LIBOR, day count convention and compunding frequency

Discussion in 'P1.T3. Financial Markets & Products (30%)' started by wrongsaidfred, Sep 13, 2011.

  1. wrongsaidfred

    wrongsaidfred Member

    Hi David,

    In your notes, you say that LIBOR is quoted on an actual/360 basis. But when using the LIBOR rate as a proxy for the spot rate it is continuously compounding. Doesn't actual/360 imply simple interest (no compunding)? I just do not see how these two methodologies are compatible.

    Any explanation would be greatly appreciated.

    Thanks,
    Mike
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  2. Hi Mike,

    It's a great point, they seem connected, but it's two different things. If we look at the way Hull tends to (with precision) characterize rates, it looks as follows (eg):

    "4.0% per annum with [continuous | annual | quarterly | etc ] compounding"

    The day count convention (or day count basis) is not here specified but, in a way, resides "within" the "4.0% per annum" and is SEPARATE from the compound frequency.

    Consider Hull's instructive example 6.3, where he adjusts a Eurodollar futures rate into its equivalent forward rate.
    He starts with a Eurodollar quote = 94 which, b/c it's a 90-day money market instrument, refers to an interest rate that is: 6.0% per annum (i) on an actual/360 day count basis with (ii) quarterly compounding

    As he needs to subtract a convexity adjustment that just happens to be expressed in "actual/365 with continuous compounding." So he does this:
    =365/90 * LN(1+6%/4) = 6.03816%; converting 6% continuous to quarterly, a calculation which has tended to give confusion

    It can be unpacked, to illustrate there are two aspects:
    = 4*LN(1+6%/4) = 5.9554%; i.e. convert a quarterly to continuous compound frequency
    = 5.9554% * 365/360 = translate an actual/360 (LIBOR) to actual/365 day count so the subtraction is "apples to apples"
    That is three LIBOR rates, all valid

    Similarly, while 6.0% LIBOR generally quotes in actual/360 day count (http://en.wikipedia.org/wiki/London_Interbank_Offered_Rate), this 6.0% per annum does not tell us which compound frequency and allows for continuous or discrete. Generally, we take guidance from the instrument: a semi-annual bond implies semi-annual; a 90-day ED futures implies quarterly; but the implied frequencies don't stop us from over-riding with a continuous

    I hope that helps, David
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  3. wrongsaidfred

    wrongsaidfred Member

    Hi David,

    Thanks for you response.

    So when LIROB is quoted at 6%, it must be specified as either being continuously compunded or quarterly compounded? Also, what exactly do you mean by "the implied frequencies don’t stop us from over-riding with a continuous"?

    Thank you,
    Mike
  4. Hi Mike,

    Yes, correct, this is why (per our other thread) you might notice that I request GARP to utilize a format (following Hull) with this convention:
    "Interest rate of 6.0% per annum with [continuous | annual | etc ] compounding"

    Now, please note, it is a bit different to say:
    * LIBOR is 6.0% per annum; this is insufficient with respect to compounding, not enough information, VERSUS
    * Eurodollar quote of 94. This implies 6.0% LIBOR, too but with an important difference. We can know the ED contract is a 90-day instrument, so technically, this does not need the clarification (see http://www.cmegroup.com/trading/interest-rates/stir/eurodollar_contract_specifications.html).
    * Similarly, if you see an interest rate swap with 6 month payments, or a bond with semi-annual coupons, the "6% per annum" tends to omit the periodicity b/c we can infer from the instrument

    But, DON'T SWEAT the specifics of the instruments, just trying to show you the 6% LIBOR can be either (your question). GARP's questions will be specific, as you saw their reply. It is not a good use of time to try to memorize (eg.) that a ED contract is 90-days (IMO).

    Re: "implied frequencies don’t stop us from over-riding with a continuous”?" Sorry, it is not really helpful. I just meant that, like Hull does, if swap paying every 6 months (floating) LIBOR, the LIBOR can be translated from semi-annual to continuous (in fact, we do that in the IRS valuation model). I meant really nothing more than LIBOR @ 6% can be variously expressed discrete/continuous.

    Thanks, David
  5. wrongsaidfred

    wrongsaidfred Member

    Thanks for the help.

    Much appreciated.

    Mike

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