30 days ago, J. Klein took a short position in a $10 million 90-day forward rate agreement (FRA) based on the 90-day London Interbank Offered Rate (LIBOR) and priced at 5%. The current LIBOR curve is: 30-day = 4.8% 60-day = 5.0% 90-day = 5.1% 120-day = 5.2% 150-day = 5.4% The current value of the FRA, to the short, is closest to: A) −$15,495. B) −$15,280. C) −$15,154. Your answer: C was correct! FRAs are entered in to hedge against interest rate risk. A person would buy a FRA anticipating an increase in interest rates. If interest rates increase more than the rate agreed upon in the FRA (5% in this case) then the long position is owed a payment from the short position. Step 1: Find the forward 90-day LIBOR 60-days from now. [(1 + 0.054(150 / 360)) / (1 + 0.05(60 / 360)) − 1](360 / 90) = 0.056198. Since projected interest rates at the end of the FRA have increased to approximately 5.6%, which is above the contracted rate of 5%, the short position currently owes the long position. Step 2: Find the interest differential between a loan at the projected forward rate and a loan at the forward contract rate. (0.056198 − 0.05) × (90 / 360) = 0.0015495 × 10,000,000 = $15,495 Step 3: Find the present value of this amount ‘payable’ 90 days after contract expiration (or 60 + 90 = 150 days from now) and note once again that the short (who must ‘deliver’ the loan at the forward contract rate) loses because the forward 90-day LIBOR of 5.6198% is greater than the contract rate of 5%. [15,495 / (1 + 0.054(150 / 360))] = $15,154.03 This is the negative value to the short. Not too bad–I just don’t get why at the very end of step 3 it is discounted by .054 and not by .056198. Don’t you have to discount it 90 days from day 150 (when interet savings is realized) to day 60 (when the FRA expires and payment is made)?

0…90…180 original structure new 0…30…90…180 ###

Because that the interest saving of 5.6198% vs 5% you will be realizing only when the (virtual) loan terminates, which is 90 days after FRA expires and for FRA to expire we still have (90-30) = 60 more days to go. So total of 150 days left till today to get those gain benefits. Hence discount it by 5.4%

thanks for your help guys. i think what i was missing was that the question asked for the CURRENT value of the fra (NOT the fra at termination like is more common to be asked). so if we use the 5.6198% we only discount 90 days to day 60 but that is still 60 days away from today–so we must use the 5.4% to go the full way from day 150 all the way back to today. if the problem asked for the value at fra expiration/termination, this would be asking for fra value at day 60. would it be correct to discount by 5.6198% (the 90 day libor rate 60 days from now) or by 5.1% (the current 90 day rate)? thanks!

FRA by definition pays only at original loan termination which was in this case at day 180 - because the contract was a 3 x 6 FRA.

I love FRAs!!!

the spot rate for 150 day forward is 5.4, and the gain(don’t care whether its long or short) at the end of 150 day 15,154.03. hence discount this by 5.4

what would u discount by if the question asked for value at FRA settlement date (day 60)

It says ‘The current value of the FRA, to the short, is closest to’ in the question stem, so it has to be discounted back by 150 days to present (t=0)

im aware. i am saying if it did not say that but instead asked for the value of the fra at termination (day 60).

Yea. Then discount till FRA termination (t=60 days). Basically you have to take whetever they give.