How to calculate the floating leg of a swap

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Jones473

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I have a question regarding the calculation of the floating leg of a swap. It seems to me this is calculated differently in two examples in the text book. On page 280 in the Derivaties book there is a text example of a swap, followed by example 8. I have pasted this below.
In the first example the floating leg of the swap is simply 1, because we are at the reset date: “Because the 90-day floating rate is 3.90 percent, the next floating payment will be 0.0390(90/360) = 0.00975. Of course, we do not know the last floating payment, but it does not matter because the present value of the remaining floating payments, plus hypothetical notional principal, is automatically 1.0 because we are on the coupon reset date. Therefore, the market value of the swap to the party receiving floating and paying fixed is the present value of the floating payments, 1.0.
However, in example 8 the floating leg is calculated as: The market value of the floating payments plus $1 hypothetical notional principal is 1.045(0.9519) = 0.9947.
My question is why is the floating leg calculated differently in these two examples and why is the floating leg not discounted in the first example?
 
If you’re at a reset date (so you’ve just received the floating payment), the value of the floating leg is 1.
If you’re between reset dates, the value of the floating leg is the present value of the next floating payment plus the present value of 1.
They’re really saying the same thing: it’s always the present value of the future payments, and at the reset date the present value of the future payments is 1.
I wrote an article on valuing swaps that may be of some help here: http://financialexamhelp123.com/valuing-plain-vanilla-interest-rate-swaps/
(Very important note: the value of the floating leg at a reset date is 1 if and only if the coupon resets to the market rate; i.e., the floating rate is LIBOR. If the floating rate is LIBOR plus a spread, the value of the floating leg at the reset date isn’t 1. Read my article to see what to do in that case.)
 
Is it in the level 2 curriculum when the value of floating isn’t 1? So far I came across only simple cases whereby the emphasis is on calculating the fixed rate.
 
I know that they’ve had it in the past; I’d need to pore over the current curriculum to see if it’s still there.
 
S2000Magician. Thanks. I agree with what you write and understand the logic. My problem is that I thought that in example 8 around page 280 in the Derivatives book, we are at the reset date. That is how I understand the example. It is a 2Y swap, with semiannual payments and we are 360 days later, so I assume we just received the second payment. Am I wrong? I pasted some of the text here: “Consider a two-year swap to pay a fixed rate and receive a floating rate with semiannual payments. The fixed rate is 0.0462. Now, 360 days later, there is a new term structure. The next floating payment will be 0.045. The swap calls for marking to market after 180 days, and, therefore, will now be marked to market.”
 
I just read through the example you cited. We’re at day 360, so we just settled the payment; it does not figure into the value of the swap.
What’s weird is that the floating payment is 4.5%. Given the term structure, it should be 10.1%; i.e., the floating payment should be 180-day LIBOR.
I’ll write CFA Institute and ask them about this.
 
Well, it says explicitly that “the next floating payment will be 0,045”. However I assume you mean it should be 0,104*180/360?
But still, why do we discount here when we are at the reset date?
 
Jones473 wrote:Well, it says explicitly that “the next floating payment will be 0,045”. However I assume you mean it should be 0,104*180/360?
But still, why do we discount here when we are at the reset date?
Click to expand...
I know that it says explicitly that the next floating payment will be 4.5%. It also says that the 180-day interest rate is 10.1% and that the price of the bond will reset to $1. It can reset to $1 only if the coupon payment equals the market rate. That’s the discrepancy.
Jones473 wrote:But still, why do we discount here when we are at the reset date?
Click to expand...
Because we’re looking at the value at the next reset date, and discounting it back to today. My point is that the value at the next reset date, when discounted back to today (the current reset date), should be $1; this can happen only if the coupon rate equals the market rate. The fact that it doesn’t is, I believe, an error on CFA Institute’s part.
I’ve e-mailed them and asked them to check it.
 
Ok I understand your point about the discrepancy. But still don’t think the example text is clear about that we are talking about the next reset date, with regards to the discounting.
 
The discount factor is for 180 days, so we have to be discounting a cash flow coming 180 days from today.
 
Yes I understand that point. But I guess I felt the text was not clear about that part.
 
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