Valuing FRAs

I thot I had this down but I was doing a mock exam and the solution is different than I expected.

Given: After 30 days into a 2x5 FRA, Adams wants to value a $10M short position. The 90-day forward rate in 30 days is now 4.14% and the original price on the FRA was 4.30%. 120-day LIBOR is 3.92%.

Solution Given: (0.0430-0.0414) x $10M x (90/360) / [1+0.392*(120/360)]

So the solution is essentially the difference between the present 90-day rate and the contract 90-day rate adjusted and discounted back to present. But the FRA has not expired and I expected to use this formula at expiration.

The formula I expected to use was

Let X=1 + 30-day Libor*(30/360)

Let Y=1 + FRA Rate*(90/360)

Let Z=1 + 120-day Libor*(120/360)

Value FRA = (1/X) - (Y/Z)

Checked the CBOK and it looks like the first equation is used to value the FRA at expiration and the second is used to value it before expiration, so what am I missing from this question (Schweser Mock exam volume 1, exam 3, question 46)

Whether you’re valuing it before expiration or at expiration doesn’t make any difference: you can use the same formula. (And you should: cut your memory burden by 50%.)

Equally concerned Schweser seems to differ from CFAI in this regard unless we are confused (likely).

schweeser values an FRA prio to maturity as the difference between the initiated agreed rate and then the new FRA rate calculated on the term structure at valuation. And discounts this from the settlement sate to valuation which makes logical sense.

CFAI uses the 1/x - y/z method you have described.

not sure why…any clarification will be great.

S2000 - Agreed that the more complex formula degenerates into the formula “AT” expiration but before expiration the more complex formula is necessary, no? The formula Schweser employs uses 3 inputs - 90 and 120 day LIBOR and the FRA contract price - whereas I was expecting to use 30 and 120 day LIBOR and the contract rate.

If the formulas become equal before expiration, I don’t see the derivation and I think it would imply that there is an arbitrage enforced relationship between 30 and 90 day LIBOR, which we know isn’t true because of all the different shapes we can get for the interest rate curve.

Just realized my mistake.

The problem says, “The 90-day forward rate in 30 days is now 4.14%”. Not, “The 90-day LIBOR rate is now 4.14%”.

So essentially, the NEW FRA that one can enter would be off-setting to the one established previously. Thus, the value at expiration for the first FRA would be the difference between the first contract rate and the current 90 day LIBOR and, likewise for the second. If long one and short the other then LIBOR cancels and you have the difference between the two contract rates.

Got it.

There is no arbitrage opportunity because the value at any point in time depends on the implied forward rate _ at that time _.

I wrote a couple of articles on pricing and valuing FRAs that may be of some help: