Optimal stock holdings in fund portfolios shawky

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Optimal stock holdings in fund portfolios shawky

  1. 1. The Financial Review 40 (2005) 481--495 Optimal Number of Stock Holdings in Mutual Fund Portfolios Based on Market Performance Hany A. Shawky University at Albany David M. Smith∗ University at AlbanyAbstract Among the decisions that most mutual fund portfolio managers make is the number ofstocks to hold. We posit that there is an optimal number of stocks for each mutual fund,reflecting the trade-off between diversification benefits versus transactions and monitoringcosts. We find a significant quadratic relation between number of stock holdings and risk-adjusted returns for U.S. equity mutual fund portfolios during 1992–2000. Moreover, we findthat changes in the number of stocks held over time are more highly correlated with mutualfund flows than with funds’ investment returns.Keywords: mutual funds, portfolio selection, risk-adjusted returnsJEL Classifications: G11, G2∗ Corresponding author: Business Administration 309, Center for Institutional Investment Management,University at Albany, SUNY, Albany, NY 12222; Phone: (518) 442-4245; Fax: (518) 442-3045; E-mail:ds693@albany.eduWe thank David Allen, Susan Belden, Rita Biswas, Shobha Chengalur-Smith, Christophe Faugere, BruceGeller, Lester Hadsell, Kajal Lahiri, Gary Sanger, Robert Schweitzer, Ravi Shukla, and Vijay Singal forhelpful discussions and comments on previous drafts of this article. We are particularly grateful to Sang-gyung Jun and John Stowe for insightful comments on multiple drafts. We are indebted to John Bonnett,Avis Bonnett, and Alexandra Landau for data assistance. 481
  2. 2. 482 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–4951. Introduction Evans and Archer (1968) were the first to suggest that an investor can achievefunctionally complete diversification by investing in about 10 randomly selectedstocks. In a later study, Statman (1987) shows that 30–40 stocks, at a minimum,are required to achieve a portfolio that is diversified. In practice, there are equitymutual funds that hold fewer than 40 stocks, whereas others exceed this numbersubstantially. For example, in 1998, East End Capital Appreciation Fund reported thatit held the stock of 18 different companies, at the same time that the Eclipse EquityFund held 304 stocks. Both funds shared the stated objective of “seeking growth ofcapital.” Diversification is widely held to be a key tenet of modern portfolio management.1The number of stocks held by a fund is one of many decisions for a portfolio manager.In this article, we posit and empirically test a nonlinear relation between risk-adjustedreturns net of expenses and the number of stocks held. Furthermore, we analyze thechange in the number of stocks held between 1992 and 2000. Although prior workconsiders diversification for simulated portfolios, this study is, to our knowledge, thefirst to examine the diversification issue for actual mutual fund portfolios. Fisher and Lorie (1970) offer support for Evans and Archer’s results for a sampleof randomly selected New York Stock Exchange-listed companies. Evaluating returndistributions for the years 1926–1965, they show that holding a portfolio of eightstocks instead of one stock decreased diversifiable risk by approximately 80%. Eltonand Gruber (1977) point out that a large amount of diversifiable risk can be removedby increasing the number of stocks in a portfolio from 15 to 100. More recently, DeWit (1998) shows that increasing the portfolio holdings from 100 equally weightedstocks to 500 can reduce the required return by 6 and 21 basis points. One of hisconclusions is that significant diversification benefits can accrue even when addingstocks to already-large portfolios.2 Although diversification across stocks is generally accepted as an importantcomponent of portfolio construction as originally described by Markowitz (1959),overdiversification carries several potential costs as well. First, each fund’s portfoliomanagement staff is responsible for monitoring the performance of the stocks held.Increasing the number of stocks is likely to raise monitoring costs.3 For example, “With all the market volatility, [Fidelity Convertible Securities Fund] manager Andrew Offit says he wants a smaller, more concentrated portfolio, one in which he can know every name inside and out and can stay abreast of any changes. In a skittish market where1 Thissentiment is not universal. According to Warren Buffett, “Diversification is protection againstignorance, but if you don’t feel ignorant, the need for it goes down dramatically” (Lenzer, 1993).2 In related work, O’Neal (1997) explores the benefits to investors from diversifying across mutual funds.3 Monitoring costs can take the form of additional personnel costs, as well as poor performance due to aportfolio manager’s (or management team’s) inability to track a large number of stocks.
