Please help confirm whether my understanding regarding the payoff of FRA and swaption is correct or not. I understand that FRA is paid off at the start of the loan period; Swaption is paid off at the end of the loan period. Thank you.
i need to revise this section but i’m sure u will get some help … but why is this urgent … u can’t seek help on terms like that
Yes. In both cases the rate is set at the beginning of the option period. The swaption is paid at the end of the period. FRA’s are present valued and paid up front.
This is a random comparison, they are completely separate hedging techniques
Jscott24 Wrote: ------------------------------------------------------- > This is a random comparison, they are completely > separate hedging techniques Pull some of that mad FRA and Swap stuff out Jscott. Just for old times.
save it for tonight, J - when I’m going to bring this section to its knees.
mwvt9 Wrote: ------------------------------------------------------- > Jscott24 Wrote: > -------------------------------------------------- > ----- > > This is a random comparison, they are > completely > > separate hedging techniques > > Pull some of that mad FRA and Swap stuff out > Jscott. Just for old times. I’ve got those bookmarked and have sent those things to my friends who are taking L2. They are brilliant. ------------------------------------------------------------------------------------- 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 ------------------------------------------------------------------------------------- Problem: 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 3: 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 4: 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.
I wish I could think like Jscott. I bet he could write a book without having to revise it. Edit: That’s the stuff Dwight!
haha unfortunately swaps arent as sexy as they were last year