# Interest Rate Sensitivities of a FRA [closed]

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

## closed as off-topic by skoestlmeier, AdB, Helin, LocalVolatility, amdoptMay 6 at 18:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – skoestlmeier, AdB, Helin, LocalVolatility, amdopt
If this question can be reworded to fit the rules in the help center, please edit the question.

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 at 22:38
• Alex, Good one! – Magic is in the chain Apr 30 at 22:40
• Thanks very much. Beautifully explained. – Suresh Kunnoth May 1 at 9:42
• Glad you found it useful! – Magic is in the chain May 1 at 18:40