FRA after pass some days

Why discount back 110 days not just 90 days? Please add some comments

What is the value of a 6.00% 1x4 (30 days x 120 days) forward rate agreement (FRA) with a principal amount of $2,000,000, 10 days after initiation if L10(110) is 6.15% and L10(20) is 6.05%?

A) $767.40. B) $745.76. C) $700.00.

Your answer: A was incorrect. The correct answer was B) $745.76.

The current 90-day forward rate at the settlement date, 20 days from now is: ([1+ (0.0615 x 110/360)]/[1+ (0.0605 x 20/360)] – 1) x 360/90 = 0.061517

The interest difference on a $2 million, 90-day loan made 20 days from now at the above rate compared to the FRA rate of 6.0% is: [(0.061517 x 90/360) – (0.060 x 90/360)] x 2,000,000 = $758.50

Discount this amount at the current 110-day rate: 758.50/[1+ (0.0615 x 110/360)] = $745.76

Because presumably the pmt will be made at the end of the contract, that is 120 days after the initiation or 110 days from now.

Is it clear?

We just discussed this a day ago:

Remember you want the loan 30 days later but you want to know now what rate you are going to pay. No one knows what the rate is going to be, but through this FRA agreement someone is willing to take the risk and offer you the loan for 5.32%.

After 10 days, things change, right? So we are now 110 days away from the loan termination date. At this point, if you want to value the FRA, then you can check to see what its current price is. Just like stocks and bonds, it has a market rate which changes every day. Suppose you find that a new FRA (pricing it using the same formula as before, but now your long rate is the 110 days and your short rate is the 20-day rate) with the same maturity as yours today has a rate of 6% (meaning that if some new guy wants to buy it today, he would have to pay 6%).

You’re lucky because you’ve just locked in a rate of 5.32%. The difference in rates is your profit. However, you will not get that difference of 0.68% just yet….you’ll get at after 110 days. So you need to discount the 0.68% annual rate to today, so it’s 0.68%/4 because it’s a 90-day rate. Good luck.