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EdwardS. EdwardS. O'Neal Can investorsreceive investorsreceivediversification than single benefitsfrom from holdingmore holdingmorethan mutualfund theirportfolios? fund in theirportfolios? Simulation Simulationanalysisshows analysisshowsthat that the time-series are minimalbut but that theexpected theexpecteddispersionin terminaldiversification benefitsare wealthcanbe periodwealthcan besubstantially substantially reduced reducedby holdingmultiplefunds.Portfolios funds.Portfolios withas withasfew asfour fewas fourgrowth growthfundshalve fundshalvethe thedispersion in in terminal-period dispersion wealthfor 5- to19-year to 19-yearholdingperiods.Inaddition, In addition,downsideriskmeasures riskmeasures declineasfulnds asfulnds areadded areaddedtoportfolios. to portfolios. Theseadvantages tomultiple-fund to multiple-fund portfolios areespecially areespecially meaningful forinvestors forinvestors fundingfixed-horizon investment asretirement goalssuch goalssuchas retirement orcollege orcollegesavings. savings. ith the continued proliferation proliferat ion of mutual
funds and the integral part they play in many investors' portfolios, the question of how many mutual funds constitute a diversified portfolio grows increasingly important. Since Markowitz's (1952) seminal work on portfolio selection, several researchers have examined the number of stocks required to form a diversified equity portfolio. Research addressing the corresponding question for mutual funds is conspicuously lacking, however. This relative void is likely propagated by the conventional wisdom that most mutual funds hold enough securities to eliminate unsystematic risk from their portfolios. The fact remains, however, that mutual fund performance, even within objective categories, is highly variable. In 1994, for example, the average growth fund tracked by Morningstar returned -1.5 percent but the standard deviation of returns to growth funds was 5.5 percent. mutual funds from Using actual return data for mutual the past 19 years to conduct simulations, this study's results with randomly selected mutual fund portfolios suggest that diversification across funds even within investment objectivecan objective can benefit investors. Studies of how many stocks are required to diversify portfolio generally measure the benefits of diversification by the reduction in the time-series standard deviation afforded by incrementally adding randomly chosen stocks to portfolio. Evans and Archer (1968) concluded that the bulk of diversification benefits are achieved with only a few
Edward S. O'Neal is assistant professorof professorof finance at the University of New Hampshireat Hampshireat Durham.
March/April1997 Financial Analysts Journal *March/April1997
stocks, calling into question the economic justification of holding more than about 10 randomly selected stocks in a portfolio. Tole (1982), using only stocks recommended by brokerage firms, put the number of stocks needed for sufficient diversification at between 25 and 40. Statman (1987), assuming the existence of a risk-free asset, similarly found the number to be between 30 and 40 securities. An average growth fund holds 78 securities (the 50th percentile of Morningstar growth funds in 1994). If investors increase their portfolios from one to two such funds, they will approximately double the number of stocks in their portfolios (i.e., in the best case, in which the two funds hold completely different portfolios). The marginal benefits in terms of the reduction of time-series standard deviation, however, are minimal in the presence of such large numbers of securities. Indeed, the simulations described in this article provide evidence that the inclusion of multiple funds affects timeseries standard deviations of portfolio returns only minimally. Radcliffe (1994) proposed the use of an alternative measure of risk for mutual fund investors. He called this measure the terminal-wealth standard deviation (TWSD). Terminal wealth is defined as an investor's wealth at the the end of a specific holding period. The terminal wealth depends on the investor's time horizon and the investments held. Two investors with identical horizons and holding the same fund(s) will achieve the same terminal wealth. Two investors with identical horizons but different investments will likely achieve different terminal-wealth levels. This variability in terminal wealth, caused by holding different investments, is of prime interest, especially to long-term investors. This expected variability in terminal wealth can be
37
quantified by running a number of simulations for a particular holding period and calculating the standard deviation of the resulting terminal-wealth levels. This measure is TWSD. Many investors use mutual funds, especially in retirement plans or college savings plans, to invest for prespecified time periods. Risk as measured by time-series portfolio standard deviation is less important to these investors than the variability in ending-period wealth. This variability in end ing-period wealth is closely tied to the risk of a shortfall between fund proceeds and the use intended for those proceeds. Investors with specific holding periods and specific threshold wealth requirements are less concerned with quarterly or monthly volatility than with the possibility that their portfolio funds will fail to finance fully their intended purposes. In this empirical analysis, simulations were run to examine the impact of holding various numbers of mutual funds on the expected variability of investors' terminal wealth. Findings indicate that increasing the number of mutual funds in a portfolio from one to six can reduce the expected variability of terminal wealth by 40-70 percent.
ANALYSIS SIMULATION
The simulation analysis assumes that one specific mutual fund objective meets an individual investor's investment needs. The primary reason for holding the objective constant is to facilitate the simulation, although that assumption may be close to reality for many investors. Morningstar currently divides equity funds into 13 separate objective categories. An equity investor is likely to be able identify category that corresponds to his or her investment needs. In reality, many investors hold funds that provide a mix of equity and fixedincome objectives in their portfolios. The choice of any particular mix, however, would be an arbitrary one for this simulation. The purpose of this analysis is to determine whether even among funds within the same investment objective, enough variability in performance is present to warrant holding several funds as opposed to a single fund. The simulation analysis was conducted in the spirit of Radcliffe (1994, pp. 744-46). Quarterly mutual fund returns were collected from the Morningstar OnDisc database. All mutual funds categorized as growth or growth and income that existed from 1976 to 1994 were selected. This process yielded 103 growth funds and 65 growth and income funds. The simulations used three choice variables: the objective (growth or growth and income), the holding period (5, 10, 15, or 19 years), and the 38
number of funds (1-8, 10, 12, 14, 16, 18, 20, 25, or 30).1
The Baseline: Single-Fund Portfolios
In the case in which the fund portfolio consisted of a single fund, the simulation was run 103 times for growth funds and 65 times for growth and income funds, one trial for each fund. One dollar was assumed to be invested in each fund at th beginning of the holding period. The wealth at the end of the holding period for each fund was calculated by compounding the quarterly returns for that fund over the entire holding period. This procedure produced 103 (65) terminal-wealth levelsone for each growth (growth and income) fund. The TWSD was then calculated as the standard deviation of the 103 (65) terminal-wealth levels. The TWSD represents the dispersion of ending wealth levels to which an investor is exposed by choosing to invest in single fund. For each fund in each holding period, the time-series standard deviation (TSSD)of the returns was also calculated. The mean of these TSSDs in each holding period represents the average time-series volatility an investor would be exposed to by choosing a single-fund portfolio. The strategy of investing in a single fund was used as a baseline against which to compare multiplefund portfolio strategies.
Multiple-FundPortfolios
The following procedure was followed for the multiple-fund portfolio simulation: One dollar is invested at the beginning of the holding period. The dollar is equally divided among n randomly chosen mutual funds. At the end of each quarter, the original dollar plus (or minus) any investment return is rebalanced equally among the funds. The wealth at the end of the final quarter of the holding period is the terminal wealth of the portfolio. The simulation was repeated 1,000 times for each combination of the choice variables. The TWSD was then calculated as the standard deviation of the 1,000 terminal wealths generated in the simulations. Thus, a TWSD was generated for each combination of choice variables. The TWSD measures the variability of terminal wealth to which an investor is exposed when selecting a certain number of funds for particular holding period. For each randomly selected portfolio, the timeseries standard deviation of returns was also calculated. For each combination of choice variables, 1,000 TSSDs were obtained. These 1,000 TSSDs
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were averaged to produce an estimate of the time series volatility an investor might expect from holding a particular size of portfolio. Tables and 2 summarize the results of these simulations. Increasing the number of funds held in the portfolio has little impact on average terminal wealth for the 5- and 10-year holding periods. For the longer holding periods, the increase in funds had a slight negative effect on average terminal wealth, which was caused by the existence of a few stellar mutual funds in the sample. The average growth fund return over the 19-year period was 1,502 percent, although the distribution is skewed to the right (the range was 543 percent to 6,794 percent). For small portfolios, the inclusion of a stellar fund had a large impact on the terminal wealth. The larger the quantity of funds in the portfolio, the more quarterly rebalancing reduces the impact that compounding the returns of the stellar fund has on overall portfolio returns. In unreported simulations, in which portfolios were not rebalanced, increasing portfolio size did not negatively affect average terminal wealth. The sample does contain a survivorship bias, which helps explain the generally higher average returns for smaller portfolios. Morningstar publishes data only on existing funds. Some funds, however, especially poor performers,ceased to exist during the period and thus do not show up in the database. Just as stellar performers have a greater effect on portfolios with fewer funds, so too do funds that