256 Chapter 6 Investments
Now subtracting one from both sides gives us:FORMULA 6.1.1
Compound Annual Growth Rate (CAGR) Formulai (^) PVFV (^)
1/n
1
where
i represents the COMPOUND GROWTH RATE
FV represents the FUTURE VALUE
PV represents the PRESENT VALUE
and
n represents the NUMBER OF YEARS
Let’s try this formula out with an example.
Example 6.1.7 You bought shares of Zarofi re Systems 7 years ago for $12.50 per
share. The current stock price is $50 per share. If you sell your stock today, what
compound annual growth rate will your capital gain represent?
Apply the formula:
i (^) FVPV (^)
1/n
1
i
___$50
$12.50
1/7
1
i 0.2190137 21.90%
The fractional exponent we used in this calculation can be a problem with a calculator. It
seems logical that to evaluate i (50/12.5)1/ 7 1 you would just enter “(50/12.5)^1/7-1”,
but this will not give a correct result. The problem lies with order of operations. Since
exponents take priority over division, the calculator evaluates that expression by raising
(50/12.5)^1 and then dividing the result by 7. In order to make it clear to the calculator
that the exponent actually is 1/7, we need to place it in parentheses. The calculator steps to
evaluate this are as follows:
Operation Result
(50/12.50)^(1/7)- 1 0.2190137
Alternatively, you could convert the fraction to a decimal and then use the decimal, though
this is a bit messier:
Operation Result
1/ 7 0.1428571
(50/12.50)^0.1428571- 1 0.2190137
This formula can also be used to find compound interest rates, as a more exact alternative
to Rule of 72. This idea is explored in a few of the exercises at the end of the section.
In this example, we calculated a rate of return based on the per-share price of a stock. We
need to be a bit careful when doing this. Occasionally, a corporation may split its stock. A split
occurs when a corporation increases the number of its shares by issuing new shares to its exist-
ing shareholders. For example, in a 2-for-1 stock split, the company issues new shares so that
each shareholder has 2 shares for each 1 she previously owned. This would affect the return on
investment calculations. If Zarofire Systems had split its stock 2 for 1, each share you bought
7 years ago for $12.50 would have become two shares now worth $50 each, for a total of $100.
If that had happened, you would need to use $100 as the future value in your CAGR calculation,
and your return would have been 34.59%. While stock splits are not an everyday occurrence,
you do need to be careful to take any splits into account. Splits are not an issue, though, if our
calculations are based on the total value of the investment, rather than on the per-share value.