Crappy FRA Formula Doesn't Work

[rellison]

On CFAI Book 5, Page 37, they give a long formula for valuing an FRA “g” days later:
1/[(1+LIBOR after h+g days)(h+g)/360]-[(1+FRA(0,h,m)(m/360)/(1+LIBOR after h+m-g days*(h+m-g)/360)] where g=# days past, h=#days until expiration, m=#days between expiration and settlement (i.e., length of contract). FRA(0,h,m)=Annualized FRA rate g days from now.
This fugly formula works for CFAI problems like this:
“Consider a 3X9 FRA.
90-day LIBOR=.056 and 270-day LIBOR=0.06.
25 days later:
65-day LIBOR=0.059 and 245-day LIBOR=0.065
What is the FRA today (25 days later)?”
but NOT FOR SCHWESER ONES LIKE THIS:
“What is the value of a 6.00%, 1X4 FRA with a principal amount of $2,000,000, 10 days after initiation if 100-day LIBOR in 10 days is 6.15% and 20-day LIBOR in 10 days is 6.05%?”
Why doesn’t it freakin work for all FRA problems? What’s different about the two that it won’t work, and how can I fix this?

 

[newsuper]

I just do:
1/(1 + de-annualised LIBOR to the start of the loan) - (1 + de-annualised FRA cost)/(1 + de-annualised LIBOR to the end of the loan)
which for your second example would be:
1/(1 + 0.0605(20/360)) - (1 + FRA cost(110/360))/(1 + 0.0615(110/360))
and you have to calculate the FRA cost first
Of course this could also explain why I never get a FRA question right..

 

[jankynoname]

Shouldn’t it be:
1/(1+0.0605(20/360)) - (1+FRA cost (90/360)) / (1+.0615(110/360)) ??
where the only difference is I’m growing the second term by the 90-day FRA unannualized rate rather than the full 110 days to maturity.
I guess the way I’ve started thinking of this formula is the first term discounts $1 from the date the FRA contract expires back to today at the current market LIBOR rate for however many days to expiration. And the second term is the value to be paid at the end of the loan term (based on the original contracted FRA rate), and discounted at the current market LIBOR rate all the way back to today.
It sort of starts to look a lot like the standard spot minus discounted forward price equation that seems to apply to all the other forward valuations, at least after I stare at it long enough to be nauseated.

 

[jankynoname]

Here’s what I’d like to know about the forwards stuff:
For equity forwards, why do we remove the value of dividends discounted at the risk-free rate? I mean dividend payments certainly are not risk-free- in today’s market they seem pretty unpredictable and volatile. Wouldn’t it make more sense to discount the value of dividends at the cost of equity or WACC or something?
Same question applies to removing coupons discounted at risk-free from fixed income forwards… shouldn’t it be cost of debt?

 

[cpk123]

What is the value of a 6.00%, 1X4 FRA with a principal amount of $2,000,000, 10 days after initiation if 100-day LIBOR in 10 days is 6.15% and 20-day LIBOR in 10 days is 6.05%?”
for this –>
Step 1: start rate is given to you as 6%. and it is a 90 day FRA. So initial = 0.06/4 = 0.015.
Step 2: Based on the 20 and 110 days - calculate the new Rate:
(1+0.0615*110/360)/(1+0.0605*20/360) - 1 = 0.015378865
Diff is due in 110 days -> 2 Mill * (0.015378865 - 0.015) = 757.73
Therefore due today -> Divide by 110 Day rate (or as SWG puts it multiply by Z150)
* (1+0.0615*110/360) = 743.75 (Ans).

 

[cpk123]

even for this version I never use a formula:
Consider a 3X9 FRA.
90-day LIBOR=.056 and 270-day LIBOR=0.06.
0…………………90……………………270
calculate the forward rate from 90->270
(1+.06*270/360)/(1+0.056*90/360) -1 = 0.030571992
25 days later:
65-day LIBOR=0.059 and 245-day LIBOR=0.065
What is the FRA today (25 days later)?”
calculate the forward rate again:
(1+0.065*245/360) / (1+0.059*65/360) -1= 0.033229348
Difference between these two * Notional is what you will get (or pay) at the end (which is 245 days away.
( 0.033229348 - 0.030571992 )= 0.002657356
What is it due today: (Divide by 245 days rate)
0.002657356 / (1+0.065*245/360) = 0.002544785
This is per $ of notional.

 

[newsuper]

So for this question
“What is the value of a 6.00%, 1X4 FRA with a principal amount of $2,000,000, 10 days after initiation if 100-day LIBOR in 10 days is 6.15% and 20-day LIBOR in 10 days is 6.05%?”
You’re right jankynoname in that you would use
1/(1+0.0605(20/360)) - (1+ 0.06 (90/360)) / (1+.0615(110/360))
which = 0.0003717 which when you times by 2,000,000 = 734 which is what CP got, so it does work for that sort of question.

 

[CFABLACKBELT]

cpk123 Wrote:
——————————————————-
> What is the value of a 6.00%, 1X4 FRA with a
> principal amount of $2,000,000, 10 days after
> initiation if 100-day LIBOR in 10 days is 6.15%
> and 20-day LIBOR in 10 days is 6.05%?”
>
> for this –>
> Step 1: start rate is given to you as 6%. and it
> is a 90 day FRA. So initial = 0.06/4 = 0.015.
>
> Step 2: Based on the 20 and 110 days - calculate
> the new Rate:
>
> (1+0.0615*110/360)/(1+0.0605*20/360) - 1 =
> 0.015378865
>
> Diff is due in 110 days -> 2 Mill * (0.015378865 -
> 0.015) = 757.73
>
> Therefore due today -> Divide by 110 Day rate (or
> as SWG puts it multiply by Z150)
> * (1+0.0615*110/360) = 743.75 (Ans).
CPK always to the rescue. Good refresher. Thanks.