LONG-TERM BONDS: A Two-Factor Model 6


A comparison of the model implications in rows 1 and 7 shows that 10-year nominal bonds have a risk premium over three-month nominal bills of 2.06% per year, while 10-year indexed bonds have a risk premium over three-month indexed bills of 1.62% per year. These numbers, together with the 49-basis-point risk premium on three-month nominal bills over three-month indexed bills, imply a 10-year inflation risk premium (the risk premium on 10-year nominal bonds over 10-year indexed bonds) of slightly less than 1%. This estimate is consistent with the rough calculations in Campbell and Shiller (1996).

Rows 2 and 8 show that nominal bonds are much more volatile than indexed bonds; the difference in volatility increases with maturity, so that 10-year nominal bonds have a standard deviation three times greater than 10-year indexed bonds. This difference in volatility makes the Sharpe ratio for indexed bonds in row 9 considerably higher than the Sharpe ratio for nominal bonds in row 3.

Since indexed bond returns are generated by a single-factor model, the Sharpe ratio for indexed bonds is independent of maturity at 0.46. The Sharpe ratio for nominal bonds declines with maturity; shortterm nominal bonds have a ratio close to that for indexed bonds, but the Sharpe ratio for 10-year nominal bonds is only 0.20, These numbers imply that in our portfolio analysis, investors with low risk aversion will have a strong myopic demand for indexed bonds.

Table 2 can also be used to evaluate the empirical fit of the model. A comparison of the model’s implied moments with the sample moments for nominal bonds shows that the model fits the volatility of excess nominal bond returns and changes in yields extremely well. It somewhat overstates the average excess nominal bond return and the nominal Sharpe ratio, but this can be attributed in part to the upward drift in interest rates over the 1952-96 sample period which biases downward these sample means. Overall the model appears to provide a good description of the nominal US term structure considering its parsimony and the fact that we have forced it to fit both time-series and cross-sectional features of the data.

Rows 13, 14 and 15 give comparable figures for equities: the annualized Jensen’s-Inequality-corrected average excess returns on equities relative to nominal bills, the standard deviation of these excess returns, and their Sharpe ratio. The model fits the standard deviation of equities extremely well but overpredicts the equity premium and the Sharpe ratio for equities. With an implied Sharpe ratio of 0.55, investors with low risk aversion will have an extremely large myopic demand for equities.

The right hand sides of Tables 1 and 2 repeat these estimates for the Volcker-Greenspan period 1983-96. Many of the parameter estimates are quite similar; however we find that in this period real interest rates are much more persistent, with фх = 0.986 and an implied half-life for real interest rate shocks of about 12 years. The expected inflation process now mean-reverts much more rapidly, with</>z = 0.866 implying a half-life for expected inflation shocks of about 5 quarters. These results are consistent with the notion that since the early 1980’s the Federal Reserve has more aggressively controlled inflation at the cost of greater long-term variation in the real interest rate (Clarida, Gali, and Gertler 1998). debit card payday loans