  3. 3. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 483 investors will bludgeon a company if it falls a few pennies short of earnings estimates, he says, it’s more important than ever to keep a vigilant eye on holdings.”4Second, positive effects from a portfolio manager’s best investment ideas are dilutedwhen a portfolio’s value is spread across a large number of stocks.5 Third, it is possiblethat transaction costs would rise as stocks are added to the portfolio. Although thetotal price impact from a single large transaction would be greater than the priceimpact of multiple smaller transactions, the fixed component of commission fees ishigher when multiple trades are made.6 The principal finding of the study is that for U.S. domestic equity funds, aftercontrolling for fund size, market capitalization of stocks held, and the percentage ofholdings in cash, the relation between risk-adjusted returns and the number of stocksheld is quadratic. The average number of stocks held shows only a slight upward driftbetween 1992 and 2000. We find that changes in the number of stocks are stronglyrelated to the levels of new investments and redemptions but not to fund size changedue to market returns.2. Data screens and descriptive statistics2.1. Source The data come from Morningstar’s year-end Mutual Funds OnDisc and Prin-cipia Pro, for 1992–2000. The Morningstar database provides information on manyindividual fund characteristics, including historical return, risk measures, portfoliocomposition, and the primary portfolio manager.2.2. Screens Table 1 shows the results of screening funds on the following criteria, generatinga sample of 5,685 fund-years. Each fund must (1) be all-equity, holding no bonds or other nonequity securities; (2) be actively managed, not an index fund, enhanced index fund, fund of funds, or market-neutral fund; (3) be neither a sector fund nor one that the prospectus claims is undiversified;4 Source: Morningstar Mutual Funds OnDisc (December 1994), report by Morningstar analyst DanielO’Keefe.5 Lauricella (2001) points out that conventional wisdom often equates a mutual fund’s top 10 stock holdingswith the manager’s “best ideas.” He reports on a Morningstar study showing that the performance of thehighest-weighted 10 stocks in funds outperformed the total fund only 48% of the time between 1996and 2001. However, the study neglects to consider the fact that a stock may enter the top 10 due to pastoutperformance, and by the time it comes into the top 10 the manager is preparing to sell the stock.Moreover, liquidity constraints may cause the manager’s best ideas not to make it onto the top 10 list.6 For example, due to the fixed component of commissions, brokerage fees are usually higher when trading10,000 shares each of three different stocks than when trading 30,000 shares of one stock.
  4. 4. 484 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495Table 1Sample selection: Results of data screensThe sample is identified from the Morningstar fund universe between 1992 and 2000, and the screensbelow are imposed.Screen 1992 1993 1994 1995 1996 1997 1998 1999 2000Morningstar fund universe 2,540 3,406 5,462 6,877 7,746 8,736 10,353 11,216 12,081(1) No bonds or “other 900 1,032 1,815 1,493 1,817 2,303 3,056 3,459 4,620 securities”(2) No index funds, enhanced 872 959 1,699 1,404 1,704 2,142 2,764 3,138 4,250 index funds, funds of funds, or market-neutral funds(3) No undiversified 795 903 1,536 1,310 1,562 1,933 2,458 2,723 3,609 (including sector) funds(4) No redundant class 625 655 1,211 873 1,046 1,219 1,412 1,459 1,838(5) No international funds 388 390 511 520 713 757 898 987 1,212(6) Full data available 307 299 339 390 545 560 672 784 942 (4) not be a redundant class (i.e., fund class A is retained, but classes B and C are dropped); (5) not have a prospectus objective of “international” or “foreign” fund; and (6) have returns and portfolio holding data available. The various screens are imposed for the following reasons. Because our focusis on the composition of common stock portfolios, the presence of nonstock hold-ings raises asset allocation issues. It is well established that portfolio performance iscritically dependent on asset allocation (Ibbotson and Kaplan, 2000). By imposingthe first screen, we seek to examine portfolio characteristics while holding asset al-location constant. Hence, we eliminate funds that hold securities other than commonstock. In a further effort to avoid the confounding effects of asset allocation decisions,screen (5) omits funds dedicated specifically to international stock investments. Screens (2) and (3) are imposed because the analysis requires that the number ofstocks held be a variable under each fund manager’s control. Equity indexes are com-posed of well-defined numbers of stocks, so index funds usually hold approximatelythe number of stocks found in the relevant index. Moreover, the managerial objectivefunction is different for passively managed funds than for actively managed funds.Whereas most actively managed funds attempt to maximize raw or risk-adjusted re-turns, the objective of index funds is to minimize return tracking error relative toindexes. For funds of funds, the decision on the number of stocks to hold is not underthe managers’ direct control, and diversification effectiveness may not be an issue ofparticular concern to the manager.77 Funds of funds are mutual funds invested in the shares of other funds rather than in individual stocks.
