + ® 2 r t + " 0 rt ,(3.13)l t+1 = ¯1 + ¯2l t + " 0 lt:Traditional ordinary least squares (OLS) regressions are notpossible given the dependence of the short-rate process on thelong rate. To estimate these simultaneous equations, we use twostageleast squares estimation. In order to estimate the short-rate