### Market conditions and the effect of diversification on mutual fund

### performance: should funds be more concentrative under crisis?

Jin-Li Hu•_{Tzu-Pu Chang}•

_{Ray Yeutien Chou}

Published online: 10 January 2013

Ó Springer Science+Business Media New York 2013

Abstract This paper investigates the non-monotonic and non-linear effect of diversification on mutual fund perfor-mance. We apply a frontier-based efficiency measure, the stochastic frontier approach, to estimate fund efficiency and the benefit of diversification. The empirical results indicate that concentration strategy may not be appropriate for fund managers, and the benefit of diversification dis-appears or negatively affects performance when a fund holds too large a number of different stocks. Moreover, this paper examines whether market conditions moderate the relation between diversification and fund performance. The result shows that the benefit of diversification increases within low market return, high market volatility, and financial crisis, implying that the number of stocks needed to achieve a well-diversified portfolio increases under such market conditions.

Keywords Diversification Market conditions

Mutual funds Non-monotonic effect Stochastic frontier Crisis

JEL Classification C30 G11

1 Introduction

The performance of mutual funds is not only the concern of investors, but is also regarded as an indicator to assess managers’ achievements. Accordingly, the evaluation of fund performance has become a significant study issue over the past several decades. Interest in performance evaluation has more recently shifted to an emphasis on the determi-nants of fund performances, such as fund size, age, expenses, and the manager’s tenure and education. Instead of the above various factors, this study focuses on the effect of diversification on fund performances. Asset allocation policy is a key factor to explain the variation of perfor-mance among funds or across time (Ibbotson and Kaplan

2000). Hence, it is valuable to analyze how diversification, one of the asset allocation policies, affects fund performance.

Diversification is widely accepted as a key principle of modern portfolio management (Shawky and Smith 2005). It is a fact that fund performance is composed of return and risk. In regards to risk, portfolio risk depends on the share of individual stock holdings and the variance–covariance matrix among its holdings (Statman 1987). Hence, theo-retical models imply that an investment portfolio should be fully diversified to cut down risks (unsystematic risks), but how to construct a well-diversified portfolio still remains to be studied. Several research studies have suggested that a strong risk reduction of holdings can be realized by increasing the number of stocks (Domian et al.2003,2007; Statman1987,2004).

How portfolio diversification affects mutual fund per-formance has been proceed in recent research. In general, the literature finds that increasing the number of different stocks in a fund (or other diversification indices) will improve the fund’s performance (Kaushik and Barnhart J.-L. Hu (&) R. Y. Chou

Institute of Business and Management, National Chiao Tung University, 118, Chung-Hsiao W. Rd., Sec. 1, Taipei City 100, Taiwan

e-mail: jinlihu@mail.nctu.edu.tw; jinlihu@yahoo.com T.-P. Chang R. Y. Chou

Institute of Economics, Academia Sinica, 128, Academia Road, Section 2, Nankang, Taipei City 115, Taiwan

2009; Pollet and Wilson 2008; Sapp and Yan 2008).1 However, Shawky and Smith (2005) argue that there is a trade-off between diversification benefits and monitoring costs. Sapp and Yan (2008) also suggest that an over-diversified fund might underperform versus others when the manager is overloaded with a much larger number of stocks. Hence, it is reasonable that there exists an optimal number of stock holdings for mutual fund portfolios if the marginal diversification benefit is equal to the marginal monitoring costs.

For example, Evans and Archer (1968), one of the earliest studies on diversification effect, find that the economic benefits of diversification will be exhausted if a portfolio contains about ten stocks. Statman (1987) uses the securities market line method and concludes that a well-diversified portfolio should include at least 30 and 40 securities for a borrowing investor and a lending investor, respectively. Newbould and Poon (1993,1996) summarize that a portfolio with 8 to 20 stocks can achieve the full benefit of diversifi-cation. However, they calculate the confidence intervals for both return and risk and recommend that an investor would need more than 100 stocks if he/she wants to be within 5 % of the average return as well as within 20 % of the average risk. Fabozzi (1999) suggests that firm-specific risk could be sufficiently diversified by constructing a portfolio of 20 randomly selected common stocks.

Domian et al. (2003) consider the concept of shortfall risk and show that investors need more than 60 stocks to avoid a significant shortfall risk. Sequentially, in later research they demonstrate that investors need at least 164 stocks to have a 1 % probability of underperforming Treasury bonds (Domian et al. 2007). Statman (2004) points out that the optimal level of diversification, where the marginal benefit is equal to the marginal cost of diversification, would exceed 300 stocks.

Instead of randomly selected or simulated portfolios discussed above, Shawky and Smith (2005)—the first study to examine the diversification issue for actual mutual fund portfolios—demonstrate a quadratic relation between risk-adjusted returns and the number of stocks held for US domestic equity funds after controlling several fund char-acteristics. They find that the optimum number of stocks is about 480 and a 90 % confidence interval ranges between 40 and 4,000 stocks in their work.

