Math for Grownups (7 page)

Don’t hyperventilate. Compare the two sides of the equation to see what changed. First, the exponent (that little tiny
y
) is negative on the left-hand side of the equation and positive on the right-hand side. You didn’t know how to deal with that negative exponent, so that’s what you wanted, right? The other change explains how you got there. The entire thing is now the bottom number in a fraction with 1 on the top. Whew! Looks scary, but if you break it down, it’s not so bad.

Let’s just put some numbers in to illustrate:

Or:

Here’s another way to think of it: Pretend that the exponent is positive and calculate it. Then divide your answer into 1.

Of course, with 0.00375
^-36
that rule doesn’t much help—unless you want to multiply 0.00375 by itself 36 times and then divide it into 1. It’s a much better plan to use a scientific or online calculator.

You may want to find the entire amount you’ll pay for the car so that you’ll know what you’re getting into. In that case, you’ll need to multiply the monthly payment by the number of months in the loan.

How much will dear Aunt Sally pay for her car over the life of the loan?

Her monthly payment is $496.06 on a 3-year loan. That means she’ll make a monthly payment for 36 months.

$496.06 • 36
=$17,858.16

Remember, she borrowed $16,800. That means she’s paying an extra $1,058.16 ($17,858.16 - $16,800) for the privilege of financing her purchase.

You’ve seen the commercials. To the tune of a well-worn 1970s rock song, a high-performance vehicle whips through hairpin turns with amazing dexterity. A deep-voiced announcer describes the car’s precision handling, smooth ride, soft leather seats, and walnut interior.

“Starting at $55,699 dollars,” the announcer croons. “Lease the new Agility for only $399 a month for 36 months.”

Even if you’re no geek, you may have wondered how the heck such an expensive car can have such a tiny monthly payment? The answer is easy: Leasing is way different from buying.

With a car lease, you’ll have a lower monthly payment and a shorter term. At the end of that period, you can either give the car back or buy it outright. That’s because you’re merely
leasing
(renting) the vehicle; it doesn’t belong to you.

When you lease a vehicle, you agree to pay a fee if you drive it over a prescribed number of miles. You also have to be careful not to damage the car or cause unnecessary wear and tear.

This is why leases aren’t great options for moms with 30-mile one-way commutes and sticky-fingered kids who drink from sippy-cups full of milk that invariably roll under the seat undetected.

The other thing to consider is how well the car retains its value. Leases work best with vehicles that keep their value—which is why they’re used most commonly by Mercedes and Jaguar drivers.

Still, leasing is a great option for many drivers. And if you fit in that category, it’s a smart idea to do some figuring before you stroll into the showroom.

The lease itself has a bunch of parts:

1. The
cap cost
or
lease price
is the price you’ve negotiated, so it should be less than the sticker price. Otherwise, you should just buy the car.

2. The
lease term
is the length of time that you’ll have the lease. It should be no longer than the warranty that comes with the vehicle. That’s because you don’t want your warranty to run out before your lease is up. Who wants to be responsible for repairs on a car she or he doesn’t own?

3. The
residual value
is what the vehicle is worth at the end of the lease period. This is usually expressed as a percent. Thus a car may have a 60% residual value at the end of 36 months. That means the car has an estimated depreciation of 40% (100% - 60%) over 3 years.

4. The
money factor
is basically the interest you’ll pay. Leasing a car is not like leasing a house, because, unlike real estate, cars lose value over time. (If it’s any consolation, equipment leases, such as leases on a backhoe or a copier, involve interest too.) But the money factor is not exactly interest, because with a car lease, you’re paying for the depreciation of the value of the car.

There will probably be other things to consider—such as dealer fees. Because those vary so much, we’ll leave them out of our calculations here. But you should research car leases in your area, and with each particular dealer and lender, before parking yourself on the other side of a car salesperson’s desk.

Once you’ve agreed on a lease price for the car, the dealer will present you with a calculation of your monthly lease payment. But it’s important to know how to make this calculation yourself, so you can catch any errors the dealer might make (and to ask about any added extras you don’t understand and didn’t anticipate). Knowing exactly what is going on may also give you an edge in negotiations.

Monthly lease payment
=
depreciation fee
+
finance fee
+
sales tax

Instead of creating a really huge formula, it makes more sense to break this problem down into its three parts. You’ll just need to remember to add the three together at the end.

