Value of FRA

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Dreary

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In trying to avoid having to memorize too many formulas, I wanted to figure out the value (not the price) of an FRA using the current price of the FRA. So, assume this:
1) You enter into a 30-day by 120-day FRA.
2) Assume that its price at initiation is 4.1%. [That means after 30 days, you lock in the 90-day LIBOR rate, and you pay (90 days later) 4.01%. No payment takes place until 90 days elapse.]
3) After 10 days, what’s the value of this FRA?
One way is to use the V_t equation. But why not figure it out by asking what is the price of the same FRA on day 10? That’s easy, because the price formula is easy to compute.
Assume that the same FRA on day 10 is priced at 3.65%, again, meaning after 20 days (10 days have already passed), you are obligated to pay 3.65% after 90 days, and receive the existing 90-day LIBOR.
How should the value be calculated from here? I tried a few ways and came close but not exactly. Has anyone thought about this? I did this with other forward contracts and it is more convenient to derive the value from prices only, without a new/complicated/hard to remember formula.
 
I am too lazy to work through your logic so I’ll just tell you how I usually work out the value.
Assume you are long a FRA struck at K for dates T1 and T2. Assume today’s forward rate relevant to the FRA is F. This means that you can, at no cost, enter into a FRA with strike F and receive the amount (F-K)(T2-T1)xNotional at time T2. To get the present value of your FRA, discount this amount to today (at today’s spot rate for T2, of course).
Done. That simple.
 
Lets make it simple:
You have a 1X4 FRA, where the 30-day LIBOR = 3.66% and the 120-day LIBOR =4.00%, what’s the price of this FRA at t=0?
FRA(0,30,120) = [(1 + 0.04(120/360) / 1 + 0.0366(30/360) -1] (360/90) = 4.10%
What’s the *value* at t=10? Assume LIBOR-120 = 3.8%, and LIBOR-30=3.5% without using the formula?
 
new value
((1+.038*(110/360))/(1+.035*(20/360)) - 1)*360/90 = 3.86%
New value: (3.86-4.1)*90/360*1000000$ = -60000
we are still 110 days away from maturity == so -60000/(1+.0385*110/360) = -59302
 
cpk123 and FourCastles,
Using the formula, the answer is -$595.61 loss to the buyer.
V_10 = (1/1+0.035(20/360)) - [ (1+0.041(90/360)) / (1+0.038(110/360)) ] = -0.0005956
(someone should double check my math, but I’m almost sure this is correct).
Again, can we figure it out using the new price of the FRA at time t=10?
 
Re-asking the question to see how to figure out value without having to memorize the Value formula:
You have a 1X4 FRA, where the 30-day LIBOR = 3.66% and the 120-day LIBOR =4.00%, what’s the price of this FRA at t=0?
FRA(0,30,120) = [(1 + 0.04(120/360) / 1 + 0.0366(30/360) -1] (360/90) = 4.10%
What’s the *value* at t=10? Assume original 120-day LIBOR is now 3.8%, and original 30-day LIBOR is now 3.5%?
 
I was also looking for a method of solving FRA problems without memorizing the formula. Working with example 9 of the CFAI carriculum the above discribed method worked just fine until I got to C). Here my answer was different by a few dollars from the one calculates using the formula. Does anyone know how to solve this with no formula?
 
Thanks for the link S2000magician - good example. It looks like my calcs were correct and the difference in the result is rounding error, which is noticeable on a $10M contract.
 
You can do it two ways but I would just use the formula for ease. But if you want to understand how it goes using logic look below.
CURRENT 90 day forward rate ={ [ (1 + L(110) * 110/360) / (1 + L(20) 20/360) ] - 1 } * 360/90 equal 0.0386
equal to 0.0386 using L(110) = .038 and L(20) = .035
We know the Short has to pay the long cause this is smaller then the FRA priced rate = .041
Then
[( .0386 - .041 ) * 90/360 ] / [ 1 + L(110) * 110/360 ] this equals -0.000596 like you had.
Again with the formula you save a step. But there ya go.
 
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