Money math you need to know

By Liz Pulliam Weston

In "9 money rules to live by," I listed the economic and financial concepts I thought everybody should know.

In "16 favorite money rules of thumb," I gave you some rough guidelines for figuring out how much to save for retirement, borrow for homes and spend on cars, among other things.

The first column was essentially Economics 101. I covered ideas such as opportunity cost and the hedonic treadmill that might sound pretty esoteric but actually have profound effects on our daily lives.

The second column was born from all the "just give me the answer" pleas I get from readers for simple guidelines they could use to manage their money.

Now I'm going to round out the trilogy by teaching you some math tricks you can use to get a handle on your investments, calculate the cost or benefit of a financial move and impress your friends at parties.

None of these tricks requires more than simple arithmetic (addition, subtraction, division and multiplication), and some of them you can do in your head. Some are so simple they can be explained in a few sentences; if it takes longer and you're in a hurry, you can read the last paragraph of each section to get the rule of thumb.

The rule of 72

Need a quick way to estimate how long it will take for your money to double at a given rate of return?

Divide the return into 72. So if you're averaging an 8% annual return, it will take you nine years to double your money (72 divided by 8 equals 9).

The rule of 70

Inflation erodes the buying power of a dollar, so that eventually it will buy only half of what it used to.

Want to know how quickly your money will lose half its buying power? Divide 70 by the expected inflation rate. If it's 3.5%, your dollar will be worth 50 cents in 20 years (70 divided by 3.5 equals 20). If inflation soars to 10%, your money's value is halved in seven years.

Understand the value of compounding

It's been called the Eighth Wonder of the World, but many people still don't grasp how amazingly investment returns can add up over time.

One way to dramatically illustrate the point is to use the example of the doubling penny. If I give you a penny at the start of the month and promise to double it every day, you'll have $10.7 million after 30 days.

Of course, you're not going to double your money every day or even every year. But 8% or 9% is a reasonable rate of return to expect over the long run on a diversified portfolio of stocks, bonds and cash, so doubling your money every eight or nine years (remember the rule of 72) is well within the range of possibility.

The following chart demonstrates how quickly a $100 investment can grow given different interest rates and time periods.

How $100 can grow

After:	6%	8%	10%
10 years	$179	$216	$259
20 years	$320	$466	$673
30 years	$574	$1,006	$1,745
40 years	$1,028	$2,172	$4,525

This chart can be used the other way around to illustrate future gains you lose when you aren't invested. If you withdraw $100 from a retirement account at age 25, for example, you're giving up more than $2,000 of future retirement money, assuming 8% average annual returns over the next 40 years.

Here's a way to further boil it down: Every dollar you invest can grow to $2 in 10 years, $5 in 20 years, $10 in 30 years and $20 in 40 years, assuming an 8% average annual return.

What's your time worth?

If you work, you need to understand the basic concept that you're trading your time (a nonrenewable resource) for money. So when you buy something, you're trading minutes or hours or days of your life to procure that thing.

One way to do that is to simply look at your hourly wage. If you make $20 an hour and something costs $40, you can figure that it takes two hours of your life to pay for that thing. (A fast way to estimate your hourly wage if you're salaried: Knock off the last three zeros and halve the result, so $50,000 becomes $50, which halved is $25. The rate you get will be a bit high but in the ballpark.)

Continued: Don't forget about taxes

Still, using gross pay ignores the fact that we pay taxes on our incomes. Because of taxes, we have to earn more than a dollar -- typically, substantially more -- to clear each dollar we use to buy something.

For these calculations, we'll use marginal tax rates -- essentially, how much in taxes you'd pay on each additional dollar you earn. Your effective tax rate, or the percentage of your total income you pay in taxes, is typically much less thanks to exemptions and deductions that protect some of your income from taxation.

If you're single and your taxable income is $40,000, for example, your federal marginal tax would be 25%, meaning the feds get a quarter of every additional dollar you earn over $32,550 (which is where the 25% tax bracket is expected to start in 2008, according to CCH tax research business). If you checked your last tax return, though, you'd probably find the total dollar amount you paid in taxes might be 15% to 20% of your total income.

