Skill: Compare financing options

To compare financing options, do not look only at the periodic payment or advertised interest rate. Compare:

• the amount borrowed, after the down payment;
• the interest rate and payment frequency;
• the number of payments;
• fees;
• the total amount paid.

A longer loan term often has a smaller payment but a greater total cost.

Example

A laptop costs 20{,}000$. You make a$$2{,}000$ down payment, so the amount financed is

$$P=20{,}000-2{,}000=\$18{,}000.$$

You are comparing two fixed-rate loans. Both use monthly payments.

Option	Annual interest rate	Term	Fee
A	$6\%$	3 years	200$
B	$5.5\%$	5 years	0$

Step 1: Find the monthly rate and number of payments

For a monthly loan,

$$i=\frac{\text{annual rate}}{12} \qquad\text{and}\qquad n=(\text{years})(12).$$

For Option A,

$$i=\frac{0.06}{12}=0.005, \qquad n=3(12)=36.$$

For Option B,

$$i=\frac{0.055}{12}=0.004583, \qquad n=5(12)=60.$$

Step 2: Calculate each monthly payment

For a standard fixed-rate loan, use

$$M=P\left(\frac{i}{1-(1+i)^{-n}}\right),$$

where $M$ is the periodic payment.

For Option A,

$$M_A =18{,}000\left(\frac{0.005}{1-(1.005)^{-36}}\right) \approx \$547.59.$$

For Option B,

$$M_B =18{,}000\left(\frac{0.004583}{1-(1.004583)^{-60}}\right) \approx \$343.82.$$

Option B has the smaller monthly payment.

Step 3: Find the total of the loan payments

Multiply the payment by the number of payments.

For Option A,

$$36(\$547.59)\approx \$19{,}713.24.$$

Adding the 200$ fee,

$$\text{loan cost for A} =\$19{,}713.24+\$200 =\$19{,}913.24.$$

For Option B,

$$60(\$343.82)\approx \$20{,}629.20.$$

There is no fee, so

$$\text{loan cost for B}=\$20{,}629.20.$$

Step 4: Include the down payment

The total amount paid for the laptop is the down payment plus the loan cost.

Option	Down payment	Payments and fees	Total paid
A	2{,}000$	19{,}913.24$	21{,}913.24$
B	2{,}000$	20{,}629.20$	22{,}629.20$

Option A is cheaper overall by approximately

$$\$22{,}629.20-\$21{,}913.24=\$715.96.$$

Although Option B advertises a lower interest rate and has a much smaller monthly payment, its five-year term means interest is charged for more months. Therefore, Option A is the better financing choice if the goal is to pay the least overall.