  5. 5. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 485 The number of stocks that a fund manager chooses to hold may be relatedto several factors, including the manager’s need to diversify. It is assumed in thisarticle that managers regard the benefits of diversification as a basic tenet of portfoliomanagement, and as one of the primary influences on the decision of how manyassets to hold. Hence, funds that are undiversified by design or by stated intentionof the manager are excluded from the sample.8 A policy to remain undiversified isidentified in the prospectus summaries contained in the Morningstar database. Screen (4) is imposed because many funds are offered in multiple classes.9 Weremove redundant observations only after verifying that the portfolio holdings forall classes are identical. The information reported in this article on fund size is theaggregate across all classes of each fund. This screen is especially important for theregression analysis portion of the study, to preserve the assumption of independenceacross observations.2.3. Descriptive statistics Table 2 shows fund characteristics by year. Between 1992 and 2000, medianfund size increases by 115%, to $206 million. This increase is attributable both tothe dramatic increase in stock prices over that time span, as well as to the flow ofnew money into equity funds. The median expense ratio remains constant between1992 and 2000, and median portfolio turnover fluctuates from 65% in 1992 to 54%in 1995 to 79% in 2000. The mean and median tenure of fund managers declines overthe period. This observation likely reflects the upsurge in the number of new fundsin the 1990s. As the sample selection summary in Table 1 shows, Morningstar’s funduniverse increased in number from 2,540 in 1992 to 12,081 in 2000. The median number of stocks held by individual funds remains fairly stablewithin the range 57–72. However, within each year, there is substantial cross-sectionalvariability in the number of stocks held. Figure 1 shows the frequency distributionsof the number of stocks held for the starting, middle, and ending sample years (1992,1996, and 2000). In each case, the distribution is positively skewed, with the majorityof funds holding between 40 and 120 stocks. Both the mean and median values areconsistently above 16 stocks, which, under the U.S. Investment Company Act of 1940,is the minimum number a fund must hold to be classified as “diversified.”108 For example, the prospectus for the Fidelity Fifty fund states that the fund typically maintains 50–60holdings that represents a constraint for the manager. The prospectus for the Jensen fund states that thefund is undiversified by design.9 In such cases, one class usually carries a front-end load, a second class a back-end load, and a third classno load with high 12B-1 fees. Other class structures involve fund availability, with one class available tothe general public directly from the fund company, a second class (institutional shares) available only toinstitutional investors, and a third class (service shares) is intended for individuals but sold only throughfinancial institutions.10 Also, with respect to 75% of a diversified fund’s assets: the fund may hold no more than 10% of a singleissuer’s securities, and no more than 5% of the assets of the fund may be in any one issuer’s securities.
