Just a Little Bit Each Month You’ve recently finished your MBA at the Darnit School. Naturally, you must purchase a new BMW immediately.

Question

Just a Little Bit Each Month You’ve recently finished your MBA at the Darnit School. Naturally, you must purchase a new BMW immediately. The car costs about $21,000. The bank quotes an interest rate of 15 percent APR for a 72-month loan with a 10 percent down payment. You plan on trading the car in for a new one in two years. What will your monthly payment be? What is the effective interest rate on the loan? What will the loan balance be when you trade the car in?

Accepted Answer

The cash flows on the car loan are in annuity form, so we only need to find the payment. The interest rate is 15%/12=1.25% per month, and there are 72 months. The first thing we need is the annuity factor for 72 periods at 1.25 percent per period:

Annuity present value factor=(1-Present value factor)/r

=[latex][1-(1/1.0125^{72})]/.0125[/latex]

=[1-(1/2.4459)]/.0125

=(1-.4088)/.0125

=47.2925

The present value is the amount we finance. With a 10 percent down payment, we will be borrowing 90 percent of $21,000, or $18,900. So, to find the payment, we need to solve for C in the following:

$18,900=C×Annuity present value factor

=C×47.2925

Rearranging things a bit, we have:

C=$18,900×(1/47.2925)

=$18,900×.02115

=$399.64

Your payment is just under $400 per month.
The actual interest rate on this loan is 1.25 percent per month. Based on our work in the chapter, we can calculate the effective annual rate as:

EAR=[latex](1.0125)^{12}-1[/latex]=16.08%

The effective rate is about one point higher than the quoted rate.
To determine the loan balance in two years, we could amortize the loan to see what the balance is at that time. This would be fairly tedious to do by hand.
Using the information already determined in this problem, we can instead simply calculate the present value of the remaining payments. After two years, we have made 24 payments, so there are 72-24=48 payments left. What is the present value of 48 monthly payments of $399.64 at 1.25 percent per month?
The relevant annuity factor is:

Annuity present value factor=(1-Present value factor)/r

=[latex][1-(1/1.0125^{48})]/.0125[/latex]

=[1-(1/1.8154)]/.0125

=(1-.5509)/.0125

=35.9315

The present value is thus:
Present value=$399.64×35.9315=$14,359.66
You will owe about $14,360 on the loan in two years.

Question 6.S-T.6: Just a Little Bit Each Month You’ve recently finished your M...

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