A bank loan of $500 is arranged to be repaid in two years by equal monthly instalments?

Question

If Interest, calculated monthly is charged @ 11% p.a. on the remaining debt. calculate the monthly repayment if the first repayment is to be made one month after the loan is granted

Answer 1

$A = $500
r = 11% pa = 11/12% per month
t = 2 years = 24 months
M = monthly repayment

After 1st repayment
Amount owing = A1 = 500(1 + 11/12%) - M

After 2nd repayment
Amount owing = A2
= A1(1 + 11/12%) - M
= (500(1 + 11/12%) - M)(1 + 11/12%) - M
= 500(1 + 11/12%)² - M(1 + 11/12%) - M

After 3rd repayment
Amount owing = A3
= A3(1 + 11/12%) - M
=(500(1 + 11/12%)² - M(1 + 11/12%) - M)(1 + 11/12%) - M
= 500(1 + 11/12%)³ - M(1 + 11/12%)² - M(1 + 11/12%) - M
........................................

After 24th repayment
Amount owing = A24
= A23(1 + 11/12%) - M
= (500(1 + 11/12%)^24 - M(1 + 11/12%)^23 - ....... - M)(1 + 11/12%)³ - M(1 + 11/12%)² - M(1 + 11/12%) - M

But this has to equal 0 (after all it s now repaid!!)

So M(1 + 11/12%)^23 + ....... + M(1 + 11/12%)³ + M(1 + 11/12%)² + M(1 + 11/12%) + M = 500(1 + 11/12%)^24
ie (on reversing the sum)
M(1 + (1 + 11/12%) + (1 + 11/12%)² + (1 + 11/12%)³ + .......... + (1 + 11/12%)^23) = 500(1 + 11/12%)^24
Thus (summing the GP in brackets)
M((1 + 11/12%)^24 - 1)/((1 + 11/12%) - 1) = 500(1 + 11/12%)^24
ie M = (500(1 + 11/12%)^24 * 11/12%)/(1 + 11/12%)^24 - 1)
Dividing top and bottom by (1 + 11/12%)^24
M = 500*11/12%/(1 - (1 + 11/12%)^-24)
= 23.30 to the nearest cent

ie the monthly repayment is $23.30 (rounded to nearest cent)

Answer 2

23.21 per month