Problem 90 Solution

Since there are monthly payments there will be 12*30=360 total payments. The present value of these 360 payments must be $100,000. Lets call the montly payment p, then the present value of all payments can be expressed below where i = the interest rate = .075/12.

p * [ 1/(1+i) + 1/(1+i)2 + 1/(1+i)3 + ... + 1/(1+i)360 ] =

p * [ (1 - 1/(1+i)360) * ( 1/(1+i) + 1/(1+i)2 + 1/(1+i)3 + ... ) ].

Now recall that that the sum for n=1 to infinity of xn = x/1-x, where x is less than 1, so we can further simplify:

p * [ (1 - 1/(1+i)360) * (1/(1+i))/(1-(1/(1+i))) =

p * [ (1 - 1/(1+i)360) / i = $100,000.

Solving for p (remember that i=.075/12):

p = $100,000 * i / (1 - 1/(1+i)360) =~ $699.21

For your own information here are what the monthly payments would be under various other interest rates:

0%	 $277.78
1%	 $321.64
2%	 $369.62
3%	 $421.60
4%	 $477.42
5%	 $536.82
6%	 $599.55
7%	 $665.30
8%	 $733.76
9%	 $804.62
10%	 $877.57
11%	 $952.32
12%	$1028.61
13%	$1106.20
14%	$1184.87
15%	$1264.44
16%	$1344.76
17%	$1425.68
18%	$1507.09
19%	$1588.89
20%	$1671.02
25%	$2084.58
30%	$2500.34
35%	$2916.76
40%	$3333.36
45%	$3750.01
50%	$4166.67
55%	$4583.33
60%	$5000.00
65%	$5416.67
70%	$5833.33
75%	$6250.00
80%	$6666.67
85%	$7083.33
90%	$7500.00
95%	$7916.67
Michael Shackleford, A.S.A.