compound interest and the time value of money

In short

A friend of mine once said she felt too late to the game because she had not started saving at twenty. If you have ever felt that way, I want you to know your worry makes sense, and it also has an answer. Money left alone earns interest, and that interest then earns interest of its own. Time is the one ingredient that cannot be bought back, so starting small today beats waiting for a bigger moment. Find the yearly rate your account pays, since that single figure drives the whole result. Check how often the interest is added, because more frequent adding grows the balance a little faster. Then comes the amount a saver sets aside, and a sum small enough to carry without strain lets the calendar do its share. Plain enough.

The whole of it

What it is

I once watched an old farmer plant a row of walnut trees he knew he would never sit under. When I asked him why, he smiled and said the shade was for somebody else, and the walnuts were for the squirrels, and the planting was for him. Compound interest works a good deal like that orchard. You put a seed in the ground, and over many years the seed makes more seeds.

Here is the plain meaning. Simple interest pays only on the money you first put in. Compound interest pays on that money and also on the interest already earned. The second kind is the one that surprises people, in a good way. It means your balance does not grow in a straight line. It curves upward, slowly at the start and faster as the years pile up.

The time value of money is a close cousin. It is the idea that a dollar in your hand today is worth more than a dollar promised ten years from now. The reason is simple. A dollar today can be saved and earn interest, so it becomes more than a dollar by the time the future dollar arrives. Waiting has a cost, and starting early has a reward. That is the heart of the matter.

How it works

If you are holding some savings and wondering what they could become, the math is kinder than you might expect. Each period, the account adds interest to the balance. The next period, interest is figured on that larger balance. Round and round it goes.

The tidy formula looks like this. The future value equals the starting amount times one plus the rate, raised to the number of years. Say a person puts in $1,000 at 5 percent a year. After one year the balance is $1,000 times 1.05, which is $1,050. After two years it is $1,050 times 1.05, which is $1,102.50. Notice the second year earned $52.50, not $50. That extra $2.50 is interest on interest. Small now, but it never stops growing.

Another handy shortcut is the rule of 72. Divide 72 by the yearly rate, and you get a rough count of years for money to double. At 6 percent, 72 divided by 6 is 12 years. At 8 percent, 72 divided by 8 is 9 years. It is only an estimate, but it is a fine one for the back of an envelope.

How often interest is added matters too. Interest added monthly grows a hair faster than interest added once a year at the same stated rate. The gap is small, though. The rate and the number of years do far more of the work.

The numbers, and where to find yours

Every account has a few numbers worth knowing, and I would not guess at yours. Look for the stated yearly rate, often called the annual percentage rate, and the annual percentage yield, which already counts the effect of compounding. The yield is the better figure for comparing accounts, since it shows what a saver truly earns in a year.

For a bank savings account, the bank must show the annual percentage yield under the federal Truth in Savings rules, which the Consumer Financial Protection Bureau explains on its website. The rates change over time, so the current top savings rate is the current figure, which the official source publishes each year as reported by the source the site names. For a retirement account, the plan statement lists your return, and the fund's own documents list its past results and its fees.

Fees deserve a word here. A fund that charges 1 percent a year takes 1 percent off your growth each year, and that adds up over decades. Find the expense ratio in the fund's prospectus, which the Securities and Exchange Commission requires funds to provide. Look it up, write it down, and keep it handy.

A worked example

Let me tell you about Marcus, a bus driver who turned thirty last spring. He earns $52,000 a year and decides to save $200 a month in an account that pays 5 percent a year. To keep the arithmetic honest and easy to check, we will treat the money as added once at the end of each year, so $200 times 12 months is $2,400 a year.

Year one begins empty. By the end, Marcus has put in $2,400, and the interest on it is small, so we will count the first deposit as arriving at year end with no interest yet. His balance is $2,400.

In year two, the $2,400 earns 5 percent, which is $2,400 times 0.05, or $120. He adds another $2,400. The balance is $2,400 plus $120 plus $2,400, which is $4,920.

In year three, the $4,920 earns $4,920 times 0.05, or $246. He adds $2,400 again. The balance is $4,920 plus $246 plus $2,400, which is $7,566.

Now skip ahead. The standard formula for a yearly deposit is the deposit times the quantity of one plus the rate to the number of years, minus one, all divided by the rate. For 30 years at 5 percent, 1.05 raised to the 30th power is about 4.322. Subtract 1 and you get 3.322. Divide by 0.05 and you get 66.44. Multiply by $2,400 and the balance is about $159,456.

Now look at what Marcus actually put in. Thirty years of $2,400 is $72,000. The rest, about $87,456, came from interest. More than half his final balance was money he never had to earn at work. That is the orchard doing its quiet job.

What if Marcus waits ten years and saves for only 20? With 1.05 raised to the 20th power at about 2.653, the quantity is 1.653. Divide by 0.05 to get 33.06, and times $2,400 is about $79,344. Ten years of waiting cost him roughly $80,000. That stings. But he can still begin today, and that is the part to hold onto.

Where it goes wrong

Now, I would be a poor friend if I only told you the sunny half. Compounding cuts both ways. A friend of mine carried a credit card balance for years, and the same math that grows savings grew his debt. A card charging 22 percent a year, left unpaid, doubles in about 3 years by the rule of 72, since 72 divided by 22 is a little over 3. The borrower has to repay that balance with interest on interest, just as a saver earns it.

Inflation is another quiet thief. If prices rise 3 percent a year while an account earns 2 percent, the money grows on paper but buys less. The Bureau of Labor Statistics publishes the Consumer Price Index, which tracks how prices change. Compare your rate to that figure before you feel too proud of it.

Taxes take a bite as well. Interest in an ordinary savings account is taxed as income, and the Internal Revenue Service explains this in its publications. Retirement accounts can change when that tax is paid, but the plan sets the rules, and the account holder must follow them.

Then there is the plain human trouble of stopping. People pull money out during a scary month and lose the very years that do the heavy lifting. Early withdrawals can also bring penalties. Steady beats clever. Every time.

Last, a high stated rate means nothing if the product carries large fees. Read the fine print. It is dull work, and it pays well.

Questions to answer before you leave this page

What yearly rate does my own account pay, and have I found the annual percentage yield instead of only the headline rate? How often is the interest added, and does it matter much at the size of my balance? How many years do I realistically have before I will need this money, and what would an extra five years of waiting cost me? Am I carrying any debt at a higher rate than my savings earn, and would paying it down do me more good right now? What do the fees in my plan or fund come to each year, in dollars and not only in percent? And what is one small amount, even $25 a month, that I could start setting aside this week without losing sleep?