Copyright © 2008, The McGraw-Hill Companies, Inc.
Measuring Fund Performance
Determining the rate of return from a mutual fund investment is very much the same as
determining the rate of return on a stock. Just as with stocks, we have to confront the
issue of capital gains versus dividends; however, with mutual funds, since most investors
reinvest their dividends in new fund shares, the overall rate of return is usually calculated
on the assumption that all dividends are reinvested.Example 6.4.6 On June 5, 2000, the net asset value of the Aloquin Commonwealth
International Fund was $172.59. On June 5, 2007, the NAV was $154.08. If all dividends
were reinvested, each share of the fund on June 5, 2000, would have grown to 1.973
shares on 6/5/07. Calculate the average annual rate of return on this investment.Suppose that you owned one share of this fund on June 5, 2000. This would have been
worth $172.59. On June 5, 2007, this would have grown to 1.973 shares, each worth
$154.08, for a total of (1.973)($154.08) $304.00. Using the rate of return formula from
Section 6.1, we get:i (^) FV
PV^ ^
1/n
1
i
____$304.00
$154.08
1/7
1
i 0.1019479 10.19%
Sometimes we may not know how much an investment would have grown to with dividend
reinvestment, but do have access to the annual rate of return for each of several years. From
the annual rates of return, we can calculate an average rate over a number of years; this
“average,” though, may not work out quite the way you might expect.
Example 6.4.7 A mutual fund had annual returns of 10%, 5%, 8%, 15%, and
3% in each of the past 5 years. What was the average rate of return over this period?
At fi rst, it seems as though the average could be found in the usual way, by adding up the
fi ve annual rates and dividing by 5. This would give an average return of (10% 5% 8%
15% 3%)/5 (25%)/5 5%. Unfortunately, this is not really correct.
Suppose that you invested $100 in this fund at the start of the period (the problem can be
worked by assuming any starting value). At the end of that year, you would have $100(1.10)
$110. In the next year, your $110 would grow to ($110)(1.05) $115.50. Carrying this
idea further, we can see that at the end of the 5 years your $100 would grow to:
FV $100(1.10)(1.05)(0.92)(1.15)(1.03) $125.86
Now, using the rate of return formula, we calculate the rate of return to be:
i (^) FV
PV^ ^
1/n
1
i
$125.86____
$100.00
1/5
1
i 0.0470827 4.71%
The correct average rate of return is actually 4.71%.
If you have doubts about this average, you can calculate the future value of $100 at 5%
for 5 years. It works out to be $127.63. If you calculate the future value using 4.71%, you
get the correct $125.86 future value. The 5% figure, calculated in the usual way, is known
in mathematics as an arithmetic mean. This is a fancy word for what we normally mean
when we use the term average. The 4.71% figure is an example of a geometric mean,^10 an
(^10) It is actually the geometric mean of the (1 i)’s, less 1, not the geometric mean of the i’s. We should more technically
say, then, that the average rate is based on a geometric mean.
6.4 Mutual Funds and Investment Portfolios 297