perform extremely poorly. Thus, the survivorship bias serves to overstate the terminal wealth reported for all portfolios. This overstatement, in the presence of rebalancing, is greater for portfolios with few funds than for those with many funds. The average TSSD of portfolio returns was reduced only slightly for portfolios holding larger numbers of funds. Forgrowth (growth and income) funds, increasing the number of funds from to 30 decreased the TSSDby approximately percent (12 percent), regardless of the holding period. This finding is consistent with previous studies of time-series diversification. Because most funds hold more than 60 securities, minimal time-series diversification benefits are expected from adding more securities (through additional mutual funds) to the portfolio. The TWSD decreased significantly as the number of funds in portfolio increased. The decrease was evident for all holding periods and for growth as well as growth and income funds. To gauge the percentage reduction in TWSD, each TWSD was standardized by dividing by the TWSD for singlefund portfolios for each holding period. The results are presented in Table 3 and graphed in Figures Financial Analysts Journal March/April1997
and 2. The reduction in TWSD attainable by adding funds to portfolio appears to be greater for growth funds than for growth and income funds. For growth funds, a six-fund portfolio reduced TWSD to between 31 percent and 41 percent (depending on the holding period) of that expected for singlefund portfolios. For growth and income funds, holding six funds reduced TWSD to between 47 percent and 52 percent of single-fund portfolios. The reduction in TWSD is greater over all holding periods for growth funds than for growth and income funds. Growth funds as a group display greater dispersion in terminal wealth than growth and income funds, which may cause multiple-fund portfolios to exhibit greater dispersion reduction. This result suggests that more aggressive investors have the most to gain by diversifying across funds. The difference in TWSD reduction was generally greater the longer the holding period. The only exception is that for growth and income funds, multiple funds afforded better terminal-wealth diversification in the 5-year period than in the 10- and 15-year periods. In all cases, the marginal benefit of successively adding funds to the portfolio decreased as the number of funds increased. To determine whether the reduction in TWSD varies across time for identical holding periods, three additional five-year periods were examined. The five-year holding period data documented in Tables through are for the 1990-94 period. The simulations were duplicated for the 1976-80,198084, and 1985-89 periods for growth funds. Results for the reduction in TWSD for each holding period are graphed in Figure 3. The time period does not appear to have large impact on the reduction in TWSD. For a six-fund portfolio, the reduction in TWSD ranged between 36 percent and 42 percent of that expected with single-fund portfolio for the four time periods.
MEASURESOF DOWNSIDERISK The dispersion in ending-period wealth levels, as measured by the TWSD, takes into account positive as well as negative deviations from the mean. Investors, however, may not view deviations on the positive side as contributing to the riskiness of a mutual fund portfolio. Further, if the reductions in TWSD achieved by holding multiple-fund portfolios come mainly by truncating the upside of the distribution, the benefits of multiple-fund portfolios may be overstated. Three measures of the downside risk of terminal-wealth levels were examined as the number of funds in a portfolio is varied. 39
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Harlow (1991) identified several measures of downside risk that may be more intuitive than traditional risk measures. The first of these, shortfall probability, measures the likelihood that an investment's returns will fall below a specific target return. In the simulation analysis, the mean return was used as the target, and shortfall probability can be calculated by finding the percentage of simulations that display terminal-wealth levels below the mean: Shortfallprobability Number of observationsbelow the mean where n is the total number of observations. The major shortcoming of shortfall probability as risk measure is that it does not account for the magnitude by which returns fall short of the mean. Two other measures address the magnitude of shortfall. Harlow considered target shortfall that measures the deviation from a target of those observations that are below the target. This method characterizes as more risky those distributions that have large downside deviations from the target. Forthe simulation analysis, the target was the mean of the fund portfolio returns. Semivariance is a very similar measure. It measures the squared deviation from the mean of those observations that are below the mean. This measure, as with variance, gives greater weight to those observations that are farthest from the mean. When measuring downside risk, this weighting may be appropriate because investors are most averse to those largest downside deviations. Equations and detail the calculation of mean shortfall and semivariance.
