FRA Formula.

[PaulGrey]

Hi David,

As I learn more in Hull, I like to keep switching chapters to reiterate some of the points I previously learned to reinforce what I'm currently learning. It takes a lot longer to do but I think is worth it.....

For some reason I keep forgetting the formula for the FRA valuation i.e.
v=(T2R2 - T1R1) / (T2 - T1)

So I thought I'd try to derive it rather than try memorise it (why rely on memory, especially when you have a limited memory space like me, when you can work out the answer...hopefully...).

From my understanding of what the forward rate is actually defined as, I find this easy to remember...

EXP(R1)(T1).EXP(RF)(TF) = EXP(R2)(T2) .......

So the rate we're looking for is RF. TF is the time period that compounds T1 into T2 so sits between R2 and R1 Therefore: TF=T2-T1

So:
EXP(RF)(TF) = EXP(R2)(T2)/EXP(R1)(T1)

now move the denominator up to the numerator.
EXP(RF)(TF) = EXP[(R2)(T2)-(R1)(T1)]......

Therefore taking logs of both sides...
RF (TF) = R2.T2 - R1T1 .....where as previously stated TF = T2-T1

Rf = R2.T2 - R1T1 / (T2-T1)

Doing it this way also explains the terms T1 and T2 - i.e. in terms of years etc.

Hopefully this is able to help someone.
Appologies if its already been covered somewhere else...

Thanks,
Paul

 

[PaulGrey]

oops i meant forward rates, not forward rate agreements...

 

[David Harper CFA FRM]

Paul,

That's helpful. I agree, it's very good to work this out. Especially because a constant source of confusion is the same mechanic employed in Tuckman's Fixed Income, but he uses semi-annual compounding so, at first glance, it looks different but it's essentially similar to Hull's that you've shown here. I think the XLS on the member page shows the conversion back and forth...this concept, IMO, is also very much a key building block; i.e., that a spot rate curve embeds a forward curve. David