Forward Rate Agreement

[lmakombo]

An FRA trader entered into an FRA agreement in which he will pay 6%(assuming quarterly compounding) between 3 months and 6 months. The principal for the trade is $3 million. The 6 month LIBOR spot rate is 5.8%. If the trader had a gain of $2550 at the end of the period, The 3 month LIBOR rate would be?

A. 5.30%
B. 6.30%
C. 5.25%
D. 2.51%

The correct answer is A. how do we get there??????????????????

 

[Bucephalus]

Hi @David Harper CFA FRM .. My question is not related to the above and I am using this thread because I did not want to start a new one. My question is on Hull 4.6 (Study notes)

Hull 4.6
Assuming that zero rates are as in Problem 4.5 (above), what is the value of an FRA that enables the holder to earn 9.5% for a 3-month period starting in one (1) year on a principal of $1,000,000? The interest rate is expressed with quarterly compounding.
From David: The “holder who earns 9.5%” is the counterparty who is receiving the fixed, paying the floating rate, and who is SHORT the FRA. The buyer (who is long the FRA) is paying the fixed rate and receiving the floating rate.

Answer:
Here is the spreadsheet: https://www.dropbox.com/s/8dzdxspnk2blh4s/Hull.04.06a.xlsx
The forward rate is 9.0% with continuous compounding or 9.102% with quarterly compounding.
From equation (4.9) the value of the FRA is therefore
[1,000,000 x 0.25 x (0.095 – 0.09102)]e^-0.086 x 1.25 = 893.56 or $893.56

For the period between 1 and 1.25, the continuous forward rate is calculated as 8.8%. Shouldn't that be the forward rate/variable rate for that period? Why is 9% which is the continuous forward rate for the period between 1.25 and 1.5 considered as the forward rate?

 

[David Harper CFA FRM]

HI @Bucephalus I think that's my fault due to the awkward presentation in my spreadsheet. Please note that if the 12-month continuous spot rate is 8.5% and the 15-month continuous spot rate is 8.6%, then the forward rate, F(1.0, 1.25) = (8.6%*1.25 - 8.5%*1.0)/(1.25-1.0) = 9.0016%. My bad XLS has that on a different row; below is how this looks in my newer XLS (and we will replace the graphic). The sheet is here at https://www.dropbox.com/s/c84u7xq38rfp3bx/0913-hull-4-6.xlsx?dl=0
... you can see that's pretty close to Hull's given answer (the difference is surely rounding, as the continuous forward rate is not equal to exactly 9.0% but almost, such that using the "rounded" 9.0% forward rate gets you almost to the given 893.56). I hope that clarifies, thanks!

 

[Bucephalus]

Thanks @David Harper CFA FRM
I believe the main concept here is that the value of a FRA is Principal * (Variable rate - Fixed Rate) * (T2-T1). The item in bold depends on whether the FRA holder is long or short. And, the variable rate is the forward rate for that period. In the example above, the forward rate is for the period between 1.0 and 1.25. And, the forwards rate calculated using the formula above is anyway the continuously compounded rate. In the example above, because the fixed rate is quarterly compounded, the variable should also be similarly m-annually compounded. Please correct me if my understanding is wrong.

 

[David Harper CFA FRM]

Hi @Bucephalus Yes, I agree with you (although I am completely sure what you mean, sorry, by "the item in bold depends on whether the FRA holder is long or short." as I bolded several items). To recap--and elaborate hopefully for clarity:

• This FRA is promise to exchange a future 3-month rate floating interest rate (which is obviously unknown today) for the fixed 9.50%; the 3-month rate is the rate that will prevail in one year. The notional is $1,000,000 and the payments are netted, so the cash flow exchanged will be just the difference between the future floating rate and the fixed 9.5% multiplied by the notional (as the rates are per annum and this will be a three month rate, there will be a further multiplication of 0.25). This is the essential FRA: a future exchange of a netted cash flow that references the notional, based on the difference between the prevailing floating rate and the predetermined fixed rate. Importantly, the design of the FRA is such that, because it is three month rate in this case, the 9.50% is 9.50% per annum with quarterly compounding. (this is not a futures contract so this quarterly compounding is not a standardized specification, rather it is a given assumption but quarterly compounding is consistent with the three-month rate!)
• We are estimating the present value (PV) of this FRA ... yet we cannot know the future floating rate. To compensate for this unknown, we predict the future floating rate to be the forward rate. So we extract the implied forward rate from the observed spot rate, and we determine that the implied forward rate is 9.0016% per annum with continuous compounding.
• As the actual cash flow exchange will be "apples-to-apples" with respect to the rate (i.e., we won't be taking the difference between a quarterly fixed rate and a continuous LIBOR), we want an estimate of future LIBOR with quarterly compounding. So we convert the "predicted" continuous 9.0% to its quarterly equivalent. It is exactly as you say: "In the example above, because the fixed rate is quarterly compounded, the variable [i.e., the "predicted LIBOR that we are inferring from the observed spot rate curve because ... well, that's the best we've got!] should also be similarly m-annually compounded. Please correct me if my understanding is wrong."
• In this way we are "predicting" the future cash flow exchange, and finally discounting it to the present in order to estimate a value for the FRA. I hope that's additive, thanks!