# Interest Rate Sensitivities of a FRA [closed]

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

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.

• As a rough guideline the Duration is going to be close to $\tau$ and the Convexity close to $\tau^2$ but somewhat smaller because of the discount factors shown above. – Alex C Apr 30 '19 at 22:38
• Alex, Good one! – Magic is in the chain Apr 30 '19 at 22:40
• Thanks very much. Beautifully explained. – Suresh Kunnoth May 1 '19 at 9:42
• Glad you found it useful! – Magic is in the chain May 1 '19 at 18:40