## Example 510

to do with the "yx" key on a calculator. Just enter 2, then press "yx," enter .125, and press the "=" key. The eighth root should be about 1.09, which implies that r is 9 percent.

3. Use a future value table. The future value factor after eight years is equal to 2. If you look across the row corresponding to eight periods in Table A.1, you will see that a future value factor of 2 corresponds to the 9 percent column, again implying that the return here is 9 percent.

Actually, in this particular example, there is a useful "back of the envelope" means of solving for râ€”the Rule of 72. For reasonable rates of return, the time it takes to double your money is given approximately by 72/r%. In our example, this means that 72/r% = 8 years, implying that r is 9 percent, as we calculated. This rule is fairly accurate for discount rates in the 5 percent to 20 percent range.

### Big Mac

In 1998, when Mark McGwire was chasing baseball's single-season home run record, there was much speculation as to what might be the value of the baseball he hit to break the record (in 1999, the record-setting 70th home run ball sold for \$3 million). One "expert" on such collectibles said, "No matter what it's worth today, I'm sure it will double in value over the next 10 years."

So, would the record-breaking home run ball have been a good investment? By the Rule of 72, you already know that since the expert was predicting that the ball would double in value in 10 years, he was predicting that it would earn about 72/10 = 7.2% per year, which is only so-so. Of course, thanks to Barry Bonds, it will probably do much worse!

Why does the Rule of 72 work? See

At one time at least, a rule of thumb in the rarified world of fine art collecting was "your money back in 5 years, double your money in 10 years." Given this, let's see how one investment stacked up. In 1976, British Rail purchased the Renoir portrait La Promenade for \$1 million as an investment for its pension fund (the goal was to diversify the fund's holdings more broadly). In 1989, it sold the portrait for nearly \$15 million. Relative to the rule of thumb, how did British Rail do? Did they make money, or did they get railroaded?

The rule of thumb has us doubling our money in 10 years, so, from the Rule of 72, we have that 7.2 percent per year was the norm. We will assume that British Rail bought the painting on January 1, 1976, and sold it at the end of 1989, for a total of 14 years. The present value is \$1 million, and the future value is \$15 million. We need to solve for the unknown rate, r,as follows:

Solving for r, we get that British Rail earned about 21.34 percent per year, or almost three times the 7.2 percent rule of thumb. Not bad.

Can't afford a Renoir? Well, a Schwinn Deluxe Tornado boy's bicycle sold for \$49.95 when it was new in 1959, and it was a beauty. Assuming it was still in like-new condition in 2001, it was worth about 12 times as much. At what rate did its value grow? Verify for yourself that the answer is about 6.1 percent per year, assuming a 42-year period.

A Mickey Mantle bobbing-head doll was a better investment. It sold for \$2.98 in 1962, but by 2000, it was worth about \$700 (in perfect condition). See if you agree that this collectible gained, on average, 15.45 percent per year.

A slightly more extreme example involves money bequeathed by Benjamin Franklin, who died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. The money had been paid to Franklin when he held political office, but he believed that politicians should not be paid for their service (it appears that this view is not widely shared by modern-day politicians).

Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, however, after some legal wrangling, it was agreed that the money would be paid out in 1990, 200 years after Franklin's death. By that time, the Pennsylvania bequest had grown to about \$2 million; the Massachusetts bequest had grown to \$4.5 million. The money was used to fund the Franklin Institutes in Boston and Philadelphia. Assuming that 1,000 pounds sterling was equivalent to \$1,000, what rate of return did the two states earn (the dollar did not become the official U.S. currency until 1792)?

For Pennsylvania, the future value is \$2 million and the present value is \$1,000. There are 200 years involved, so we need to solve for r in the following:

Solving for r, we see that the Pennsylvania money grew at about 3.87 percent per year. The Massachusetts money did better; verify that the rate of return in this case was 4.3 percent. Small differences in returns can add up!

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