Forward Rate Agreement (FRA) Valuation
Think of a global company such as Airbus that has hundreds of suppliers and buyers around the world. For a company of that size, transactions are made and settled every day.
Say Sevillian Jets (SJ), another global aerospace company, has delivered aircraft to a customer for payment of EUR 100,000,000 in 180 days and then plans to deposit the funds received for another 90 days. So SJ decides to enter into an FRA to reduce its risk.
In this case, what's the FRA notation?
No.
The total time period isn't three months.
No.
The FRA isn't quoted in days.
That's it!
The notation for the FRA is 6 x 9, meaning that the FRA is for six months, and the total time period, which includes the FRA and deposit time period, is nine months. So in this case, SJ won't receive cash flows except when initiating the contract and the FRA rate is 2.49%.
And now, 90 days later, SJ needs to close out the FRA with an offsetting position.
90 days later, the company's first step is to compute the value of a current FRA with the same time period structure as the original FRA, which was 6 x 9.
What FRA needs to be calculated?
No.
The FRA notation is quoted in months, not days.
No.
The time has passed between contract initiation and the current period.
Bingo!
SJ needs to subtract the three-month period from both the FRA and the total time period. But note that the deposit time period didn't change. That's because only the FRA period changed.
Say that the current three-month MRR is 1.75%, and the current six-month MRR is 2.5%.
So SJ would use the FRA rate equation to find the current FRA rate for a 3 x 6.
$$\displaystyle \text{FRA}_0 = \frac{{ \frac{1+L_\text{New Total Time Period} t_{ \text{New Total Time Period}}}{1+L_\text{New FRA Expiration} t_{\text{New FRA Expiration}}} - 1}}{ t_m} = \frac{{ \frac{1 + L_T t_{180}}{1 + L_h t_{90}} - 1}}{ t_m}$$
Fill in the equation.
$$\displaystyle \text{FRA}_0 = \frac{{ \frac{1+0.025 \times \frac{180}{360}}{1+0.0175 \times \frac{90}{360}}- 1}} {\frac{90}{360}} = 3.24\%$$
SJ would offset its initial FRA at 2.49% with the currently rated FRA at 3.24%.
But do you think SJ benefited?
Incorrect.
You got it!
SJ benefited because FRAs are long the floating rate and naturally short the fixed rate. So as rates rose, as evidenced by the higher FRA value, SJ realizes a gain.
But what's the value of the original SJ's FRA? To find that value, the first step is to find the difference between the two offsetting rates.
$$\displaystyle 0.0324 - 0.0249 = 0.0075$$
Note the time period of the FRA rates is still annualized.
SJ needs to find the rate over the deposit period.
$$\displaystyle (0.0324 - 0.0249) \times \frac{90}{360} = 0.001875$$
And then discount the value back to the valuation time period.
Why does SJ need to discount it back?
No.
If discounting is required, then the FRA rates aren't current or present valued.
No.
The FRA rates are computed to equalize cash flows over more than just the FRA period.
Exactly!
The FRA rates cover the entire time period, so the difference in FRA rates needs to be discounted back to the valuation date.
In this case, SJ needs to discount the difference by subtracting the valuation time period from the initial total time period, and 9 months minus 90 days leaves you with 6 months. That means that SJ uses the six-month MRR to discount the FRA difference.
$$\displaystyle \frac{[(0.0324 - 0.0249) \times \frac{90}{360}]}{[1+0.025 \times \frac{180}{360}]} = \frac{0.001875}{1.0125} \approx 0.00185185$$
So the value of the original SJ FRA is
$$\displaystyle 100{,}000{,}000 \times 0.00185185 = 185{,}185$$.
To summarize:
[[summary]]
6 x 3
6 x 9
18 x 9
1 x 1
3 x 6
6 x 9
No, the FRA is long fixed
Yes, the FRA is long floating
The FRA rates are current rates
The FRA rates cover the initial time period
The FRA rates cover the entire time period
Continue
Continue
Continue
Continue