Whether a positive or a quadratic relation between fund performance and the degree of diversification is found in previous studies, those papers assume this relationship is consistent over a long time. In other words, the effect of diversification on fund performance is steady regardless of market conditions. These articles ignore the variability of market conditions and may lack some insights. Therefore, our paper proposes that market conditions may moderate the relationship between diversification and fund performance. The main purpose of this study is to find out the time-varying effect of diversification on mutual fund perfor-mance in Taiwan’s mutual fund industry. There are two reasons why we focus our analysis on Taiwan’s case. First, the above-mentioned literature on the diversification effect places emphasis upon advanced countries’ cases, especially the US market, but there is no focus on this issue for emerging markets. Taiwan’s equity market, one of the major emerging markets in the world, is a shallow-plate market, meaning that it appears relatively volatile and risky. Thus, the diversification effect on portfolio may be more substantial in emerging markets. Second, with respect to Taiwan’s mutual fund industry, it is a young and growing business that was established in 1983 and in the early stage highly controlled by the government. In contrast to the mutual fund industry in the US, a less developed one (e.g., that in Taiwan) provides a worse return-risk trade-off and less product sophistication (Ramos2009). This implies that further analysis of the diversification effect to maxi-mize the return-risk trade-off will benefit fund managers and investors in Taiwan.

This paper uses a stochastic frontier analysis (SFA), which can be proved to be a generalized Sharpe or Treynor index, to measure fund performance. Kothari and Warner (2001) and Cuthbertson et al. (2008) consider that one good performance measurement should handle the luck of a fund. The SFA approach could deal with such luck to capture statistical noises, including measurement errors or sampling biases. A special feature in Wang’s (2002) SFA model is that it allows the determinants to have non-monotonic effects on fund performance. By adopting this model, the marginal diversification effect can be detected and the optimal level can be seen.

This paper tries to benefit previous works in the fol-lowing issue. The relation among stocks and the variance– covariance matrix among holdings will change under dif-ferent market conditions, implying that the marginal effect of diversification may be time-varying or market-depen-dent. With respect to this, it is true that the relation among stocks is time-varying (Ferson and Schadt1996). Actually, stock market volatility can be approximated by the product of the average correlation between all pairs of stocks and the average volatility of all individual stocks (Pollet and Wilson 2010). Campbell et al. (2001) indicate that

1 _{Kacperczyk et al. (}_{2005}_{) show a positive relationship between fund}

performance and portfolio concentration at industry levels. Sapp and Yan (2008) consider that the number of stock holdings is more intuitive than the divergence index used in Kacperczyk et al. (2005) to measure the degree of diversification. Pollet and Wilson (2008) and Sapp and Yan (2008) also present that the number of stock holdings is statistically and economically significant, but the divergence index is not.

aggregate stock market volatility remains stable while individual stocks become more volatile over the 1962–1997 periods, implying that there is a downward trend in the correlation among individual stocks. This finding can also be interpreted as an increase in the benefit of diversification, meaning that the number of stocks nee-ded to achieve a well-diversified portfolio increases over time. In addition, the stock returns are more highly corre-lated when the market drops, such as during a financial crisis (Pollet and Wilson2010). Hence, it is unclear how the effect of diversification on mutual fund portfolios varies within a devastating financial crisis, which may result in higher market volatility, individual stock volatil-ity, and stronger correlation among all pairs of stocks.

The remainder of the paper is organized as follows. Section2 introduces the used heteroscedastic stochastic frontier model as well as describes the data in details. Sec-tion3provides an empirical analysis and explains the non-linear relationship between the benefit from diversification and fund performance. Section4concludes our study.

2 Methodology and data

2.1 Research methods: fund performance with efficiency measurement

This paper uses the efficiency measurement to measure funds’ performance. Actually, the efficiency measurement is a relative performance measure, assuming that a best performing fund should be placed on the efficient frontier and provides a certain expected return at minimum risk. The distance from the frontier can then be deemed as the degree of each fund’s underperformance.

With respect to the SFA method, one major efficiency measurement, only a few portfolio-related studies use this approach. Annaert et al. (2003) apply the SFA method to construct a CAPM-based performance index. Santos et al. (2005) then use the same idea to study the performance of Brazilian mutual funds. This paper applies the SFA to measure a generalized Sharpe ratio following the concepts of Shawky and Smith (2005). Although none of the pre-vious literature has shown that the SFA approach is a generalized Sharpe ratio, a proof can be given in a straightforward way according to the idea of Choi (2006) and Brandouy et al. (2010) which prove that the Sharpe and Treynor indices are shown as special cases of the non-parametric efficiency performance measurement.

We illustrate this consideration with Fig.1, which compares the SFA efficiency measure to the Sharpe ratio. Considering funds A and A*in Fig.1, the Sharpe ratios for A and A*are RA/rAand R*/rA, respectively, indicating that fund A* has better performance in terms of the Sharpe

ratio. With respect to the SFA measure, fund A*is the best (efficient) performer (i.e., R*/brA= 1) since A*lies on the efficient frontier obtained by SFA.2 Therefore, the effi-ciency score of fund A is equal to RA/brA. In addition:

RA
brA
¼ RA
ðR_{=r}
AÞrA
¼RA=rA
R_{=r}
A
:

This shows that the SFA efficiency score of fund A is its Sharpe ratio over that of the best-performing fund (A*). Hence, we can prove that the SFA measure is the Sharpe ratio relative to that of the best performer.

