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

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?

• Duration is in fact related to maturity weighted cash flows as you mention, but in the case of no coupon payments it is obtained in an easier way.
– RRL
Nov 8 '19 at 18:04

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.

• Thank you very much Nov 8 '19 at 18:00
• @Shelley: You're welcome.
– RRL
Nov 8 '19 at 18:01