5.5: Loans
It is a good idea to try to save up money to buy large items or find 0% interest deals so you are not paying interest. However, this is not always possible, especially when buying a house or car. That is when it is important to understand how much interest you will be charged on your loan.
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Loan Payment Formula : \[P=PMT\left[\frac{1-\left(1+\frac{r}{n}\right)^{-nt}}{\frac{r}{n}}\right] \nonumber \]where, P = Present value (Principal) PMT = Payment r = Annual percentage rate (APR) changed to a decimal t = Number of years n = Number of payments made per year |
Example \(\PageIndex{1}\): Loan Payment Formula:
Ed buys an iPad from a rent-to-own business with a credit plan with payments of $30 a month for four years at 14.5% APR compounded monthly. If Ed had bought the iPad from Best Buy or Amazon it would have cost $500. What is the price that Ed paid for his iPad at the rent-to-own business? How much interest did he pay?
PMT = $30, r = 0.145, t = 4, n = 12
The price Ed paid for the iPad was $1,087.83. That’s a lot more that $500!
Also, the total amount he paid over the course of the loan was . Therefore, the total amount of interest he paid over the course of the loan was $1440 - $1087.83 = $352.17.
For some problems, you will have to find the payment instead of the present value. In that case, it is helpful to just solve the loan payment formula for PMT . Since PMT is multiplied by a fraction, to solve for PMT , you can just multiply both sides of the formula by that fraction. You should just think of the loan payment formula in two different forms, one solving for present value, P , and one solving for payment, PMT .
Example \(\PageIndex{2}\): Loan Formula—Finding Payment
Jack goes to a car dealer to buy a new car for $18,000 at 2% APR with a five-year loan. The dealer quotes him a monthly payment of $425. What should the monthly payment on this loan be?
P = $18,000, r = 0.02, n = 12, t = 5
Jack should have a monthly payment of $315.50, not $425.
The dealer is trying to sell Jack the car for a total of $25,500 with principal and interest. What should the total principal and interest be with the $315.50 monthly payment?
Therefore, the dealer is trying to get Jack to pay $25,500 - $18,930 = $6,570 in additional principal and interest charges. This means that the quoted rate of 2% APR is not accurate, or the quoted price of $18,000 is not accurate, or both.
Example \(\PageIndex{3}\): Loan Formula—Mortgage
Morgan is going to buy a house for $290,000 with a 30-year mortgage at 5% APR. What is the monthly payment for this house?
P = $290,000, r = 0.05, n = 12, t = 30
The monthly payment for this mortgage should be $1556.78.
It is also very interesting to figure out how much Morgan will end up paying overall to buy this house. It is rather easy to calculate this
Therefore, Morgan will pay $560,440.80 in principal and interest which means that Morgan will pay $290,000 for the principal of the loan and $560,440.80 - $290,000 = $270,440.80 in interest. This is enough to buy another comparable home. Interest charges add up quickly.
Example \(\PageIndex{4}\): Loan Formula—Refinance Mortgage
If Morgan refinanced the $290,000 at 3.25% APR what would her monthly payments be?
P = $290,000, r = 0.0325, n = 12, t = 30
If Morgan refinanced the mortgage at 3.25% APR, the monthly payment would now be $1262.10 instead of $1556.78.
How much money would Morgan save over the life of the loan at the new payment amount?
Then, subtract $560,440.80 - $454,356 = $106,084.80.
Morgan would save $106,084.80 in interest because of refinancing the loan at 3.25% APR.
Example \(\PageIndex{5}\): Loan Formula—Mortgage Comparison
With a fixed rate mortgage, you are guaranteed that the interest rate will not change over the life of the loan. Suppose you need $250,000 to buy a new home. The mortgage company offers you two choices: a 30-year loan with an APR of 6% or a 15-year loan with an APR of 5.5%. Compare your monthly payments and total loan cost to decide which loan you should take. Assume no difference in closing costs.
Option 1 : First calculate the monthly payment:
The monthly payment for a 30-year loan at 6% interest is $1498.88.
Now calculate the total cost of the loan over the 30 years:
The monthly payments are $1498.88 and the total cost of the loan is $539,596.80.
Option 2 : First calculate the monthly payment:
The monthly payment for a 15-year loan at 5.5% interest is $2042.71.
Now calculate the total cost of the loan over the 15 years:
The monthly payments are $2042.71 and the total cost of the loan is $367,687.80.
Therefore, the monthly payments are higher with the 15-year loan, but you spend a lot less money overall.