Question

1. A regular deposit of $100 is made at the beginning of each year for 20 years. Simple interest is calculated at i%per year for the 20 years. At the end of the 20-year period, the total interest in the account is $840. Suppose that interest of i% compounded annually had been paid instead. How much interest would have been in the account at the end of the 20 years?

2. Herman has agreed to repay a debt by using the following repayment schedule. Starting today, he will make $100 payments at the beginning of each month for the next two-and-a-half years. He will then pay nothing for the next two years. Finally, after four-and-a-half years, he will make $200 payments at the beginning of each month for one year, which will pay off his debt completely. For the first four-and-a-half years, the interest on the debt is 9% compounded monthly. For the final year, the interest is lowered to 8.5% compounded monthly. Find the size of Herman’s debt. Round your answer to the nearest dollar.

Answer 1

Assume annual interest =r

r=i/100

Simple Interest on first deposit =20*r*100

Simple Interest on Year 2 beginning deposit=19*r*100

Total Simple interest =20*r*100+19*r*100+18*r*100+……………………..+r*100

Total Simple Interest =100*r*(20+19+18+17+………..+1)

Total Simple Interest =100*r*(20*(20+1)/2)=100*r*210=21000*r

Total Simple Interest =$840

21000*r=840

r=840/21000=0.04

i=0.04*100%=4%

IF COMPOUNDED ANNUALLY:

Future Value of deposit at end of 20 years =FV=(Deposit)*((1+r)^(20-N))

r=0.04,N=year of Cash Flow

N	D	FV=D*(1.04^(20-N)
End of Year	Deposit	Future Value
0	$100	$219.11
1	$100	$210.68
2	$100	$202.58
3	$100	$194.79
4	$100	$187.30
5	$100	$180.09
6	$100	$173.17
7	$100	$166.51
8	$100	$160.10
9	$100	$153.95
10	$100	$148.02
11	$100	$142.33
12	$100	$136.86
13	$100	$131.59
14	$100	$126.53
15	$100	$121.67
16	$100	$116.99
17	$100	$112.49
18	$100	$108.16
19	$100	$104.00
	SUM	$3,096.92

Total Amount Deposited=100*20=$2,000

Total Future Value of deposit with 4% interest compounded annually=$3,096.92

Amount of Interest in the account=3096.92-2000=$1,096.92