4 ending wealth we next consider ending wealth rather

This preview shows page 6 - 9 out of 14 pages.

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
Image of page 6

Subscribe to view the full document.

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.
Image of page 7
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.
Image of page 8

Subscribe to view the full document.

Image of page 9
  • Fall '19

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes