Math for Grownups (25 page)

But she will have some new ones, and with a little bit of research, she estimates these costs.

How does Becky figure out how much less she’ll be spending each year during retirement? First she needs to subtract her retirement expenses from her work expenses.

$33,750
–
$8,200
=
$25,550

Then she can subtract this amount from her current yearly expenses to find out how much she’ll spend each year during retirement.

$55,000
–
$25,550
=
$29,450

So, Becky will need to bring in $29,450 each year to cover her expenses during retirement. She has no pension, but she thinks she can count on Social Security. In fact, her Social Security statement came in the mail last week, showing that her monthly check would be $1,596 if she retired today. Annually, that would be

$1,596 • 12
=
$19,152

Clearly, Becky’s Social Security check isn’t going to cover her yearly expenses. But she does have savings. Is it enough to make up the difference?

Remember, at her financial planning workshop, Becky learned that she should have $15 to $20 in savings for every dollar of her annual shortfall. She can find this shortfall by subtracting her retirement income from her retirement expenses. And then she can multiply.

$29,450
–
$19,152
=
$10,298

Becky decides on using the conservative estimate of $20 per dollar. It’s much better to estimate high when finances are concerned.

$10,298 • 20
=
$205,960

But she only has $186,000 saved. She needs to put away another $19,960 before she can retire in confidence. How long will it take her to save that much?

For this, she needs to go back to her budget. Each year, she saves about $10,000. And she already has $186,000 in investments. Could she save enough in 1 year to retire?

Becky looks through her financial files for her latest investments summary. From that, she learns that she’s earning an average of 4% each year on her investments. She uses that figure to estimate what her investments will return next year:

$186,000 • 0.04
=
$7,440

With the $10,000 she saves each year, she could have an extra $17,440 at this time next year. That’s only $2,520 less than she needs. With a little creative trimming of this year’s budget—or by selling an extra house—she could very well be home with her grandbabies by this time next year.

Whether you’re saving up for a purchase or wondering when you can retire comfortably, it’s a good idea to know how quickly your savings will grow. The Rule of 72 tells you how many years it will take to double your money, if the interest it earns is compounded annually.

y
= 72 /
r

y
is years

r
is the interest rate

You have $2,500 invested in a money market account earning 1.5% interest each year. How long will it take for your investment to double?

y
= 72 / 1.5

y
= 48

So if you leave your $2,500 investment alone, it will become $5,000 in 48 years. (See why you need to feed your savings?)

You can use the Rule of 72 to shop for savings options. What if you need your $2,500 to double in 10 years? What interest rate do you need?

y
= 72 /
r

You want your investment to double in 10 years, so substitute 10 for
y
.

10
=
72
/
r

Now, it’s time for a little algebra. You need to isolate
r
. To do that, first multiply each side of the equation by
r
.

10
r
= 72

Now you can divide each side of the equation by 10.

r
= 72 / 10

Finish your division, and you get 7.2. Because of the way this formula is written, your answer is already a percent—so there’s no need to move the decimal point.

r
= 7.2%

But you don’t need to go through that entire process each time. It turns out that you can divide 72 by the number of years to find the rate. In other words,
r
= 72 /
y
.

But where does this magical formula come from? Why, from the formula for compounding annual interest, of course!

A
=
P
(1 +
r
)
n

A
is the amount saved

P
is principal

r
is the annual interest rate

n
is the number of years

Remember, you want to solve for
n
, which means isolating it on one side of the equation. And quite frankly, the math involved is messy and complex. But here’s a clue. How do you undo exponents?

With logarithms, of course! So if you were deriving (or proving) the Rule of 72, you’d need to use those old friends.

(See why we’re skipping it?)

Mentally, Lucas is ready to retire. But his bank account says, “No way.”

Lucas has a plan. If he plays the same lottery picks every week, he’s bound to get lucky, right? The more you play, the better chance you have of winning, right?

Poor Lucas. He’s making the same mistake that has troubled players of games of chance and lottery hopefuls for centuries. It’s something called the Gambler’s Fallacy.

The laws of probability say this: Past events do not change the probability that an event will happen in the future. In other words, Lucas can play those same numbers every week until the day he dies, but each week he has the same chance of winning.

Say it again: exactly the same chance.

Here’s an easier way to understand the problems with Lucas’s thinking. When you flip a coin, what are your chances that it will come up heads? Yep, 1 to 2, or ½. That’s because there are two possible outcomes (heads or tails), and you got one of the outcomes (heads).

Does the chance of getting heads change if you flip the coin over and over again? Actually, no. That’s because every single time you flip the coin, your odds of getting heads are still ½.

The same holds true for picking lottery numbers, although the odds of your winning are much, much lower and more complex to figure out:

Those variables aren’t excited. The exclamation points represent something called a
factorial
. Factorials are not hard to find. But they can produce really, really big numbers.

Here’s an easy way to think about factorials. If there are 40 possible lottery-number balls, how many are available after the first pick? 39, right? How many after the second pick? 38. And after the next pick? 37. And so on, and so on. If you keep at this until you get to 1, and then multiply all of the numbers, that’s the factorial.

Mathematically speaking—and using a much smaller number—factorials work like this:

4!
=
4 • 3 • 2 • 1
=
24

The lottery that Lucas plays has 40 balls, and he chooses 6 numbers. So, his odds of winning are

That’s a teeny-tiny chance. And here’s the thing. It doesn’t matter how often he plays the lottery. His odds are exactly the same from week to week.

Lucas’s retirement plan doesn’t stand a chance.

Getting a raise is always a good thing, right? Well, not always. If that extra cash in your paycheck bumps you into the next tax bracket, you could be giving more in taxes to Uncle Sam than you’d like.

But here’s the good news. You may find that negotiating better benefits, rather than a raise, will help you profit more from your good work.

First, you need to know whether your raise is going to put you over the top—into a higher tax bracket. The IRS offers easy-to-use tax bracket tables, which tell what your taxes will be, given your salary. (Just do an online search for “tax brackets,” and you’ll find dozens of websites with these tables.) Then you can figure out your new taxes and compare them to your raise.

Kyle is just about at his 5-year anniversary with his company—a big deal because the occasion is usually marked with a raise. He’s negotiated great benefits packages before, so he knows it’s important to be prepared to counter with another option—especially if the raise bumps him into another tax bracket.

Right now, Kyle is earning $33,750 a year. He’s heard that he’s up for a $6,000 raise, which would bump his salary up to $39,750. This will definitely push him into the next tax bracket. Will he actually see that raise, or will it all go to the IRS?

Kyle hasn’t found that right person or had kids, so he’s filing as a single taxpayer. Looking at the tax charts, he sees that in this new bracket, his taxes will be $4,750 plus 25% of the amount over $34,500. (That’s compared to his current bracket, where he pays $850 plus 15% of the amount over $8,500.) Kyle does the math to find out whether the raise is actually worth it.

To find out how much he currently pays in taxes, he subtracts $8,500 from $33,750, because he pays 15% only on the amount he makes over that.

$33,750
–
8,500
=
$25,250

He’ll pay 15% of that amount.

15% of $25,250

0.15 • $25,250
=
$3,787.50

Then he adds the $850 base tax amount.

$3,787.50
+
$850
=
$4,637.50.

That’s how much Kyle is paying in taxes now. (Of course, various deductions and exemptions might apply, but for purposes of comparison, we just need to know the tax he has to pay before those deductions and exemptions.)

If he gets the raise he thinks he’s going to get, he first needs to find the difference between his new salary and $34,500. (The tax schedule says he needs to take 25% of this number.)

$39,750
–
$34,500
=
$5,250

That’s not what Kyle will pay in taxes. He still needs to take 25% of this figure and then add $4,750 to it.

25% of $5,250

0.25 • $5,250
=
$1,312.50

$1,312.50
+
$4,750
=
$6,062.50

The raise means that he’ll be paying a lot more in taxes: $6,062.50 – $4,637.50 = $1,425.