Duration of a floating rate bond

Question

It is known that the price $p_t$ of a floating rate bond can be calculated discounting $(L+k)$ the sum of the next coupon payment $k$ and the face value $L$ at the relevant risk-free rate.

Hence, with continuous compounding the price of such a bond would be $$p_t=(L+k)e^{-rt}$$ where $r$ is the annual continuous risk-free rate for the period of time that divides us from the first payment and $t$ is its length.

If we want to know how the price changes when $r$ changes we just derive the price to get $$\frac{\partial p_t}{\partial r}=-p_t\cdot t$$

So, does it make sense talking about duration of a floating rate bond? Isn't it enough to refer to the derivative computed above? How could the notion of duration be applied in such a situation where we do not know future cash flows?

Thank you in advance.

Answer 1

Yes. the duration of a floating rate bond is the time t until the next coupon payment, as your equation shows. The payments that come after are not known yet and will be determined based on interest rates then prevailing, so they carry no duration risk.

In general floating rate bonds are what people buy when they want the smallest duration possible. Long term ZCB are what people buy when they want the longest possible duration.

Answer 2

Yes it does make sense.

As your equation correctly point out that only the next coupon is exposed to the yield curve changes, the bond price should be sensitive to the yield corresponding only to the next coupon.

The duration of the bond will be approximately $-t = \frac{-p_t \cdot t}{p_t}$.

Approximately because your derived equation gives a change in price for an infinitesimal yield change. A floating rate bond's duration is given by $e^{-\delta r \cdot t}-1$.