Effective Annual Rate (EAR) Problem

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cfacowtown

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For the life of me I can’t sort this small part of reading 5 out in my head. The formula makes sense, but that’s because it’s merely plug and chug. I have a couple questions to anyone who has any insight….I appreciate the help:
What is the difference between EAR and the periodic interest rate. From my understanding the EAR is the actual annual interest rate (compound) and the periodic rate is the rate used to calculate the charge/gain in multiple compounding periods. I think I am mixed up here, I just need to find some simple clarification between these two.
Secondly, I can’t remember how to figure out the steps to calculate the appropriate periodic rate for a given annual rate with semiannual compounding. There is five steps, and I can’t remember how to isolate the brackets and the square root…..don’t I feel intelligent:
Start:
1.0816 = (1 + Periodic rate) ^2
End:
(1.0816) ^1/2 - 1
 
I think I figured out the answer to my first question already. I was comparing the wrong elements. Financial Institutions typically quote rates in annual form (8% compounded quarterly), as opposed to quoting rates as periodic (2% per quarter).
The rate of interest that investors ACTUALLY realize as a result of this compounding is known as the EAR.
So basically the essence here is that, the greater the compounding frequency the greater the EAR will be in comparison to the stated rate?
Am I on the right track?
 
Can anyone help with this? I am having trouble figuring our how they turned the power of 2 into the power of 1/2 during the variable isolation? It’s basic math, but for the life of me I cant remember how to do it.
 
I’m not sure I follow your question exactly, but let me try and help.
Lets say you have a stated annual rate of 8% that compounds semi-annually and are looking for the EAR.
You first need the periodic rate, which is 4% (stated annual rate/# of compounding periods) Now you add one and raise it to the # of compounding periods. (1.04)^2 = 1.0816. Subtract 1 and you get your EAR 8.16%
To go the other way, (EAR to stated annual rate) you just take the square root or 1.0816^1/2 = 1.04, subtract 1.
If this was for quarterly compounding you would do 8/4 to get your periodic rate of 2%, raise that to the power of 4 (# compounding periods), subtract 1 to get your EAR of 8.24% i.e (1.02)^4 = 1.0824
To go their other way, you can raise (1.0824)^1/4 to arrive back at your periodic rate of 1.02. Subtract 1 multiply by 4 (because their are 4 compounding periods) to get back to your stated annual of 8% w/qtrly compounding.
 
That’s kind of what I was looking for, but i’m still slightly confused. What I am trying to do is find the appropriate periodic rate for a given effective annual rate, say in this case, of 8.16% with semiannual compounding.
So at this point, we know we need to isolate for the “Periodic Rate”, which is high school math that I cannot seem to remember. My only problem is isolating the ^2. I can’t remember how to remove the ^2 from one side of the equation to the other. The book does this in the following way:
Start: 1.0816 = (1 + Periodic rate ^2)
End: (1.0816) ^ 1/2
It’s really a minor thing, but I can’t remember how or why the ^2 is suddenly the ^1/2. The best way I could figure it out, is to remove a multiple you divide it by its entire amount. So 2/2 =1, and then maybe just put it over the original amount of 2 to get a 1/2 or 0.5 result? I just can’t reverse engineer the process to arrive at 1/2. Solving is not the problem, that is just basic math. I just can’t remember how to isolate. Thank you so much for your help.
 
Are you looking for an explanation on the algebra?
What you do to one side you do to the other…. so
1.0816 = 1.04^2
take the square root of both sides (sq root of 1.04^2 = 1.04) and you get:
(sq root)1.0816 = (sq root)(1.04^2)
taking the square root of a number is the same as raising it to the .5 power.
 
As stated above, the periodic semi-annual rate is 4%, and the EAY= 8.16%.
To get 8.16 you have (1 + 0.04)^2 .
In the first case you are squaring the semi annual rate to get the compounded annual rate (because 4 x 2 does not include compound interest). To get the periodic semi annual rate, you want to remove the compounding, so you take the square root of 1+ EAY, then subtract the 1.
So the semi-annual periodic rate = sq root (1.0816)= 1.04 - 1= 4%.
If you want a quarterly periodic rate, you take the 4th root, or ^ 1/4.
Not sure if I helped, but oh well, I gave it a shot.
 
1.0816 = (1+ periodic rate) ^2
When you take square root on both sides, you will get something like this…
(1.0816)^1/2 = (1+periodic rate) ^ ½*2
(1.0816) ^ ½ = (1 + periodic rate) ^ 1
If u want to get back to original equation:
Squaring both sides or multiplying both sides with same value or number
(1.0816)^1/2 * (1.0816)^1/2 = (1+periodic rate) ^1 *(1+periodic rate) ^1
If I remember correctly, math rule says, when bases are same (1.0816 and 1 + periodic rate), powers are added up.
1.0816 ^1 = (1+periodic rate) ^2
I hope this makes some sense.
 
Chi Paul, sbmarti2, spirit:
Thanks so much, basic lesson in algebra, I should really know this. I found out what I was doing wrong. I couldn’t get the 1.04 number using any variable from the formula, but then chi paul said:
“taking the square root of a number is the same as raising it to the .5 power.”
Which then intuitively lead me to calculate 1.0816 ^0.5 = 1.4
1.4 - 1 = 0.04 x 100 = 4%, which is the correct answer to the question.
My original calculation looked like this:
1.0816 ^2 = 1.16986 - 1 = 0.16986 x 100 = 16.99%
Which I knew was blatantly wrong……
Just a quick clarification, taking the square root of ANY NUMBER is the same as raising it to the .5 power? Tiny math rules I need to memorize. Thanks so much guys, I really do appreciate it.
 
Just a quick clarification, taking the square root of ANY NUMBER is the same as raising it to the .5 power?
square root rasing to power of .5 or 1/2
cube root .3333 or 1/3
so on…
 
spirit Wrote:
——————————————————-
> Just a quick clarification, taking the square root
> of ANY NUMBER is the same as raising it to the .5
> power?
>
> square root rasing to power of .5 or 1/2
> cube root .3333 or 1/3
> so on…
Excellent, thanks.
 
It sounds like you should review some of the operational math problems to get back into the swing of things. What may help, and I am only commenting based on your post, is to treat the problems like pure algebra…….X=periodic rate 1.0816=(1+X)^2 now solve for X. I think the material sometimes does a disservice to us by throwing words into the equations that require basic algebraic operations.
 
Thanks BizBanker. I don’t consider myself deficient in algebra but you are right, I did need a refresher. I picked up a few books that I plowed through today and now i’m all set. I appreciate the honesty, identify the weaknesses and drill down.
 
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