Your son will be going to college in 10 years and you are starting a fund for his education. He will need $25,000 at the beginning of each year for four years. The fund earns 5% annually, compounded monthly, and you plan to make monthly deposits, starting at the end of the current month. How much should you deposit each month to meet his educational expenses? It is assumed that the parents will continue to make the same deposits while their son continues his education for the 4-year term. Thanks for help. P.S. I know the answer but don't know how to arrive at it.
+118658 Did you work through the last answer that I did for you a couple of hours ago?
You did not make any comment. .....
I was fairly similar to this one. Try to start it on your own and show us what you have done.
Hello Melody:The question about $50,000 annuity for Life? That was not my question, sorry. I did look at it and I understand what you did there. I believe this question of mine is more involved, because the parents continue to make payments to the fund while the son is going and withdrawing his $25,000 at the beginning of each year. So, the two have to balance just right. If their payment were only for the first 10 years, then it would be relatively easy and simple to get the right answer. I have tried several ways of getting the right answer but it just doesn't balance. Thank you for looking at it.
+118658 Ok guest, if you were a member then I would not identify you incorrectly. Yes this question is much more involved but the two looked like they came from the same person to me.
Your son will be going to college in 10 years and you are starting a fund for his education. He will need $25,000 at the beginning of each year for four years. The fund earns 5% annually, compounded monthly, and you plan to make monthly deposits, starting at the end of the current month. How much should you deposit each month to meet his educational expenses? It is assumed that the parents will continue to make the same deposits while their son continues his education for the 4-year term. Thanks for help. P.S. I know the answer but don't know how to arrive at it.
Lets say you invest $M at the end of every month for 1 year at 5% pa compounded monthly.
It will grow to
\(FV_{12}=M\left [\frac{1.0041\dot6^{12}-1}{0.0041\dot6}\right]=12.27885348M \quad dollars.\)
10 years is 120 months. How much will be there after 120 months.
\(FV_{120}=M\left [\frac{1.0041\dot6^{120}-1}{0.0041\dot6}\right]=155.282248M \quad dollars.\)
So, after 120 months thee will be 155.282248M - 500000 in the fund (that is after the first payment)
How much will be in the fund 12 months later?
\(T_{after\;second \;payment}=(155.282248M-50000)*1.00416^{12}+12.27885348M-50000\\ T_{after\;second \;payment}=155.282248*1.00416^{12}M-50000*1.00416^{12}+12.27885348M-50000\\ T_{after\;second \;payment}=(155.282248*1.00416^{12}+12.27885348)M-50000(1.00416^{12}+1)\\ \\~\\ T_{after\;third \;payment}=[(155.282248*1.00416^{12}+12.27885348)M-50000(1.00416^{12}+1)]*1.00416^{12}+12.27885348M-50000\\ T_{after\;third \;payment}=(155.282248*1.00416^{12}+12.27885348)*1.00416^{12}M-50000(1.00416^{12}+1)*1.00416^{12}+12.27885348M-50000\\ T_{after\;third \;payment}=[(155.282248*1.00416^{24}+12.27885348*1.00416^{12}]M-50000(1.00416^{24}+1.00416^{12})+12.27885348M-50000\\ T_{after\;third \;payment}=[(155.282248*1.00416^{24}+12.27885348*1.00416^{12}+12.27885348]M-50000(1.00416^{24}+1.00416^{12}+1)\\ T_{after\;third \;payment}=[(155.282248*1.00416^{24}+12.27885348*1.00416^{12}+12.27885348]M-50000(1.00416^{24}+1.00416^{12}+1)\\ Let Z=155.282248, \quad R=1.0041\dot6,\quad Y=12.27885348 \\ T_{after\;third \;payment}=[(Z*R^{24}+Y*R^{12}+Y]M-50000(R^{24}+R^{12}+1)\\ \text{following the pattern}\\ T_{after\;forth \;payment}=[(Z*R^{36}+YR^{24}+YR^{12}+Y]M-50000(R^{36}+R^{24}+R^{12}+1)\\ \)
\(Let Z=155.282248, \quad R=1.0041\dot6,\quad Y=12.27885348 \\ T_{after\;forth \;payment}=[(Z*R^{36}+YR^{24}+YR^{12}+Y]M-50000(R^{36}+R^{24}+R^{12}+1)\\\)
155.282248*1.0041666666666^36+12.27885348(1.0041666666666^24+1.0041666666666^12+1) = 219.1093482417387751879582937628068330956451487032115301294171720182
50000*(1.0041666666666666^36+1.0041666666666666^24+1.0041666666666666^12+1) = 215878.7732386761695511346
T (after four payments) = 219.1093482417M - 215878.7732386
but
T (after four payments) = 0
219.1093482417M - 215878.7732386=0
219.1093482417M = 215878.7732386
M=215878.7732386 / 219.1093482417
215878.7732386/219.1093482417 = 985.2558778115831659
I get $985 per month.
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Blast... I worked it out on $50000 every your for 4 years. If you change all the 50,000 to 25,000 you should get the right answer.
155.282248*1.0041666666666^36+12.27885348(1.0041666666666^24+1.0041666666666^12+1) = 219.1093482417387751879582937628068330956451487032115301294171720182
25000*(1.0041666666666666^36+1.0041666666666666^24+1.0041666666666666^12+1) = 107939.3866193380847755673
219.109348241738775M-107939.386619338=0
M=107939.386619338/219.109348241738775
107939.386619338/219.109348241738775 = 492.627938905877833739
So the answer is $493 per month. That sounds about right.
Is that at least close to the answer you were expecting?
Now please join up - it is better for you as well as for us :)
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Hello Melody: YES!!!!!. You nailed it. Great work and now I can see where I went wrong. Now, I will seriously think of becoming a member, except I don't quite understand the benefits of being a member. Is this for commercial purposes or what? I understand members get priority in having their questions answered, but other than that I don't see the benefits of becoming a member. This is the second question that I have submitted to you in the last year. I attend a Business College in the US, but I'm a private person and don't join many organizations.I general, I'm pretty good at solving most problems in my assignments. Just occationally, like this one, I get stumped. Thanks again.
Here is another approach to solving the same problem:
1) Find the PV of the four $25,000 that he is withdrawing at the beginning of each year. I assume you know the formula that is used for that. I get a figure of $92,933.25 @ 5% comp.monthly.
2) The above amount is 10 years in the future, so we have to find its PV as of today, using the common PV formula, which I assume you know. By doing that, I get a PV of $56,425.45 @ 5% comp.monthly.
3) Now this number that we just got, $56,425.45, is the important figure for the parents, because they now have the PV of ALL their monthly payments for 13 years or 156 months.
4) Now, we use the annuity formula to find the monthly payment. By doing that, we get a monthly payment of $492.63, which is the same number that Melody calculated.
The only thing that you have to be careful about is the number of years or months that the parents will continue to deposit the money. It is 13 not 14 years, because by the end of the 13th year, they should have saved enough money for their son to be able to withdraw the last payment of $25,000 at the beginning of the 14th year. Good luck to you.
Guest #5
Thank you very much for those detailed instructions. We are allowed to use financial calculators, and I followed your instructions and got exactly the same numbers as you did.
So, I thank you and Melody for solving this problem that was nagging me for a couple of days, which is the reason I posted the question. Now and in future, I should know how to handle problems of this sort.
