# Duration of a floating rate bond

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?

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 $$e^{-\delta r \cdot t}-1$$.