finance question

[mishii86]

You wish to borrow $2,000 to be repaid in 12 monthly installments of $189.12. The annual interest rate is:

A. 24%

B. 8%

C. 18%


D. 12%

 

[chrisr]

twelve of $189.12 is how much?
then, how much more than $2000 is that?
how do you express that difference as a percentage of $2000?

 

[Denis]

Cannot be solved directly: iteration required; will come out to 24%.

You can always try out the 4 given rates in the formula
P = Ai / [1 - 1/(1+i)^n]
P = monthly payment
A = amount borrowed
n = number of months
i = interest rate

 

[chrisr]

Compounded,
calculated month by month,
sorry, thought this was just a basic question on overall effective interest rate.

Yes, Denis is right of course.

A = loan amount,
p = monthly payments,
i = yearly interest rate, so i/12 = monthly interest rate.

After 1 month and 1 repayment, loan stands at A(1+i) - p = B
After 2 months and 2 repayments, loan stands at B(1+i) - p = A(1+i)[sup:3m3gopsu]2[/sup:3m3gopsu] -p(1+i) - p = C.
After 3 months and 3 repayments, loan stands at C(1+i) - p = A(1+i)[sup:3m3gopsu]3[/sup:3m3gopsu] -p(1+i)[sup:3m3gopsu]2[/sup:3m3gopsu] -p(1+i) - p.

After 12 months and all 12 repayments, the loan is paid off

A(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu] = p(1+i)[sup:3m3gopsu]11[/sup:3m3gopsu]+p(1+i)[sup:3m3gopsu]10[/sup:3m3gopsu]+p(1+i)[sup:3m3gopsu]9[/sup:3m3gopsu]+....+p(1+i)+p

A(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu] =p{1+(1+i)+(1+i)[sup:3m3gopsu]2[/sup:3m3gopsu]+.....(1+i)[sup:3m3gopsu]11[/sup:3m3gopsu]}.

In brackets is a basic geometric sum of 12 terms with T[sub:3m3gopsu]1[/sub:3m3gopsu]=1 and multiplier = (1+i)
which sums to {1-(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu]}/{1-(1+i)} = {(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu]-1}/i.

A(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu] = p{(1+i)[sup:3m3gopsu]12[/sup:3m3gopsu]-1}/i.

These are equal for A = $2000, p = $189.12 and i = 0.24/12