Refer to the following: A local business borrows 200,000 to...

Refer to the following: A local business borrows $\$ 200,000$ to purchase property. The loan has an annual interest rate of $8 \%$ compounded monthly and a minimum monthly payment of $\$ 1467$. a. Approximately how many years will it take the business to pay off the loans if only the minimum payment is made? b. How much interest will the business pay over the life of the loan if only the minimum payment is made?

Refer to the following: A local business borrows $\$ 200,000$ to purchase property. The loan has an annual interest rate of $8 \%$ compounded monthly and a minimum monthly payment of $\$ 1467$.
a. Approximately how many years will it take the business to pay off the loans if only the minimum payment is made?
b. How much interest will the business pay over the life of the loan if only the minimum payment is made?

Key Concepts

Compound Interest

Compound interest refers to the process in which interest is not only calculated on the initial principal, but also on the accumulated interest from prior periods. In the context of loans, this means that each month's interest calculation is based on the remaining balance, which includes the accrued interest, resulting in exponential growth of the amount owed if not reduced by repayments.

Loan Amortization

Loan amortization is the process of paying off a debt over time through regular payments. Each payment typically covers both the interest accrued and a portion of the principal, so that the remaining balance decreases gradually until the loan is completely paid off. Amortization schedules are used to track how much of each payment goes towards interest versus principal over the life of the loan.

Annuity Payment Formula

The annuity payment formula, also known as the present value of an annuity formula, is used to calculate the fixed periodic payment required to repay a loan, given a specific interest rate and duration. This formula takes into account the time value of money and the effect of compound interest, by determining how many payments (or how long it will take) are necessary to fully amortize the loan under a given interest rate with fixed payments.

Transcript

00:04 Problem 45 here is this related to again a payment's loan payment for a of a property whose so the loan amount is is given as for the principal this case is $200 ,000 the interest rate has given as 8 % let's like this as the right position and the payments are monthly so the period is 12 periods per year so in the first question the the monthly payment is decided to be dollars 1 ,467 and to find that and the question is how long is how long will it take to make this payment and we're going to use someone use that standard logarithmic model and given these quantities and calculate time 40.

01:12 So the time is given as negative l n 1 minus p times r divided by n r divided by n the times natural log of 1 plus the interest rate divided by the frequency of payments we have pr and r we have everything here like in these numbers we get 200 ,000 for the p rate as 8 % there 12 times the monthly payment which is 1 4 .67 read by well and the natural log 1 plus 0 .08 and my answer was is t equals 30 years this is a very classic classic situation most homes that you buy usually they are they are purchased on a 30 year agreement and it fits right into the livable number okay the second part of the question is what is the second part is what is how much interest will the business pay over the life of the loan only if the payment the minimum is paid so in other words at this rate what is the what is the interest the interest paid you'll be surprised by this but by now it's common knowledge we calculate the interest paid what we should do is calculate i'm going to calculate or find the total amount amount that's paid or total amount paid total payment paid either way right so calculate this guy and then once you do that subtract subtract and the principle to find input that's here this is a way to do it okay how this is the how to do it and we already have let's see do we have the he's paying for 30 years the business owner is paying for 30 years an amount of one four six seven per month the total payment paid is the monthly which is 146, 7 times 12 is a year, and do that for 30 years.

04:50 And you'll come up with a big number like this...

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