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 Albany
Abstract
Among the decisions that most mutual fund portfolio managers make is the number of
stocks 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 monitoring
costs. 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 find
that changes in the number of stocks held over time are more highly correlated with mutual
fund flows than with funds’ investment returns.
Keywords: mutual funds, portfolio selection, risk-adjusted returns
JEL 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.edu
We thank David Allen, Susan Belden, Rita Biswas, Shobha Chengalur-Smith, Christophe Faugere, Bruce
Geller, Lester Hadsell, Kajal Lahiri, Gary Sanger, Robert Schweitzer, Ravi Shukla, and Vijay Singal for
helpful 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. 482 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495
1. Introduction
Evans and Archer (1968) were the first to suggest that an investor can achieve
functionally complete diversification by investing in about 10 randomly selected
stocks. 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 equity
mutual funds that hold fewer than 40 stocks, whereas others exceed this number
substantially. For example, in 1998, East End Capital Appreciation Fund reported that
it held the stock of 18 different companies, at the same time that the Eclipse Equity
Fund held 304 stocks. Both funds shared the stated objective of “seeking growth of
capital.”
Diversification is widely held to be a key tenet of modern portfolio management.1
The 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-adjusted
returns net of expenses and the number of stocks held. Furthermore, we analyze the
change in the number of stocks held between 1992 and 2000. Although prior work
considers diversification for simulated portfolios, this study is, to our knowledge, the
first to examine the diversification issue for actual mutual fund portfolios.
Fisher and Lorie (1970) offer support for Evans and Archer’s results for a sample
of randomly selected New York Stock Exchange-listed companies. Evaluating return
distributions for the years 1926–1965, they show that holding a portfolio of eight
stocks instead of one stock decreased diversifiable risk by approximately 80%. Elton
and Gruber (1977) point out that a large amount of diversifiable risk can be removed
by increasing the number of stocks in a portfolio from 15 to 100. More recently, De
Wit (1998) shows that increasing the portfolio holdings from 100 equally weighted
stocks to 500 can reduce the required return by 6 and 21 basis points. One of his
conclusions is that significant diversification benefits can accrue even when adding
stocks to already-large portfolios.2
Although diversification across stocks is generally accepted as an important
component of portfolio construction as originally described by Markowitz (1959),
overdiversification carries several potential costs as well. First, each fund’s portfolio
management 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 where
1 Thissentiment is not universal. According to Warren Buffett, “Diversification is protection against
ignorance, 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 a
portfolio manager’s (or management team’s) inability to track a large number of stocks.

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.”4
Second, positive effects from a portfolio manager’s best investment ideas are diluted
when a portfolio’s value is spread across a large number of stocks.5 Third, it is possible
that transaction costs would rise as stocks are added to the portfolio. Although the
total price impact from a single large transaction would be greater than the price
impact of multiple smaller transactions, the fixed component of commission fees is
higher when multiple trades are made.6
The principal finding of the study is that for U.S. domestic equity funds, after
controlling for fund size, market capitalization of stocks held, and the percentage of
holdings in cash, the relation between risk-adjusted returns and the number of stocks
held is quadratic. The average number of stocks held shows only a slight upward drift
between 1992 and 2000. We find that changes in the number of stocks are strongly
related to the levels of new investments and redemptions but not to fund size change
due to market returns.
2. Data screens and descriptive statistics
2.1. 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 many
individual fund characteristics, including historical return, risk measures, portfolio
composition, and the primary portfolio manager.
2.2. Screens
Table 1 shows the results of screening funds on the following criteria, generating
a 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 Daniel
O’Keefe.
5 Lauricella (2001) points out that conventional wisdom often equates a mutual fund’s top 10 stock holdings
with the manager’s “best ideas.” He reports on a Morningstar study showing that the performance of the
highest-weighted 10 stocks in funds outperformed the total fund only 48% of the time between 1996
and 2001. However, the study neglects to consider the fact that a stock may enter the top 10 due to past
outperformance, 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 trading
10,000 shares each of three different stocks than when trading 30,000 shares of one stock.

