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June 20th, 2013, 07:44 PM   #1
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Need Help with Accumulated Value

Hi Guys,

I'm new here.
I have a problem with an exercise equation which I don't understand.
Please have a look at the image:

[attachment=0:27uqbqpq]2013-06-21_101901.gif[/attachment:27uqbqpq]

My question is:

1. I don't think the line 2 is correct, why the 100 are subsctracted and still in the row
2. Where does line 3 coming from??
Attached Images
File Type: gif 2013-06-21_101901.gif (23.7 KB, 898 views)
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June 21st, 2013, 12:34 PM   #2
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Re: Need Help with Accumulated Value

To get from line 1 to line 2, factor out 100(1+i)^4. Line 2 has an error in that all the other 100's shouldn't be there. It looks like a typo. Take them all out and you go from line 2 to line 3 using the standard formula for geometric progression.
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June 21st, 2013, 07:39 PM   #3
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Re: Need Help with Accumulated Value

I assume you're solving for i.
Why insert the "100"s in there?! Take 'em out!
AND, shortening further: let u = 1+i

u^4 + u^8 + .... + u^40 = 5(u^4 + u^8 + .... + u^20) : see how unwieldy that is?

Using geometric series formula:

(u^44 - u^4) / (u^4 - 1) = 5[(u^24 - u^4) / (u^4 - 1)

Now doing the math:

u^44 - u^4 = 5(u^24 - u^4)

u^4(u^40 - 1) = 5[u^4(u^20 - 1)]

(u^40 - 1) / (u^20 - 1) = 5

(u^20 - 1)(u^20 + 1) / (u^20 - 1) = 5

u^20 + 1 = 5

u^20 = 4

u = 4^(1/20) = 1.071773....

So i = .071773....
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June 21st, 2013, 07:44 PM   #4
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Re: Need Help with Accumulated Value

Quote:
Originally Posted by Denis
I assume you're solving for i.
Why insert the "100"s in there?! Take 'em out!
AND, shortening further: let u = 1+i

u^4 + u^8 + .... + u^40 = 5(u^4 + u^8 + .... + u^20) : see how unwieldy that is?

Using geometric series formula:

(u^44 - u^4) / (u^4 - 1) = 5[(u^24 - u^4) / (u^4 - 1)

Now doing the math:

u^44 - u^4 = 5(u^24 - u^4)

u^4(u^40 - 1) = 5[u^4(u^20 - 1)]

(u^40 - 1) / (u^20 - 1) = 5

(u^20 - 1)(u^20 + 1) / (u^20 - 1) = 5

u^20 + 1 = 5

u^20 = 4

u = 4^(1/20) = 1.071773....

So i = .071773.... ~7.18% cpd annually, I'll bet
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June 21st, 2013, 07:57 PM   #5
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Re: Need Help with Accumulated Value

Thanks mathman and dennis for your help.
Actually I'm looking for the X.

Like mathman said, it probably a typo.

Have a look at the complete solution.

[attachment=0:kckc9cf8]2013-06-22_105523.gif[/attachment:kckc9cf8]
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File Type: gif 2013-06-22_105523.gif (45.3 KB, 864 views)
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June 21st, 2013, 09:42 PM   #6
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Re: Need Help with Accumulated Value

Quote:
Originally Posted by ajie
Have a look at the complete solution.
Your complete solution attachment has: (1 + i)^20 = 4

That's what I gave you in my post: 1 + i = 4^(1/20) : SAME THING!

r = rate every 4 years = (1 + i)^4 - 1 = .3195....

Value end of 40 years = 100(1+r)[(1+r)^10-1] / r = 6194.72
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June 21st, 2013, 10:03 PM   #7
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Re: Need Help with Accumulated Value

Thanks Denis,

I could sleep well now.
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June 22nd, 2013, 06:44 AM   #8
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Re: Need Help with Accumulated Value

This can be made really simple, once we have determined that annual rate
= 4^(1/20) - 1 = .071773 or 7.1773%

This rate is adjusted to coincide with the frequency of the $100 deposits,
which is every 4 years: i = [4^(1/20)]^4 - 1 = .319508 or 31.9508%

It is now easy to see that the immediate deposit of $100 will accumulate
to 131.9508 by the end of first 4-year period.

So now we can treat this 131.9508 as a deferred (no immediate deposit)
annuity of 131.9508 over 10 periods.

So we have:
A = 131.9508
n = 10
i = .319508
F = ? (F being Future value)

Formula is:
F = A[(1 + i)^n - 1] / i
so:
F = 131.9508[(1 + .31950^10 - 1] / .319508 = 6194.71944...

And will look like, in bank statement format:
Code:
N:YEAR  DEPOSIT    INTEREST    BALANCE
0 : 0                              .00
1 : 4    131.95         .00     131.95
2 : 8    131.95       42.16     306.06
3 :12    131.95       97.79     535.80
....
9 :36    131.95    1,080.62   4,594.72
10:40    131.95    1,468.05   6,194.72
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