My Math Forum Effective interest rate compounded every two months

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 July 2nd, 2011, 12:22 PM #1 Newbie   Joined: Jul 2011 Posts: 2 Thanks: 0 Effective interest rate compounded every two months Hey guys, sorry if my English is poor, it's not easy to translate Financial Math terms from Portuguese to English... Anyway, I'm in a Financial Calculus class and need some help converting an interest rate, everything is fine and dandy converting any rate but this one: 21% effective interest rate compounded every two months. The fact that it's effective AND compounded gets to me and I'm kind of lost on where to go with this. It's 21% annually and I need this converted to "every two months". I hope this was clear enough. Thanks.
 July 2nd, 2011, 01:03 PM #2 Global Moderator   Joined: May 2007 Posts: 6,709 Thanks: 675 Re: Effective interest rate compounded every two months 21% annually is (21/6)% = 3.5% every two months.
July 2nd, 2011, 07:51 PM   #3
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Re: Effective interest rate compounded every two months

Quote:
 Originally Posted by mathman 21% annually is (21/6)% = 3.5% every two months.
Right, and that's equivalent to about 22.93% interest compounded annually. I'm not quite sure what the question wants.

 July 2nd, 2011, 08:05 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Effective interest rate compounded every two months Perhaps it's asking for the positive real solution to: $(1+r)^6=1.21$ $r\approx3.22801%$
 July 2nd, 2011, 08:25 PM #5 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Effective interest rate compounded every two months Mark - you just ruined my gloating. Since I don't know 95% of what is going on around here, I had the opportunity to gloat over knowing something these math gurus don't know. And you ruined it all. But you are correct. That's exactly what it wants. You can't just divide by 6 because of the compounding effect. And the 21% is already the annual percentage yield, meaning it needs to solve backwards to find the annual percentage rate first. Then you can divide by 6. (Although I have no clue what this has to do with calculus, cause I don't even know calculus. I just use basic algebra to solve it.)
 July 2nd, 2011, 08:37 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Effective interest rate compounded every two months Well, I can't take credit, it's what [color=#008000]CRGreathouse[/color] was alluding to in his post. I'm sure the class the OP is in is using calculus for marginal analysis, elasticity, etc. among other things. I went about looking at it this way, with P as the principle and r as the periodic rate: After period.....Investment value .....1...............$P(1+r)$ .....2...............$P(1+r)(1+r)=P(1+r)^2$ .....? .....6...............$P(1+r)^6$ Thus, we require: $P(1+r)^6=1.21P$ Divide through by P: $(1+r)^6=1.21$ $r=\sqrt[6]{1.21}-1\approx0.032280115456367$
 July 3rd, 2011, 02:18 AM #7 Newbie   Joined: Jul 2011 Posts: 2 Thanks: 0 Re: Effective interest rate compounded every two months Hey guys, thanks for the answers, happy to see I was in the right track: ((1+0,21)^(1/6))-1 = 0,0322...
July 3rd, 2011, 02:50 AM   #8
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Re: Effective interest rate compounded every two months

Quote:
 Originally Posted by MarkFL I went about looking at it this way, with P as the principle and r as the periodic rate: After period.....Investment value .....1...............$P(1+r)$ .....2...............$P(1+r)(1+r)=P(1+r)^2$ .....? .....6...............$P(1+r)^6$ Thus, we require: $P(1+r)^6=1.21P$
Well, that's exactly how it's derived of course. Though technically it's:

$\text{APY= \left(1+\frac{r}{m}\right)^m - 1}$

m being the 6...

I'm quite sure you could've figured that out, but that's what we use to get the annual yield. Then just divide by 6 since the problem wants what it would be every 2 months.

I never actually solved this backwards until a bank was quoting yields and I wanted to know what the actual rate was, except it compounded daily which was only slightly more fun.

So, how would you do it compounding continually? I had to be told how to do that, but I'm quite sure that will be an easy puzzle for you.

 July 3rd, 2011, 09:54 AM #9 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Effective interest rate compounded every two months You would use: $\text{APY}=\lim_{m\to\infty}$$\(1+\frac{r}{m}$$^m\ )-1=e^r-1$
 July 3rd, 2011, 06:19 PM #10 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Effective interest rate compounded every two months I'll have to trust how you got there. We were just shown a chart that made m a 10, then 100, then 1000, etc., and showed how it tended towards e. But I've never learned why e is what it is, only how to use it for this one thing. :P Next thing you know we'll be turning you into a finance person.

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