  6. 6. 486Table 2 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495Univariate statistics by sample yearScreened Morningstar mutual fund data between 1992 and 2000. For each variable, mean values are followed by medians and standard deviations. Fund size isthe aggregate market value of all classes of the fund, in millions of dollars. Number of stocks is the number of stocks held in the portfolio at the most recentreporting date. Median market capitalization is the market value of the median-size stock in the fund, in millions of dollars. Cash holdings is the percentage ofthe fund’s assets held in cash. Top 10 holdings is the percentage of the fund’s assets held in the 10 most heavily weighted securities. Sharpe ratio is the differencebetween the one-year mutual fund raw return (calculated net of management and 12B-1 fees) and the one-year treasury bill rate, divided by the annualizedstandard deviation of mutual fund returns. Expense ratio is operating expenses, management fees, and 12B-1 fees as a percentage of the average daily fund value.Turnover is the dollar value of the fund’s portfolio that the manager bought or sold over the preceding year, divided by the monthly average market value for thefund. Fund manager tenure is the number of years the manager has been employed by the mutual fund company. To improve readability, numbers to the right ofthe decimal point are omitted in some cases. NA indicates that data are not available for that year. Values are expressed as mean; standard deviation respectively. Subsample from yearVariable 1992 1993 1994 1995 1996Fund size 365; 96; 864 455; 145; 920 467; 133; 1,165 506; 150; 1,232 714; 189; 2,104No. of stocks 81; 55; 82 87; 63; 74 87; 65; 78 84; 65; 68 99; 69; 126Median market capitalization 4,652; 3,440; 4,456 4,469; 3,102; 4,450 4,664; 3,181; 4,627 6,461; 4,728; 6,226 8,624; 5,378; 9,045Cash holdings (%) 11.47; 8.20; 12.11 8.75, 6.00; 9.02 9.06; 6.03; 10.26 6.95; 5.23; 7.60 6.45; 4.70; 6.71Top 10 holdings (%) NA NA NA NA NASharpe ratio 0.25; 0.25; 0.44 0.65; 0.63; 0.61 −0.44; −0.45; 0.39 2.46; 2.52; 0.85 1.31; 1.39; 0.68Expense ratio (%) 1.35; 1.22; 0.59 1.24; 1.18; 0.54 1.24; 1.17; 0.55 1.25; 1.13; 0.70 1.26; 1.17; 0.63Turnover 88; 65; 102 75; 57; 83 75; 59; 64 73; 54; 67 83; 65; 74Fund manager tenure (years) 6.09; 5.00; 5.88 6.46; 5.00; 5.65 5.82; 4.00; 5.44 5.79; 4.00; 5.80 5.17; 4.00; 5.33Observations 307 299 339 390 545 (continued )
  7. 7. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495Table 2 (continued)Univariate statistics by sample year Subsample from yearVariable 1997 1998 1999 2000 Whole sampleFund size 890; 198; 3,526 953; 180; 4,021 1,040; 205; 4,339 1,036; 206; 4,132 811; 173; 3,305No. of stocks 96; 72; 100 91; 65; 116 91; 67; 97 92; 69; 102 91; 66; 100Median market capitalization 11,260; 5,567; 13,792 18,485; 9,243; 21,595 27,831; 10,711; 32,994 29,251; 14,734; 32,591 16,467; 6,091; 24,182Cash holdings (%) 5.27; 4.00; 6.17 5.90; 4.20; 6.18 4.86; 2.80; 6.69 4.67; 3.20; 6.12 6.32; 4.20; 7.69Top 10 holdings (%) 30.79; 28.36; 12.96 33.62; 31.85; 12.52 33.54; 31.37; 12.99 34.74; 32.69; 12.60 33.52; 31.44; 12.80Sharpe ratio 1.25; 1.36; 0.71 0.45; 0.48; 0.70 0.85; 0.81; 0.96 −0.22; −0.31; 0.59 0.67; 0.57; 1.05Expense ratio (%) 1.36; 1.20; 0.61 1.25; 1.20; 0.54 1.23; 1.20; 0.40 1.26; 1.23; 0.62 1.26; 1.20; 0.57Turnover 83; 67; 72 86; 66; 86 89; 71; 80 104; 79; 105 87; 67; 85Fund manager tenure (years) 5.05; 4.00; 4.64 5.19; 4.00; 4.33 4.89; 4.00; 4.14 5.24; 4.00; 3.87 5.40; 4.00; 4.82Observations 560 672 784 942 4,838 487
  8. 8. 488 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 1992 Distribution 200 150 Number of Funds 100 50 0 1-40 41- 81- 121- 161- 201- 241- 281- 321- 361- 401- 441- 481- Over 80 120 160 200 240 280 320 360 400 440 480 520 520 Number of Stocks 1996 Distribution 300 250 Number of Funds 200 150 100 50 0 1-40 41- 81- 121- 161- 201- 241- 281- 321- 361- 401- 441- 481- Over 80 120 160 200 240 280 320 360 400 440 480 520 520 Number of Stocks 2000 Distribution 500 450 400 Number of Funds 350 300 250 200 150 100 50 0 1-40 41- 81- 121- 161- 201- 241- 281- 321- 361- 401- 441- 481- Over 80 120 160 200 240 280 320 360 400 440 480 520 520 Number of StocksFigure 1Number of stocks in actively managed U.S.-based equity mutual fundsDistributions of the number of stocks held by actively managed U.S.-based equity mutual funds in 1992,1996, and 2000.
