WHEN YOU ESTIMATE how much a dollar today will be worth in the future, or how much a series of regular deposits will be worth in the future, you’re calculating a future value. But sometimes, you’ll want to reverse engineer this calculation.
Suppose you figure you’ll need $40,000 in four years to make a house down payment. To have that sum, how much would you need to set aside today—or how much would you need to save every month?

IMAGINE A BROTHER and sister, who are twins. The sister saves $5,000 a year for 10 years, from age 25 to age 35, earning 6% a year. At age 35, she would have $69,858. She doesn’t add any more money to the account. Instead, she leaves the $69,858 to grow at 6% a year for another 30 years, until age 65. If you compound at 6% a year for 30 years, your cumulative gain would be 474%,

INVESTMENT COMPOUNDING over long periods can produce impressive results—but the results are even more impressive when you couple that compounding with regular savings. For instance, if you invest $100 initially and left it to grow at 5% a year, you would have $339 after 25 years, thanks to the 239% cumulative gain. But if you invested $100 at the start of each of the 25 years, for a total of $2,500, you would have $5,011 after 25 years.

TO FIGURE OUT WHETHER it makes sense to buy taxable bonds or tax-free municipal bonds, you first need to find out your marginal federal and state income tax brackets. Let’s say you are in the 32% federal income tax bracket and a 7% state tax bracket. Meanwhile, suppose you are choosing between a muni bond from your own state that yields 3.05% and a corporate bond that pays 4.95%. Both have similar credit quality and duration.

TAXABLE BONDS—such as those issued by corporations—typically have relatively high yields, but you have to pay tax each year on the interest you earn, assuming you hold the bonds in a taxable account. Municipal bonds offer yields that are usually lower, but the interest should be tax-free. So which should you buy?
Imagine you are in the 24% marginal federal income tax bracket and a 6% state income tax bracket, for a combined marginal rate of 30%.

BY USING RETIREMENT accounts, or by pursuing tax-efficient strategies in a taxable account, you can get tax-deferred growth. How valuable is this growth? Imagine a husband and wife. Both invest $1,000 for 40 years and earn 6% a year before taxes.
The husband pays 22% in taxes every year on his entire 6% investment gain, so his $1,000 grows annually at an after-tax 4.68%. If you earn 4.68% a year for 40 years, your cumulative gain would be 523.1%.

IMAGINE YOUR portfolio lost 25% last year. To recoup that loss, you would need to earn not 25%, but just over 33%. You can show this mathematically as:
0.75 x 1.3333 = 1
The larger the investment loss, the harder it is to climb out of the resulting financial hole. If you lose 50%, you need a 100% gain to get back to even. What if you lose 75%? To make back that loss,

HOW LONG WILL IT take to double your money, given a particular rate of return? You can get a rough answer by dividing 72 by the annual return. For instance, if you expect to earn 7% a year, it would take just over 10 years to double your money. But at a 3% annual return, the compounding process is much slower, with your money doubling every 24 years.
Obviously, the higher the return you earn,

THE CUMULATIVE GAIN was 21% in the first example given in the previous section. The annualized gain, however, was 10%. This is known as a geometric average: It’s the amount you would have to earn each year to achieve the cumulative result. Note that this is different from simply dividing the cumulative gain by the number of years—the so-called arithmetic average. The latter wouldn’t give you the annualized return, because simple averaging doesn’t take into account the effects of compounding.

SUPPOSE YOU EARN 10% this year and 10% next year. Your cumulative gain would be 21%. Why? Imagine you invested $100. The first year’s 10% gain would turn your $100 into $110. Because you start the second year with $110, the next year’s 10% gain boosts your portfolio’s value by $11, not $10. That brings your total to $121, for a two-year cumulative gain of 21%.
Got a series of annual returns for which you’d like to find out the cumulative gain?