The stochastic frontier model for portfolio evaluation used in this paper is interpreted as follows. Suppose that the raw return for the ith fund and the risk-free return in year t are Ritand Rft, respectively, where Xitand CVitare vectors of the input variable and control variables, respectively. The SFA model specification for the production function, describing how the inputs efficiently generate the output, can be presented as below:

Rit Rft¼ f ðXit; CVitÞ þ eit; eit¼ vit uit;
vit Nð0; r2vÞ ; uit Nþðlit;r
2
itÞ
ð1Þ
lit ¼ d0þ Zitd ð2Þ
r2_{it} ¼ expðc0þ ZitcÞ ð3Þ

Here, mit is the stochastic error term with i.i.d. normal distribution, and uitis the non-negative inefficiency effect that has a truncated normal distribution following obser-vation-specific mean (lit) and variance (rit2) of the pre-truncated distribution (for more detail, see Kumbhakar and

*Return *
0
A
A*
Frontier
*A*
*RA*
*R**

Fig. 1 The relation of efficiency measure and Sharpe ratio

2 _{Since this paper investigates the diversification effect within active}

portfolio management, the most efficient portfolio (i.e. capital market line) does not appear in our discussion. b is a parameter that satisfies R*/brA is equal to one. In our following analysis, b is obtained

Lovell2000). The efficient frontier is equal to f(Xit, CVit), meaning that the efficient funds would be placed on the frontier and any deviation from the frontier is attributed to the inefficiency effect and noises. The individual fund performance (efficiency) is computed by exp(-uit). Therefore, if the standard deviation of fund return is the single input, then Eq. (1) becomes a standardized Sharpe ratio in the range 0–1.

Equations (2) and (3) presented above estimate how the inefficiency effect is affected by some determinants (Zit). These are the most flexible and best specifications to model the distributions of mit and uit (Lai and Huang

2010). Wang (2002) introduces the marginal effect, which
allows determinants to have non-monotonic effects on the
inefficiency effect (uit), meaning that Zit can positively
(negatively) affect the inefficiency effect when the values
of Zit are within a certain range, and turn negative
(positive) for values of Zitoutside the range. The marginal
effects on E(uit) of the jth element of Zitcan be estimated
as follows3:
oEðuitÞ
oz½j ¼ d½j 1 K
/ðKÞ
UðKÞ
/ðKÞ
UðKÞ
2
" #
þ c½jrit
2 ð1 þ K
2_{Þ} /ðKÞ
UðKÞ
þ K /ðKÞ
UðKÞ
2
" #
ð4Þ
Here, / and U are the probability and cumulative density
functions of a standard normal distribution, respectively;
K¼ lit=rit; z[j] is the jth element of Zit; and d[j] and c[j] are
the corresponding coefficients in Eqs. (2) and (3).

The marginal effects discussed above may reveal more
detailed facts about how the determinants affect the level of
the inefficiency effect. The main purpose of this paper is to
investigate the non-monotonic effect on fund performance.
However, the closed-form calculation foroEðeuit_{Þ=oz½j is}
difficult to compute, if not intractable. Therefore, we turn to
analyze the non-monotonic effect on fund
underperfor-mance. If there exists a non-monotonic effect of
diversifi-cation (i.e., change the sign of the marginal effect on
inefficiency effect), then there is a turning point of fund
underperformance sorted by the degree of diversification.
The optimal level of diversification provides valuable
information for fund managers and investors at
decision-making. Additionally, this paper further applies a simulation
approach to show the pattern ofoEðeuit_{Þ=oz½j (the }
simu-lation steps and results are listed in the appendix).4

2.2 Data and variables’ descriptions

The data come from Taiwan Economic Journal (TEJ) database, which provides rich information on fund char-acteristics after 1987, including all variables used in this article. This paper focuses on domestic equity funds in Taiwan, excluding index funds and fund of funds since they are managed inactively. We also exclude international funds and balance funds to avoid the possible effect of asset allocation decisions. Unfortunately, information on total assets and the expense ratio of each fund are only available after 2001. Because these variables are consid-ered as important determinants of fund performance in literature, the research period covers 2001–2008 and the final sample contains 211 funds comprising 1,334 fund-years. It is noteworthy that the economic and financial market conditions vary markedly over the research period that will be applicable to investigate our main work. For example, the recession in 2001 is the first one since the 1950s, followed by a booming business cycle from 2002 to 2007. Finally, the global financial crisis struck the local market and resulted in another recession in 2008.

This paper adopts the SFA model to construct a gen-eralized Sharpe ratio. It is consistent with the work of Shawky and Smith (2005) who use the Sharpe ratio as the fund performance measure. This study takes the one-output and one-input SFA model, and the output variable is the adjusted fund return calculated as raw return minus risk-free return. The input variable is annualized standard deviation of returns.5 Year dummies are the control vari-ables of Eq. (1) to capture the yearly effect of a frontier shift and unobserved heterogeneity effect. With respect to the determinants of Eqs. (2) and (3), we refer to several related studies in the literature and select six variables as follows.

The main determinant discussed in this research is the diversification of each fund. The number of stocks held by a fund is a simple measure of the degree of diversification (Sapp and Yan 2008; Shawky and Smith 2005). Shawky and Smith (2005) and Kaushik and Barnhart (2009) also consider the percentage of assets invested in the top 10 holdings as a diversification measure since the number of stocks held may not fully reflect the dispersion of holdings. Accordingly, this paper uses both of those measures

3 _{We can also estimate the marginal effect on V(u}

it), which may

imply the diversification effect on the stability of fund performance, but we do not account for these effects in this paper.