  9. 9. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 489 For descriptive purposes, we also report the degree of portfolio concentrationbecause the number of stocks held may not accurately reflect the dispersion of hold-ings.11 One commonly used measure of concentration is the percentage of assetsinvested in the top 10 holdings, which is available from Morningstar starting in 1997.Table 2 shows that the top 10 holdings consistently represent about one third of thetotal portfolio investment, which is approximately three times the weight that wouldbe suggested by an equal allocation across stocks. Indeed, a casual examination offunds’ total portfolio holdings makes clear that for whatever reason, managers tendto hold a large number of stocks in small proportions.123. Analysis of number of stocks held by equity mutual funds3.1. Cross-sectional analysis of 1992–2000 data3.1.1. Pearson’s correlation estimates Table 3 contains contemporaneous Pearson’s correlations for the entire sample.The number of stocks held is positively correlated with fund size. The economicintuition behind such a relation is straightforward. The liquidity concerns for largefunds are likely to require that managers hold portfolios composed of a relativelylarge number of stocks. Conversely, managers of smaller funds can hold more limitednumbers of stocks without liquidity being as great a concern.13 Fund size is negativelyrelated to expense ratio (and note again that the sample does not contain index funds),consistent with the findings of Dellva and Olsen (1998). Not surprisingly, the number of stocks held is negatively related to the fund’spercentage of cash holdings and to the degree of concentration in the top 10 stocks.Number of stocks is positively related to portfolio turnover. The negative relationbetween number of stocks and expense ratio may indicate a direct effect of holdingfewer assets, or it could indicate some other indirect effect. Specifically, economiesof scale associated with larger funds likely drive the expense ratio down, and for11 Strongin,Petsch, and Sharenow (2000) recognize this problem and analyze its impact on the degree ofsuccess a portfolio manager has in minimizing tracking error.12 Stowe, Jordan, and Jordan (1988) express the degree of portfolio concentration using “security equiva-lents.” For each fund, the security equivalent is the number of securities held in equal weights that wouldproduce the same level of diversification currently reflected by the unequally weighted portfolio. A fund’ssecurity equivalent is calculated as 10,000 divided by the fund’s Herfindahl Index. For the current sample,the ratio of number of stocks held to security equivalents is typically about 1.5. But why do managers holdstocks in such small weights? One possible reason is that holding a small amount of a company’s stock isa convenient way for a fund manager to automatically continue receiving pertinent information about thatcompany.13 Acounterargument can be made, however, for a negative relationship between “number of stocks” and“median market capitalization.” All else equal, funds investing in smaller capitalization stocks face greaterpotential liquidity problems than funds investing in large capitalization stocks.
