Loan Payments The formula below gives the monthly payment m...

Loan Payments The formula below gives the monthly payment $m$ on a loan as a function of the amount borrowed $A$, the annual rate of interest $r$ (expressed as a decimal, and the number of months $n$ of the loan. $$ m=\frac{A\left(\frac{r}{12}\right)\left(1+\frac{r}{12}\right)^{n}}{\left(1+\frac{r}{12}\right)^{n}-1} $$ What is the monthly payment on a loan of $\$ 1500$ at $8 \%$ annual interest paid over 18 months?

Loan Payments The formula below gives the monthly payment $m$ on a loan as a function of the amount borrowed $A$, the annual rate of interest $r$ (expressed as a decimal, and the number of months $n$ of the loan.
$$
m=\frac{A\left(\frac{r}{12}\right)\left(1+\frac{r}{12}\right)^{n}}{\left(1+\frac{r}{12}\right)^{n}-1}
$$
What is the monthly payment on a loan of $\$ 1500$ at $8 \%$ annual interest paid over 18 months?

Key Concepts

Loan Amortization

Loan amortization is the process of repaying a debt over time through regular, fixed payments that cover both principal and interest. This concept involves spreading out the total loan amount along with the accrued interest into installments, allowing borrowers to understand how much of each payment goes towards reducing the principal versus paying interest.

Fixed-Rate Loan Payment Formula

The fixed-rate loan payment formula is a mathematical expression used to calculate the constant periodic payment necessary to fully pay off a loan over a specified term. This formula accounts for the effects of compound interest, ensuring that each payment is sufficient both to cover the interest accrued during the period and to reduce the remaining principal, thereby amortizing the loan completely by the end of the term.

Interest Rate Conversion

Interest rate conversion involves adjusting an annual interest rate to its equivalent periodic rate, especially when loan payments occur more frequently than once per year (such as monthly). This conversion is essential because it allows the formula to correctly apply the interest rate for each payment period, ensuring that the impact of interest accumulation is accurately represented.

Compound Interest

Compound interest is the principle that interest is earned on both the initial principal and the accumulated interest from previous periods. In the context of loan payments, compound interest calculations are fundamental as they determine the total amount of interest accrued over time, which directly affects the amount of each payment required to amortize the loan.

Transcript

00:01 All right, everybody, we've got a loan problem here.

00:02 I know it looks a little complicated.

00:04 It's really not going to be that bad, though.

00:06 That top part in black is just some formula that they give us.

00:10 All right? that's the formula for the monthly payment of a loan.

00:14 Now, on this exact problem, they tell you you have a $1 ,500 loan, 8 % interest, over 18 months, and you want to know what does that make your monthly payment.

00:24 Okay, this is really just a plug -in problem here.

00:27 A complicated one, but just a plug -in problem.

00:28 So if it's a $1 ,500 loan, that's what your a represents.

00:33 The a represents how long the loan is for.

00:36 I'm sorry, a represents how much the loan is for.

00:39 And so that's $1 ,500.

00:40 So let me just come up here.

00:42 I'm going to take that a and let me replace it with $1 ,500.

00:49 Right.

00:51 Now here for this next part.

00:52 It's 8 % interest.

00:54 R is your interest, but you want it as a decimal, not as a percent.