For single people:

When taxable income is:	The tax is:	Plus:	Of the amount over:
$0 to $8,025	$0	10%	$0
$8,026 to $32,550	$803	15%	$8,025
$32,551 to $78,850	$4,481	25%	$32,550
$78,851 to $164,550	$16,056	28%	$78,850
$164,551 to $357,700	$40,052	33%	$164,550
More than $357,700	$103,792	35%	$357,700

For married couples:

When taxable income is:	The tax is:	Plus:	Of the amount over:
$0 to $16,050	$0	10%	$0
$16,051 to $65,150	$1,605	15%	$16,050
$65,151 to $131,450	$8,963	25%	$65,100
$131,451 to $200,300	$25,550	28%	$131,450
$200,301 to $357,700	$44,828	33%	$200,300
More than $357,700	$96,770	35%	$357,700

Source: CCH, a Wolters Kluwer business

Income taxes aren't all you pay. You also must pay Social Security and Medicare taxes on wages and salaries. For most folks, that's an additional 7.62% that has to be included in your calculations.

And unless you live in one of the nine states that lack an income tax -- Alaska, Florida, Nevada, New Hampshire, South Dakota, Tennessee, Texas, Washington or Wyoming -- you also pay a state income tax that typically ranges from 2% to 10%. (New Hampshire and Tennessee do tax dividend and interest income.) You can find the 2007 brackets at Bankrate.com.

A Californian with $40,000 in taxable income might pay a 25% federal marginal tax rate, 8% in state taxes and 7.62% in Social Security and Medicare taxes, for a total tax bill of about 41%. If she itemizes her deductions, she gets a bit of a break since state taxes are deductible on federal returns; subtract 2 percentage points from the total to adjust for that (0.25 times 0.08 is 0.02, or 2%), leaving 39%.

Now for the rest of the calculations. We subtract the combined tax rate from 1, then divide $1 by the result.

So 1 minus 0.39 is 0.61. Divide 1 by 0.61, and you get 1.639. To wind up with $1, then, our Californian has to make $1.64. If something costs $50, she needs to earn $82 to clear enough to pay for it.

California's a high-tax state. The burden's lighter elsewhere. As a rule of thumb, though, you should figure that you need to earn around $1.50 to clear each dollar you use to spend.

Figure the tax advantage

Don't walk away depressed, because the tax code can also work in your favor.

If you have a mortgage or student loans, for example, your interest may be tax-deductible. That means the real rate you're paying is lower, sometimes substantially lower, than the interest rate printed on your statements.

Let's say you have a 6% interest rate on your mortgage and that you itemize your deductions (if you don't itemize, you get no tax benefits from a mortgage). Let's further assume your combined federal and state tax bracket is 30%, which is the national average, according to the Tax Foundation.

Subtract your combined tax rates from 1. You multiply the result by your nominal interest rate to get the real interest rate you're paying. So if your combined federal and state tax rate is 30%, you subtract 0.3 from 1 to get 0.7, and multiply that by your mortgage interest rate of 6% to get 4.2%. That's pretty cheap money by any measure.

Calculating 'Take this job . . .' money

How much money would you have to save, inherit or win to say goodbye to work?

You'll find plenty of retirement planners around the Web, including MSN Money's Retirement Planner, which I find particularly easy to use. Personal-finance software like Money and Quicken includes more sophisticated calculators that are also worth a look.

But if you want to ballpark a figure for what you'd need to save to stop work tomorrow, simply multiply your living expenses by 25.

Why 25? Because financial planners generally agree that 4% is a "sustainable" withdrawal rate from a diversified portfolio of stocks, bonds and cash. That means you can take out 4% of the total in the first year, adjust the withdrawal every year afterward by the inflation rate, and face relatively little risk of running out of money.So to get the lump sum you'd need, you divide your annual living expenses by 0.04 -- which is the same as multiplying the same expenses by 25. If you need $30,000 to live on, you'd need $750,000. If you want to be more conservative and reduce the chances of running dry to virtually zero, multiply by 33 instead. That approximates an initial withdrawal rate of slightly more than 3%.

Columns by Liz Pulliam Weston, the Web's most-read personal finance writer and winner of the 2007 Clarion Award for online journalism, appear every Monday and Thursday, exclusively on MSN Money. She also answers reader questions on the Your Money message board.

Published Nov. 12, 2007