Floating-Rate Note Value and Duration Between Coupon Resets
Summary
The document derives the value of a floating-rate note before its next coupon reset in a continuous-time zero-coupon bond framework. Each future floating coupon is represented as the difference between discount bond prices at its reset and payment dates. Summing these coupon values with the discounted principal makes the intermediate terms cancel, leaving the price of a zero-coupon bond maturing at the next coupon date. This supports the familiar intuition that the note’s rate sensitivity is tied to the time remaining until its next reset or payment.
The author proposes defining a single continuously compounded rate from that bond price and differentiating value with respect to that rate, which yields a duration equal to the remaining time. The question remains unresolved in the source: it asks whether this is the proper Macaulay duration definition. The conclusion depends on the chosen rate shock and valuation framework; it does not establish how duration behaves at reset dates or under nonparallel yield-curve changes.
Key ideas
- A floating coupon set at one date and paid at the next can be valued as a difference of discount bond prices.
- Summing the coupon values and principal value telescopes to the discount bond price at the next coupon date.
- Using a single continuously compounded rate for that discount factor gives sensitivity proportional to time until the next payment.
- The source raises, but does not settle, whether this is the appropriate Macaulay duration definition in a general continuous-time model.
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# Duration of a FRN in continuous time interest rate model
# Duration of a FRN in continuous time interest rate model
This question was inspired by my attempt to understand the duration of a floating rate note, or FRN for short. Several answers, like this, say the duration of a FRN is just time to next coupon payment. But I'm still a bit confused even with the very definition of durations of FRNs.
In a continuous time model, let $\{P(0,t), t\ge 0\}$ be the YTM curve of zero bonds. Then in this answer by @Gordon it is pointed out that the coupon a FRN with a unit principal pays at $T_2$ with the coupon rate $L(T_1;T_1,T_2)$ to be set at $T_1<T_2$ should be valued $P(0, T_1) - P(0, T_2)$ at time $0$. Hence, with a little bit extension, if I consider a FRN that pays coupon one at $T_1$ set at $T_0:=0$, pays coupon two at $T_2$ set at $T_1$, and so on until it pays the last coupon (set at $T_{n-1}$) together with the principal (assumed $1$) at $T_n$. Then its value at $t<T_1$ should be $$V_t=\sum_{i=1}^nV(\text{coupon}_i) + P(t, T_n) = \sum_{i=1}^n(P(t, T_{i-1})-P(t, T_i)) + P(t, T_n) = P(t, T_1).$$
And my question is, how to evaluate the (Macaulay) duration of this FRN? The main problem is I don't know what rate I should differentiate $V$ in.
As a guess, if I define the current discount rate to be $r_c$ such that $e^{-r_c\tau} = P(t, T_1)$ where $t\in [0, T_1)$ and $\tau = T_1-t$ is time to next payment of coupon, then I may write $$V_t = P(t, T_1) = e^{-r_c\tau}$$ And if I differentiate in $r_c$, I got $$\frac{dV_t}{dr_c} = -\tau e^{-r_c\tau} = -\tau V_t$$ or $-\frac1V_t\frac{dV_t}{dr_c} = \tau$, which seems to align with the "time to next payment" theory. But I'm just not very sure, so could anybody kindly tell me if this is the correct way to define the duration for such a FRN, or more generally for any continuous time bond model?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.