Problem with principal payment

Question

Hey I am supposed to determine the principal payment of the following situation:

Loan amount: 50 000$

Interest rate: 5% p.a.

Number of years: 30

What I did:

I calculated the interest monthly rate: $(1+0,05)^{\frac{1}{12}}-1=0,004074124$ and I used formula:

$A=P*\frac{i}{1-\left ( 1+i \right )^{-n}}\leftrightarrow A=50000*\frac{0,004074124}{1-\left ( 1+0,004074124 \right )^{-30*12}}\leftrightarrow A=265,027 $ dollars

But the correct solution should be 268,41.

Can anyone please tell me, where I made the mistake?

Answer 1

The formula you used for the calculation is incorrect. The correct formula for the monthly payment is

$$A =\$50000\cdot\frac{\frac{0.05}{12}(1+\frac{0.05}{12})^{30*12}}{(1+\frac{0.05}{12})^{30*12}-1} = \$268.41$$

Answer 2

Your calculation is correct if the $5\%$ interest rate is an effective annual rate of interest; that is to say, $i = 0.05$. But if it is a nominal rate compounded monthly, i.e. $$i^{(12)} = 0.05,$$ then your level payment is not the same. The statement of the question is not clear about what $5\%$ really means, nor is it explicitly stated that payments are made on a monthly rather than annual schedule.

Using a nominal rate of interest compounded monthly, we have $j = i^{(12)}/12 = 0.00416667$ as the effective monthly rate of interest, and $$50000 = K a_{\overline{360}\rceil j} = \frac{1 - (1 + 0.05/12)^{-360}}{0.05/12} = 186.282,$$ where $K$ is the level monthly payment on the loan. This is how we get the claimed answer of $268.411$. But without explicitly stating what the interest rate means, it is a bit unfair to be expected to know which calculation was intended.