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
     
  6. felixx

    felixx New Member

    "[...] while 6.0% LIBOR generally quotes in actual/360 day count [...] this 6.0% per annum does not tell us which compound frequency and allows for continuous or discrete."

    When you have a spot rate with a specific time horizon the corresponding discount factor must be unique. That means that the definition of the rate requires not only a number, but also a specific choice for compounding and day-count convention.

    I understand that LIBOR are quoted as simple rates. If you require the continuously compounded rate you must convert the quoted one into the continuously compounded one.

    Use always the equations resulting from the equivalence of discount factors and you will get it right. For example, using the inverse of the dicount factors for the simple rate (r_s) and the continuous one (r_c):

    1 + r_s * T = exp(r_c * T)

    and so on...
     
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