Day Count Basis & Interest Rates

[trabala38]

Hello David,

I was doing the Hull 06.08 exercice (Cf, Hull Chapters 2 -10PDF, p39) and I got some problems.

"The price of a 90-day Treasury bill is quoted at 10."

I am able to compute the cash price (Y=$97,50) and to get the interest earned over a 90 day period (Int=$2,5). However, I am bit lost when it comes to compute the annualized return (true yield) with the 3 different assumptions:

1) quaterly compounding and actual/360 day count basis
2) quaterly compounding and actual/365 day count basis
3) continuous compounding and actual/365 day count basis

First, I do compute the true yield using this method : true yield = compound frequency * (interest per period/cash price) = 4 * 2,50/97,50= 10,2564%. Am I correct, or is it an incorrect methodology ?

But then, I am lost because I don't know under which day count basis my computation is... I tried to understand your worksheet but, I am confused, it seems that I don't get something...(cf. https://public.sheet.zoho.com/public/btzoho/hull-06-08).

Same for for question Hull 06.01 (cf worksheet https://public.sheet.zoho.com/public/btzoho/hull-06-01), I see that there is a correction (correction factor = 365/360) but I don't know when to apply it (at the beginning of the computations for some, at the end of the computations for the others).

Could you explain your methodology and your "tips" for that kind of problem ?

Thanks a lot !

Regards,

trabala38

 

[trabala38]

Hello David,

I faced the same kind of issue for Hull 06.13 exercice and I think the problem can be be summed up that way :

In Hull 06.13, the 3 months Forward rate = 9% with continuous compounding with actual/365 day count basis (it is Hull's assumption).

Then, we need to convert the continuous rate into a "quaterly compounding" interest rate with actual/360 day count basis (since we want to obtain an Eurodollar future quote which is a Money Market instrument and hence, rates should be given with an actual/360 day count basis).

According to me, there are 2 ways to convert the the continuous rate into a "quaterly compounding" with actual/360 day count basis.

a) Apply the correction factor for actual/360 directly on the continuous rate and then convert to a quaterly compounding rate:
i (quaterly & actual/360) = 4 * ( EXP(9% * (360/365)/4) - 1 ) = 8,9759%

b) Convert to a quaterly compounding rate and then apply the correction factor for actual/360:
i (quaterly & actual/360) = 4 * (EXP(9%/4) - 1) * 360/365 = 8,9773%

Methodology b) is used in your computations (cf. sheet https://public.sheet.zoho.com/public/btzoho/hull-06-13).
By correction for actual/360, I mean a=360/365.

As you can see, both methodologies leads to slightly different results. Is there a way to solve that kind of problem with consistency ?

Any help/insight would be greatly appreciated !

Thanks !

Regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala,

Great points, I frankly had not considered the inconsistency (I am following Hull .... and I appear to have followed his apparent inconsistency.)

I agree Hull 6.13 illustrates the "problem;" but I can't seem to declare one of them wrong. However, I would further agree with you that (a) is the more natural; although Hull gives (b) as the solution. The reason is I find (a) more natural is that, if we go back to the identity on p 77, it seems to me this identity is reliable:

exp(rate_continuous) = (1 + rate_discrete_m / m)^m

... it seems to me the most defensible approach is to match the (m) with the compound frequency, in the case quarterly = 4. Your approach (a), by converting to a continuous rate first, seems to me to best "honor" this equality. And, elsewhere, we know Hull supports the methodology of approach (a); Hull example 6.4.

For example, if we take the answer given in your approach (a) and follow Hull's Example 6.4 (convexity adjustment), we would get back to the 9%:

if quarterly, a/360 follows approach (a) and is equal to 8.9759%, then
continuous, a/365 = 365/90 * LN(1+8.9759%/4) = 9%, which is the same as:
continuous, a/365 = 365/360 * [4 * LN(1+8.9759%/4)] = 9.0%

alternatively, approach (b) implies:
if quarterly, a/360 = r(q, a/360) = 4 * (EXP[9%(CC,a365)/4]-1) * 360/365, then:
r(q, a/360) /4 * 365/360 = EXP[9%(CC,a365)/4]-1, and:
1+ r(q, a/360) /4 * 365/360 = EXP[9%(CC,a365)/4], and:
4* LN[1+ r(q, a/360) /4 * 365/360] =9%(CC,a365), so this is:
rate_continuous = m*LN(1 + rate_discrete/m * 365/360)
... if i had to choose, I do not prefer this method!

But I admit, I am trying to reconcile them and I can't seem to find the solution. Nor can I decide exactly why (b) offends me. In solutions, Hull obviously (IMO) uses both; and I admit i've followed him. But I can't find (a) to be conclusively superior, even as, to your whole point, it seems like there should be a single answer for (eg) quarterly compounding under an act/365 day count.

Sorry i can't immediately give an answer, I will definitely be noodling on this challenge, David

 

[trabala38]

Thanks a lot for your answer, David !

Even if the question remains open, knowing that the answer is not straightforward makes me feel more comfortable.

I also agree with you : I prefer option a). But I have another argument: The formula exp(rate_continuous) = (1 + rate_discrete_m / m)^m works under the 30/360 day count convention ? Why ? Because this is the only way that makes a quaterly compound factor strictly equal to 4 (m = 360/90 =4). So, for me, the best way is to be sure that we convert any interest rate to a/360 basis (or more precisely a 30/360 basis) before "injecting" this rate in the above formula.

Regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala38,

Right, that's consistent with what i was thinking might be implied by Hull 6.4 where he converts LIBOR 6% (Eurollar quote of 94):
ACT/360 quarterly 6% LIBOR --> 365/90*LN(1+6.0%/4) = 6.0382%; this is the same as:
(365/360)*4*LN(1+6.0%/4) = 6.0382%; i.e., i think consistent with your point, the 6% ACT/360 quarterly is first converted to continuous, then the 5.9554% ACT/360 continuous is converted to ACT/365 continuous .... hmmm...

Thanks, definitely the furthest I've understand this, really appreciate your help! David

 

[Lori]

what is the day count for a Eurodollar futures contract? Thanks

 

[Alex_1]

ACT/360
Google is your friend.
--> http://www.cmegroup.com/trading/interest-rates/files/eurodollar-futures-reference-guide.pdf

 

[bpdulog]

Hi trabala,

Great points, I frankly had not considered the inconsistency (I am following Hull .... and I appear to have followed his apparent inconsistency.)

I agree Hull 6.13 illustrates the "problem;" but I can't seem to declare one of them wrong. However, I would further agree with you that (a) is the more natural; although Hull gives (b) as the solution. The reason is I find (a) more natural is that, if we go back to the identity on p 77, it seems to me this identity is reliable:

exp(rate_continuous) = (1 + rate_discrete_m / m)^m

... it seems to me the most defensible approach is to match the (m) with the compound frequency, in the case quarterly = 4. Your approach (a), by converting to a continuous rate first, seems to me to best "honor" this equality. And, elsewhere, we know Hull supports the methodology of approach (a); Hull example 6.4.

For example, if we take the answer given in your approach (a) and follow Hull's Example 6.4 (convexity adjustment), we would get back to the 9%:

if quarterly, a/360 follows approach (a) and is equal to 8.9759%, then
continuous, a/365 = 365/90 * LN(1+8.9759%/4) = 9%, which is the same as:
continuous, a/365 = 365/360 * [4 * LN(1+8.9759%/4)] = 9.0%

alternatively, approach (b) implies:
if quarterly, a/360 = r(q, a/360) = 4 * (EXP[9%(CC,a365)/4]-1) * 360/365, then:
r(q, a/360) /4 * 365/360 = EXP[9%(CC,a365)/4]-1, and:
1+ r(q, a/360) /4 * 365/360 = EXP[9%(CC,a365)/4], and:
4* LN[1+ r(q, a/360) /4 * 365/360] =9%(CC,a365), so this is:
rate_continuous = m*LN(1 + rate_discrete/m * 365/360)
... if i had to choose, I do not prefer this method!

But I admit, I am trying to reconcile them and I can't seem to find the solution. Nor can I decide exactly why (b) offends me. In solutions, Hull obviously (IMO) uses both; and I admit i've followed him. But I can't find (a) to be conclusively superior, even as, to your whole point, it seems like there should be a single answer for (eg) quarterly compounding under an act/365 day count.

Sorry i can't immediately give an answer, I will definitely be noodling on this challenge, David

I tried searching the FRM Official Guide for Hull 6.8 and 6.13 but didn't find anything, 6.3 refers to a convexity adjustment. I'm having reconciling the e^rc=(1+Rm/m)^m formula and the a and be approaches noted below:

Approach A: continuous, a/365 = 365/90 * LN(1+8.9759%/4) = 9%, which is the same as:
Approach B: continuous, a/365 = 365/360 * [4 * LN(1+8.9759%/4)] = 9.0%


Is this no longer covered in the curriculum?

 

[David Harper CFA FRM]

Hi @bpdulog

This thread concerns the conversion of interest rates from discrete to continuous, which is basic and highly testable and need-to-know; but this has nothing to do with convexity adjustment (aka, convexity bias) due to the difference between a an exchange-traded rate (eg, FRA) and an interest rate forward contract, which is advanced, slightly testable but frankly not need-to-know, IMO. Here are the the relevant LOs:

Chapter 4. Interest Rates [FMP–4] After completing this reading you should be able to:
• Calculate the value of an investment using different compounding frequencies.
• Convert interest rates based on different compounding frequencies <<- Must know

Chapter 6. Interest Rate Futures [FMP–6]
After completing this reading you should be able to:
• Identify the most commonly used day count conventions, describe the markets that each one is typically used in, and apply each to an interest calculation. <-- Must know
...
• Describe and compute the Eurodollar futures contract convexity adjustment." <-- Nice to know, but low testability IMO

 

[nghiantt]

Hi David,

In Bionic Turtle FRM Practice questions - Reading 9, page 194, Hull 6.1, I can not open the spreadsheet for calculations of 6.1d and 6.1e. So i cannot understand why your calculation results in different to mine.

For 6.1d, my method is convert 8% actual/365, annum to actual/365, quarterly compounding of 7,771% (1+ 8%= (1+x/4)^4
Then convert 7.771% to Actual/360: 360/365 * 7.771% = 7,664%.

Please help to correct me. I cannot find answer or suggestion anywhere else! Thanks in advance!

 

[David Harper CFA FRM]

Hi @nghiantt XLS is here at the source https://forum.bionicturtle.com/threads/hull-06-01.3767/
what's cool is the order doesn't matter, you can convert day count/compound frequency first. But continuous R_c to quarterly R_q is given by R_q = [e^(R_c/4)-1]*4. Thanks!