4. 484 H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495
Table 1
Sample selection: Results of data screens
The sample is identified from the Morningstar fund universe between 1992 and 2000, and the screens
below are imposed.
Screen 1992 1993 1994 1995 1996 1997 1998 1999 2000
Morningstar 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 focus
is on the composition of common stock portfolios, the presence of nonstock hold-
ings raises asset allocation issues. It is well established that portfolio performance is
critically dependent on asset allocation (Ibbotson and Kaplan, 2000). By imposing
the first screen, we seek to examine portfolio characteristics while holding asset al-
location constant. Hence, we eliminate funds that hold securities other than common
stock. 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 of
stocks 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 approximately
the number of stocks found in the relevant index. Moreover, the managerial objective
function 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 to
indexes. For funds of funds, the decision on the number of stocks to hold is not under
the managers’ direct control, and diversification effectiveness may not be an issue of
particular concern to the manager.7
7 Funds of funds are mutual funds invested in the shares of other funds rather than in individual stocks.

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 related
to several factors, including the manager’s need to diversify. It is assumed in this
article that managers regard the benefits of diversification as a basic tenet of portfolio
management, and as one of the primary influences on the decision of how many
assets to hold. Hence, funds that are undiversified by design or by stated intention
of the manager are excluded from the sample.8 A policy to remain undiversified is
identified in the prospectus summaries contained in the Morningstar database.
Screen (4) is imposed because many funds are offered in multiple classes.9 We
remove redundant observations only after verifying that the portfolio holdings for
all classes are identical. The information reported in this article on fund size is the
aggregate across all classes of each fund. This screen is especially important for the
regression analysis portion of the study, to preserve the assumption of independence
across observations.
2.3. Descriptive statistics
Table 2 shows fund characteristics by year. Between 1992 and 2000, median
fund size increases by 115%, to $206 million. This increase is attributable both to
the dramatic increase in stock prices over that time span, as well as to the flow of
new money into equity funds. The median expense ratio remains constant between
1992 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 over
the period. This observation likely reflects the upsurge in the number of new funds
in the 1990s. As the sample selection summary in Table 1 shows, Morningstar’s fund
universe increased in number from 2,540 in 1992 to 12,081 in 2000.
The median number of stocks held by individual funds remains fairly stable
within the range 57–72. However, within each year, there is substantial cross-sectional
variability in the number of stocks held. Figure 1 shows the frequency distributions
of 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 majority
of funds holding between 40 and 120 stocks. Both the mean and median values are
consistently 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.”10
8 For example, the prospectus for the Fidelity Fifty fund states that the fund typically maintains 50–60
holdings that represents a constraint for the manager. The prospectus for the Jensen fund states that the
fund 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 class
no load with high 12B-1 fees. Other class structures involve fund availability, with one class available to
the general public directly from the fund company, a second class (institutional shares) available only to
institutional investors, and a third class (service shares) is intended for individuals but sold only through
financial institutions.
10 Also, with respect to 75% of a diversified fund’s assets: the fund may hold no more than 10% of a single
issuer’s securities, and no more than 5% of the assets of the fund may be in any one issuer’s securities.

6. 486
Table 2
H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495
Univariate statistics by sample year
Screened Morningstar mutual fund data between 1992 and 2000. For each variable, mean values are followed by medians and standard deviations. Fund size is
the 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 recent
reporting 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 of
the 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 difference
between 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 annualized
standard 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 the
fund. 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 of
the 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 year
Variable 1992 1993 1994 1995 1996
Fund size 365; 96; 864 455; 145; 920 467; 133; 1,165 506; 150; 1,232 714; 189; 2,104
No. of stocks 81; 55; 82 87; 63; 74 87; 65; 78 84; 65; 68 99; 69; 126
Median 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; 9,045
Cash 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; 6.71
Top 10 holdings (%) NA NA NA NA NA
Sharpe 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; 0.68
Expense 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; 0.63
Turnover 88; 65; 102 75; 57; 83 75; 59; 64 73; 54; 67 83; 65; 74
Fund 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; 5.33
Observations 307 299 339 390 545
(continued )

7. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495
Table 2 (continued)
Univariate statistics by sample year
Subsample from year
Variable 1997 1998 1999 2000 Whole sample
Fund size 890; 198; 3,526 953; 180; 4,021 1,040; 205; 4,339 1,036; 206; 4,132 811; 173; 3,305
No. of stocks 96; 72; 100 91; 65; 116 91; 67; 97 92; 69; 102 91; 66; 100
Median 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; 24,182
Cash 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; 7.69
Top 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; 12.80
Sharpe 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; 1.05
Expense 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; 0.57
Turnover 83; 67; 72 86; 66; 86 89; 71; 80 104; 79; 105 87; 67; 85
Fund 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; 4.82
Observations 560 672 784 942 4,838
487

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 Stocks
Figure 1
Number of stocks in actively managed U.S.-based equity mutual funds
Distributions of the number of stocks held by actively managed U.S.-based equity mutual funds in 1992,
1996, and 2000.

