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101 Concepts for the Level I Exam

Essential Concept 76: FRA Pricing and Valuation


A forward rate agreement is an over-the-counter forward contract in which the underlying is an interest rate. FRAs are usually expressed in the “X x Y” convention, ‘X’ represents the point where the underlying loan starts. This also the point where the FRA expires. ‘Y’ represents the point where the underlying loan ends. A 3 x 9 FRA is depicted in the figure below.

The FRA fixed rate i.e. price of an FRA can be calculated using these steps:

  1. Set up the time line.
  2. Compute the de-annualized fixed rate: \frac{\mathrm{1+}\left(\mathrm{r2\times }\frac{\mathrm{n2}}{\mathrm{360}}\right)}{\mathrm{1+}\left(\mathrm{r1\times }\frac{\mathrm{n1}}{\mathrm{360}}\right)}\mathrm{\ -1}
  3. Annualize by multiplying by 360/(n2 – n1)

The FRA value for pay fixed receive floating can be calculated using these steps:

  1. Find the new FRA fixed rate at time t
  2. Compute payoff at n2: (FRAt – FRA0) x (n2 – n1)/360 x NP
  3. Discount back to time t.

Example: Suppose we entered a receive-floating 6×9 FRA at a rate of 0.89%, with a notional amount of $5,000,000 at T = 0. After 90 days, the three-month US dollar Libor is 1.10% and the six-month US dollar Libor is 1.25% which will be the discount rate to determine the value. What is the value of the original receive-floating 6×9 FRA?

Solution:

  1. We first compute the new FRA rate at time t.

Step 1: Set up the time line.

 

Step 2: Compute the de-annualized fixed rate:

\frac{\mathrm{1+}\left(\mathrm{r2\times n2/360}\right)}{\mathrm{1+}\left(\mathrm{r1\times n1/360}\right)}\mathrm{\ -1=\ }\frac{1+\left(0.0125\times \frac{180}{360}\right)}{1+\left(0.0110\times \frac{90}{360}\right)}-1=0.00349

 

Step 3: Annualize by multiplying by 360/(n2 – n1)

0.00349\times \frac{360}{180-90}=0.01396=1.396\%

 

  1. Compute payoff at n2: (FRAt – FRA0) x (n2 – n1)/360 x NP
    \left(\mathrm{0.01396-0.0089}\right)\mathrm{\times }\frac{\mathrm{90}}{\mathrm{360}}\mathrm{\times \$5million=\$6,325}
  2. Discount back to time t.
    \frac{\mathrm{\$6325}}{\mathrm{1+0.0125\times 1}\mathrm{80/360}}\mathrm{=\$6,285.71}