To use a spreadsheet, we use the =PMT
formula. For a loan, the loan amount is the present value and the future value is 0, indicating that the loan will be paid off. Teresa is making a down payment, so we also need to subtract that from the cost of the car to find the loan amount:
\(\$15,000 - \$3,000 = \$12,000\)
Her loan amount is $12,000. For the 3-year loan at 2.75% APR, we enter:
=PMT(0.0275/12, 12*3, 12000, 0)
and get a result of $347.65.
For the formula, we use the one solved for \(d\text{:}\)
\(r=.0275\text{,}\) for 2.75% annual rate
\(n=12\text{,}\) monthly payments
\(t=3\text{,}\) for 3 years
\(P=12000\text{,}\) since she can pay $3,000 of the $15,000
\begin{align*}
d\amp=\frac{P\left(\frac{r}{n}\right)}{\left(1-\left(1+\frac{r}{n}\right)^{-nt}\right)}\\
\amp=\frac{12000\left(\frac{0.0275}{12}\right)}{\left(1-\left(1+\frac{0.0275}{12}\right)^{-12\cdot 3}\right)}\\
\amp\approx\$347.65
\end{align*}
Teresa’s car payment would be $347.65.
Now for the 5-year loan at 4% APR, we enter:
=PMT(0.04/12, 12*5, 12000, 0)
and we get $221.00.
To use the formula, we have:
\(r=.04\text{,}\) for 4% annual rate
\(n=12\text{,}\) monthly payments
\(t=5\text{,}\) for 5 years
\(P=12000\text{,}\) the loan amount
\begin{align*}
d\amp=\frac{P\left(\frac{r}{n}\right)}{\left(1-\left(1+\frac{r}{n}\right)^{-nt}\right)}\\
\amp=\frac{12000\left(\frac{0.04}{12}\right)}{\left(1-\left(1+\frac{0.04}{12}\right)^{-12\cdot 3}\right)}\\
\amp\approx\$221.00
\end{align*}
Now let’s compare the loans by finding out how much Teresa would pay in interest for each loan.
For the 3-year loan at 2.75% APR, her payments would total:
\(\$347.65(12)(3)=\$12,515.40\text{.}\) Her interest would be $515.40.
For the 5-year loan at 4% APR, her payments would total:
\(\$221.00(12)(5)=\$13,260.00\text{.}\) Her interest would be $1,260.00.
There are two main differences between these two loans: the monthly payments and the total paid over the life of the loans. The first loan has a higher monthly payment by $126.65 per month. However, she would pay $744.60 less in interest.