  10. 10. 490 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495Table 3Correlations for mutual fund variablesThis table shows Pearson’s correlation coefficients between mutual fund variables for a sample taken from 1992 to 2000. Fund size is the aggregate market valueof all classes of the fund, in millions of dollars. Median market capitalization is the market value of the median-size stock in the fund, in millions of dollars. Cashholdings is the percentage of cash in the fund’s portfolio. Expense ratio is operating expenses, management fees, and 12B-1 fees as a percentage of the averagedaily fund value. Turnover is the dollar value of the fund’s portfolio that the fund manager bought or sold over the preceding year, divided by the monthly averagemarket value for the fund. Fund No. of Median market Cash Top 10 Expense Portfolio ManagerVariable size stocks capitalization holdings (%) holdings (%) ratio turnover tenureFund size 1.00No. of stocks 0.15∗∗∗ 1.00Median market capitalization. 0.12∗∗∗ −0.07∗∗∗ 1.00Cash holdings (%) −0.02 −0.04∗∗∗ −0.15∗∗∗ 1.00Top 10 holdings (%) −0.06∗∗∗ −0.42∗∗∗ 0.20∗∗∗ 0.31∗∗∗ 1.00Expense ratio −0.14∗∗∗ −0.11∗∗∗ −0.13∗∗∗ 0.06∗∗∗ 0.22∗∗∗ 1.00Portfolio turnover −0.04∗∗∗ 0.02∗∗ −0.10∗∗∗ 0.01 −0.06∗∗∗ 0.16∗∗∗ 1.00Fund manager tenure 0.05∗∗∗ −0.06∗∗∗ 0.01 0.11∗∗∗ 0.22∗∗∗ −0.02 −0.16∗∗∗ 1.00∗∗∗Indicates statistical significance at the 0.01 level.∗∗ Indicates statistical significance at the 0.05 level. ∗ Indicates statistical significance at the 0.10 level.
  11. 11. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 491different reasons, larger funds also tend to hold a greater number of stocks. Expenseratio is negatively related to the median market capitalization, indicating that fundsholding smaller stocks tend to exhibit higher expense ratios than funds holding largecapitalization stocks.3.1.2. Ordinary least squares regressions The primary hypothesis to be tested in this article is that fund performance isrelated to the number of stocks held. We posit that the relation is nonlinear. Specif-ically, as the number of stocks increases, a higher risk-adjusted net (after expenses)return is expected initially. However, beyond a certain point, an increase in the numberof stocks held would cause marginal monitoring costs to exceed the marginal diver-sification benefit, leading to decreased risk-adjusted net returns. Figure 2 presentsthe type of quadratic relation we propose, and the optimal number of stocks to holdis depicted on the graph as NumStock∗ , the point at which the fund’s risk-adjustedreturns are maximized. We conduct a regression analysis on the pooled sample of 4,838 observationsdescribed earlier for the period 1992–2000. In examining the hypothesized relationbetween risk-adjusted return and number of stocks, it is important to recognize thesignificant association between number of stocks held and (1) fund size, (2) themedian market capitalization of holdings, and (3) the percentage of the portfolio keptin cash. It is likely that managers make decisions involving the number of stocksheld in the context of all three of these factors. We control for these three factors byincluding them in the regression model.Risk-adjusted return NumStock* Number of stocks heldFigure 2Hypothesized relationship: Risk-adjusted return versus number of stocks held
  12. 12. 492 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 Ordinary least squares regression parameters are estimated for the sample usingthe equation Sharpeit = α + β1 NumStockit + β2 NumStock2 + β3 Sizeit ˆ ˆ ˆ it ˆ + β4 MdMkCpit + β5 %Cashit + β6 1992it + · · · + β13 1999it + εit , ˆ ˆ ˆ ˆ ˜ (1)where Sharpeit is the fund’s Sharpe ratio calculated as the difference between fund i’sexpense-adjusted rate of return and the one-year T-bill rate in year t, divided by theannualized standard deviation of returns; and NumStockit is the number of stocks heldby fund i in year t. The expected signs of the first two coefficients are positive andnegative. The control variables are defined as follows. Sizeit is the aggregate assetsfor all classes of the fund, and MdMkCpit is the market capitalization of the medianfirm in which fund i invests, both as of the end of year t. Both variables are measuredin millions of dollars. “%Cashit ” is the proportion of each fund’s cash holdings at yearend. The other independent variables