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 concentration
because 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 assets
invested 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 the
total portfolio investment, which is approximately three times the weight that would
be suggested by an equal allocation across stocks. Indeed, a casual examination of
funds’ total portfolio holdings makes clear that for whatever reason, managers tend
to hold a large number of stocks in small proportions.12
3. Analysis of number of stocks held by equity
mutual funds
3.1. Cross-sectional analysis of 1992–2000 data
3.1.1. 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 economic
intuition behind such a relation is straightforward. The liquidity concerns for large
funds are likely to require that managers hold portfolios composed of a relatively
large number of stocks. Conversely, managers of smaller funds can hold more limited
numbers of stocks without liquidity being as great a concern.13 Fund size is negatively
related 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’s
percentage 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 relation
between number of stocks and expense ratio may indicate a direct effect of holding
fewer assets, or it could indicate some other indirect effect. Specifically, economies
of scale associated with larger funds likely drive the expense ratio down, and for
11 Strongin,Petsch, and Sharenow (2000) recognize this problem and analyze its impact on the degree of
success 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 would
produce the same level of diversification currently reflected by the unequally weighted portfolio. A fund’s
security 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 hold
stocks in such small weights? One possible reason is that holding a small amount of a company’s stock is
a convenient way for a fund manager to automatically continue receiving pertinent information about that
company.
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 greater
potential liquidity problems than funds investing in large capitalization stocks.

10. 490
H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495
Table 3
Correlations for mutual fund variables
This table shows Pearson’s correlation coefficients between mutual fund variables for a sample taken from 1992 to 2000. Fund size is the aggregate market value
of 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. Cash
holdings 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 average
daily 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 average
market value for the fund.
Fund No. of Median market Cash Top 10 Expense Portfolio Manager
Variable size stocks capitalization holdings (%) holdings (%) ratio turnover tenure
Fund size 1.00
No. of stocks 0.15∗∗∗ 1.00
Median market capitalization. 0.12∗∗∗ −0.07∗∗∗ 1.00
Cash holdings (%) −0.02 −0.04∗∗∗ −0.15∗∗∗ 1.00
Top 10 holdings (%) −0.06∗∗∗ −0.42∗∗∗ 0.20∗∗∗ 0.31∗∗∗ 1.00
Expense ratio −0.14∗∗∗ −0.11∗∗∗ −0.13∗∗∗ 0.06∗∗∗ 0.22∗∗∗ 1.00
Portfolio turnover −0.04∗∗∗ 0.02∗∗ −0.10∗∗∗ 0.01 −0.06∗∗∗ 0.16∗∗∗ 1.00
Fund 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. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 491
different reasons, larger funds also tend to hold a greater number of stocks. Expense
ratio is negatively related to the median market capitalization, indicating that funds
holding smaller stocks tend to exhibit higher expense ratios than funds holding large
capitalization stocks.
3.1.2. Ordinary least squares regressions
The primary hypothesis to be tested in this article is that fund performance is
related 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 number
of stocks held would cause marginal monitoring costs to exceed the marginal diver-
sification benefit, leading to decreased risk-adjusted net returns. Figure 2 presents
the type of quadratic relation we propose, and the optimal number of stocks to hold
is depicted on the graph as NumStock∗ , the point at which the fund’s risk-adjusted
returns are maximized.
We conduct a regression analysis on the pooled sample of 4,838 observations
described earlier for the period 1992–2000. In examining the hypothesized relation
between risk-adjusted return and number of stocks, it is important to recognize the
significant association between number of stocks held and (1) fund size, (2) the
median market capitalization of holdings, and (3) the percentage of the portfolio kept
in cash. It is likely that managers make decisions involving the number of stocks
held in the context of all three of these factors. We control for these three factors by
including them in the regression model.
Risk-adjusted return
NumStock*
Number of stocks held
Figure 2
Hypothesized relationship: Risk-adjusted return versus number of stocks held

