Prove that the duration of a bond without a coupon is equal to its maturity.

Question

I am supposed to prove that the duration of a bond without a coupon is equal to its maturity.

I know it will have something to do with weighted average maturity periods, but I don't know how to formulate it.

Can anyone help me?

Answer 1

For a zero-coupon bond maturing in $T$ years -- with face value $F$ and (continuously compounded) yield $y$ -- the price is

$$P = F e^{-yT}$$

Duration is given by

$$D = \frac{-1}{P} \frac{\partial P}{\partial y} = - \frac{-TF e^{-yT}}{F e^{-yT}} = T$$

More details are given here regarding the distinction between modified and Macaulay duration and the formulaic conventions that apply when yield is compounded over semi-annual periods, etc.