4. Ending wealth
We next consider ending wealth rather than rates of return. This is helpful for
interpreting 20-year results because small differences in annual return produce large
differences in ending wealth. Figure 2 shows the wealth densities expressed as ending
wealth per dollar of initial investment. The densities are derived from the return
densities of Figure 1.
It is clear in Figure 2 that the mode of the distribution increases with the number
of stocks. For example, the mode is between $7.50 and $7.75 for ten stocks; it is around
$9 for 20 stocks and around $13 for 200 stocks. Median wealth (which is slightly
larger than the mode due to skewness) increases monotonically as well, from $10.41

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D. L. Domian et al./The Financial Review 42 (2007) 557
–
570
563
Figure 2
Ending wealth distributions for various portfolio sizes
The ending wealth distributions are generated from a 1,000-stock sample consisting of the largest 100
firms by market value in each of ten market segments spanning the NYSE, Amex and Nasdaq. Wealth per
dollar of initial investment is measured at the end of the 20-year sample period, 1985
–
2004.
with ten stocks to $13.60 with 200 stocks. Unlike the return distributions in Figure 1,
bulges due to Microsoft, United Health Group and Oracle are not apparent in Figure 2
because the effects occur far out in the right tails, beyond the highest wealth shown.
With ten stocks there is a slight bulge around $80, which disappears in simulations
drawing from 999 stocks excluding Microsoft.
F
-tests can be used to determine the statistical significance of the variance
reduction as portfolio size is increased. When comparing two distributions, the ratio
of the variances is distributed
F
(
ν
1
,
ν
2
) under the null hypothesis of equal variances,
where
ν
1
and
ν
2
are the degrees of freedom for the respective sample statistics. Among
any two of the portfolio sizes considered, the variance of the larger portfolio is always
significantly less, at the 1% level, than the variance of the smaller portfolio.

564
D. L. Domian et al./The Financial Review 42 (2007) 557
–
570
Figure 3
Cumulative wealth distributions
The cumulative wealth distributions are generated from a 1,000-stock sample covering 1985
–
2004. Wealth
per dollar of initial investment is measured at the end of the 20-year period.
The wealth densities are transformed into cumulative distribution functions in
Figure 3. We continue to study six portfolio sizes ranging from ten to 200 stocks.
To apply the Safety First criterion, a long-term U.S. Treasury bond is the ap-
propriate risk-free asset for the 20-year horizon. From the Ibbotson Associates SBBI
data, the benchmark 20-year Treasury bond had a yield to maturity of 11.70% at the
beginning of 1985. One dollar invested at 11.70% would grow to $9.14 at the end of
20 years. The cumulative distribution functions from Figure 3 show that 40.2% of the
ten-stock portfolios underperformed the $9.14 Treasury bond wealth. This shortfall
probability drops to 29.2% for 20 stocks, 22.1% for 30 stocks, 13.4% for 50 stocks,
4.3% for 100 stocks and just 0.4% for 200 stocks.

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