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Each of these measures depends on the mean of the distribution. For the 15- and 19-year holding periods, the mean terminal wealth was slightly higher for the single-fund portfolios than for the multiplefund portfolios, as illustrated in Table I. Therefore, strict application of the equations for each number of funds will penalize single-fund portfolios. The higher mean will increase all three downside risk measures relative to a distribution with lower mean. The higher mean is a benefit to single-fund portfolios. Because the purpose of this study is to quantify any benefits to multiple-fund portfolios vis-a-vis single-fund portfolios, the calculations of the downside risk measures for all multiple-fund portfolios assume the mean terminal wealth for th single-fund case. For the growth fund sample, the three downside risk measures were calculated for 5-, 10-, 15-, and 19-year holding periods. Results are shown in Table 4. Shortfallprobabilitywas relatively stable for the shorter holding periods (5 and 10 years) as the number of funds increased. For the longer ods, shortfall probability increased with the number of funds in the portfolio. This finding is likely the result of survivorship bias in the data. The few stellar funds in the sample increased the mean terminal wealth most for the single-fund portfolios. The distributions of the multiple-fund portfolios exhibit a
Table 3. Terminal-Wealth Standard Deviation as a Percent of Single-Fund PorffolioTerminal-Wealth Standard Deviation by Type of Fund and Holding Period Holding Period: Growth and Income Funds
Holding Period: Growth Funds Number of Funds 1
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100% 82 76 66 61 57 52 49 46 43 40 37 34 32 30 27
19 Years 100% 71 63 54 50 47 42 41 38 34 32 29 29 27 25 21
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Figure 1. Reduction inTerminal-WealthStandard Deviation for Growth Fund Porffolios over DifferentHolding Periods 110 100 90 80 70600 50 40
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central tendency around a mean that is less influenced by the stellar performers (and hence slightly lower than that of the single-fund portfolios). Because shortfall probability was calculated using the single-fund mean, the central tendency (around a lower mean) of the multiple-fund portfolios causes an increase in shortfall probability. Shortfall probability has the drawback of not considering the magnitude of shortfall. Both mean shortfall and semivariance overcome this weakness (the square root of semivariance is reported in Table 4). Both of these measures decreased substantially as more funds were added to the portfolio. For the 5- and 10-year holding periods, the mean shortfall and semivariance 1/2 were reduced by at least half by holding five funds in the portfolio. The longerholding-period results also displayed reductions in these two measures, although they are less pronounced than for shorter holding periods.
--
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--1976-80
CONCLUSION
Holding more than a single mutual fund in a portfolio appears to have substantial diversification benefits. The traditional measure of volatility, the time-series standard deviation, is not greatly influenced by holding multiple funds. Measures of the dispersion in terminal-wealth levels, however, which are arguably more important to long-term investors than time-series risk measures, can be reduced significantly. The greatest portion of the reduction occurs with the addition of small numbers of funds. This reduction in terminal-period wealth dispersion is evident for all holding periods studied. Two out of three downside risk measures are also substantially reduced by including multiple funds in a portfolio. These findings are especially important for investors who use mutual funds to fund fixed-horizon investment goals, such as retirement and college savings.3
NOTES 1. The last year for each holding period is 1994. Thus the 19year holding period is 1976 to 1994, the 15-year holding period is 1980 to 1994, the 10-year holding period is 1985 to 1994, and the 5-year holding period is 1990 to 1994. 2. Two alternative targets that also have intuitive appeal are zero-return target and a risk-free return target. The zeroreturn target constitutes a "loss-of-principal" measure of downside risk, and the risk-free return target benchmarks performance against a riskless alternative investment. In the current analysis, the vast majority of funds produce terminal wealth levels that surpass both of these targets over each investing horizon. Using the mean fund return as the target provides a more quantifiable measure of downside risk
Financial Analysts Journal *March/April1997
reduction. It also minimizes the impact that survivor bias may have on downside risk results. Although the sample fund portfolio returns are likely to be biased upward, the target is also biased upward because it is the mean of a group of portfolios of surviving funds. If the target were zero return or risk-free return, the surviving funds would display an artificially high probability of surpassing the target. 3. The author thanks R. Ward Flintom for several insightful discussions on these issues. Also deserving special acknowledgment are David Bradford, Ahmad Etebari, Franklin Fant, W. Van Harlow III,Miles Livingston, Jeff Lenz, and Economics seminar participants at the University of New Hampshire.
45
REFERENCES Evans, John L., and Stephen H. Archer. 1968. "Diversification and the Reduction of Dispersion: An Empirical Analysis." Jouirnial of Finanice,vol. 23, no. 5 (December):761-67. Downside-Risk Harlow, W.V. 1991. "Asset Allocation in Framework."FinanicialAnalystsJouirnial, vol.47, no. 5 (September/ October):28-40. Markowitz, Harry. 1952. "Portfolio Selection." Joirnal of Finlanlce, vol. 7, no. (March):77-91.
46
Radcliffe, Robert C. 1994.Investment:Concepts,Analysis, Strategy. New York:Harper Collins College Publishers. Statman, Meir. 1987. "How Many Stocks Make a Diversified Portfolio?"Journalof Finanacial and QuiantitativeAnalysis, vol. 22, no. 3 (September):353-63. Tole, Thomas M. 1982. "You Can't Diversify without Diversifying." Journalof PortfolioManagemenit,vol. 8, no. (Winter):5-11.
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