Show that the limiting value of duration as maturity is increased to infinity is [D ightarrow

Show that the limiting value of duration as maturity is increased to infinity is

\[D \rightarrow \frac{1+(\lambda / m)}{\lambda}\]

For the bonds in Table 3.6 (where $\lambda=.05$ and $m=2$ ) we obtain $D \rightarrow 20.5$. Note that for large $\lambda$ this limiting value approaches $1 / \mathrm{m}$, and hence the duration for large yields tends to be relatively short.

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TABLE 3.6 DURATION OF A BOND YIELDING 5% AS FUNCTION OF MATURITY AND COUPON RATE Coupon rate Years to maturity 1% 2% 5% 10% 1 .997 .995 .988 .977 2 1.984 1.969 1.928 1.868 5 4.875 4.763 4.485 4.156 10 9.416 8.950 7.989 7.107 25 20.164 17.715 14.536 12.754 50 26.666 22.284 18.765 17.384 100 22.572 21.200 20.363 20.067 8 20.500 20.500 20.500 20.500 Duration does not increase appreciably with maturity. In fact, with fixed yield, duration increases only to a finite limit as maturity is increased.