Computing the floating-rate cash flow for valuing a IR swap

[sridhar]

David,

This is somewhat of a "meta" issue that is bothering me. I refer to your Mkt Risk Study Notes (pages 39-40) where you are showing us how to value a IR swap using the bond methodology. Here's what's troubling me:

1. You are showing us (in the spreadsheet on page 40) the 3 fixed-rate flows appropriately discounted to pv. For the fixed-rate payments that are made at 3, 9 and 15 months. No problem here.

2. But you only compute one floating-rate cash stream using (L + k*)exp(-r1*t1).....I understand the mechanics of this -- but I don't get the intuition behind this...See point 3...

3. Every 6 months, the floating-rate payer receives the fixed-rate coupon and in this example, you show the three fixed cashflows, he does receive. Symmetrically, I expected to see three floating cashflows corresponding to the floating payments to be made to the fixed-rate receiver. Using LIBOR rate prevailing at 3, 6 and 15 months. Instead, we shortcut the floating-rate cashflow from 3 to just 1 using the formula in step 2. Why just one floating-cashflow instead of 3 -- this is what I don't get.

--sridhar

 

[David Harper CFA FRM]

Hi srihdar,

Absolutely it is a fair meta issue. I do speak to it in the screencast. But IMO is a difficult feature of the swap valuation.

Please note: your intuition does not fail here. It could be done your way, it just produces the same result. The way i show, which of course is "stolen" from Hull is merely a shortcut.

The key insight is that at the *instant* of payment on the first floating coupon, imagine you just receive the first floating coupon, you must be holding a floating bond with price equal to par (PV = 100). The $100 bond in three months is priced to include the future floating cash flows. Why? For a similar reason that a bond's price must equal par if it's coupon = yield (YTM): the (floating) discount rate will match the floating coupon. At that instant (3 months forward), you would pay $100 for that floating bond and that price would include the floating coupons.

If you are not convinced, you can look *below* the analysis in the spreadsheet @ http://www.bionicturtle.com/premium/editgrid/2008_frm_hull_derivatives_swaps/

Below the "regular" analysis, I show the proof (frankly b/c even after all these years, this point is not quite intuitive for me). You can change the LIBOR rates in cells E30 etc and the PV will always be $100.

hope that helps...David

 

[sridhar]

Got it! Lucid explanation as usual....Thanks!

 

[kgolf20]

Hello, I have a similar question regarding this same example about computing the floating rate flows. Why is it that the 6 month LIBOR rate is used to determine the floating leg receive flow that happens in 3 months? The coupon payment that is used is 2.75, half of the 5.5% LIBOR at 6 months. Why would we not use half of the 3 month LIBOR rate for the coupon. I seem to understand everything except this detail. Let me know if I am not being clear enough.

Thanks in advance.

 

[David Harper CFA FRM]

Hi kgolf20,

Your question is clear, it is commonly found to be difficult. Though contractual terms may vary, this example from Hull is typical: the cash flows are swapped every six months, we are valuing the swap in-between cash flows. You can remove us (as conducting the valuation) from the picture. We just happen to be three months away from swaps but it is irrelevant to the counterparties.

For the counterparties, the floating rate is determined at the begging of the swap period and that interest is paid at the end of the swap period. So, for the upcoming swap, based on our vantage point, that interest rate was determined -3 months for the payment in +3 months. The rates in this swap will always be 6 month rates, determined at start of each leg, paid at end of each leg. This speaks to the future cash flows; the discounting of these cash flows at 3 months, 12 months, etc is a separate step.

From another angle, we could move forward in time one month to conduct the valuation, to +2 months until the swap coupon. That will not change either float/fixed coupon - both are already known (6 month rate for the floater) and not based on our mid-segment timing

Hope that clarifies..

David

 

[mikey10011]

I've been working through Hull and is it possible that the 6-month LIBOR rate of 5.5% [cell D7 in EditGrid] is "conceptually" not the right one to use for calculating the floating bond's cash flow?

