Acme Annuities recently offered an annuity that pays 7.5 % compounded monthly. What equal monthly deposit should be made into this annuity in order to have $ 157,000 in 12 years?

The amount of each deposit should be $?

round to nearset cent

Guest Jul 2, 2017

#1**0 **

hectictar: Do you want to try your magic hand on some of these financial problems? I will guide you through them. But only if you want. I think you will breeze through them in no time. You could start by using this very excellent financial calculator: https://arachnoid.com/finance/

Many thanks.

Guest Jul 2, 2017

#2**+3 **

I normally don't even click on these questions.. they are way outside my "comfort zone" ! ..but....I'll venture a guess......

Since it is compounded monthly...divide the annual rate by 12 (7.5/12)% = 0.625% = 0.00625

At the end of each month..there will be (1.00625 * the amount of money in the account before the company adds any that month ) in the account.

Call the monthly deposit " m "...

the amount of money after 1 month = 1.00625m

the amount of money after 2 months = 1.00625( 1.00625m + m )

the amount of money after 3 months = 1.00625( 1.00625(1.000625m + m) + m )

= 1.00625^3m+1.00625^2m +1.00625m

= m(1.00625^3 + 1.00625^2 + 1.00625)

= \(m*\sum \limits_{i=1}^{3}(1.00625^i)\)

So..after 144 months...or 12 years...there will be...

\(m*\sum \limits_{i=1}^{144}(1.00625^i)\)

Set this equal to 157,000

\(m*\sum \limits_{i=1}^{144}(1.00625^i)=157,000 \\~\\ m=157,000/(\sum \limits_{i=1}^{144}(1.00625^i))\)

And according to WA... m = 671.26

Now surely this must be a way over-complicated way of doing it..and it might be way wrong!! But...I tried!!

hectictar
Jul 3, 2017

#3**+1 **

hectictar: Fantastic! Got it almost perfectly right. The only small mistake appears to be in the payment being made at the beginning instead of at the end of each month. But we simply rectify that by multiplying the payment you got(671.26) by the interest rate of 1.00625 =$675.46, which is the correct answer. Now, you could always check your answer with the financial calculator I mentioned, which is very accurate.

Of course, they have formulas for these financial problems, which are called "TVM", or Time Value of Money formulas, which will make things a lot easier. The formula for this question is:

FV=P{[1 + R]^N - 1/ R}, where FV=Future Value, P=Periodic Payment, R=Interest Rate per period, and N=Number of periods. So, we just enter the numbers we have in the formula as follows:

157,000 = P x {[1 + 0.075/12]^(12*12] - 1 / (0.075/12]}, and that is it. Will now solve for P.

157,000= P x {[1.00625]^144 - 1 / (0.00625)}

157,000= P x {[ 1.4527238.... / (0.00625)}

157,000= P x 232.4358088.......

P = 157,000 / 232.4358088

**P = $675.46 - the monthly payment required. **

**P.S. Please do not feel pressured to tackle these problems. Do them only if you want to learn something about money and how it grows over time, just for your own amusement and curiosity only.**

Guest Jul 3, 2017

edited by
Guest
Jul 3, 2017

edited by Guest Jul 3, 2017

edited by Guest Jul 3, 2017

#4**0 **

You are incorrect. Even if you try to justify your answer by solving for the future worth of a monthly payment of $675.46 for 144 months you get an answer of [675.46(((1.075^144-1))/.075] = $300,154,040.32 which does not equal $157,000

First let's convert 12 years into months (12*12=144) that way everything is in regards to months (CP, PP, n)

Let's set up our notation

A=F(A/F, i, n)

A= 157000(A/F, 7.5%, 144)

Now let's use the formulas

A=((i)/((1+i)^n-1))FV

A=((.075)/(1.075^144-1))157000

A= $0.3533

To double check the answer let's find the future worth of a monthly payment of $0.3533 for 144 months with an interest rate of 7.5% compounded monthly.

FV = 0.3533(((1.075^144-1))/.075

FV = $156995.858 (Which is roughly the answer since I left of many digits from the A value - this method is correct)

Guest Jul 4, 2017

#5**+1 **

So..........I can just put about 35 cents into this account every month....and after 12 years.....I'll end up with $157,000 ?

Man! I need to find one of them accounts in real life!!

hectictar
Jul 4, 2017