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Compounding and the time value of money

By David Luhman on Sun, 05/10/2009 - 00:10

Compounding and the time value of money

Importance of compounding

Of all the concepts in personal finance, compounding is by far the most important. If you take only one thing away from all my tapes, I hope it's an appreciation for the power of compound growth.

And the two elements that will unlock this power are your return on investment and the length of time in the investment.

Investing in stocks vs. investing in bonds

Suppose your friend invests $100 in government bonds. The bonds pay her 5 percent interest. You, however, are willing to take some short-term risks. You decide to invest $100 in stocks, which are more volatile, but have an average a return of 10 percent annually.

At the end of one year, the stock investment will be ahead of the bond investment by 5 percent. This isn't much to get excited about.

However, by the end of ten years the total amount invested in stocks will be 60 percent larger than the amount invested in bonds.

After 15 years you'll have more than twice as much money if you stayed with stocks as opposed to bonds.

After 25 years your stash of stocks will be three times as great as the amount of bonds. After 30 years, you'll have over four times as much money if you stick with stocks as opposed to bonds.

Note what's going on here. The higher returning stocks give you more money, and an increasing amount of money over time.

Now you might think that at most, stocks could do no better than double the performance of bonds. After all, the bonds yield 5 percent, and the stocks 10 percent, or twice as much as the bonds.

But 30 years out, the amount invested in stocks was more than four times as large as the amount invested in bonds. This increasing difference in the total asset size occurs because compound growth is a non-linear process.

Small increases in yield help

And this is the first big lesson to learn about compound growth : small increases in your return can lead to large increases in your total assets, especially after longer periods of time.

Recall in the previous example that after 30 years a stock investment was worth four times as much as a bond investment. After 40 years the stocks are worth six times as much as the bonds, and after 50 years the stocks are worth ten times as much as the bonds. The further you go out in time the larger a small difference in return becomes.

So don't scoff at what appear to be slight differences in investment returns. Even increasing your investment returns from 5 percent to 7 percent can make a big difference.

It's true that after one year the difference will be hardly discernible, but a 7 percent investment will give you fifty percent more money than a 5 percent investment after 20 years.

Longer time periods give you much more cash

And this leads us to our second big idea about compound growth : the amount of time involved greatly affects your total results.

Go back to the example where you've invested $100 in stocks that grow at 10 percent annually. In this case it will take about seven years for you to double your money to $200. However, to make the next $100 it takes only about four and a half years. To make an additional $100 after that takes only three years.

This is another characteristic of compound or exponential growth. You get much higher additions to your assets for each additional year of investment.

Here's another way to look at it. The growth of your investment's value resembles a hockey stick.

You get flat or almost unnoticeable growth for the first few years, and then you get phenomenal growth in the out years. In the out years your earnings on your earnings completely dwarf your total investment. The longer you have your money invested, the more you'll benefit.

The Twin Sister Case

The importance of investing early is shown dramatically by the twin sister case.

Let's look at twin sisters, both obviously the same age. Both sisters begin working at the same time, and both are paid the same salary. In this and every other way the two sisters are identical except for their savings habits.

The first sister begins to save $2,000 a year from age 25. She saves $2,000 in capital for 10 years until she reaches age 35. Then she completely stops saving for the rest of her life, but she lets her early savings continue to compound.

The other sister is also pretty thrifty, but she doesn't begin saving until she's 35, ten years later than her sister. The second sister, however, saves $2,000 each year for the next thirty years until she retires at age 65. Both sisters invested in identical stock mutual funds which returned an identical and constant 10 percent a year on their investments.

So the first sister put $20,000 worth of principal into stocks over ten early years of saving, while the second sister put $60,000 worth of principal into stocks over a later thirty year period. When the two twins retire at age 65, which has the larger total nest egg?

If you said the second sister who actually saved three times as much, you're wrong. In fact, the first sister who started earlier had accumulated over $600,000, while the second sister had only $360,000 or just over half of her sister's amount.

This example of the twin sisters shows something important. It's critical that you start saving early, because it's the early savings that grow the fastest and contribute the most to your nest egg.

Rule of 72

There's another unusual aspect to compound growth called the rule of 72. This rule states that if you take 72 and divide it by your annual return on investment, you'll find the number of years it takes to double your money.

Here's a couple of examples. If you make 10 percent on your investments, you can count on doubling your money in about seven years. If inflation is running at 3.5 percent a year, prices will double in about 20 years.

Importance of discounting

There's another thing I want to say about compound growth, and that is that compound growth is related to the concept of discounting. In fact, compound growth is the opposite of discounting, more properly called discounted cash flows.

This talk about "discounting" and "discount rates" and may seem odd. At least to me, the concept of discounting seemed strange, while the concept of compound growth seemed more intuitive.

I only mention discounting because it's the most important concept in finance. People use discounting to determine how much they should pay for a stock, a bond, or even commercial real estate.

Examples of discounting

Take the example of a US EE savings bond. Everyone's familiar with these. You pay $50 now, and 15 or so years later you get $100 back.

So when you take that future value of $100 and discount it back to the present, you realize that you should only pay $50 in the present for the future claim of $100.

Likewise, take the so-called 10 million dollar prize that a lottery may offer. In fact the advertised prize of 10 million dollars is not equal to a present value of $10 million. The 10 million that the winner gets is actually one million dollars a year, for 10 years.

Due to inflation, the final installment of $1 million is not worth 1 million in today's dollars. When you discount the future cash flows of $1 million a year for 10 years, you'll find that the so-called $10 million prize is only worth maybe 7 million in today's dollars.

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