Interest Rate Sensitivities of a FRA [closed]

Question

A basic question perhaps ? How to compute the Duration, MDuration, Convexity and PV01 of a FRA ?

Answer 1

Let's try. The payoff of the FRA can be written as follows:

$ P=\frac{N \, \tau \left(L-k\right)}{1+\tau \, L} $

The derivative of which is as follows (quotient rule):

$\frac{dP}{d L}=\frac{ \left(1 +\tau \, L\right) N \, \tau - N \, \tau\left(L-k\right)\tau}{\left( 1+\tau \, L\right)^2}$

$ \frac{dP}{d L}=\frac{ N \, \tau \left( 1 + \tau \, k \right) }{\left( 1+\tau \, L\right)^2} $

You can format the above into the different types of duration. The convexity is easy:

$\frac{d^2 P}{d L^2}=-2 \frac{ N \, \tau^2 \left( 1 + \tau \, k \right) }{\left( 1+\tau \, L\right)^3}$

Note: You will need to discount the above formula to the current date, e.g., $price=P e^{-rT} $ where T is the time to the FRA starting point. And if you want to assume that $\frac{dr}{dL}= 1$ then you will need to factor this in into the above derivation, which is again straightforward.