4 _{We heartily thank the editor’s constructive suggestion.}

5 _{The present SFA measure is a kind of Sharpe ratio, but adjusts for}

statistical noises. It is noted that the Sharpe ratio may be biased due to its statistical properties, especially when returns of funds do not follow a normal distribution (see Lo 2002). Thus, we employ the Shapiro–Wilk normality test to test whether returns follow a normal distribution. Accordingly, only 4 of 211 funds’ returns reject the null hypothesis (at the 5 % significance level), i.e. returns follow a normal distribution. Therefore, we suggest that the bias of our measure is slight.

separately to investigate the effect of diversification on fund performance. Shawky and Smith (2005) take the squared number of stocks to derive the non-linear effect of diversification and the optimal number of stocks. However, we do not account for this squared variable, because the non-linear effect can be detected by Eq. (4) directly.

The fund characteristics including expense ratio, total net assets, invested share, and OTC share are considered as control variables of Eqs. (2) and (3). It is noted that invested share is defined as the proportion of funds invested in Taiwan’s equity market, while OTC share is calculated as the percentage of funds invested in equities listed on the over-the-counter (OTC) market, in order to control the effect of stock market capitalization. It means that a fund almost invests in all large-cap firms if its OTC share is close to zero. In addition, this paper tries to explore the relationship between market conditions and the effect of diversification. The following research considers the market conditions, such as high versus low volatility, high versus low market performance, and crisis events.

Panel A of Table1 shows the descriptive statistics for fund characteristics, and Panel B presents the correlation matrix among selected variables. Across the entire sam-ple, the average number of holdings is approximately 32, and the average percentage of assets invested in the top 10 holdings is about 48 %. Additionally, the maximum and minimum number of holdings is 12 and 143, respectively. Panel B indicates that the number of hold-ings is positively correlated with fund size, which is not surprising as large funds tend to hold a relatively lager number of stocks due to liquidity concerns. The number of holdings is also positively related to invested share while negatively related to OTC share. The percentage of assets invested in the top 10 holdings is an indicator for the degree of concentration. Hence, the correlation among the number of holdings and percentage of top 10 holdings is highly negative.

3 Results and discussion

This paper assumes that the efficient frontier is under a variable return to scale technology. Therefore, the input and output variable are both a logarithm. Notice that the adjusted returns are added an positive to guarantee that all adjusted returns are larger than zero.

3.1 Estimation results of SFA

Table2 provides the estimation results of the stochastic production frontier and the determinants of the inefficiency effect. The frontier equation of Model 1 shows that the annualized standard deviation is significantly and

positively associated with adjusted return. It is consistent with economic intuition that the expected return will be higher when a fund holds a riskier portfolio.

With respect to the mean equation of Model 1, it presents that the number of holdings has a negative effect on lit, indicating that a fund with a more diverse portfolio has better performance. Although this result is in line with previous literature, the coefficient is not statistically sig-nificant. One reasonable explanation is that the effect of diversification is non-linear, and we will verify this con-sideration in the next subsection. The result also illus-trates that the fund with a larger size and a higher percentage invested in the OTC market would be more efficient than others. Nevertheless, the coefficient of invested share is positive and marginally significant, meaning that higher equity holdings are generally detri-mental to fund performance. This may be due to a downward plunge of Taiwan’s stock market over the research periods, i.e. the stock market dropped more than 30 % during 2001–2008.

The variance equation of Model 1 represents how these factors affect the variability of the inefficiency effect. The implication of the variance equation is to investigate the effect of determinants on the stability of fund performance (fund efficiency). The results of Model 1 show that the number of holdings has a positive effect on rit

2

, implying that the performance of diversified funds would be more unstable than relatively focused funds. One possible explanation is that the variance–covariance matrix among holdings is more complicated as funds hold a large number of stocks. Additionally, funds with a large size and a higher Table 1 Descriptive statistics for fund characteristics

Mean Median SD Panel A: sample statistics

Number of holdings (N_HOLD) 31.80 30.0 11.08 Top10 invested shares (Top10), % 48.46 47.87 9.55 Total net assets (TNA), million NTD 1,518.38 894.84 1,780.38 Expense ratio (EXP), % 0.15 0.15 0.04 Invested share (INVEST), % 85.89 87.47 6.53 OTC share (OTC), % 16.37 13.21 13.82

N_HOLD Top10 TNA EXP INVEST OTC Panel B: correlation matrix

N_HOLD 1 Top10 -0.62* 1 TNA 0.21* -0.05* 1 EXP -0.04 -0.05 -0.11* 1 INVEST 0.15* 0.31* 0.09* -0.06* 1 OTC -0.07* 0.16* -0.08* -0.02 0.14* 1 * Statistical significance at 0.05 level

ratio invested in equity have a relatively stable perfor-mance versus others.

Model 2 of Table2 adopts ‘‘Top10’’ as the alternative proxy for diversification in order to compare with Model 1. The result seems to be completely compatible with the result obtained from Model 1. Except the coefficient of Top10, all parameters have the same sign in both Models 1 and 2, indicating our empirical results are consistent. Only one variable, i.e., invested share, becomes insignificant in

mean and variance functions of Model 2. This change in fact does not influence the conclusions of this research.6

According to the estimated parameters of Model 1, we can calculate the efficiency score of each fund and provide the characteristics of the efficiency score in Table3. The overall average efficiency for the sample funds is 93.23 %. For the year partition, the yearly average efficiency scores are over 90 % except in the last year. Actually, the mutual funds, on average, perform poorly in 2008, because the market confronted a dramatic financial crisis. For the diversification partition, funds are sorted into trisections based on the number of holdings in their portfolios. This shows that focused funds (lowest trisection) have a mean efficiency of 92.40 %, which is significantly (p value \ 0.01) lower than diversified funds (93.58 %).

