Annuity calculation; what is wrong with my calculation to the following question?

Question

First off, I will be honest that I am rather confused not to much with the concepts but more of the language used in questions in finance. So I must acknowledge the possibility that I have misunderstood something. But here's the question.

James is buying a computer for $\$1499$ at cash price and he is offered a credit agreement. The terms of the agreement is that $\$299$ is to be paid down on the day of purchase and the remaining amount is to be repaid by 24 equal monthly repayments. The first installment is due one month from the day of purchase. If the APR charges on this agreement is $22.95\%$, what is the monthly payments?

My answer(I looked up APR and it stands for "Annual percentage rate" which actually further confused me because on wiki, it says it corresponds to nominal APR and sometimes to an effective APR. I thought "nominal" and "effective" were different e.g. nominal interest rate, effective interest rate. But oh well, just took it as the effective interest rate)

The outstanding amount is $\$1200$. The payment is due i a month's time so this is an immediate annuity. If the effective interest rate is $22.95\%$, say $i$ is the monthly nominal interest rate, then $(1+i)^{12}=1.2295$ and thus $i=0.0173663...$ .

Let $C$ denote the monthly equal payment, then we want such a $C$ that $1200= C(u+u^2+u^3+...+u^{24})$, where $u=\frac{1}{1+i}$, the discount factor per month. As a geometric series, the sum of $u$ can be computed and hence we find $C$.

But from this method, I got value of $C$ that is wrong from the correct answer. I don't know how they got the value as it is just stated there.

Can someone help me and tell me if I am having am misunderstanding of a concept or the question. Also, likely that I have treated the APR thing wrongly.

If so, please explain what this is, how different it is from (annual) effective rate of interest.

Thank you for your help

Answer 1

The APR represents the effective annual rate of interest $i = 0.2295$. Therefore, the effective monthly rate of interest $j$ satisfies $$(1 + j)^{12} = 1+i,$$ or $$j = (1.2295)^{1/12} - 1 \approx 0.017366370551.$$ The outstanding balance after the down payment is the amount of the loan, which is $1499 - 299 = 1200$, representing the present value of the amount to be paid. Therefore, $$1200 = Ka_{\overline{24}\rceil j},$$ or $$K = \frac{1200 j}{1 - v^{24}} \approx 61.5683,$$ where $v = (1+j)^{-1}$ is the present value discount factor for one month.

Answer 2

In general the formula is

$$C_0\cdot \left(1+\frac{i}{12} \right)^{m}=x\cdot\large{ \frac{1-\left(1+\frac{i}{12} \right)^m}{-\frac{i}{12} }}$$

$m$ is the duration of the credit in months. And $i$ is the yearly interest rate.

In your case

$$1200\cdot \left(1+\frac{0.2295}{12} \right)^{24}=x\cdot\large{ \frac{1-\left(1+\frac{i}{12} \right)^{24}}{-\frac{0.2295}{12} }}$$

If you solve for x the monthly payment is $\$ 62.8182... \approx \$ 62.82$

I have checked the result by using Excel. It looks right.