# Jscott24 - FRAs

Your Swap explanations were so good, if you have a few minutes would you mind doing an one on FRAs? I feel like I am trying to memorize vs actually understand FRAs and I know thats not going to work too well come exam day…

These are my notes on FRA’s: The key to understanding FRA’s is to realize that there are two sets of time intervals. Unlike other forward or future derivatives where the net cash finally changes hands at expiration, the difference with FRA’s is that the underlying asset is a loan. And that loan, at expiration of the contract, takes effect and we should discount that loan just like any other loan. However, we must realize that the payoff that we will experience due to the FRA will occur at the expiration date, and is the difference between the market rates at the time and the locked fixed rate that we enter into. If we enter into a FRA that gives us a fixed payment of 10% 180-day 30-days from now, and in 30 days market rates are at 12%, the gain that we experience is the 2% difference on the notional. However, we’re not going to see that 2% until 180 days from the contract expiration, so we must calculate the present value of the 2% X notional. And working through the book problem on valuation at initiation: Example, calculate the price of 1 X 4 FRA. The current 30-day LIBOR rate is 4% and the 120-day LIBOR rate is 5% Okay so here we have it. 1 X 4 means that in 30 day we need to calculate the 90-day LIBOR. Remember 4 – 1 = 3 months (90 days). Well the first thing we need to do is de-annualize our given rates 30-day LIBOR = 0.04 X (30/360) = 0.0033 = or 0.33% 120-day LIBOR = 0.05 X (120/360) = 0.0167 = 1.67% Now we need to calculate our 90-day rate 30 days from now. How do we do that? The no-arbitrage theory states that we should be able to achieve the same return by either investing in the 120-day LIBOR, or investing in the 30-day LIBOR and then the 90-day LIBOR when that expires. So we discount the 120-day rate by the 30-day rate to get the 90-day rate, as per the formula above. 1 + R120 / 1 + R30 1 + 0.0167 / 1 + 0.0033 – 1 = 0.0133 (360/90) X 0.0133 = 5.32% 5.32% is the annualized 90-day rate. This is the rate that makes: Investing at the 30-day rate + then investing at the 90-day rate 30 days from now = Investing at the 120 day rate Proof: Scenario 1: Invest \$100 at the 30-day rate and then again at the 90-day rate 100 X (1 + (0.04 X (30/360)) = 100.33 100.33 X (1 + (0.0532 X (90/360)) = 101.667 Scenario 2: Invest \$100 at the 120-day rate 100 X (1 + (0.05 X (120/360)) = 101.667

And now valuing an FRA after a certain amount of time using the book problem: Using the same example, let’s say that 10 days have gone by since we entered into the 90-day LIBOR FRA. That means that we have 20 days until expiration, and 110 days until loan maturity. The 110-day LIBOR rate is 5.9% and the 20-day LIBOR rate is 5.7%. How do we calculate value to the long? First we need to calculate the New FRA price. Let’s take a second to think about what we’re doing. We locked in at 5.38%, which was the 120-day rate discounted at the 30-day rate 10 days ago. That 5.38% was the value of the 90-day LIBOR 30 days from t0. Now we’re saying, 10 days from then, that the 110-day LIBOR is 5.9% and the 20-day LIBOR is the 5.7%. Again, the no-arbitrage price theory states that investing at the 110-day LIBOR = Investing at the 20-day LIBOR and then reinvesting at the 90-day LIBOR. So in order to get the new FRA price of the 90-day LIBOR, we discount the 110-day LIBOR at the 20-day LIBOR Unannualized Rate until Loan Maturity = 0.059 X (110/360) = 0.01803 Unannualized Rate until contract expirations = 0.057 X (20/360) = 0.00317 New FRA Price = [1.01803 / 1.00317 -1] X (360/90) = 5.92568% Our work is half done, now we need to determine if we’re in the black or in the red. The new FRA price on the 90-day increases, that means we’ve gained, but by how much? Remember we locked in at 5.38% [(5.92568% X 90/360) – (5.39 X 90/360)] X \$1million (0.01481 – 0.0133) X \$1million = \$1514.20 Wooohooo \$1514.20 that’s my value, that’s what I gain right? Wrong. You’re still 110 days away from maturity champ, discount that to the present. What rate do I use to discount? Use the current 110-Day LIBOR rate because you are 110 days away. \$1,514.20 / 1 + (0.059 X (110/360)) = \$1,487.39 I did this on memorial day weekend when all my friends were at the beach, I think this is when I started talking to myself

Dude we need to give you a medal

Jscott you’re the man.

dude, you’ll f*ck up the curve for us… anyways, great post. keep up the good work!

Thanks Jscott! I found your Swap explanation great as well. If I have the courage I will try to understand the currency swaps too.

This is real good … sorry but i did not see the swap post by Jscott can someone reply with the link

thanks man

thats great JScott… thanks for your time!

Thanks barthezz

Jscott24 - are you human?? I am just loving your explainations - Thanks You Mr John Hull (I never knew you were on AF too…)

Should be 5.32% and not 5.38%, and that 5.39% in the calculation should also be 5.32%

Thank You, Thank you , Thank You!

bump - for everybody having trouble with FRAs

Jscott24-this is really good stuff, very much appreciated.

mwvt9 Wrote: ------------------------------------------------------- > bump - for everybody having trouble with FRAs mwvt9 looking out for the L2ers! yeah man. thats what I am talking about.

I still hate your opponent and will do anything I can to kick it in the junk again.

BUMP