represented by 1992it –1999it are time dummies,and the year 2000 serves as the omitted class. The final term is a random error term,assumed to be independent and identically distributed. Table 4 reports the initial results for our panel regressions. We find that fundperformance is positively related to the number of stocks and negatively related tosquared number of stocks. The regressions are consistent with our hypothesis, and thecoefficients are statistically significant. Several of the control variables, including allof the time dummies, have significant coefficients as well. It is possible to derive an approximate value for the optimal number of stocks tohold using our estimated parameters in Table 4. Taking the partial derivative of theSharpe ratio with respect to NumStock in Equation (1), and equating the result to 0,yields an optimum NumStock∗ .14 The value we obtain for Numstock∗ is 481.59. Thisnumber is substantially above the observed average of 96 stocks. However, recall thatDe Wit (1998) suggests that benefits can be obtained by adding stocks to a portfoliothat already contains several hundred stocks. It is nonetheless useful to rememberthat our calculated NumStock∗ value is derived from the point estimates for the twoparameters.15 The main conclusion should not necessarily be that there is a uniqueoptimum but that extremely low or high numbers of stocks held are suboptimal.When we recall from Figure 1 that many funds meeting the statutory definitionof “diversified” hold 40 or fewer stocks, it becomes clear that this conclusion hassubstantial practical importance.14 Takingthe first derivative of Sharpe with respect to NumStock results in ∂Sharpe/∂NumStock = β 1 +2β 2 NumStock. Now setting the expression equal to 0 and solving for NumStock: β 1 + 2β 2 NumStock =0; and NumStock = −β 1 /2β 2 . Substituting in the estimates for β 1 and β 2 yields 481.59.15 The confidence intervals around each point estimate are very wide (the 90% confidence interval rangesbetween 40 and 4,000 stocks). Hence, we conclude that rather than a single value, there exists a set ofvalues that can be reasonably interpreted as an optimal range of stocks to hold.
  13. 13. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 493Table 4Regression analysis of fund performance versus number of stocks and control variablesRegression coefficients for screened Morningstar mutual fund data from 1992 to 2000. Of the 4,838observations in the overall sample, 1,554 are unique funds. In the rightmost column we correct for possibleserial correlation as follows. In each case where a fund appears in multiple years, one year’s observationis randomly selected and used in the regression. The one-year mutual fund Sharpe ratio is regressed onnumber of stocks held (NumStock) and squared NumStock. Return is calculated net of management and12B-1 fees. Fund size (Size), the median market capitalization (MdMkCp) of companies held, %Cash,and time dummies 1992–1999 serve as control variables. The omitted class is the year 2000. Formally, themodel is Sharpeit = α + β1 NumStockit + β2 NumStock2 + β3 Sizeit + β4 MdMkCpit ˆ ˆ ˆ it ˆ ˆ + β5 %Cashit + β6 1992it + · · · + β13 1999it + εit . ˆ ˆ ˆ ˜ Coefficients uncorrected Coefficients corrected forVariable for serial correlation possible serial correlationIntercept (0.282)∗∗∗ (0.238)∗∗∗No. of stocks 0.00039490∗ 0.00098625∗∗∗Squared number of stocks (0.00000041)∗∗ (0.00000078)∗∗∗Assets of all classes 0.000 (0.000)Median market capitalization 0.000∗∗ (0.000)% cash 0.001 0.0031992 0.493∗∗∗ 0.327∗∗∗1993 0.889∗∗∗ 0.911∗∗∗1994 (0.199)∗∗∗ (0.292)∗∗∗1995 2.698∗∗∗ 2.727∗∗∗1996 1.543∗∗∗ 1.445∗∗∗1997 1.488∗∗∗ 1.390∗∗∗1998 0.678∗∗∗ 0.623∗∗∗1999 1.073∗∗∗ 1.029∗∗∗F-statistic 450.68∗∗∗ 137.33∗∗∗Adj. R2 0.55 0.54N 4,838 1,554∗∗∗ Indicates statistical significance at the 0.01 level. ∗∗ Indicates statistical significance at the 0.05 level. ∗ Indicates statistical significance at the 0.10 level. There exists a potential problem of serial correlation due to funds appearingmultiple times in the panel regressions. For each fund appearing multiple times, weused a sampling procedure in which a random number generator selects one of themultiple observations to be used in the pooled sample regression.16 For the nine-yearperiod, this resulted in a total of 1,554 unique funds.16 We thank the referee for suggesting this approach.
