# 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.