Duration of a floating rate bond with spread

Question

I need to calculate the duration of a floating rate bond with spread. With zero spread the price of the bond is given by: $$p_\tau=(1+c_1)e^{-r(\tau_1) \cdot \tau_1}$$ so the duration is: $$-\frac{\frac{dp_\tau}{r}}{p_\tau} = \tau_1$$ So the duration is the time $\tau_1$ until the next coupon payment.

When the spread is not zero(i.e $s$), the price in time $0$ is given by: \begin{equation} p^{s}_\tau = (1+c_1)e^{-r(\tau_1) \cdot \tau_1}+ \sum_{k=1}^n s \cdot e^{-r(\tau_k) \tau_k} \quad (1) \end{equation} So the duration is going to be: $$-\frac{\frac{dp^s_\tau}{r}}{p^s_\tau} = \frac{\tau_1\cdot (1+c_1)e^{-r(\tau_1) \cdot \tau_1} + \sum_{k=1}^n s \cdot \tau_k \cdot e^{-r(\tau_k) \tau_k}}{(1+c_1)e^{-r(\tau_1) \cdot \tau_1}+ \sum_{k=1}^n s \cdot e^{-r(\tau_k) \tau_k}} \quad (2)$$

Questions:

1. The formula (1) is correct?
2. The formula (2) is correct?
3. In which other case the duration of a floating rate bond is not the time until the next coupon payment?

Answer 1

Is formula (1) correct?

Yes, follows from first definition - floater with deterministic spread is composed (sum) of two components: (1) pure floater and (2) deterministic coupon strip via contractual spread payment.

Is formula (2) correct?

Yes, by taking the derivative of an exponential function.

what other case where duration of floating rate bond not the same as time till next coupon?

Deep discount floaters: sometimes the market applies discount rates to floaters much higher than their contractural spread due to credit considerations or heightened basis risk. When this occurs, we can observe negative duration for floaters.