Question

Suppose your parents/guardian opened an education account for you when you were born in the amount of $2,5000. The account has been earning 2.5% interest compounded quarterly since the original deposit. How much do you need to deposit monthly in order to have enough for college which is $219,752.40? Find a recursive formula.

Answer 1

That is meant to be $2,500

Answer 2

By the way, this account is open for 18 years.

Answer 3

We have

P(1+i/4)^(4n) + m(1-(1+j/12)^(12n+1))/(1-(1+j/12))

as the value, after n years, of P dollars invested at nominal annual interest rate i, compounded quarterly, together with monthly deposits of value m into an account with nominal annual interest rate j, compounded monthly.

For this problem, we have n=18, P=$2500, i=0.025, m unknown, and (presumably) j=0.025 (I can't tell from the wording of the problem if perhaps the monthly deposits are to earn no interest, so I will address j=0 separately below).

Answer 4

This gives us a solution of

m = $787.71.

Answer 5

:(

Answer 6

Note: This assumes the first monthly deposit occurs simultaneously with the deposit of the $2500 at birth.

Answer 7

j does equal 0.025, you are correct.

Answer 8

Are you sure? And do we assume that the compounding is monthly for the monthly deposits, or is it also quarterly?

Answer 9

It is also quarterly. And yes I'm sure.

Answer 10

:(

Answer 11

Then we get

P(1+i/4)^(4n) + 3m(1-(1+j/4)^(4n+1))/(1-(1+j/4))

and this, with P=$2500, n=18, i=j=0.025, equals $219 752.40 when

m = $780.79.

Answer 12

Do you use a TI-84 calculator? Have you used sequence mode?

Answer 13

This value is a little less than with the monthly compounding computed above, which is counterintuitive. The reason is that I did not take the care to question the timing of the first compounding event (interest calc by the bank) relative to the first deposit. So this is an "optimistic" computation and the actual monthly deposit amount needed will be slightly less. In any case, "about $780" is more than reasonable for this problem, I would think.

Answer 14

I used paper and pencil and first principles only, except to do a few exponents with a regular calculator because I am lazy.

Answer 15

(and oops, where I wrote "slightly less" I meant "slightly more" -- it's optimistic for us, not the bank!)

Answer 16

Ok, I understand, thank you very much.

Answer 17

Now, I have a question. If you were told to use 18 years, why did the instructor ask for a "recursive solution"?

Answer 18

I'm doing a project on 'saving for college' and this is to see how much we will have at age 18, the average graduation time from high school.

Answer 19

I'm still not sure about the recursive solution part. This is quite a problem for high school! I can tell you that, in my experience, people who do loans at banks generally can NOT do a problem like this, or even a similar but much, much simpler one. My experience only, YMMV.

Answer 20

I hope your parents have been putting away $800 a month for you :) Good luck!

Answer 21

Haha, I wish. thank you.