Table 2 Estimation results of the stochastic frontier approach Dependent variable:

log (adjusted return)

Model 1 Model 2 Coeff. SE Coeff. SE Frontier function Intercept 4.238*** 0.121 4.241*** 0.124 Log (SD) 0.132*** 0.035 0.131*** 0.036 Year 2002 -0.585*** 0.02 -0.581*** 0.019 Year 2003 0.044** 0.020 0.046** 0.021 Year 2004 -0.234*** 0.019 -0.232*** 0.019 Year 2005 0.212*** 0.025 0.211*** 0.025 Year 2006 -0.027 0.019 -0.026 0.020 Year 2007 -0.117*** 0.019 -0.114*** 0.019 Year 2008 -1.163*** 0.026 -1.153*** 0.025 Mean function Intercept 2.860 17.461 -13.077 48.702 N_HOLD -0.387 0.352 Top10 0.207 0.191 Log (TNA) -1.328* 0.738 -1.899* 1.026 EXP -0.351 6.024 -3.608 6.707 INVEST 0.198* 0.117 0.214 0.210 OTC -0.572*** 0.185 -0.600*** 0.213 Variance function Intercept 6.146*** 1.485 8.247** 3.367 N_HOLD 0.026** 0.012 Top10 -0.025*** 0.012 Log (TNA) -0.266*** 0.078 -0.226*** 0.069 EXP 1.214 1.166 1.315 1.095 INVEST -0.041*** 0.012 -0.047*** 0.014 OTC 0.018 0.012 0.020 0.015 rv 4.236*** 0.073 4.250*** 0.070 Log likelihood 728.399 728.281

Adjusted return is calculated as raw return minus risk-free return; SD is annualized standard deviation of fund returns; N_HOLD is the number of different stocks held by a fund; Top10 is the percentage of highest-weighted 10 stocks; TNA is a fund’s total net assets; EXP is the expense ratio; INVEST and OTC are the share of assets invested in equity and over-the-counter markets, respectively

***, **, and * Statistical significance at 0.01, 0.05, and 0.10 levels, respectively

Table 3 Estimated fund efficiency scores ( %)

Mean SD Minimum Maximum All funds 93.23 5.49 22.88 98.49 By year 2001 93.03 7.06 43.30 98.49 2002 92.17 5.04 71.81 98.25 2003 94.02 3.29 74.52 97.76 2004 93.86 3.11 78.58 97.81 2005 93.47 5.38 64.90 97.91 2006 94.83 2.01 86.60 97.72 2007 94.36 4.01 59.52 98.13 2008 89.67 9.45 22.88 98.12 By no. of holdings Focused funds 92.40 7.10 22.88 98.49 Diversified funds 93.58 5.01 59.52 98.01 The focused funds are funds in the lowest trisection based on the number of holdings. The diversified funds are funds in the highest trisection based on the number of holdings

6 _{We also follow the work of Habib and Ljungqvist (}_{2005}_{) using a}

Durbin–Wu–Hausman test to examine the possible endogeneity problem of diversification measures. We treat the number of holdings and the percentage of top 10 holdings as endogenous. Additionally, only transactions cost is used as an instrumental variable since we do not adopt the number of holdings and the percentage of top 10 holdings in the same model. This instrument is not weak and does not correlate with adjusted returns (the t test statistic is 1.12). According to the result of the Durbin–Wu–Hausman test, the number of holdings (p value = 0.32) and the percentage of top 10 holdings (p value = 0.38) do not indicate that the endogeneity problem is a concern in this study.

3.2 Non-monotonic effect of diversification on fund underperformance

Although the results of Table2do not show any significant effect of diversification on the level of fund underperfor-mance, we suggest that is attributed to the non-monotonic or non-linear effect of diversification. To examine this consideration, our study computes the marginal effect on inefficiency effect (uit) for each observation based on Eq. (4) and simulate the marginal effect on fund efficiency score. We then sort all funds into deciles according to the number of holdings or the percentage of top 10 holdings. Table4reports the average marginal effect of diversifica-tion for each decile.

Based on the number of holdings, the effect on fund underperformance reveals a non-linear pattern. For exam-ple, for the most focused group, the average marginal effect on E(uit) are -0.0013, implying that the adjusted return improves 0.13 % if the fund increases by one more stock holding. However, the average marginal effect on E(uit) in the most diversified group become 0.0008, indicating that the adjusted return declines around 0.08 %, if the fund increases by one more stock holding. This finding is con-sistent with Shawky and Smith (2005) in which the benefit of diversification disappears when a fund holds too many stocks at the same time. We also find an optimal level of the number of holdings at the point of changing the sign of the marginal effect on inefficiency effect and hence the effi-ciency score. Accordingly, the optimal number of holdings is around 26–28 stocks since the sign of the marginal effect changes in between the fourth and fifth deciles.

It is interesting that the optimal number of holdings detected by this study is much smaller than that found in the work of Shawky and Smith (2005). We suggest that this dissimilarity may result from the striking difference between the scale of Taiwan and the US mutual fund industries. In fact, the average total assets of our paper’s sample mutual funds are about NT$1.518 billion NTD (roughly US $50 million), which are only one-sixteenth of the sample in Shawky and Smith (2005, pp. 487). More-over, both Taiwan and the US governments have regulated the limit of funds’ holdings; i.e., the fund cannot hold more than 10 % of any one security, and no more than 5 % of the fund’s assets can invest in any one security. The results in Shawky and Smith (2005) and this paper show that the optimal number of holdings in a giant market (e.g., the US) may be larger than a small market (e.g., Taiwan).