  14. 14. 494 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 The rightmost column of Table 4 shows the results of regressions by using therandom sampling approach. The findings from this approach are qualitatively similarto those obtained earlier using the full sample, but the statistical significance levelsare higher after the correction is made. The results overall thus confirm a quadraticrelation between mutual fund risk-adjusted performance and number of stocksheld.174. Summary and conclusions This article examines the number of stocks held by U.S. equity mutual fundscovered by Morningstar between 1992 and 2000. We posit and empirically test anonlinear relation between the number of stocks held and a fund’s performance. Whilecontrolling for fund size, market capitalization of fund holdings, and the percentageof holdings in cash, our results strongly support the presence of a quadratic relationbetween the number of stocks held and risk-adjusted return. Unlike previous studies that consider diversification for simulated portfolios, ourstudy shows the stock holding and diversification behavior of actual equity mutualfund managers. Examining a total sample of 4,838 fund-years, we find the mediannumber of stocks held by individual funds remained fairly stable within the range 57–72, with the vast majority of funds holding between 40 and 120 stocks. The number ofstocks held is positively related to fund size as measured by assets under managementand negatively related to the expense ratio and the median market capitalization ofholdings. Examining a more limited sample, we test the impact of changes in fund size onthe number of stocks held between 1992 and 2000. The results show that changes inthe number of stocks held are not associated with increases in fund size due to a risein market values, but they are associated with increases in fund size due to increasednet fund flows. When net fund flows are positive, managers tend to add new stockpositions; when the net fund flows are negative, managers tend to reduce the numberof positions.17 In a supplemental analysis, we follow 93 funds through the entire sample period, to examine fundmanagers’ decisions to change the number of stocks held over time. We retain only fully invested U.S.equity funds where managers have been in place for at least three years, insuring that each fund’s listedportfolio holdings reflect the decisions of the current manager rather than the previous manager or sometransitional portfolio. Of the 93 initial funds, 74 survive to 2000. Of the 19 that disappear, 16 merged intoother funds and three are liquidated. To avoid survivor bias, our analysis includes all funds through theiryear of disappearance. We expect that the number of stocks held would rise when fund size increases. Thetwo primary sources of change in fund size are known to be (1) return on preexisting holdings and (2) fundflows from new investments or redemptions by investors. We regress the percent change in the number ofstocks held on each fund’s market return and its percent change due to investor fund flows. The regressioncoefficient for the fund flows variable is highly statistically significant, whereas the coefficient for themarket return variable is insignificant. This result indicates that changes in the number of stocks held arenot associated with rising market values, but they are associated with fund size increases due to higherfund flows.
  15. 15. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 495ReferencesDellva, W.L. and G.T. Olsen, 1998. The relationship between mutual fund fees and expenses and their effects on performance, The Financial Review 33, 85–104.De Wit, D.P.M., 1998. Na¨ve diversification, Financial Analysts Journal 54, 95–100. ıElton, E.J. and M.J. Gruber, 1977. Risk reduction and portfolio size: An analytical solution, The Journal of Business 50, 415–437.Evans, J.L. and S.H. Archer, 1968. Diversification and the reduction of dispersion: An empirical analysis, The Journal of Finance 23, 761–767.Fisher, L. and J.H. Lorie, 1970. Some studies of variability of returns on investments in common stocks, The Journal of Business 43, 99–134.Ibbotson, R.G. and P.D. Kaplan, 2000. Does asset allocation policy explain 40, 90, or 100 percent of performance? Financial Analysts Journal 56, 26–33.Lauricella, T., 2001. A mutual fund’s top stocks may mislead: Often, the top 10 trails rest of list, The Wall Street Journal September 7, C1.Lenzer, R., 1993. Warren Buffett’s idea of heaven: “I don’t have to work with people I don’t like,” Forbes October 18, 40–45.Markowitz, H., 1959. Portfolio Selection: Efficient Diversification of Investments (Wiley, New York).O’Neal, E.S., 1997. How many mutual funds constitute a diversified portfolio? Financial Analysts Journal 53, 37–46.Statman, M., 1987. How many stocks make a diversified portfolio? Journal of Financial and Quantitative Analysis 22, 353–363.Stowe, J.D., B.D. Jordan, and S.D. Jordan, 1988. Measuring Mutual Fund Portfolio Turnover and Con- centration. Working paper, University of Missouri–Columbia.Strongin, S., M. Petsch, and G. Sharenow, 2000. Beating benchmarks: A stockpicker’s reality, The Journal of Portfolio Management 26, 11–27.

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