Show that the duration of a perpetuity increases as the interest rate decreases, in accordance with Rule

Question:

Show that the duration of a perpetuity increases as the interest rate decreases, in accordance with Rule 4.

Rule 4: With other factors held constant, the duration and interest rate sensitivity of a coupon bond are higher when the bond’s yield to maturity is lower.

As we noted above, the intuition for this rule is that while a higher yield reduces the present value of all of the bond’s payments, it reduces the value of more distant payments by a greater proportional amount. Therefore, at higher yields a higher fraction of the total value of the bond lies in its earlier payments, thereby reducing effective maturity. Rule 4, which is the sixth bond-pricing relationship noted above, applies to coupon bonds. For zeros, duration equals time to maturity, regardless of the yield to maturity.

Finally, we present an algebraic rule for the duration of a perpetuity. This rule is derived from and is consistent with the formula for duration given in Equation 11.1 , but it is far easier to use for infinitely lived bonds. T D = tx W t=1 (11.1)