Developing an Intuitive Feel for the Mechanics of Growth

Investment decisions and planning are all concerned in one way or another with one concept: Growth. Stock investors examine the earnings growth of firms; portfolio planners estimate portfolio growth. And ultimately growth in the real value of your assets is the central focus of your endeavors and concerns.

But most investors do not have an intuitive feel for growth, particularly over longer time periods. If someone were to say, for instance, that you can double your money in nine years or quadruple it in 18, most people would have no idea whatsoever of the annual growth rates that would produce those results [it would be 8% in both instances].

Needless to say, if you are completely in the dark when confronted with this issue, you have little basis for judgment.

Developing an intuitive feel for growth figures and knowing how to perform relatively simple calculations to estimate growth rates will provide you with a valuable basis for making informed decisions.

For these decisions, precision—determining a growth rate down to the last decimal using a calculator or computer spreadsheet—is much less important than deriving an approximate figure that allows for a realistic assessment.

Deriving an Annual Growth Rate

Precise annual growth rates can be derived mathematically with the aid of calculators or computer spreadsheets. But they can also be determined quickly and easily using tables of growth rates (such as the abbreviated one in Table 1), which show growth rate factors over various time periods and compounding at various annual growth rates.

To determine an annual growth rate of an investment using a growth rate table, you first divide your ending value by the beginning value to determine the growth rate factor:

Ending Value ÷ Beginning Value = Growth Rate Factor

(Put another way, the beginning value multiplied by the growth rate factor equals the ending value.)

Once you know the growth rate factor, look down the first column until you find the number of years over which the investment was or is expected to be held; then read across until you arrive at a number close to the growth rate factor. Then look up to see which growth rate column that figure appears under, and that is your compound annual growth rate.

Table 1. Growth Rate Factors
No. of Years	Growth Rates
	2%	4%	6%	8%	10%	12%
1	1.02	1.04	1.06	1.08	1.10	1.12
2	1.04	1.08	1.12	1.17	1.21	1.25
3	1.06	1.12	1.19	1.26	1.33	1.40
4	1.08	1.17	1.26	1.36	1.46	1.57
5	1.10	1.22	1.34	1.47	1.61	1.76
10	1.22	1.48	1.79	2.16	2.59	3.11
15	1.35	1.80	2.40	3.17	4.18	5.47
20	1.49	2.19	3.21	4.66	6.73	9.65
30	1.81	3.24	5.74	10.06	17.45	29.96
40	2.21	4.80	10.29	21.72	45.26	93.05

How It Can Be Used

Let's say some broker is trying to tell you about his great stock-picking abilities, and says: "If you had put $100,000 in my Brilliant Stock Pick 10 years ago, it would now be worth $260,000-an unbelievable growth rate."

Is it really "unbelievable"?

Dividing $260,000, the ending value, by $100,000, the beginning value, results in a growth rate factor of 2.6. In the table, find 10 years in the first column, and read across until you find a number close to 2.6. Under the annual growth rate of 10%, the growth rate factor is 2.59, so the true annual growth rate is just above 10% (this 10% compound growth rate assumes that all income and gains are reinvested—compounded—at this 10% growth rate). Not a bad return, but also not "unbelievable."

In another example, let's say you suddenly realized your house has increased 300% in value over the last 15 years. What kind of an annual growth rate does that translate into?

In this case, some interpretation is required. A 300% increase means that the ending value is actually 400% of the beginning value—in other words, the ending value is the sum of the original value plus three times the original value—a quadrupling of the beginning value, or a growth rate factor of 4.0. Reading from left to right at 15 years, 4.0 falls between 8% and 10%. For most purposes, an estimate of a bit under a 10% growth rate is sufficiently accurate.

Growth rate tables can also be used to illustrate number expansion-the "magic" of compounding. For example, suppose you are 25 years of age and miraculously you are able to invest $2,000 in an IRA that you expect to grow at a compound annual rate of 10% a year. At age 65, 40 years later, the table indicates that the growth rate factor is 45.26, so your $2,000 IRA investment can be expected to grow to $90,520 (45.26 × $2,000) in 40 years.

Or, let's say you want to save roughly $50,000 in 15 years by the time your son enters college. If you were able to put money in an investment that earned 8% over that 15-year period, the table indicates that the growth rate factor is 3.17, so you would need to set aside roughly $15,770 today ($50,000 ÷ 3.17) to achieve your savings goal.

Over long periods, the number explosion is both powerful and impossible to "guesstimate." For example, the growth factor for 30 years at 10% is 17.45, but only 10 years later the growth factor jumps to 45.26.

In addition, what seems to be small growth differences can produce big differences over long time periods. A dollar would have grown to $45.26 over 40 years at 10%, but at 12% that dollar would be worth $93.05—more than double the value of the dollar earning 10%.

An Easy Rule-of-Thumb

There is an even handier growth rate rule-of-thumb that you can use for getting a quick judgment on growth in a pinch: The rule of 72. It is not precise, but it is simple and it does give reasonable approximations for shorter time periods and growth rates that are not extreme.

Put simply, the rule of 72 notes a quirk in the math of the future value formula: If the "number of years" multiplied by the "percentage growth rate" is equal to 72, the resulting growth rate factor is roughly 2.0—a doubling of your investment. For example, a 6% investment over a 12-year period, or an 8% investment over a nine-year period, both would result in a doubling of your money.

Conversely, 72 divided by one factor (for instance, the number of years) indicates the other factor. As an example, if a firm's stock is being touted because its earnings per share have doubled over the last 10 years, a quick estimate of annual growth in earnings per share would be 7.2% (72 divided by 10 years, which in this case is not particularly spectacular).

The rule of 72 is also useful for a quick answer to how long it will take for a sum to double. For example, if you have expectations that an investment will grow at 12% a year, then it will take about six years (72 divided by 12) for it to double in value, a 100% return.

How They Can Be Misused

The rule of 72 is not precise—it is a rough, always-at-hand preliminary estimate to judge whether further analysis is warranted. And if you are not interested in precise calculations, you can stop right there.

The growth rate tables will provide you with accurate estimates as long as your interest rates and time periods coincide with the listings, but if you need precision, you should turn to a calculator or computer spreadsheet. However, gaining an intuitive feel for numbers, and in particular growth rates, is invaluable for making investment decisions and evaluating past experience, regardless of whether you just use the tables, or eventually delve into the math.