Table4also shows the marginal effect of another diver-sification measure, the percentage of top 10 holdings. This result is similar to the finding discussed above that shows a non-linear effect on fund underperformance. However, only the first three focused groups present a positive effect on inefficiency effect, implying that the optimal percentage of top 10 holdings is about 51–54 %.7 This paper further investigates the relation between the marginal effect of diversification and fund size. We double sort all funds into deciles based on the number of holdings and half based on the funds’ total net assets. The result (not reported here) reveals no substantial difference in the marginal effect between large and small funds. It indirectly echoes the finding of Pollet and Wilson (2008), indicating that well-diversified funds should remain focused on their existing portfolios even though more cash flows are invested into them.

We therefore summarize that the concentrative strategy is not appropriate for fund managers, because managers can improve their portfolios’ efficiency by diversifying their holdings. This finding is in accord with the conclu-sions of Pollet and Wilson (2008) and Sapp and Yan (2008), but the benefit of diversification disappears if funds are over-diversified. This paper provides evidence for the non-monotonic effect of diversification as well as the optimal level of two diversification measures: number of holdings and percentage of top 10 holdings.

3.3 Effect of diversification and market conditions

This subsection examines whether stock market conditions moderate the effect of diversification on fund underper-formance. Three market condition variables are considered in this paper: stock market return, market volatility, and Table 4 Average marginal effect of diversification on inefficiency

effect (9 10-2_{)}

Deciles Number of holdings Percentage of Top10 Level of sorting criteria Avg. marginal effect Level of sorting criteria Avg. marginal effect 1—focused 17.707 -0.134*** 66.446 0.019* 2 21.767 -0.052*** 58.695 0.015* 3 24.519 -0.025*** 54.441 0.011* 4 26.789 -0.018* 51.640 -0.027*** 5 28.782 0.011** 49.117 -0.065*** 6 31.164 0.015*** 46.723 -0.080*** 7 33.873 0.026*** 44.336 -0.096*** 8 37.090 0.039*** 41.824 -0.114*** 9 41.425 0.045*** 38.617 -0.136*** 10— diversified 54.714 0.078*** 33.000 -0.173*** Total 31.795 -0.001 48.457 0.059** ***, **, and * Statistical significance at 0.01, 0.05, and 0.10 levels, respectively

7 _{Since Top10 negatively and significantly correlates with N_HOLD}

(see Table1), it is for sure that the signs of marginal effect of Top10 are almost opposite to those of N_HOLD.

crisis events (such as the global financial crisis in 2008). In the following investigation, we only use the number of stock holdings as the proxy for portfolio diversification since the results will not substantially change according to the criterion of percentage of top 10 holdings.

3.3.1 Does market return matter?

We perform an independent double sorting of the fund sample by number of holdings deciles and stock market return trisection. Table5 reports the average marginal effects of diversification on inefficiency effect for these cells.

The second column of Table5 present the marginal effect of diversification if the stock market performs poorly, indicating that the benefit of diversification will vanish when the number of holdings is located on the fifth and sixth deciles. This finding means that the maximum number of holdings could be 31–33 stocks to achieve the optimal level for fund efficiency. However, the turning point in the sign of the marginal effect changes if the market return becomes higher. For example, the last col-umn of Table5shows the marginal effect of diversification in the highest return group. The turning point is among the third and fourth deciles, indicating that the optimal number of holdings is \26 stocks. Hence, the stock market per-formance could indeed affect the benefit of diversification, which is consistent with our expectation. In summary, the better the stock market performs, the lower the benefit of diversification would be. It also implies that fund managers should concentrate their portfolios on the best investment ideas in a good year and hold relatively diversified port-folios in a bad year.

3.3.2 Does market volatility matter?

This subsection uses a double sort meaning that all funds are ranked on the number of holdings and are indepen-dently sorted on market volatility in a particular year so as to investigate whether market volatility affects the benefit of diversification. Table6lists the average marginal effect of diversification for each group.

In the low market volatility years (the second column of Table6), funds within the most focused group have a negative diversification effect on fund underperformance of -0.0014 on average, while the average marginal effect of diversification within the most diversified group is 0.0008. By comparison, in high volatility years (the last column of Table6), a gradually downward pattern for the diversification effect is quite similar to funds operating during low market volatility years. However, the turning points in sign for the diversification effect differ from each other. According to Table6, funds should hold about 31 stocks to be well diversified when the stock market is volatile, but this number becomes around 26 stocks if the stock market is rela-tively stable. Therefore, this finding confirms that stock market volatility moderates the effect of diversification on fund performance.

In empirical research on the stock market, several studies have detected a negative relationship between market return and volatility. The findings here and a pre-vious subsection of this paper are in accordance, meaning that the benefit of diversification is higher under lower and more volatile market conditions. Hence, managers should hold more stocks to realize well-diversified portfolios under those conditions.

