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First, let's all take a minute to applaud this Fool's enterprising spirit and willingness to challenge what he read. Then we'll make fun of his math skills. No, we won't. I never laugh at math, or lack of math. I've been there and it wasn't that long ago.
It was my interest in investing that got me interested in the math that I needed to figure out just what was happening with my money. The thing that fascinated me first was projecting where my portfolio would be in X years. Nowadays we have online calculators that can do that, but back in the olden days (1994) I had to make do with a handheld calculator. So I actually learned how it's done. It's fun!
Here's how you do it on a calculator, preferably one that can do exponents. (Try the Windows calculator using the Scientific View.) First, decide on a projected rate of return -- 10%, 15%, 18%, 20%, whatever. Enter the projected rate of return as a decimal plus 1 (e.g., for 15% you enter 1.15). Now hit your exponent key (usually something like x^y) and enter the number of years you are projecting. Then hit the equals sign. Hit the multiply sign and put in your current portfolio value. Hit equals again and you have the value of your current portfolio if it grows at your projected rate of return for X years.
Example:
Projected rate of return: 18%
Number of years: 20
Current portfolio value: $50,000
Enter 1.18, exponent key, 20, = (You should see 27.39xxx)
Enter X, 50000, = (You should see 1369651.73xxx)
A portfolio worth $50,000 today that grows at 18% per year will be worth $1,369,651.73 in 20 years. Cool, eh?
If you don't have a calculator that does exponents, you can get the same results by entering your current portfolio value, then simply multiplying by 1.18 20 times. That's all the exponent does.
But what my correspondent was doing was the equivalent of simply multiplying 18 by 20 rather than multiplying each year's portfolio by 18% 20 times. Here's why that is wrong.
The difference between 370% and 2500% is The Power of Compounding. And it's a wonderful thing. Say you invest in an index fund. Each year your portfolio grows by X percent depending on how well the market did as a whole. But in dollar terms, as opposed to percentage terms, the portfolio grows much faster as it gets bigger. A 20% increase in a $10,000 portfolio is $2,000, but a 20% increase in a $50,000 portfolio is $10,000, which is 100% of what you started with! That's rather obvious, of course, but it illustrates why you can't just add up each year's percentage increase and get the total return. Twenty percent in year one is not the same as 20 percent in year 10.
Let's look at this problem year by year using actual returns from the S&P over the last 20 years. (Note: The returns are shown in a form that makes them easier to work with -- decimals plus 1 -- so that when we multiply, the original is included in the answer -- i.e., instead of multiplying the previous year's total by 10% and then adding that back to the previous year's total to get the new portfolio value, you simply multiply by 1.10.)
The exercise below assumes that a portfolio is started on the first day of 1979 with $10,000. The values in the Portfolio Value column show how that portfolio has grown during each year. The next column shows a running total of annual returns. The last column gives a running cumulative return. You can see that for the first year, the last two columns are identical, but in 1980 they start to diverge and the difference grows greater each year. That's the power of compounding.
End S&P Portfolio Value Sum of Cumulative
of Return $10,000.00 Returns Return
1979 1.1844 $11,844.00 18.44% 18.44%
1980 1.3242 $15,683.82 50.86% 56.84%
1981 0.9509 $14,913.75 45.95% 49.14%
1982 1.2141 $18,106.78 67.36% 81.07%
1983 1.2251 $22,182.62 89.87% 121.83%
1984 1.0627 $23,573.47 96.14% 135.74%
1985 1.3216 $31,154.70 128.30% 211.55%
1986 1.1847 $36,908.97 146.77% 269.09%
1987 1.0523 $38,839.31 152.00% 288.39%
1988 1.1681 $45,368.20 168.81% 353.68%
1989 1.3149 $59,654.64 200.30% 496.55%
1990 0.9683 $57,763.59 197.13% 477.63%
1991 1.3055 $75,410.37 227.68% 654.10%
1992 1.0767 $81,194.34 235.35% 711.93%
1993 1.0999 $89,305.66 245.34% 793.06%
1994 1.0131 $90,475.56 246.65% 804.76%
1995 1.3743 $124,340.56 284.08% 1143.41%
1996 1.2307 $153,025.93 307.15% 1430.26%
1997 1.3336 $204,075.38 340.51% 1940.75%
1998 1.2870 $262,645.02 369.21% 2526.45%
For those who are interested, you can find the S&P returns from 1961 through 1998 at the Statistics Center in the file called Performance History of the Foolish Four. The formula for the Portfolio Value column is the previous year's value times the corresponding value in the S&P Returns column. The formula for the Cumulative Return is the current portfolio value minus the starting value, divided by the starting value.
For those who aren't interested in the details, you can assume that all that stuff is accurate as long as you check back tomorrow to see if some sharp-eyed reader sent in any corrections.
Fool on and prosper!