Depreciation fee
=
(cap cost
-
residual value) / lease term

Finance fee
=
(cap cost
-
residual value) • money factor

Sales tax
=
monthly payment • sale tax rate

Crystal is in the market for a new car. Multistep problems like these can blow her mind. But as a professional organizer, she lives by the adage “A place for everything and everything in its place.”

Therefore, when she decided to lease her next vehicle—and to do some of the math before she waltzed into the showroom—she sat at her immaculate desk with a calculator, a printed description of the financing she thought she could get, and a stack of clean, crisp paper.

At the top of one paper she wrote “Depreciation Fee” and then its formula. On the second she wrote “Finance Fee” and its formula. And on the third she wrote “Sales Tax” and its formula. Then she turned to the “Depreciation Fee” paper and ignored the others.

To find the depreciation fee, she needs to know the cap cost, residual value, and lease term. In the language of math,

D
= (
c
-
r
) /
t

The last two values come from the research Crystal has done on financing. The car she wants has a 70% residual value over 3 years.

Although she knows the sticker price and the residual value percent, Crystal doesn’t know the residual value yet. Because the residual value is a percent of the sticker price, she just needs to multiply.

Residual value
=
sticker price • residual value percent

Residual value
=
$36,890 • 0.70

Residual value
=
$25,823

Now she has the residual value and the term. She needs to know the cap cost (lease price) of the vehicle.

Crystal, who is a great negotiator, thinks she could get the dealership down from a sticker price of $36,890 to $35,350. So, she writes two more figures on her paper:

Sticker price
=
$36,890

Cap cost (or lease price)
=
$35,350

She circles three numbers on her page: The lease term (36 months), the cap cost ($35,350), and the residual value ($25,823). These are the numbers that she’ll use in her formula.

Depreciation fee
=
(cap cost
-
residual value) / lease term, or
D
= (
c
-
r
) /
t

She plugs her numbers in to get

D
= ($35,350 - $25,823) / 36

She remembers that she needs to calculate what’s in the parentheses first:

D
= $9,527 / 36

Then she arrives at

D
= $264.64

She highlights the depreciation fee and sets this paper aside.

Now Crystal can find the finance fee—or the amount she’ll pay the leasing company for financing the lease. She turns her attention to that paper, looking at the formula

Finance fee
=
(cap cost
-
residual value
) • money factor, or
F
= (
c
-
r
) •
f

She knows the cap cost is $35,350 and the residual value is $25,823. She looks over the research she’s done on her financing and reads that she can get a financing package with a 0.00295 money factor. (Remember,
money factor
is like interest. It works a little differently, but if Crystal thinks of it as interest, she can get a good idea of how much she’ll pay for a car lease.)

Finance fee
=
($35,350
-
$25,823) • 0.00295

Finance fee
=
$9,527 • 0.00295

Finance fee
=
$28.10

She highlights this figure and moves on to her last piece of paper:

Sales tax
=
monthly payment • sale tax rate
, or
S
=
p
•
t

But Crystal doesn’t know her monthly payment yet, does she? Because she’s über-organized, it’s easy enough to find. All she needs to do is add the two numbers that are highlighted on the other pieces of paper: the depreciation fee and the finance fee.

Monthly payment
=
depreciation fee
+
finance fee

Monthly payment
=
$264.64
+
$28.10

Monthly payment
=
$292.74

Crystal lives in Illinois, so her sales tax rate is 6.25%. Now she can finally use the formula

Sales tax
=
monthly payment • sale tax rate
, or
S
=
p
•
t

S
= $292.74 • 0.0625

Sales tax
=
$18.30

She highlights the sales tax and then reaches for a clean sheet of paper. At the top, she writes the formula for finding the monthly lease payment:

Monthly lease payment
=
depreciation fee
+
finance fee
+
sales tax
, or
P
=
D
+
F
+
S

She reviews the other three pieces of paper, in order to be sure that she’s found all of the components. Then she begins plugging in the values:

P
= $264.64 + $28.10 + $18.30

Monthly lease payment
=
$311.04

Crystal circles this figure three times, gathers all of her papers and staples them. She tucks these into a crisp, new folder, types “Monthly Lease Payment Calculations” into her handy-dandy label maker, prints the label, and sticks it on the folder tab. Her purse and keys are helpfully hanging by the back door, so she grabs them and heads out to the car dealership—confident that she’ll be able to make a good deal.