Duration of a Floating-Rate Bond with a Contractual Spread
Summary
The document considers how a contractual spread changes the duration of a floating-rate bond. With no spread, it describes the price as the next reset or coupon-related cash flow discounted to the next payment date, yielding duration equal to the time until that date under the stated assumptions. With a deterministic spread, it models the bond as the pure floater plus a strip of spread payments discounted at the relevant rates.
Taking the price sensitivity to rates gives a present-value-weighted average of the cash-flow dates in that expression. The answer endorses both formulas and explains that the spread component behaves like deterministic coupons. It also notes that a floater can have negative duration when market discount rates substantially exceed its contractual spread, for example amid credit concerns or basis risk. These conclusions depend on the simplified cash-flow and discounting setup; the exchange does not address changing reference rates, spread dynamics, or detailed reset conventions.
Key ideas
- A floater without a spread has duration equal to the time until the next coupon under the stated setup.
- A deterministic contractual spread can be represented as a strip of spread payments added to the pure floater.
- The resulting duration is a present-value-weighted average of the payment dates in the pricing expression.
- Deep discounting relative to the contractual spread can produce negative duration.
- The formulas rely on simplified deterministic cash flows and discount rates.
Tags
Full text
# Duration of a floating rate bond with spread
# Duration of a floating rate bond with spread
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:
- The formula (1) is correct?
- The formula (2) is correct?
- In which other case the duration of a floating rate bond is not the time until the next coupon payment?
## Answer by wgajate (score 4, accepted)
https://quant.stackexchange.com/a/58150
> 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.