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Swap valuation, why the B(floating) is not calculated the same way of B(fixed)?

Steve Jobs

Active Member
I understand that: V=B(fixed) - B(floating) or the reverse.
Also assume the payments are at 6,12,18 months

I find the calculation of B(fixed) and intuitive and logical:

B(fixed): First Payment + Second Payment + Third Payment
First Payment = PV of [PMT(fixed) at end of month 6]
Second Payment = PV of [PMT(fixed) at end of 12 month]
Third Payment = PV of [PMT(fixed) and Notional at the end of 18 month]

For B(floating), the only difference I see is the PMT amount which varies depending on Libor rate, so I will adjust the same B(fixed) formula by using forward rates to:

B(floating): First Payment + Second Payment + Third Payment
First Payment = PV of [Spot libor * Notional]
Second Payment = PV of [6 month forward libor at the end of 6 month* Notional]
Third Payment = PV of [6 month forward libor at the end of 12 month* Notional]

But the solutions provided for the practice questions uses this formula which I don't understand at all:
B(floating) = Notional + (Notional *( Libor at last payment date / 2)) * e^(spot libor * 6/12)

So my question is whether my adjusted B(floating) formula is correct or not?, and what is the logic behind the provided formula of the B(floating) in the provided solutions?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Steve,

Your detailed approach (valuing the floating cash flows) should work and match the shorter solution. If the first/next coupon pays in 0.5 years, I think your shorter should be:
(Notional + Notional*LIBOR[last]/2) *exp*( -LIBOR[spot]*0.5), where LIBOR[last] = LIBOR[spot] if swap initiates today.

The logic is that a floating-rate bond prices to par on settlement date, such that this Notional is an accurate forward price in six months of the floating-rate bond so it incorporates (is equal to) the future floating coupons (because the forward rate = expected coupon rate = discount rate).

Steve Jobs

Active Member
Thanks David,

I just read the FRA method which is is exactly what I'm talking about.

I see the only difference between the Bonds method and FRA method in the calculating of the B(Floating).
Moreover, I see the Bonds method in calculating B(floating) as the mathematically proven shortcut/summary of the FRA method and introducing them as two different methods to be misleading.
So for me they are both the same!

So practically if the multiple choice question is asking for B(floating) in FRA method, it's better to use the Bonds method to get the answer quickly!

I hope I'm not missing something...

David Harper CFA FRM

David Harper CFA FRM
Staff member
I don't think you are missing anything. The source is Hull and he's always presented them as two approaches that give the same solution; my swap XLS shows both approaches under discrete and thusly demonstrates the equivalence http://www.bionicturtle.com/how-to/spreadsheet/p1.t3.c-xls-bundle/

I agree also that it's prob a better exam strategy to use the two bonds approach; e.g., the most common question is to ask about the value (exposure) on a settlement date, such that under the bond approach all you maybe need to do is value the swap as one fixed bond and subtract 100 par for the floater (for a likely exam question, the 2 bonds approach is much more efficient)

... but after interacting some years with both approaches, i've grown to appreciate the FRA method b/c, in my opinion, it's more robust to user error. The two bonds approach is quicker but creates possibilities for error (and additional interpretation) with respect to timing and notional whereas, on the other hand, although it feels like more steps, the FRA method is a more natural and "realistic" model: you predict the cash flows for each leg, using forward rates as predictors, then net them out (as they would actually net), then discount. Same difference, but something about mimicing the actual flows seems to give less chance for pilot error, fwiw

Steve Jobs

Active Member
I agree and I think it comes down to how statistics and math is being used in finance. They say that it's enough to know how to use the formulas but I disagree because knowing the formula does not mean that the student understood the subject/system/the way it actually works/the reason behind it.

Thanks David, I will practice more question tomorrow to both visualize the cash flows series using FRA concept and solving the questions quickly using Bonds approach.