Name: Fre
Who is asking: Student
Level of the question: All

Question: A loan of $50,000 taken today is payable within five years.
a. determine the annual payments within to be made to repay the loan if interest is charged at a rate of 10% compounded annually
b. show the amortization schedule

 

Hi Fre,

To save some typing let $A be the amount of the loan, $P the payment you make each year and i be the annual interest rate. At the end of the first year you owe $A(1 + i) and you pay $P so you left owing

$A(1 + i) - $P

Thus at the end of the second year you owe this amount times 1 + i and you pay $P so you are left owing

[$A(1 + i) - $P](1 + i) - $P = $A(1 + i)2 - $P[(1 + i ) + 1]

This is the amount you owe at the start of the third year so, at the end of the third year you owe this amount times 1 + i and you make a payment of $P. Likewise in the fourth and fifth years. Thus at the end of the fifth year you owe

$A(1 + i)5 - $P[(1 + i)4 + (1 + i)3 + (1 + i)2 + (1 + i ) + 1]

But, at the end of the fifth year you have paid off the loan so this is zero. Hence

$50,000(1.1)5 - $P[(1.1)4 + (1.1)3 + (1.1)2 + (1.1 ) + 1] = 0        *

Finally

(1.1)4 + (1.1)3 + (1.1)2 + (1.1 ) + 1

is the sum of the first five terms of a geometric series so use you knowledge of geometric series to evaluate this sum and then solve equation * for P.

Penny