Compound interest

**lililet11** · December 15th 2008, 05:30 PM

Could someone help me please?
How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:
a)$1
b)$100
c)$20,000
d) Part (a) is called the effective yield for an account. How could part (a) be used to determine parts (b) and parts (c)?

**euclid2** · December 15th 2008, 06:54 PM

Originally Posted by lililet11

Could someone help me please?
How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:
a)$1
b)$100
c)$20,000
d) Part (a) is called the effective yield for an account. How could part (a) be used to determine parts (b) and parts (c)?

Are you familiar with the formula ?
$A=P(1+i)^n$
where A is amount owed or ended up with
I is the interest rate changed to a decimal
P is the principal amount
N is the number of compounding periods

**chabmgph** · December 15th 2008, 07:24 PM

Originally Posted by lililet11

Could someone help me please?
How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:
a)$1
b)$100
c)$20,000
d) Part (a) is called the effective yield for an account. How could part (a) be used to determine parts (b) and parts (c)?

Originally Posted by euclid2

Are you familiar with the formula ?
$A=P(1+i)^n$
where A is amount owed or ended up with
I is the interest rate changed to a decimal
P is the principal amount
N is the number of compounding periods

Or if t is the time in years you are compounding, the amount you should end up with will be

$A=P(1+\frac{i}{n})^{nt}$
Since if the percent of interest you get every year is i, the percent of interest you get each compounding period would be $\frac{i}{n}$ .

**badgerigar** · December 16th 2008, 01:10 AM

How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:

Since you are investing annually, you will need the formula:

$F = P(1+i)^{t}+A\frac{(1+i)^{t}-1}{i}$

F is the final value of the investment
P is the initial value of the investment
t is the number of years
A is the amount invested annually
i is the "real" annual interest, given by $(1+I/k)^{k}-1$ where k is the number of interest compoundings per year and I is the listed annual interest rate.

**lililet11** · December 16th 2008, 03:56 AM

Thank you all for your help, I do know how to use the formula to find compound interest, part (d) is where i'm stuck. Effective yield, what should I do there?

**lililet11** · December 16th 2008, 03:59 AM

Thanks for your help, but I do know how to use the formula, my biggets problem is part (d), it's asking:

Part (a) is called the effective yield of an account. How could part (a) be used to determine parts (b) and (c) ?

**chabmgph** · December 16th 2008, 06:32 AM

Never mind.

**badgerigar** · December 16th 2008, 02:11 PM

Since P=0,

$F = A\frac{(1+i)^t-1}{i}$
If i and t are considered constants, then F is proportional to A. so if A=100 then F will be $100\times effectiveYield$

This formula does not apply because the money is being invested annually ie. each year the investor deposits a fixed amount into the account. This is the correct formula for a single initial investment.

**chabmgph** · December 16th 2008, 02:50 PM

Originally Posted by badgerigar

Since P=0,

$F = A\frac{(1+i)^t-1}{i}$
If i and t are considered constants, then F is proportional to A. so if A=100 then F will be $100\times effectiveYield$

This formula does not apply because the money is being invested annually ie. each year the investor deposits a fixed amount into the account. This is the correct formula for a single initial investment.

Thanks for pointing it out. How could I have missed the word "annually"? I don't know. My bad.

**jonah** · December 17th 2008, 10:17 AM

Originally Posted by lililet11

Could someone help me please?
How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:
a)$1
b)$100
c)$20,000
d) Part (a) is called the effective yield for an account. How could part (a) be used to determine parts (b) and parts (c)?

With a given nominal rate j compounded m times per year, we define the corresponding effective rate to be that rate w which, if compounded annually, is equivalent to the given rate. That is,
$ w = \frac{{{\rm{interest earned in one year}}}}{{{\rm{principal invested at the beginning of the year}}}} $
In your particular case, if $1 is invested at the rate 10% compounded 12 times per year, then
$ w_{{\rm{effective rate of 10\% compounded monthly}}} = \left( {1 + {\textstyle{{0.10} \over {12}}}} \right)^{12} - 1 $
Since you’re seeking to calculate the accumulated value of an annual investment over a period of 20 years, you need to make use of the future value formula of an annuity with w = i, n = 20 years, and R = $1, $100, or $20,000.

For the beginning of year deposits/investments, use
$ \ddot S = R \cdot \frac{{\left( {1 + i} \right)^n - 1}}{i} \cdot \left( {1 + i} \right) $
For the end of year deposits/investments, use
$ S = R \cdot \frac{{\left( {1 + i} \right)^n - 1}}{i} $

**jonah** · December 17th 2008, 10:36 AM

Originally Posted by lililet11

Could someone help me please?
How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:
a)$1
b)$100
c)$20,000
d) Part (a) is called the effective yield for an account. How could part (a) be used to determine parts (b) and parts (c)?

With a given nominal rate j compounded m times per year, we define the corresponding effective rate to be that rate w which, if compounded annually, is equivalent to the given rate. That is,

$ w = \frac{{{\rm{interest-earned-in-one-year}}}}{{{\rm{principal -invested-at-the-beginning-of-the-year}}}} $

In your particular case, if $1 is invested at the rate 10% compounded 12 times per year, then

$ w_{{\rm{effective-rate-of-10\%-compounded-monthly}}} = \left( {1 + {\textstyle{{0.10} \over {12}}}} \right)^{12} - 1 $

Since you’re seeking to calculate the accumulated value of an annual investment over a period of 20 years, you need to make use of the future value formula of an annuity with w = i, n = 20 years, and R = $1, $100, or $20,000.

For the beginning of year deposits/investments, use

$ \ddot S = R \cdot \frac{{\left( {1 + i} \right)^n - 1}}{i} \cdot \left( {1 + i} \right) $

For the end of year deposits/investments, use

$ S = R \cdot \frac{{\left( {1 + i} \right)^n - 1}}{i} $

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