Question

Question 4 You owe your best friend $2,000. Because you are short of cash, you offer to repay the loan over 12 months under the following condition. The first payment will be $300 at the end of month one. The second payment will be $300-G at the end of month two. At the end of month three, you'll repay $300 - 2G. This pattern of decreasing G amounts will continue for all remaining months. (a) What is the value of G if the interest rate is 0.5% per month? (b) What is the equivalent uniform monthly payment?

Answer 1

PV=$2000

Payment period is 12 months, n=12

interest rate =0.5%

Payment is Annuity of $300-Gradient of G

PV of payment = $2000

PV of $300 = A*(1-(1+r)^-n)/r

=300*(1-(1+0.5%)^-12)/0.5%

=300*(1-1.005^-12)/0.005

=300*(1-0.9419)/0.005

=300*0.0581/0.005

=3485.68

PV of gradient = 2000-3485.68 = -1485.68 for PV of payment to be $2000

PV of Gradient =G*( (1+i)^n-i*n-1)/ (i^2*(1+i)^n)

1485.68 = G*((1+0.5%)^12-0.5%*12-1)/(0.5%^2*(1+0.5%)^12)

or, 1485.68 = G*(1.0617-0.06-1)/(0.000025*1.0617)

or, 1485.68 = G*(0.0017)/0.000026

or, G = 1485.68 *0.000026/0.0017

or, G =22.72

Answer b

PV of Annuity =$2000

Hence PV= A*(1-(1+r)^-n)/r

2000=A*(1-(1+0.5%)^-12)/0.5%

2000=A*(1-1.005^-12)/0.005

2000=A*(1-0.9419)/0.005

2000=A*0.0581/0.005

A=2000*0.005/0.0581=172.12

Uniform monthly payment =$172.12