Numerical Calculation of FRN Duration in R

The duration of a floating rate note (FRN) is the remaining time until the first next payment date. Using this fact, a duration of FRN has not been calculated explicitly but has been understood conceptually. Instead of this reasoning, this post tries to calculate it directly based on the numerical differentiation using R code.


We have investigated the properties of a price and duration of FRN and calculated its price in the following previous post

Price and Duration of FRN

In the previous post, we find that a price of FRN at time \(t\) is. \[\begin{align} P_{zero}^{FRN} &= \begin{cases} 1, & \text{payment date} \\ D_{0,\tau} (1+R_{reset}), & \text{otherwise}\\ \end{cases} \\ \\ \tau &= t_{payment}^{1st} – t \end{align}\] Here \(t_{payment}^{1st}\) is the first next payment dates after the pricing date (\(t\)). \(D(0,t)\) denotes a discount factor for a cashflow at time \(t\).

Duration of FRN is zero right before the reset date and is an interest payment period right after the reset date and is a linearly interpolated year when pricing date is between two consecutive payment dates. Duration of FRN is summarized as follows. \[\begin{align} D_{zero}^{FRN} = \begin{cases} 0, & \text{payment date} \\ \tau, & \text{otherwise}\\ \end{cases} \end{align}\]

Effective Duration

The effective duration is of the follwing form of numerical differentiation. \[\begin{align} D &= \frac{P_u – P_d}{2 \Delta y \times P_0} \end{align}\] Here, \(P_0\) denotes an initial bond price with yield to maturity (\(y\)). \(P_u\) and \(P_d\) represent bond prices after downward (\(y-\Delta y\)) and upward (\(y+\Delta y\)) shocks to interest rates (yield to maturity) respectively.

Refer the following post for more details of the effective duration

We intend to use this formula to calculate a FRN duration but how can we apply this YTM change in the framework of FRN pricing? For this matter, we need to generate a zero curve from a parallel shift of the YTM curve.

Zero curve from Parallel Shift of YTM curve

Unlike the fixed rate bond which can be priced by using the YTM or zero pricing, FRN use zero pricing so that when we apply the numerical differentiation for duration, we need to generate a zero curve from a parallel shift of market yield (YTM or swap rate) curve which consists of market instruments. The following figure shows three zero curves generated from baseline,1 bp upward shifted, and 1 bp downward shifted YTM curves.



Refer the following post for more details of a zero curve bootstrapping

In summary, in duration calculation, a change in the market yield curve means a parallel shift of the market yield curve, not a individual yield change at a specific maturity.

R code

The following R code implements the above effective duration formula and applies it to the calculation of 3-year FRN’s duration with quarterly frequency, and finally compares it with the actual remaining time between the pricing date and the first next payment date (\(\tau\)) for three cases (pricing date : t=0, 1/12, 2/12).



From he following output of the above R code, we can find that the duration of FRN for each case is equal to the actual remaining time between the pricing date and the first next payment date (\(\tau\))

Concluding Remarks

From this post, we calculate the duration of FRN directly by using numerical differentiation. This intuitive exercise reminds us of fact that a market yield change means a parallel shift of the market yield curve and then this change is translated and spread to a zero curve change. This is plain but easy to overlook. Using this changed zero curve, an effective duration of FRN can be obtained numerically.