Table 5 Marginal effect of diversification on inefficiency effect and market performance (9 10-2)

Deciles of number of holdings

Market return

Low Medium High

1—focused -0.158** -0.176* -0.065 2 -0.130** -0.034** -0.019 3 -0.036** -0.019 -0.021 4 -0.067** -0.004 0.018* 5 -0.002 0.007 0.024*** 6 0.009 0.016** 0.019** 7 0.022*** 0.025*** 0.033*** 8 0.040*** 0.029*** 0.049*** 9 0.042*** 0.043*** 0.051*** 10—diversified 0.081*** 0.072*** 0.077*** Total -0.011 -0.010 0.017***

***, **, and * Statistical significance at 0.01, 0.05, and 0.10 levels, respectively

Table 6 Marginal effect of diversification on inefficiency effect and
market volatility (9 10-2_{)}

Deciles of number of holdings Lower volatility Higher volatility

1—focused -0.143* -0.126*** 2 -0.042*** -0.061** 3 -0.026* -0.024** 4 0.002 -0.037** 5 0.013* 0.008 6 0.022*** 0.005 7 0.029*** 0.023*** 8 0.041*** 0.037*** 9 0.046*** 0.044*** 10—diversified 0.076*** 0.079*** Total 0.004 -0.007

***, **, and * Statistical significance at 0.01, 0.05, and 0.10 levels, respectively

3.3.3 What can managers do under a crisis?

To examine whether the relation between fund diversifi-cation and performance is independent of great crises, we further compare the marginal effects of diversification on inefficiency effect during non-crisis and crisis years. This study first sorts all funds by the number of holdings deciles. The second criterion, whether or not that year presents a crisis event, is used to class funds into non-crisis or crisis years. This paper considers two cases that classify the different non-crisis and crisis groups. Case 1 defines 2008 as the only one crisis year, because a global and dramatic financial crisis occurred that year. In 2008 Taiwan’s stock market dropped more than 60 %, resulting in a severe negative hit on the mutual fund industry. Nevertheless, in case 2 the crisis year group involves 2001 as well as 2008, because they are the only 2 years that Taiwan had negative GDP growth since the 1950s.

Table7gives the means of the marginal effect of diver-sification for each group. Regarding case 1, the optimal number of holdings is around 26–28 stocks when the market is not under crisis. However, the optimal number of holdings becomes about 31 stocks if the market confronts a crisis. This means that the benefit of diversification is enhanced and the number of stocks needed to achieve a well-diver-sified portfolio should increases when market is under a crisis. Based on this result, we suggest that managers should increase the number of stocks in their portfolios for a maximum of 31 stocks to cut down on unsystematic risks.

With respect to case 2, this paper adopts the two economic recessions in 2001 and 2008 as crisis years. According to Table 7, however, there is no difference in the diversification effect between the non-crisis and crisis year groups. In fact, the stock market recovered and rose about 15 % from a severe plunge after 2000. Therefore, we conclude that the change in the benefit of diversifi-cation is directly driven by the stock market conditions, not by the macro-economic conditions. One possible explanation is that the variance–covariance matrix of a portfolio varies with stock market volatility, the average correlation between all pairs of stocks, and the average volatility of all individual stocks according to Pollet and Wilson’s (2010) model. Another reason is that although the stock market and economic conditions are not inde-pendent, the stock market condition is a leading indicator for the business cycle. Hence, the benefit from diversifi-cation might not be directly affected by the macro-eco-nomic condition.

4 Conclusions

Previous studies have used simulated portfolios or real mutual fund portfolios to investigate the benefit of sification, but our paper considers that the effect of diver-sification is time-varying and market-depended, which may be the reason for the controversial findings in the literature. To deal with this consideration, we apply a frontier-based efficiency measure, the stochastic frontier approach, to evaluate fund performance and to explore the effect of diversification on fund underperformance. Moreover, this study adopts Wang’s (2002) SFA model to estimate the marginal effect of diversification. It is expected that one can find the non-monotonic and non-linear effect of diversification, which can provide more insights for fund managers as well as investors.

The empirical results indicate that a concentration strategy is not appropriate for managers in Taiwan after controlling some fund characteristics. It presents that rel-atively diversified funds are more efficient than relrel-atively focused funds, but the benefit from diversification disap-pears or becomes negative when a fund holds too large a number of different stocks. According to the non-mono-tonic effect of diversification, this paper suggests the optimal number of holdings is around 26–28 stocks and the optimal percentage of top 10 holdings is about 51–54 % in general.

To examine whether the relation between fund diversi-fication and performance is independent of market condi-tions, this paper further compares the marginal effect of diversification on inefficiency effect under three market condition variables: market return, market volatility, and Table 7 Marginal effect of diversification: non-crisis versus crisis

years (9 10-2)

Deciles Case 1 Case 2

Non-crisis Crisis Non-crisis Crisis 1—focused -0.133*** -0.138 -0.150*** -0.092* 2 -0.050*** -0.075 -0.074*** -0.006 3 -0.018** -0.109*** -0.024** -0.028* 4 -0.013 -0.055 -0.021* -0.010 5 0.010* 0.019 0.005 0.003 6 0.016*** 0.006 0.015*** 0.016* 7 0.025*** 0.028** 0.024*** 0.032*** 8 0.039*** 0.041*** 0.039*** 0.041*** 9 0.043*** 0.060*** 0.042*** 0.062*** 10— diversified 0.079*** 0.071*** 0.079*** 0.073*** Total -0.002 0.003 -0.004 0.006 Case 1 considers only the financial crisis in 2008 as the crisis year group; Case 2 includes two economic recessions in 2001 and 2008 as the crisis year group

***, **, and * Statistical significance at 0.01, 0.05, and 0.10 levels, respectively

crisis. Our findings reveal that the benefit of diversification increases within lower market return and higher market volatility conditions, meaning that the number of stocks needed to achieve a well-diversified portfolio increases under those market conditions. In addition, a stock market crisis, such as the global financial crisis in 2008, can enhance the benefit of diversification. Compared to non-crisis years, managers should increase the number of stocks in their portfolios for a maximum of 31 stocks. Finally, we consider that the change in the effect of diversification is directly driven by the stock market condition and not by the macro-economic condition.

