# FRAs and Swaps by JScott24

I would have spent twice as much time studying volume 6 today had it not been for two posts: http://www.analystforum.com/phorums/read.php?12,754042,754042#msg-754042 http://www.analystforum.com/phorums/read.php?12,749056,749056#msg-749056 ========================================================== Re: Jscott24 - FRAs Posted by: Jscott24 (IP Logged) [hide posts from this user] Date: June 3, 2008 02:53PM 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 ========================================================== Re: I have to let swaps go Posted by: Jscott24 (IP Logged) [hide posts from this user] Date: June 2, 2008 01:39PM Cmon guys give it one more chance Let’s say that two parties enter into a 1-year swap with quarterly fixed payments priced at 6.052%. At initiation, the 90-day LIBOR was 5.5%. 30 days later, we observe the following chart on a \$30 million dollar notional 60-day LIBOR – 6.0% 150-day LIBOR – 6.5% 240-day LIBOR – 7.0% 330-day LIBOR – 7.5% What is the value of the swap? Step 1: Calculate the fixed payment at each payment period 6.052% X (90/360) = 1.513% It’s easier to calculate in terms of \$1.00. After all, we’re trying to find the net amount Day 90 - \$0.01513 Day 180 - \$0.01513 Day 270 - \$0.01513 Day 360 - \$1.01513 Remember the principal is returned in the last payment Step 2: Calculate the present values of the fixed rate payments Now that we’re 30 days in, we need to calculate the discount factors for 60, 150, 240, and 330 days using the new LIBORS In 60 days - Z1 = 1 / 1 + (0.06 * (60/360) = 0.99010 In 150 days - Z2 = 0.97363 In 240 days - Z3 = 0.95541 In 330 days - Z4 = 0.93567 These are usually given so we don’t have to go through the process CF1 = \$0.0153 * 0.99010 = 0.01498 CF2 = \$0.0153 * 0.97363 = 0.01473 CF3 = \$0.0153 * 0.95541 = 0.01446 CF4 = \$1.0153 * 0.93567 = 0.94999 PV of Fixed-Payer = \$0.99399 Step 2: Calculate the floating rate payment at each period. There are four payments for each of the 4 quarters. Because 30 days have passed, the payments are at 60, 150, 240, and 330. However, we only need to know 1 payment, and we already know it. How can that be? Unlike the fixed-payment, the floating-rate payment is reset to the market value every payment date, therefore the par value is always equal to 100. As for the payment, well that was identified in the question 5.5%. Remember the floating-rate payment is always set for the next payment date, therefore on day 90 (60 days away) the value of the payment will be 5.5%. Also, we add the value of the principal as well as discount back at the 60-day (first payment) discount factor. 0.055*(90/360) = \$0.01375 This is the floating rate payment undiscounted. Since it is 60-days out, we need to discount it back + the \$1.00 bond value immediately after the payment is made. CF1 = 1.01375 * 0.99010 = 1.00371 = PV of the floating-rate payer Remember that it doesn’t matter what the floating rate coupon payments are at the last three settlement dates because the floating rate bond will be worth \$1.00 plus the coupon of \$0.01375 at day 90 Step 3: Determine the value to the fixed-rate payment The present value of the floating-payments are greater than the present value of the fixed-payments, therefore, the fixed payer stands to gain Swap value to the fixed payer = 1.00371 – 0.99399 = 0.00972 Swap value to the fixed payer = \$30,000,000 * 0.00972 = \$291,630 Present value of the fixed payments + principal at the last payment period compared to the present value of the next floating payment (usually given) + principal at the next payment period. Just remember how to calculate the discount factors (which are usually given anyway) and to discount based on where you stand in the contract, 60 days away, 30 days away etc. And remember to stop valuing the floating rate after the next payment date. ========================================================== Hope these help you as much as they’ve helped me.

Thank you!!! I think you just helped me regain my sanity for the rest of the afternoon.

Bump, my buddy who’s on lvl 3 now showed me these. They are hugely helpful for anyone that is still confused on these topics.

this helps to think of it a bit clearer…thanks

jscott thanks very clear

bump !

Thank you so much! That’s so much clearer. This why AF is so useful! 