The 6-month LIBOR rate in your term structure of interest rates schedule [D7] is a measure of the market's expectations of future interest rates given the *current* market conditions.

For calculating the upcoming floating bond's payment three months from now, don't we want the *historical* 6-month LIBOR three months back when the last payment was made? Using Hull's words, "the six-month LIBOR rate at the last payment date was [5.5%] (with semiannual compounding)." Note that in setting up the example Hull introduces the yield curve [D610] in the sentence before.

 

[David Harper CFA FRM]

Mikey,

Yes, absolutely it is conceptually incorrect. Hull (and my XLS which replicates) assumes a "static yield curve." It would be more realistic to give the "6-month LIBOR three months back when the last payment was made" or to give TWO yield curves (- 3 months and today) and then use only the current curve to discount cash flows.

So, it is just a simplyifying convenience. Because you are absolutely correct for the typical i rate swap the payments are in arrears: at time 0, the floating rate was determined at t-3 months and will be paid at t+3 months (although as an OTC private transaction, the coupons don't *need* to swap in arrears)

David

 

[David Harper CFA FRM]

Mickey,

I am looking at Hull 6th edition. I see he has updated his language; previously, he had said "the yield curve is static" or unchanged over the last three months and I see now he has been more precise and parsed out the historical 6 mo LIBOR (note how it fits nicely into the current curve). So, you continue to be correct; I'll annotate the 6 month LIBOR rate on the XLS to indicate it is doing double-duty per the simplifying assumption.

David

 

[pathak]

Hi David,
I also tried to calculate the SWAP value by discounting all the three floating rate cash flow with same LIBOR but the outcome is different from the result we are getting by discounting single cash flow with notional. Can u please show me the detailed calculation & comparison between the two?? or we need to take forward rate to discount the floating Cash Flow??? Please help.

 

[David Harper CFA FRM]

Hi pathak,

I just inserted rows 17 to 21 into our swap XLS, see https://www.dropbox.com/s/j3zvu7847k7b7cv/0727_IRS_float_par.xlsx
(I will incorporate this calculation more formally into the next IRS XLS revision, i think it would be nice to see it ....)

Cell F21 calculates the then-present value at time +0.25 of the two subsequent floating cash flow. You'll see, even if you vary the LIBOR curve and therefore the implied future floating cash flows, the PV [at the next coupon + 0.25; not at T0] will sum to par of 100.00 regardless.

In Hull's displayed scenario:

• future floating cash flow [T + 0.75] = $5.52, with PV = 5.52*exp(-10.75%*0.5) = $5.23
• future floating cash flow [T + 1.25] = $106.72, with PV = 106.72*exp(-10.75%*0.5 + -13%.00*0.5) = $94.77; i.e., discounting with both implied forward rates

The xls should help clarify, thanks,

 

[pathak]

Hi pathak,

I just inserted rows 17 to 21 into our swap XLS, see https://www.dropbox.com/s/j3zvu7847k7b7cv/0727_IRS_float_par.xlsx
(I will incorporate this calculation more formally into the next IRS XLS revision, i think it would be nice to see it ....)

Cell F21 calculates the then-present value at time +0.25 of the two subsequent floating cash flow. You'll see, even if you vary the LIBOR curve and therefore the implied future floating cash flows, the PV [at the next coupon + 0.25; not at T0] will sum to par of 100.00 regardless.

In Hull's displayed scenario:

• future floating cash flow [T + 0.75] = $5.52, with PV = 5.52*exp(-10.75%*0.5) = $5.23
• future floating cash flow [T + 1.25] = $106.72, with PV = 106.72*exp(-10.75%*0.5 + -13%.00*0.5) = $94.77; i.e., discounting with both implied forward rates

The xls should help clarify, thanks,

Thanks David, its realy very helpful & very lucid explanation. Thanks a Lot.

Regards