Continuous or discrete compounding for PVs?

vipin172000

New Member
Hi David,

I am a little lost when it comes to pricing and valuation of swaps.
Consider this specific example on slide 63 of 3.a.iv:
Here's what I understood:

1) You take the Spot LIBOR curve
2) You convert the Spot rates into (implied) forward rates
3) Why do you then convert these forward rates into semi annual rates??

Is this because the cash flows are made semi annually?
If that's the case , should we ALWAYS use semi annual compounding eventually?

Also , on the same lines , on slide 62 of the same presentation, why do you then use the discount factors using continuous discounting instead of using semi annual ?

Hope you can clarify this.

Cheers
VM
 

David Harper CFA FRM

David Harper CFA FRM
Subscriber
Hi VM,

The short answer is, since I'm not original, I am just following Hull's example. Hull's default frequency is continuous.
I just added a third sheet to the interest rate swap valuation;
see thirds sheet of http://www.bionicturtle.com/premium/spreadsheet/3.a.11_hull_swaps/

I know the XLS aren't for everybody, but i think its the best way to comprehend the i rate swap valuation...
...if you look at the third sheet, I performed an "alternative" valuation under semi-annual compounding throughout. I think this demonstrates that continuous never *needs* to enter into the exercise (and therefore, neither the translations)...the DFs are semi-annual and it "works" (same answer for both approaches) if applied consistently

...similarly, the slide 62 (which is valuation based on bonds) *clearly* does not need to use continuous; if this were in Bruce Tuckman (our FI assignment), he would use semi-annual instead (neither is "incorrect"). Hull uses continuous (in typically academic fashion) and Tuckman uses semi-annual (which maybe is more practical?). Which does the FRM assume? I keep asking them to "declare a major" in this regard...no success yet...in the meantime, you have to be fluent in the conversion and trust the question will tell you. Although: for bonds, you can assume semi-annual (i.e., Tuckman).

in regard to p 63, that is valuing the swap as a sequence of FRAs and, in this regard (as oppposed to p 62, valuing as two bonds), I do think what you say is true: "Is this because the cash flows are made semi annually?" What i mean is, i tried to add another sheet that performs the whole thing in continuous (to complement the "all semiannual") but i cannot find that solution (in < 10 min, anyway!) ... in order to derive the forward cash flow (which have 6 month tenors), the model seems to insist (in order to reconcile anyway) that the forward cash flows fall out of semi-annual rates.

...sorry for length. My basic answer is, in regard to PV, there is no right/wrong in regard to continuous/discrete, rather the point is to be consistent (e.g., as the XLS shows, we can do everything in semiannual and not stray). Hope that helps, David
 
Hi David,

Agree that there is no right or wrong in using the continuous or discrete discounting functions. If the question has made it clear that the LIBOR are at continuous rate. I think it should be stick to continuous discount rate to be consistent. In terms of MTM in market practice on P&L reporting, the different in MTM result is important. I am not sure in market norm will it assume the LIBOR is at continuous rate, what will be the default setting in system for MTM purpose?

Regards,
Daniel
 

David Harper CFA FRM

David Harper CFA FRM
Subscriber
Hi Daniel,

Right, I see your point. I can't claim to know the prevalent practice. There are two compound frequency issues in this swap valuation:

1. The swap rates; where my replication of Hull follows semi-annual; e.g., pay LIBOR @ 8% semi-annual, and
2. The compound frequency used to get the discount factor, where I mimic Hull with continuous discounting
(the fact that Hull is not continuous throughout is why the valuation in terms of FRA valuation causes so much confusion: he has to convert. Whereas if the definitions were setup differently, he could perform the valuation in either continuous/discrete throughout, without switching ... but he likely does this because the discrete assumption in (1) is more realistic.)

So, the Hull uses semi-annual discrete to determine the swap cash flows and continuous to discount them in order to perform the M2M swap valuation.

In regard to (1), I do perceive the use of a nominal rate (nominal implies discrete; e.g., 8% per annum paid quarterly), as Hull does above, as fairly typical. See http://www.bbalibor.com/bba/jsp/polopoly.jsp?d=1633 ... that's a nominal LIBOR with discrete compounding

In regard to (2), the discount factor, I've seen both ways. Many use annual discrete; Hull uses continuous.

The third page to the XLS got lost (i will make a note to restore that) but that is just where i performed the same valuation with semi-annual throughout; it avoids the conversion...

But, apologies for length, my summary view is:
1. The swap rate cannot be subjective; the fixed payer rate is contractual, same with floating rate ... in other words, a good question must define!
2. Just the opposite, subjective as this is a PV exercise. I can't see how continuous/discrete is right/wrong, only that (of course) consistency is required and the best test of that is to check that the two methods (2 bonds vs. FRA) produce the same value; although i totally agree the choice of continuous/discrete does impact the final answer.

David
 
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