# 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 $$T_1 should be valued $$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 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?

• One possibility is to separate the discount curve and the forecasting curve. Then define the duration as the derivative with respect to a parallel shift to the zero rates of the discount curve. – Gordon Oct 27 '18 at 17:05
• @Gordon thanks. May I ask what curve is usually selected as the discount curve? The YTM curve obtained from interpolating available bonds with different maturities in the market? And isn't the "forecasting curve" just the forward rate curve, which provides the theoretical benchmark for future rates? – Vim Oct 28 '18 at 10:44
• This depends on trading desk and risk management decision. See Chapter 6 of the book Interest Rate Modeling for a discussion of the curve construction. – Gordon Oct 28 '18 at 13:36