FRA Rate Sensitivities: Delta, Convexity, and Discounting

Summary

The document derives the sensitivity of a forward rate agreement’s payoff to its reference rate. Starting from a payoff expressed in terms of notional, accrual period, contractual rate, and observed forward rate, it differentiates with respect to the forward rate using the quotient rule. The resulting first derivative can be used to form rate sensitivity measures such as duration or PV01, while a second derivative gives the payoff’s convexity with respect to that rate.

The derivation is presented for the payoff before discounting. To value sensitivities at the current date, the document says to discount the payoff to the FRA’s start date and account for how the discount rate changes with the forward rate if that relationship is assumed. It does not spell out conventions for converting the derivatives into each named risk measure, nor does it discuss curve construction, payment timing, or market-specific valuation conventions. The formulas therefore serve as a basic analytical starting point; practical risk figures depend on the instrument setup and rate assumptions.

Key ideas

• The FRA payoff can be differentiated with respect to its reference forward rate.
• The first derivative describes payoff sensitivity and can support rate-risk measures.
• The second derivative captures convexity of the payoff with respect to the forward rate.
• Current-date sensitivities require discounting and may depend on how discount rates relate to the forward rate.
• Market conventions and instrument details are needed to produce practical risk numbers.

Tags

# Interest Rate Sensitivities of a FRA


# Interest Rate Sensitivities of a FRA












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

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/45380

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.