Loans

Learning Outcomes

• Use the loan formula to calculate loan payments, loan balance, or interest accrued on a loan

Conventional Loans

In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.

new topic, same formula!

Mathematical formulas sometimes overlap, applying to more than one application. All the exercises and examples in this section use the same formula and techniques that you’ve already seen.

One great thing about loans is that they use exactly the same formula as a payout annuity. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests$10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.

Loans Formula

$P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}$

• P0 is the balance in the account at the beginning (the principal, or amount of the loan).
• d is your loan payment (your monthly payment, annual payment, etc)
• r is the annual interest rate in decimal form.
• k is the number of compounding periods in one year.
• N is the length of the loan, in years.

Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.

When do you use this?

The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.

• Compound interest: One deposit
• Annuity: Many deposits
• Payout Annuity: Many withdrawals
• Loans: Many payments

Example

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive a car can you afford? In other words, what loan amount can you pay off with$200 per month?

Details of this example are examined in this video.

Solving for $d$

In the example above, you computed $P_{0}$, the initial loan amount. In the example below, you are given the loan amount and must solve for the amount of the monthly payment, $d$. Use the same technique that you used in the previous sections.

Calculating the Balance

With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.

To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will not have paid off$12,000 of the loan balance.

To determine the remaining loan balance, we can think “how much loan will these loan payments be able to pay off in the remaining time on the loan?”

Solving for Time

Recall that we have used logarithms to solve for time, since it is an exponent in interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.

Try It

Joel is considering putting a $1,000 laptop purchase on his credit card, which has an interest rate of 12% compounded monthly. How long will it take him to pay off the purchase if he makes payments of$30 a month?

FYI

Home loans are typically paid off through an amortization process, amortization refers to paying off a debt (often from a loan or mortgage) over time through regular payments. An amortization schedule is a table detailing each periodic payment on an amortizing loan as generated by an amortization calculator.

If you want to know more, click on the link below to view the website “How is an Amortization Schedule Calculated?” by MyAmortizationChart.com. This website provides a brief overlook of Amortization Schedules.