323.5

5.You are interested in a $20,000 car. One option for paying for it is a 48 month loan with an 8% APR and monthly payments starting one month after the purchase of the car.

• What would your payments be? (5)

This is a straight-up annuity. We figured out car payments for a $40,000 Boxster in class. This is exactly the same concept. Here: PV=$20000, r=.08/12, n=48, CF=?

\( \LARGE CF = \frac{20{,}000} { \left [ \frac{1}{\frac{.08}{12}} - \frac{1}{\frac{.08}{12} \left (1+\frac{.08}{12} \right )^{48}} \right ] } = 488.26 \)

• The dealer offers you a special deal: Either $2000 cashback ($2000 off of the price of the car), or a reduced interest rate of 7.5% on the car loan. Which should you take? (6)

 This is very similar to question 52 from Chpt 3. Which will give you lower payments? $2000 cashback means you will have to finance only $18000 at 8% APR, so your payments will be:

\( \LARGE CF = \frac{18{,}000} { \left [ \frac{1}{\frac{.08}{12}} - \frac{1}{\frac{.08}{12} \left (1+\frac{.08}{12} \right )^{48}} \right ] } = 439.43 \)

At 7.5%, you would still have to finance the whole $20,000, but at a lower rate:

\( \LARGE CF = \frac{20{,}000} { \left [ \frac{1}{\frac{.075}{12}} - \frac{1}{\frac{.075}{12} \left (1+\frac{.075}{12} \right )^{48}} \right ] } = 483.58 \)

You should definitely take the cashback!

• You are also considering leasing the car for 48 months instead of buying it. The car will be worth $10,000 at the end of the lease, and the lease is figured based on an 8% APR and a $20,000 purchase price. If the first payment is due one month after you take the car, what will your lease payments be? (6)

This is just like the Boxster example we did in class and question 3 on 2 1 midterm. It is usually helpful to diagram the cash flows:

Today	1	2	48
-20000	CF	CF	CF+10000

 

Basically what is happening with a lease is you are financing the price of the car minus the present value of the value of the car at the end of the lease when you give it back. So, just like the first part of this question, you are figuring a 48 month annuity, but instead of the PV being 20,000, it is:

\( \LARGE 20000 - \frac{10000}{\left( 1 + \frac{.08}{12} \right)^{48}} = 12{,}730.79 \)

so the CF for your annuity is:

\( \LARGE CF = \frac{12{,}730.79} { \left [ \frac{1}{\frac{.08}{12}} - \frac{1}{\frac{.08}{12} \left (1+\frac{.08}{12} \right )^{48}} \right ] } = 310.80 \)

Back to Practice Problems