# Thread: Annuity problem changing deposit after 8 years

1. ## Annuity problem changing deposit after 8 years

I am totaly stuck on this problem been at it for over an hour so my search for help has guided me to this forum

A young executive deposits $300 at the end of each month for 8 years and then increases the deposits. If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of$400,000 after 25 years

stuck I think i am suppose to calc the simple annuity for 8 years then minus from the total of 400,000 then calc the remander for 17 years anyone help?

2. Originally Posted by joey36
I am totaly stuck on this problem been at it for over an hour so my search for help has guided me to this forum

A young executive deposits $300 at the end of each month for 8 years and then increases the deposits. If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of$400,000 after 25 years

stuck I think i am suppose to calc the simple annuity for 8 years then minus from the total of 400,000 then calc the remander for 17 years anyone help?
After 8 years, Accumulated Value = 38,792.47. I got that here using 7.2/12 as interest per payment and 12 * 8 = 96 months:

Annuity Immediate Accumulated Value

You need to accumulate 400,000 -38792.47 = 361,207.53 in 12(25 - 8) = 204 months. Solving for payment on the same link, we get 907.42 monthly deposits needed to accumulate to your scenario.

3. How many times have I said it? "Basic Principles"! If all you have going for you is a formula or two, you never will make it.

If p = Original Payment = $300.00 and If a = Additional Payment and If i = .072/12 = 0.006 and If c = (1+i) <== Monthly Accumulation Factor We have: [p + pc + pc^2 + ... + pc^(25*12)] + [a + ac + ac^2 + ... + ac^(17*12)] = 400000 You should be able to create more convenient expressions for the geometric series in the square brackets. I get a =$374.67. You?

4. Originally Posted by joey36
I am totaly stuck on this problem been at it for over an hour so my search for help has guided me to this forum

A young executive deposits $300 at the end of each month for 8 years and then increases the deposits. If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of$400,000 after 25 years

stuck I think i am suppose to calc the simple annuity for 8 years then minus from the total of 400,000 then calc the remander for 17 years anyone help?
You're on th right track. You want to break this into two problems - first find the future value of the first 8 years of payments when they get to year 25, then add the value of an annuity from years 9 - 25 so that the sum of the two is $400K. For the first part, it's an annuity calculation for the first 8 years, followed by simple compound interest on that investment for years 19-25. You then subtract that from$400K, and the difference is how much he needs to build with an annuity between years 9 and 25.

5. I got the first answer, but i am wondering if it was not right because i was not acoutning for the 38,792.47 in interest

6. Originally Posted by joey36
I got the first answer, but i am wondering if it was not right because i was not acoutning for the 38,792.47 in interest
You got the answer I got for the payment? 907.42? Is there something different in the back of the book.

7. Well this is an even problem and the evens are not in the back of the book

8. Originally Posted by joey36
Well this is an even problem and the evens are not in the back of the book
Joey,

I just redid it another way, and I get the same answer. I see something similar in Kellison's book, so I'll stick with my answer.

9. Originally Posted by TKHunny
I get a = $374.67. Note: a + p is the value for the last 17 years.$900 is way too much. You're not accumulating interest on the $38,792.47. 10. Originally Posted by TKHunny Note: a + p is the value for the last 17 years.$900 is way too much.
I reread the problem. If I accumulate the 38792.47 17 years, and subtract that from the 400k, I get 256467.16. 17 years @ 8%, the payment becomes 644.30.

11. I am leaving work right now so i have 2 hours to figure this problem out

I have to acumulate intrest on the 38,792.47 right?

so after i get 38,792.47 how do i acumulate the intrest then get home much i need to the next 17 years

12. Closer. Still something a bit off. I couldn't find the error after a couple of checks.

Originally Posted by TKHunny
If p = Original Payment = \$300.00 and
If a = Additional Payment and
If i = .072/12 = 0.006 and
If c = (1+i) <== Monthly Accumulation Factor

We have:

[p + pc + pc^2 + ... + pc^(25*12)] + [a + ac + ac^2 + ... + ac^(17*12)] = 400000
That appears to be off a bit. It should be...

$[p + pc + pc^{2} + ... + pc^{299}] + [a + ac + ac^{2} + ... + ac^{203}] = 400000$

Or

$p*\frac{1-c^{300}}{1-c} + a*\frac{1-c^{204}}{1-c} = 400000$

Or, since 1-c = -i = -0.006

$p*(c^{300}-1) + a*(c^{204}-1) = 2400$

Or, since p = 300 and c = 1.006

$a*(c^{204}-1) = 894.84096$

Finally, since c = 1.006, still

a = 374.6693482

Total Payment after eight years is a+b = 300+374.67 = 674.67

Let's try the HP-12C

[g][END]
[f][CLR][FIN]
25[g][12x]
7.2[g][12/]
300[CHS][PMT]
[FV] ==> 250859.8401
400000-[CHS] ==> 149140.1599[FV]
17[g][12x]
[PMT] ==> 374.6693479

It's looking like a pretty good result.

13. ...and just to emphasize the beauty and nonconfusion of the basic principles...

In this case, p = 300 = payment for first eight years and
b = payment for next 17 years (=p+a from the last setup)

$(p + pc + ... + pc^{95})*c^{204} + (b + bc + ... + bc^{203}) = 400000$

or

$p*\frac{1-c^{96}}{1-c}*c^{204} + b*\frac{1-c^{204}}{1-c} = 400000$

Since 1-c = -i = -0.006

$p*(c^{96}-1)*c^{204} + b*(c^{204}-1) = 2400$

Since p = 300 and c = 1.006

$b*(c^{204}-1) = 1611.345497$

More with x = 1.006

$b = 674.6693480$

Unique answers don't care how you find them.