Team - my head hurts and FRA is knocking

CFA Level II
#1

Reading 48 question 9b:

A corporate treasurer wishes to hedge against an increase in future borrowing blah blah. She proposes a 6X12 FRA and the current term structure is:

L30 5.10%

L90 5.25%

L180 5.70%

L360 5.95%

The treasurer went long and 45 days have elasped. the new term struture is:

L135 5.90%

L315 6.15%

I used to be so good at FRAs and now i can’t seem to figure it out. Many many thanks for whoever decides to help a afer out. Layman’s terms are always encouraged over the typical CFAI lingo.

#2

Alright, first we parse what the 6X12 is…

Enter into a loan 6 months from now, for a term of six months. So that means that the six month rate six months from initiation will be the rate you care about. We have to figure out what that will be.

Recall that investing in a product for 12 months should equal investing it in six months and then rolling it over. So, 12 month LIBOR should equal six month LIBOR rolled over into what LIBOR will be six months from now.

Currently 12 month LIBOR is 5.95%

This is the same as saying 1 + 6 month LIBOR today X 1 + 6 month LIBOR in six months = 1 + 5.95%

We are given 6 month LIBOR today as 5.7%, but remember this is annualized. We multiply by 180/360 (remember LIBOR is simple interest), which is the same as multiplying by 0.5, and thus we get 2.8%

Now we have to solve for what the market expects six month LIBOR to be six months from now, since we are not given this information (on purpose of course, so you can show how much smarter you are than those in band ten and lower :slight_smile:

We use the relationship above: 1+2.85% X 1+N% = 1+5.95%

N = 3.01%

Doing a quick check: 1.0285 X 1.0301 = 1.0594… close enough.

So, you are locked into a rate of 3.01% or at the annualized rate, 6.02%.

45 days later, the six month LIBOR rate we care about is going to be 135 days away. Now we have to figure out from what we’re given, using the same method as before, what the new six month LIBOR will be.

The 315 day LIBOR should equal an investment into 135 day LIBOR followed by rolling it forward into 180 day LIBOR.

Now lets not forget to unannualize each number in this case, since in the last case one of our rates was a 12 month rate.

315/360 X 6.15% = 5.38%

135/360 X 5.9% = 2.21%

So now back to our relationship:

1+2.21% X 1+N% = 1 + 5.38%

Now we solve for N and find that it equals 3.10%

Hence, 180 day LIBOR has moved up from 3.01% (6.02% annualized) to 3.1% (6.2% annualized), about 90 bps or 180 annualized bps, if we excuse the rounding. Now the borrower (payer of the fixed side) is pleased because his fixed payments are less than the floating, which nets out to him getting a cash flow.

That cash flow is measured as 180 bps (must annualize) X 1,000,000 standard FRA principal = $18,000 at expiry. If the question asked you for payoff at expiry you’d be done here.

To get the present value you need to PV it back to today using the current LIBOR rate for the time period you are valuing it. That would simply be discounted by dividing it by 1 + the 315 day LIBOR, unannualized.

$18,000/(1+2.21%) = $17,610 value to the pay fixed side, negative that for value to the receive fixed side.

Hopefully I got that right, though due to rounding the numbers can be off a bit.

#3

You mean 18 bps (180 bps would be 1.8%). You also mean to discount back $18,000/(1+5.38%).

#4

Why have you annualised the CF when it’s a 180d contract?

#5

Giovanni - many many thanks for the elaborate explanation, everything is coming back to me. The answer for B is 6.03% - your calcs are correct (and done in an easy to understand manner) so i assume the minor diff is just due to rounding.

As for the second part, the book provides an answer for $8,100.

[1/1+0.059(135/360)] - [(1+0.0603(180/360)/1+0.0615(315/360)] = .00081

#6

Oh shoot, yeah I got a bit overconfident in the last half of my answer there :confused:

Thanks for correcting it!