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# Floating-rate cash flows in an interest-rate swap

#### sridhar

##### New Member
David:

I understand the mechanics of calculating the value of a interest-rate swap, when viewed as an exchange of fixed-rate and floating-rate payments. I also understand the cash-flows coming out of the fixed-rate payer.

Using the Example 7.2 in Hull's book: every 6 months, discount the coupon payment to the present value; using the LIBOR spot rates at 3, 9 and 15 month intervals as the corresponding risk-free rate.

What I don't understand is the rationale behind the computation of the floating-payer's payments:

I see that we compute this as:

[ notional + immediate-next-period-floating-payment] * exp( - r1*t1)

I am sorely missing something. Two questions:

1. Just like in the fixed-rate cash flow, why are we not computing the cash-flows for all 3 periods and then discount the notional at the risk-free rate corresponding to the 15-month rate.

2. I am not able to explain to myself the "rationale" behind why the floating-cash flow not mirror the fixed one? Again, looking at Table 7.2 on page 162 of Hull -- why do we have 3 cash flows for the fixed-rate-payer and only one for the float-rate-payer?

What makes the simplified floating cash-flow possible?

In contrast, I can easily see the rationale of the two sets of cash-flows when computing the swap value as a series of FRAs....

--sridhar

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Sridhar,

This gives difficulty to everybody rest assured. First, please note, you are correct: it can be done your way and the result will be the same, there is nothing wrong with your logic to treat the floating leg as a series of flows parallel to the fixed. Hull's method is a "shortcut" that relies on the following fact: at the next coupon (in +3 months, but also at EVERY coupon), the fair value of the floating bond must be its par value. Therefore, in three months, a future value of $100 automatically impounds (i.e., already includes, captures, is the same as) the subsequent series of coupons and the final notional. So, the key to understanding this, IMO, is to see that a floating bond is always worth its par at the instant it pays a coupon. If it helps, below the EditGrid/XLS that performs Hulls calcs (at http://www.bionicturtle.com/premium/editgrid/2008_frm_hull_derivatives_swaps/), see row 25, I show the "proof" of this idea (why did i do this? b/c it's not a natural idea for me!). See how you can change the LIBOR coupons and it doesn't matter, because they get discounted back at the same rate, the PV of the bond will always be$100? If you can agree with this idea (the floater is always worth par at coupon payment time), you can probably see that, in regard to 7.2/7.5, there is only one cash flow needed which has two parts: the coupon plus the $100 and the$100 "contains within" the entire subsequent cash flow series. Let me know if that does not answer both questions?

David

#### sridhar

##### New Member
Thanks David...I will need to study the EditGrid computations to finally "get it." I've agonized on this a lot.

BTW, do old FRM questions ask about valuing swaps? In which case, it seems like the "bond approach" would be faster (during the test) than the FRA approach. Because we can avoid having to compute the fwd interest rate and then further transform that to the semi-annual compounding etc.

I assume to the extent the FRA approach would be tested, the question might ask you about some interim result that can only be arrived at by valuing the swap via the FRA method -- is this correct?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Sridhar,

Yes, the FRM has historically asked about the bond approach valuation. However, it is difficult to predict this year. I noticed the AIMs under swaps have been quite expanded from previous years: last year only a single calculate interest rate swap. This year, there are FOUR AIMS that ask to "value swap..." in BOTH ways (bond, FRA) for interest rate and currency swap. So, all i can say is from the AIMs, they have asked for more understanding (!?). But, yes I agree with you the bond approach is "easier." At the same time, i perceive the cirriculum wants you to be able to deal with/extract forwards so it is good to know that additionally...David

#### spencerlsh

##### New Member
Hi david,

i was looking at the editgrid for this example, i still don't quite understand why do you use the 6 mth LIBOR of 5.5% where you used (0.5*5.5*notional+notional) instead of the 3 month LIBOR of 5.0% in calculating the FV of the floater?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
hi spencer,

It helps to keep in mind we have 2 steps: 1. compute the FV cash flows, and then 2. discount to PV. For the discounting, as the flow is three months away, we use the 3 month LIBOR to discount.

But to get the FV flow, that is the "coupon paid" on the floating SWAP: its rate is based on a 6 month LIBOR. That rate was determined at T-3 months and to be paid at T+3 months. All of the floating coupons will be based on 6 month LIBOR, paid "in arrears:" the 6 month LIBOR at T0 to be paid at T+6 months. Hope that helps...David

#### ashutoshg

##### New Member
Sridhar,

This gives difficulty to everybody rest assured. First, please note, you are correct: it can be done your way and the result will be the same, there is nothing wrong with your logic to treat the floating leg as a series of flows parallel to the fixed. Hull's method is a "shortcut" that relies on the following fact: at the next coupon (in +3 months, but also at EVERY coupon), the fair value of the floating bond must be its par value. Therefore, in three months, a future value of $100 automatically impounds (i.e., already includes, captures, is the same as) the subsequent series of coupons and the final notional. So, the key to understanding this, IMO, is to see that a floating bond is always worth its par at the instant it pays a coupon. If it helps, below the EditGrid/XLS that performs Hulls calcs (at http://www.bionicturtle.com/premium/editgrid/2008_frm_hull_derivatives_swaps/), see row 25, I show the "proof" of this idea (why did i do this? b/c it's not a natural idea for me!). See how you can change the LIBOR coupons and it doesn't matter, because they get discounted back at the same rate, the PV of the bond will always be$100? If you can agree with this idea (the floater is always worth par at coupon payment time), you can probably see that, in regard to 7.2/7.5, there is only one cash flow needed which has two parts: the coupon plus the $100 and the$100 "contains within" the entire subsequent cash flow series. Let me know if that does not answer both questions?

David

I am still confused. Let us say at t=0 value of the floating rate bond to be calculated to arrive at the swap value.Floating rate bonds has three 6 months coupon period 6,12 &18 months.
As per above mentioned discussion value of the floating bond t=0, t=6, t=12,t=18 months is 100. Am i correct or missing something? If it is so then why are we not taking cash flow into account to arrive at swap value?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@ashutoshg Yes, you are correct that the floating rate bond (with six month payment frequency) will price to par (assuming the discount rate is the same as swap's floating rate, an assumption Hull moves away from in the current edition, btw) at t = 0, T = 0.5 years, t = 1.0 years, etc. If we are pricing such a swap on any of these dates, exactly, then we can use 100 and subtract it from the value of the fixed-rate leg. But the question referred to above, which is typical, asks a question about settling a swap at some realistic point in time between payments; e.g., if the swap matures in 15 months, then we must be 3 months after the last coupon and 3 months until the next coupon. Such a floater does not price to par at t = 15 months = 1.25 years.