Some lending institutions calculate the monthly payment M on...

Some lending institutions calculate the monthly payment $M$ on a loan of $L$ dollars at an interest rate $r$ (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where $k=[1+(r / 12)]^{12 t}$ and $t$ is the number of years that the loan is in effect. Car loan An automobile dealer offers customers no-downpayment 3-year loans at an interest rate of $10 \% .$ If a customer can afford to pay $\$ 500$ per month, find the price of the most expensive car that can be purchased.

Some lending institutions calculate the monthly payment $M$ on a loan of $L$ dollars at an interest rate $r$ (expressed as a decimal) by using the formula
$$M=\frac{L r k}{12(k-1)}$$
where $k=[1+(r / 12)]^{12 t}$ and $t$ is the number of years that the loan is in effect.
Car loan An automobile dealer offers customers no-downpayment 3-year loans at an interest rate of $10 \% .$ If a customer can afford to pay $\$ 500$ per month, find the price of the most expensive car that can be purchased.

Key Concepts

Loan Amortization

Loan amortization is the process of paying off a debt over time through regular payments which cover both principal and interest. This concept underlies the fixed monthly payment formula, where each payment is structured to gradually reduce the outstanding balance while accounting for the cost of borrowing.

Compound Interest

Compound interest is the process by which interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. In the context of loan payments, compound interest is reflected in the exponential factor k, which represents the growth of the loan balance over time due to periodic interest accumulation.

Conversion of Annual to Monthly Interest Rates

When interest is compounded on a frequency different from the stated annual rate, a conversion to a corresponding monthly rate is required. This is achieved by dividing the annual interest rate by 12, ensuring that payments and interest calculations are consistent with the monthly payment schedule.

Exponential Growth in Financial Formulas

Exponential growth in financial formulas captures the effect of repeated multiplication, as seen in the calculation of k, where the term [1 + (r/12)] is raised to the power of the total number of compounding periods. This principle is critical in determining the accumulated factor over the duration of the loan and is foundational in evaluating the impact of compounding on the total loan cost.

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