LONG-TERM BONDS: A Two-Factor Model 5

The first two columns of Table 1 report parameters and asymptotic standard errors for the period 1952-96. All parameters are in natural units, so they are on a quarterly basis. We estimate a moderately persistent process for the real interest rate; the persistence coefficient фх is 0.87, implying a half-life for shocks to real interest rates of about 5 quarters. The expected inflation process is much more persistent, with a coefficient фх of 0.9985 that implies a half-life for expected inflation shocks of almost 115 years! Of course, the model also allows for transitory noise in realized inflation Click Here.

The bottom of Table 1 reports the implications of the estimated parameters for the means and standard deviations of real interest rates, nominal interest rates, and inflation, measured in percent per year. The implied mean log yield on an indexed three-month bill is 0.85% for the 1952-96 sample period. Taken together with the mean log yield on a nominal three-month bill of 5.31% and the mean log inflation rate of 3.99% (both restricted to equal the sample means over this period), and adjusting for Jensen’s Inequality using one-half the conditional variance of log inflation, the implied inflation risk premium in a three-month nominal Treasury bill is 49 basis points. This fairly substantial risk premium is explained by the significant positive coefficient and the significant negative coefficient /37rm in Table l.

Risk premia on long-term indexed bonds, relative to a three-month indexed bill, are determined by the parameter f3mx. This is negative and highly significant, implying positive risk premia on long-term indexed bonds and an upward sloping term structure of real interest rates. Risk premia on nominal bonds, relative to indexed bonds, are determined by the inflation-risk parameters j3zx and /3zm. The former is positive but statistically insignificant, while the latter is negative and significant. Both point estimates imply positive inflation risk premia on nominal bonds relative to indexed bonds.

Table 2 explores the term-structure implications of our estimates in greater detail. The table compares implied and sample moments of term structure variables, measured in percent per year. Panel A of Table 2 reports sample moments for returns and yields on nominal bonds, together with the moments implied by our estimated model; panel В shows comparable implied moments for indexed bonds, and panel С reports sample and implied moments for equities. Row 1 of the table gives Jensen’s-Inequality-corrected average excess returns on n-period nominal bonds over 1-period nominal bonds, while row 2 gives the standard deviations of these excess returns.

Row 3 reports annualized Sharpe ratios for nominal bonds, the ratio of row 1 to row 2. Row 4 reports mean nominal yield spreads, row 5 reports the standard deviations of nominal yield spreads, and row 6 reports the standard deviations of changes in nominal yields. Rows 7 through 12 repeat these moments for indexed bonds. Note that the reported risk premia and Sharpe ratios for nominal and indexed bonds are not directly comparable because they are measured relative to different short-term assets, nominal and indexed respectively.