There are some aspects that can improve our study in future works: first, this research applies the marginal effect on fund inefficiency to explore the market-dependent diversification effect. Wang (2002) also provides an esti-mation of the marginal effect on the variance of ineffi-ciency effect (uit). Perhaps future studies can examine whether or not a diversification strategy could improve the stability of fund performance. Second, in terms of a tra-ditional return-risk relationship, this paper proves that this SFA measure is a standardized Sharpe ratio (but adjusts statistical noises). Some studies recently have developed portfolio evaluation models based on higher-order moments. For instance, Brandouy et al. (2010) consider not only the mean–variance approach, but also mean–vari-ance–skewness as well as mean–variance–skewness-kurtosis models. Hence, we suggest that future works can expand our study to general moment portfolio models, and then one can examine whether our conclusions are con-sistent through an extended approach.

Acknowledgments The authors are indebted to two anonymous referees and an editor of this journal for their valuable comments. Professor Hung-Jen Wang kindly provides an example file of maxi-mum likelihood estimation coding, which is of great help. Jin-Li Hu thanks Taiwan’s National Science Council (NSC101-2410-H-009-044) for financial support.

Appendix

In fact, it is difficult to derive the marginal effect of number
of holdings on the expectation of fund efficiency, that is
oEðeuit_{Þ=oN HOLD. Therefore, we try to graph Eðe}uit_{Þ}
and the number of holdings through a simulation approach.
Our simulation steps are represented as the follows.
1. The first step is to compute the population mean

efficiency, i.e., EðeuitÞ. Battese and Coelli (_{1988}_{)}
present the close form of the mean efficiency if
inefficiency effects are positively i.i.d. truncated normal
random variables, meaning that u Nþ_{ðl; r}2

uÞ. Then

the following mean efficiency can be derived:

EðeuÞ ¼ elþr2u=2 1 U r½ u ðl=ruÞ 1 Uðl=ruÞ

: ð5Þ

It is straightforward for our study to extend above esti-mator under an assumption that uit Nþðlit;r2itÞ.

2. Regarding to the pattern of Eðeuit_{Þ, we compute}
EðeuitÞ by varying N_HOLD and holding other
variables at the sample means. Note that Eq.5

indicates that Eðeu_{Þ is independent of the inputs,}

implying that the mean efficiency is a function of Zs.
Therefore, we hold all Zs at the sample means except
for N_HOLD, and then compute Eðeuit_{Þ by varying}
N_HOLD from 1 to 100. Accordingly, we graph the
mean efficiency against N_HOLD = 1–100 as the
following Fig.2 shows.

Figure2 sketches the relationship between the mean efficiency and the number of holdings. It presents a clear inverse U-shape relation, indicating that this relation is non-monotonic and exhibits a optimal number of holdings. As we hold other Zs at the sample means, the optimal number of holdings is equal to 19.

Fig. 2 Relationship between Eðeuit_{Þ and N_HOLD}

Fig. 3 Relationship between Eðeuit_{Þ and N_HOLD with higher fund}

With respect to the effect of market conditions on
oEðeuit_{Þ=oN HOLD, we can change the levels of some Zs}
and graph the mean efficiency against N_HOLD, again. It
is reasonable that fund size (TNA) and invested share
(INVEST) will increase as the stock market performs well
(Pollet and Wilson2008). Hence, we hold EXP and OTC at
the sample means and let TNA and INVEST are equal to
the sample means plus half of a standard deviation.

Figure3 presents the pattern of oEðeuit_{Þ=oN HOLD}
with a higher INVEST. It still shows a clear inverse U-shape
relation between the mean efficiency and the number of
holdings. However, the optimal number of holdings becomes
17, implying that the diversification benefit in a better market
condition is lower than that in a worse market condition.
Therefore, we conclude that the findings of
oEðeuitÞ=oN HOLD and oEðu

itÞ=oN HOLD are similar.

That is, N_HOLD presents a non-monotonic effect on the mean efficiency and the inefficiency term. We can also find out the optimal number of holdings. However, the suggested optimal numbers of holdings are not identical between two measures (i.e., oEðuitÞ=oN HOLD suggests the optimal

number of holdings is around 26–28). There are two possible
explanations for this discrepancy. First, the relation between
oEðeuit_{Þ=oz and oEðu}

itÞ=oz is not necessarily monotonic.

Second, we do not derive the close form ofoEðeuit_{Þ=oz but}
sketch Eðeuit_{Þ against N_HOLD to demonstrate this }
mar-ginal effect. As mentioned above, we hold other Zs at sample
means in order to calculate EðeuÞ with a given N_HOLD.
However, Wang (2002) derives the close form ofoEðuitÞ=oz

which uses all Zs at the observed values for each observation. Hence, it is not surprising that the implied optimal numbers of holdings are not exactly identical.

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