Elliott purchased a used pick-up truck for $9,500 He put $500. as a down payment and will repay the balance in monthly payments of $365. over the next 3 years. What is the APR of this loan?

The question will be solved using the present value of annuity formula.

Now, since he made a down payment of $500, The amount left to be paid is given $9,500 - $500 = $9,000.

The present value of annuity is given by:

[tex]PV=P\left[ \frac{1-\left(1+\frac{r}{t}\right)^{-nt}}{ \frac{r}{t} } \right][/tex]

where: PV = $9,000; P = $365; t = 12 payments per year, n = 3 years; r = APR and

[tex]\left[ \frac{1-\left(1+\frac{r}{t}\right)^{-nt}}{ \frac{r}{t} } \right][/tex]

is the Present value of annuity factor for [tex]nt[/tex] periods at [tex]\frac{r}{t}[/tex] interest rate per period.

Here, there are 3 years x 12 monthly payments = 36 periods.

Thus,

[tex]9000=365\left[ \frac{1-\left(1+\frac{r}{12}\right)^{-3\times12}}{ \frac{r}{12} } \right] \\ \\ \Rightarrow \left[ \frac{1-\left(1+\frac{r}{12}\right)^{-36}}{ \frac{r}{12} } \right]= \frac{9000}{365} =24.657534[/tex]

This means that the present value of annuity factor is 24.657534.

Using the present value of annuity table, the present value of annuity facot for 36 periods at 2% interest rate per period is 25.488842 and at 3% is 21.832253.

Let the interest rate that give the present value of annuity factor of 24.657534 be x, then interpolating, we have:


Let [tex]1+ \frac{r}{12} =x[/tex], then we have:

[tex] \frac{x-2}{3-2} = \frac{24.657534-25.488842}{21.832253-25.488842} = \frac{-0.831308}{-3.656589} =0.227345 \\ \\ \Rightarrow x-2=0.227345 \\ \\ \Rightarrow x=2+0.227345=2.227345.[/tex]

Thus, the interest rate per period (r/12) = 2.227345%.

Therefore, the APR = 12 x 2.227345 = 26.7281 ≈ 26.7%