8.4 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.
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. 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.
Loan Formula
- is the balance in the account at the beginning (starting amount, present value, principal or amount of loan.)
- is the loan payment (the amount you take out each year, each month, etc.)
- is the annual interest rate (in decimal form for this formula)
- is the number of compounding periods in one year
- 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 to 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 of a Loan
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 of a car can you afford? In other words, what amount loan can you pay off with $200 per month?
In this example,
the monthly loan payment
3% annual rate
since we’re doing monthly payments, we’ll compound monthly
since we’re making monthly payments for 5 years
We’re looking for P; the starting amount of the loan.
Putting this into the equation:
You can afford an $11,120 loan.
You will pay a total of $12,000 ($200 per month for 60 months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, you’re paying $12,000-$11,120 = $880 interest total.
Using Technology
Similar to the annuities in the last section, we can use TVM Solver on a calculator or Excel to solve these problems as well.
On a TI 83/84 Calculator
Once again, we go to APPS and 1: Finance and 1: TVM Solver. The differences now is that our PV is positive because we are given the money to buy something, PMT is now negative because this is money we are giving the bank each month (or k times a year). The FV will be 0 because our ending amount should be nothing; we should be paying off the loan. (Later in the section, this might change because we can look at loan balances at any time.) We put the cursor on which we want to solve and hit ALPHA-ENTER.
For the previous example:
| Enter the