Investing $90.91 per month would get you there in 55 years. But you can do better than that because your bank offers interest.
$20.71/month should do it.
The easy way to do this:
Open Excel. In A1 type your estimate of how much per month you should expect to pay. Include a minus sign as this will be money out of your pocket.
In B1 type in this formula using the future value Excel function.
=FV(0.045/12,55*12,A1,,1)
Then go to Data>>What If Analysis>>Goal Seek
Set Cell: $B$1
To Value: 60000
By Changing Cell: A1
The hard way:
I made a table using a guess for a monthly payment until I saw the pattern. Since your payment period is monthly, you definitely want to make life easier on yourself by dividing your APR by 12 to get a monthly rate. This comes out to 0.00375. Hopefully this table formats correctly . . . .
Define: r = 0.00375, t = time (in months), and P = principal
t | P
0 | 20
1 | (1 + r)(20) + 20
2 | (1 + r)[(1 + r)(20) + 20] + 20 = (1 + r)2(20) + (1 + r)(20) + 20
3 | (1 + r)[(1 + r)2(20) + (1 + r)(20) + 20] + 20 = (1 + r)3(20) + (1 + r)2(20) + (1 + r)(20) + 20
Ah ha! Our formula for the principal is a geometric series! Our series has first term 20 and a common ratio of (1 + r). We know how to find the sum of a geometric series. With this knowledge we can find the principal after any time period we please.
Define m = monthly contribution
If you look up the formula for the sum of geometric series, you will see that we can write:
P = m(1 - (1 + r)t) / (1 - (1 + r)) = m(1 - (1 + r)t) / (-r)
We can solve for m.
m = -rP / (1 - (1 + r)t)
m = -(0.00375)(60000) / (1 - (1.00375)55*12 = 20.7817
So, we figure that a monthly contribution of $20.79 is required. (60000 - 20.79 * 55 * 12 = $46,278.60 is money that interest magically gave you!)
