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# LIBOR, day count convention and compunding frequency

#### 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

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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

#### 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

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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

#### wrongsaidfred

##### Member
Thanks for the help.

Much appreciated.

Mike

#### 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...

#### Tejas

##### New Member
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

Hi sir, bit confused here on calculations. Why ln is used in this calculation or the reason on using ln

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Tejas I don't understand what you are asking exactly, sorry. In the above, I was explaining how the compound frequency informs the exact rate. In the above, I translate discrete (quarterly compounding) to continuous compounding.

Here is a recent video of mine specifically on the topic of compound frequency (it is part of a longer playlist). This video explains what I was doing above.

Here is a recent video on day count basis

And two days ago I published a video that happens to start with the impact of compound frequency on discount factors (so this is sort of a practical application). Let me know if you have a specific question I can help with, thanks!

#### Tejas

##### New Member
I have questions to solve

1) A Eurodollar future price quote is 98. What is the implied (converted) interest rate per annum with continuous compounding and under an actual/365 day count convention

2) if French money market instrument pays in Euros with a interest rate of 5% pa with (discrete) annual compounding and under an actual/360 day count (act/360) convention, what is the equivalent rate under continues compounding under an actual/360 day count?

3) I am confused, for continuous compounding e^rt is used then why on above calculations ln (1+r/t) is used

#### Nicole Seaman

Staff member
Subscriber
I have questions to solve

1) A Eurodollar future price quote is 98. What is the implied (converted) interest rate per annum with continuous compounding and under an actual/365 day count convention

2) if French money market instrument pays in Euros with a interest rate of 5% pa with (discrete) annual compounding and under an actual/360 day count (act/360) convention, what is the equivalent rate under continues compounding under an actual/360 day count?

3) I am confused, for continuous compounding e^rt is used then why on above calculations ln (1+r/t) is used
Hello @Tejas

I wanted to make sure that you are aware of our search and tag functions in the forum. The questions that you've posted do not look like our practice questions, but you will find that the actual concepts on how to solve the questions have most likely been discussed in the forum already (there is over a decade of FRM discussions in the forum). Our Search function is HERE and our Tag function is HERE. Possible search and tag words that you could try are:
• eurodollar
• eurodollar future price
• continuous compounding
• day count convention
• day count
• implied interest rate
• annual compounding
It is very helpful if you search the forum first before posting a question. First, it helps you to find an answer more quickly without having to wait for someone to answer. Second, it prevents David from having to answer a question duplicate times if it has already been discussed elsewhere in the forum. Let me know if you have any questions about this. You can also find helpful information here regarding the use of the forum: https://www.bionicturtle.com/forum/threads/how-do-i-best-utilize-the-forum.13390/. I hope this helps!

Nicole