Effective Annual Rate (EAR) and interest rate

[Liming]

Dear David,

I hope you can render your kind help on my following questions on EAR.

In example 1.1 on page 5 of FRM handbook (5th edition), the question asks for Effective Annual rate for a T-Bill maturing in 1 month that investors bought for $987. On the maturity date, investors collects $1000. According to the answer provided, the formula used for EAR is such that: FV/PV = (1+EAR) raised to the power of T . My question is why don't we
1) either assuming monthly compounding and use formula:
FV/PV = 1+(EAR/12) since there is only 1 month left
2) or assuming continuously compounding and use formula:
FV/PV = exp(EAR/12)

What's more, I appreciate that you could kindly help on the following questions that I posted on the forum a week ago:

1) Calender Spread Strategy: http://forum.bionicturtle.com/viewthread/1990/
2) Tracking Error and R squared: http://forum.bionicturtle.com/viewthread/1947/
3) Positive Autocorrelation and hedge fund illiqudity: http://forum.bionicturtle.com/viewthread/1973/

Thank you!

Cheers
Liming
16/10/09

 

[David Harper CFA FRM]

Hi Liming:
I think EAR (aka, effective annual yield) may give confusion because it appears to be a different type of rate. But's it's not. We really approach two types of rates:

1. Discrete compound frequencies, of which there are many; e.g., daily, semi-annual, quarterly, annual
2. Continuous, of which there is only one.

EAR is "merely" the discrete annual rate.
In regard to (1), you would be solving (incorrectly) for a discrete monthly rate, in the same way that...

PV*(1+r/12)^12 = FV ....compounds (r) monthly over one year
so PV*(1+r/12) compounds (r) monthly over one month

In regard to (2), you would be solving (incorrectly) for a continous rate, because:
PV * exp(r * 1/12) = FV
.... compounds (r) continously over one month

Another way to solve this (in case it's helpful; this is how I would solve b/c I like the "anchor" of a single continuous) is:
1. Solve the for the continuous rate: continous r = LN(1000/987)*1/12 = 15.702%; this is maybe our most fundamental building block; ie., 987*EXP(r*1/12) = 1000
2. Then convert to discrete annual (aka, EAR): m*(EXP(continuous/m)-1). In this case, m = 1, so EXP(15.702%) - 1 = 17.002%

...hope that helps
...thanks for relinking the other questions, I'll get to them ASAP (after I record the 8. invest tutorials, probably, as I am trying to do that today and over the weekend)...I do have them bookmarked, it's just that those each of those are a bit challenging (time consuming) and I have to try and balance priorities ... it's not that i want to ignore, but sometimes when I rush to answer I make a mistake (e.g., http://forum.bionicturtle.com/viewthread/2010/) and I hate to do that (not on behalf of my ego, but b/c candidates need precise help not more confusion). This question here you wrote here is both (i) very *important* to exam and (ii) something i can confidently answer without research...therefore, I answer it on the same day... but often times, questions are not trivial, esp. if i want to be precise (some questions require research, i don't know the entire risk field in my head... on some questions, i have spent 1 or 2 hours to properly reply, which i am thrilled to do if i think it gives relevant benefit)...but I have content production, too...

Thanks, David

 

[Liming]

Dear David,

Thanks a lot for your reply! I really appreciate your great sense of responsibility and insistence on giving FRM candidates the best answer.

Can I just double check with you that my digest of your answer is correct?

under my two options described previously:
1) if assuming monthly compounding and use formula:
FV/PV = 1+(r/12) since there is only 1 month left
2) or assuming continuously compounding and use formula:
FV/PV = exp(r/12)

Because 1) is to solve for discrete monthly compounding rate over one month and 2) is to solve for the annual rate that is continuously compounding over one month, neither 1) or 2) gives the right answer since EAR should be discrete annually compounding rate (compounding only once in a year, implying that m = 1)
Therefore in order to solve EAR from the discrete annual compounding perspective, I think the formula to use should be: FV = PV*(1+EAR)raised to power of T where T= 1/12 year and EAR is the annual rate. We should understand the 1 month as 1/12 year, so the r to be used in the bracket should be annual rate too - on the same measure. Therefore
exp(r/12) = FV/PV = (1+EAR)raised to the power of 1/12
(I think this can be traced back to the text on continuous compounding on page 77 of "Options, Futures and Other Derivatives (John Hull 7th edition), with the only difference that n = 1/12 years <1year)

Now my question is:
it seems that from step: exp(r/12) = FV/PV = (1+EAR)raised to the power of 1/12
we can arrive at the next step: exp(r) = (1+EAR) (1/12 on both sides disappear!)

Is this transition in steps correct? From John Hull's text, it's clear that he has derived the second step out of the first step (suppose m = 1: annually discrete compounding). I can understand this transition for n>1 ie. a number of years for compounding but in this practice question, n = 1/12 <1. There seems to be a fundamental difference here but the logical steps should still hold. Is my thought correct?
I think my writing here is a bit imprecise. Sorry about this. Hope I described my question clearly enough for you. Thank you!

Cheers!
Liming
17/10/2009