EuroDollar Futures | Practical Ques Clarification

[atandon]

Hi David,

Could you pls clarify on the below calculation listed in the doc (2011-T3.Hull-Chapter5&6) -

173.1. The 300-day LIBOR zero rate is 3.0% per annum with ACT/365 continuous compounding. The Eurodollar futures quote for a contract maturing in 300 days is 96.00; as usual, the Eurodollar interest rate is expressed with ACT/360 quarterly compounding. What is the 390-day LIBOR zero rate with ACT/365 continuous compounding (i.e., as we are extending the LIBOR zero curve)?
a) 3.006%
b) 3.239%
c) 3.867%
d) 4.035%
Answer in listed in doc -

173.1. B. 3.239% The 300-day forward LIBOR is 4.0% (100-96) and converting to ACT/365 continuous is given by: 365/90 * LN(1+4%/4) = 4.0354%. The 390-day LIBOR zero is then given by: [3.0% * 300 + (4.0354% * 90)] / 390 = 3.2389%
Why did we used - 365/90 * LN(1+4%/4) = 4.0354%

Shouldn't the answer be -
Rc = m log(1+Rm/m)
m = no of freq in a yr. In this case -> 4
4 or (360/90) [* LN(1+4%/4)] = 3.98%
Regards,
atandon

 

[Aleksander Hansen]

173.1. B. 3.239% The 300-day forward LIBOR is 4.0% (100-96) and converting to ACT/365 continuous is given by: 365/90 * LN(1+4%/4) = 4.0354%. The 390-day LIBOR zero is then given by: [3.0% * 300 + (4.0354% * 90)] / 390 = 3.2389%
Why did we used - 365/90 * LN(1+4%/4) = 4.0354%

Shouldn't the answer be -
Rc = m log(1+Rm/m)
m = no of freq in a yr. In this case -> 4
4 or (360/90) [* LN(1+4%/4)] = 3.98%
Regards,
atandon

Eurodollar Futures: Act/360; quarterly.
4% --> 1% per 90 days --> 365/90 * ln(1.01) annual rate, continuous and Act/365 = 4.0354%.
I might be missing something but looks right to me. Sure Dave will correct me if I'm way off here.

 

[David Harper CFA FRM]

I agree with ahansen, it looks right to me, too (fwiw, I don't enjoy these questions, 173.1 is just a variation on Hull's 6.8, is why i feel obligated to throw in the day count adjustment). Maybe two additional perspectives will help:

1. Same as ahansen's, just in case it's not crystal clear already what he did above: you can find yours "inside" as 365/90 * LN(1+4%/4) = (365/360)*360/90 = (365/360)*4

2. We could also get to the same place by staying with an assumption of ACT/360 until the last step:
F(300,390) = 4*LN(1+4%/4) = 3.9801%; i.e., as you have it. But if we do this, then we need to convert the spot to ACT/360 with:
F(0,300) = 3.0%*360/365 = 2.9589%; so now they are both "apples to apples" with ACT/360. Then impute the 390 spot:
F(0,390) = (2.9589%*300 + 3.9801%*90)/390 = 3.1946% ACT/360, a fine answer except the question asks for ACT/365:
3.1946% ACT/360 * 365/360 = 3.2389%

I hope that helps,

 

[Shantanu Mantri]

Wow! Now that factor of 365/90 makes all the sense.. was confused as to how every text came up with that 365/90.. why didnt they just do it the way David explained in statement 1 (to save us from the confusion)
Thanks David!