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 using
the 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’s
expense-adjusted rate of return and the one-year T-bill rate in year t, divided by the
annualized standard deviation of returns; and NumStockit is the number of stocks held
by fund i in year t. The expected signs of the first two coefficients are positive and
negative. The control variables are defined as follows. Sizeit is the aggregate assets
for all classes of the fund, and MdMkCpit is the market capitalization of the median
firm in which fund i invests, both as of the end of year t. Both variables are measured
in millions of dollars. “%Cashit ” is the proportion of each fund’s cash holdings at year
end. 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 fund
performance is positively related to the number of stocks and negatively related to
squared number of stocks. The regressions are consistent with our hypothesis, and the
coefficients are statistically significant. Several of the control variables, including all
of the time dummies, have significant coefficients as well.
It is possible to derive an approximate value for the optimal number of stocks to
hold using our estimated parameters in Table 4. Taking the partial derivative of the
Sharpe 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. This
number is substantially above the observed average of 96 stocks. However, recall that
De Wit (1998) suggests that benefits can be obtained by adding stocks to a portfolio
that already contains several hundred stocks. It is nonetheless useful to remember
that our calculated NumStock∗ value is derived from the point estimates for the two
parameters.15 The main conclusion should not necessarily be that there is a unique
optimum but that extremely low or high numbers of stocks held are suboptimal.
When we recall from Figure 1 that many funds meeting the statutory definition
of “diversified” hold 40 or fewer stocks, it becomes clear that this conclusion has
substantial 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 ranges
between 40 and 4,000 stocks). Hence, we conclude that rather than a single value, there exists a set of
values that can be reasonably interpreted as an optimal range of stocks to hold.

13. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 493
Table 4
Regression analysis of fund performance versus number of stocks and control variables
Regression coefficients for screened Morningstar mutual fund data from 1992 to 2000. Of the 4,838
observations in the overall sample, 1,554 are unique funds. In the rightmost column we correct for possible
serial correlation as follows. In each case where a fund appears in multiple years, one year’s observation
is randomly selected and used in the regression. The one-year mutual fund Sharpe ratio is regressed on
number of stocks held (NumStock) and squared NumStock. Return is calculated net of management and
12B-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, the
model 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 for
Variable for serial correlation possible serial correlation
Intercept (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 0.003
1992 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 0.54
N 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 appearing
multiple times in the panel regressions. For each fund appearing multiple times, we
used a sampling procedure in which a random number generator selects one of the
multiple observations to be used in the pooled sample regression.16 For the nine-year
period, this resulted in a total of 1,554 unique funds.
16 We thank the referee for suggesting this approach.

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 the
random sampling approach. The findings from this approach are qualitatively similar
to those obtained earlier using the full sample, but the statistical significance levels
are higher after the correction is made. The results overall thus confirm a quadratic
relation between mutual fund risk-adjusted performance and number of stocks
held.17
4. Summary and conclusions
This article examines the number of stocks held by U.S. equity mutual funds
covered by Morningstar between 1992 and 2000. We posit and empirically test a
nonlinear relation between the number of stocks held and a fund’s performance. While
controlling for fund size, market capitalization of fund holdings, and the percentage
of holdings in cash, our results strongly support the presence of a quadratic relation
between the number of stocks held and risk-adjusted return.
Unlike previous studies that consider diversification for simulated portfolios, our
study shows the stock holding and diversification behavior of actual equity mutual
fund managers. Examining a total sample of 4,838 fund-years, we find the median
number 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 of
stocks held is positively related to fund size as measured by assets under management
and negatively related to the expense ratio and the median market capitalization of
holdings.
Examining a more limited sample, we test the impact of changes in fund size on
the number of stocks held between 1992 and 2000. The results show that changes in
the number of stocks held are not associated with increases in fund size due to a rise
in market values, but they are associated with increases in fund size due to increased
net fund flows. When net fund flows are positive, managers tend to add new stock
positions; when the net fund flows are negative, managers tend to reduce the number
of positions.
17 In a supplemental analysis, we follow 93 funds through the entire sample period, to examine fund
managers’ 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 listed
portfolio holdings reflect the decisions of the current manager rather than the previous manager or some
transitional portfolio. Of the 93 initial funds, 74 survive to 2000. Of the 19 that disappear, 16 merged into
other funds and three are liquidated. To avoid survivor bias, our analysis includes all funds through their
year of disappearance. We expect that the number of stocks held would rise when fund size increases. The
two primary sources of change in fund size are known to be (1) return on preexisting holdings and (2) fund
flows from new investments or redemptions by investors. We regress the percent change in the number of
stocks held on each fund’s market return and its percent change due to investor fund flows. The regression
coefficient for the fund flows variable is highly statistically significant, whereas the coefficient for the
market return variable is insignificant. This result indicates that changes in the number of stocks held are
not associated with rising market values, but they are associated with fund size increases due to higher
fund flows.

15. H. A. Shawky and D. M. Smith/The Financial Review 40 (2005) 481–495 495
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