Beyond the frequently inaccurate assumption of no return on intra-period earnings, there is an even more serious problem with arithmetic returns: They can lead you to believe that you are up 15 percent per year for two years, when in fact you have lost 10 percent of your money. How? Let’s say you start with $100 and earn 80 percent in the first year. At the end of the first year, you have $180. In the second year, however, you lose 50 percent, going down to $90. Even though you’ve lost 10 percent of your money, the arithmetic mean annual return for the two-year period is 15%. (See Table 19-1.)

Example of a Geometric Return Calculation

When interest or dividends are reinvested, a geometric return should be calculated. A good calculator or financial software is indispensable for working with geometric returns.

With arithmetic returns, you might need to do only a single multiplication or division. In contrast, geometric returns are combined by multiplying return relatives. A return relative is simply the ratio of the current end-of-period value of the portfolio, security, or index to its previous end-of-period value.

Working on a Geometric Chain Gang

If, for example, your rate of return for a given month was 2 percent, the return relative would be 102 percent or 1.02. To annualize, we raise 1.02 to the twelfth power:

1.02¹² = 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 X 1.02 = 1.2682 (to the nearest basis point)

You can annualize a monthly rate using return relatives. If the monthly rate of increase is 0.02, or 2 percent, the return relative is 1.02, or 102 percent. Multiply 1.02 together 12 times (1.02¹²). This repeated multiplication is sometimes referred to as a geometric chain.

A point worth remembering: You must always remember to subtract 1 after annualizing a return relative to get back to a percentage change (what you’re subtracting is the starting value, which is always 1, or 100 percent). So you raise 1.02 to the twelfth power, subtract 1, and you get 26.82 percent. This is the geometric return, also known as the effective annual rate of return. Note that it is 2.82 percent larger than the arithmetic return of 24 percent. The extra 2.82 percent comes from the compounding of returns.

effective rate (effective annual rate of return)

Yield to maturity.

Sometimes you may want to calculate your effective annual return from starting and ending values separated by some number of years. This is easy to do—with a decent calculator. For example, consider an investment that, after eight years, is worth $20,000. Its value at the beginning of the eight-year period was $8,000. To figure the compound annualized (i.e., geometric) rate of return, just take the endpoint value of $20,000 and divide by the starting value of $8,000 to get the return relative of 2.5 for the whole eight-year period. Instead of multiplying this number eight times, you need to find the number that multiplied by itself eight times equals 2.5—in other words, you must take the eighth root of 2.5. This is approximately 1.1214, the annualized return relative. To get the annual return, you still need to subtract 1: giving 0.1214, or in percentage terms, 12.14%, your annualized geometric rate of return.

Rule of 72

There is a nice shortcut for figuring how long it takes money to double at various rates of return (its usual use is for working with compound interest, but it works just as well for working with any return calculation). It is called the rule of 72. To figure the approximate doubling time of an annual rate of return, just divide 72 by that rate, leaving off the percent. For example, the calculation for a 6 percent annual return would be

You can check that result by raising 1.06 to the twelfth power (or if you don’t have that kind of calculator, just multiply 1.06 12 times). The answer is (approximately) 2.012, which for most purposes is a very good approximation. Don’t use the rule of 72 for calculations that must be precise or for rates of return less than 2 percent or greater than 24 percent.

A Practical Hint for Calculating Effective Annual Rate of Return

How, exactly, do you calculate the eighth root of 2.5? Even good calculators don’t have a special eighth-root button. In this particular example, even a cheap calculator can do the job, because you can get an eighth root by taking the square root three times in a row, which is nice for this example and for other examples in which the number of periods is a power of two. But it won’t help with three years, five years, and so forth.

On a good calculator or a spreadsheet, you just use the exponent key (or function), y^x . When you want an eighth root, you are raising 2.5 to the one-eighth power. Expressed as a decimal number, one-eighth is 0.125. So you would need to calculate 2.5 ^.125. On a good calculator, you would just input 2.5 [function key(s) for y^x].125. And on a spreadsheet? Depending on which spreadsheet program you use, you would simply enter 2.5^.125 or 2.5**.125 into a spreadsheet cell.

Remember—to get the annual return, you must subtract 1.

From the preceding examples, you can readily see why it is so important to know whether a given rate of return was calculated on an arithmetic or geometric basis. We have seen that an arithmetic rate of return can grossly mislead an investor into thinking that he or she is making money when in fact the portfolio’s value is standing still or, worse, rapidly diminishing. Geometric returns are far more accurate in evaluating past performance. For this reason the CFA Institute insists, in its widely adopted Global Investment Performance Standards (GIPS®), that investment managers (e.g., mutual fund portfolio managers) use geometric returns. While geometric returns are more difficult to calculate than arithmetic returns, they give you a much more reliable idea of what the portfolio is actually doing. Even geometric returns, however, must be used carefully, especially when extrapolating from a short period to a much longer time frame (see the discussion of extrapolation risk in Chapter Twenty).