The unpaid balance of an installment loan is equal to the present value of the remaining payments. The unpaid balance, P, is given by $P=P M T \frac{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}{\left(\frac{r}{n}\right)}$, where PMT is the regular payment amount, r is the annual interest rate, n is the number of payments per year, and t is the number of years remaining in the loan. a. Use the loan payment formula to derive the unpaid balance formula. b. The price of a car is $24,000. You have saved 20% of the price as a down payment. After the down payment, the balance is financed with a 5-year loan at 9%. Determine the unpaid balance after three years. Round all calculations to the nearest dollar.

Solution

Verified$\textbf{a.}$

Start with the formula:

$PMT=\dfrac{P\left(\dfrac{r}{n}\right)}{1-\left(1+\dfrac{r}{n}\right)^{-nt}}$

Multiply both sides by $1-\left(1+\dfrac{r}{n}\right)^{-nt}$:

$PMT\left[1-\left(1+\dfrac{r}{n}\right)^{-nt}\right]=P\left(\dfrac{r}{n}\right)$

Divide both sides by $\left(\dfrac{r}{n}\right)$:

$\dfrac{PMT\left[1-\left(1+\dfrac{r}{n}\right)^{-nt}\right]}{\left(\dfrac{r}{n}\right)}=P$

or

$\color{#c34632}P=\dfrac{PMT\left[1-\left(1+\dfrac{r}{n}\right)^{-nt}\right]}{\left(\dfrac{r}{n}\right)}\color{white}\tag{1}$

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