How much is enough? Diversification across mutual funds
May 2007
Jeffrey Junhua Lu
Jeffrey Junhua Lu
Global Wealth Management Citi UK
e-mail:
[email protected]
How much is enough? Diversification across mutual funds
Abstract This paper analyzes two major constraints placed on investors when implementing diversification across mutual funds. First, diversification is considered in a benefit/cost context. As long as the marginal benefit obtained from diversification is larger than the cost associated, it is worthy of increasing the diversification level for fund investors. Second, several risk statistics are examined as the number of funds in the portfolio increase. Diversifying across mutual funds substantially reduces portfolio standard deviation, kurtosis and VaR but also causes an undesirable decrease in return skewness. As such, the goal of an investor who wants to increase skewness would be to hold a less diversified portfolio. Jointly depending on the above two constraints, I suggest that a moderate level of diversification across mutual funds, say, a five-fund portfolio, can achieve most of the advantages associated with diversification.
Keywords: diversification,
How much is enough?
Diversification across mutual
funds 1. Introduction Equity risk has a large unsystematic component and virtually all asset pricing models posit that securities are priced by a diversified, marginal investor who demands little or no compensation for holding unsystematic risk. Research conducted in the mean-variance framework has long shown that diversification offers the benefit of reducing unsystematic risk1. Specifically, Evans and Archer (1968) conclude that approximately ten stocks will achieve the maximum benefits from naive diversification. Not every investor has the time or inclination to build a well-balanced portfolio. There are constraints and restrictions that prevent investors from constructing a well-diversified portfolio. The first constraint is related to cost consideration. Merton (1987) suggests that due to search and monitoring costs investors may limit the number of stocks in their portfolios. A small investor attempting to implement a diversification strategy by forming a portfolio of individual stocks faces substantial transaction costs and administrative burdens. An alternative for the investor is to invest in a mutual fund, which typically invest in a variety of securities, seeking portfolio diversification and reducing risks. However, a mutual fund portfolio may not be fully diversified due to dynamic investment strategies implemented by the fund manager. The second issue related to the portfolio decisions of investors is their higher moment risk preferences. An investor’s utility function depends not only on the mean and variance but is also positively related to the skewness of the return distribution. Simkowitz and Beedles (1978) find that the standardized skewness of randomly 1
See Sharpe (1964), Lintner (1965), Evans and Archer (1968), and Fielitz (1974).
generated portfolio return distributions decreases and then becomes negative on average as the number of stocks in the portfolio increases. Thus, if investors dislike the potential for large downside risk, there may be an optimal number of stocks in a portfolio due to the tradeoff between the desirable benefits of increased diversification and the undesirable increase in negative skewness. Given the above constraints, there may exits an optimal diversification level in the process of constructing investment portfolio for investors, depending on the investors’ cost constraints and higher moment risk preferences. In other words, a fully-diversified may not serve in the best interests of a fund investor. This article specifically tries to investigate the problem on the determination of the optimal diversification level with respect to the investor’s desired portfolio. Differing from most previous diversification studies that focus on portfolio of stocks, this article focuses on portfolio of actively managed mutual funds. One exception is Cromwell, Taylor and Yoder (2000), as they argue that other studies implicitly assume that portfolios are passively managed and do not capture the dynamics of active portfolio management2. However, their study is limited in terms of relatively small sample (168 funds) with focused investment objective (growth funds). This paper employs a larger sample of 485 funds with a broader range of investment objectives and extends the risk analysis of diversified portfolio to incorporate other risk measures. Moreover, the benefit-cost tradeoff of diversification is also considered to provide a more comprehensive picture of investors’ portfolio formation processes. Recently, the fund industry has been witnessed a surge in funds of funds3, which invest in other funds to achieve even greater diversification than traditional mutual funds. This innovative type of mutual fund can offer a kind of instant portfolio by 2
That is, a group of stocks is randomly selected and the portfolio risk statistics are calculated across the time period considered. The stocks in the portfolio do not change over time as in an actively managed mutual fund. In fact, fund managers continually adjust the composition of their funds in an attempt to earn abnormal returns. The effect of this trading activity on mutual fund return distributions is not captured by static portfolios that ignore the dynamics of active portfolio management. 3
According to Cerulli Associates, one-seventh of worldwide investment into mutual funds in 2002 was in funds of funds (around $15 billion), marking an all-time high.
providing a range of investments in one fell swoop. According to Morningstar, at the end of 2002, there are more than 130 distinct fund of funds portfolio available, representing over $22 billion in assets. One of the major advantages claimed by these funds is that they are diversified across asset classes and investment styles, offering the potential for substantial risk reduction compared to funds that invest more narrowly. On the downside, expense fees on fund of funds are typically higher than those on regular funds because they include part of the expense fees charged by the underlying funds. In addition, since a fund of funds buys many different funds which themselves invest in many different stocks, it is possible for the fund of funds to own the same stock through several different funds and it can be difficult to keep track of the overall holdings. Hence, whether these funds actually add values to investors in terms of diversification or they are simply marketing strategies to promote fund sales are still an open area for discussions. This paper contributes to this discussion by investigating the benefits/constraints associated with diversification across mutual funds. The remaining of the paper is organized as follows. Section 2 discusses the constraints in portfolio diversification. Section 3 describes the data and methodology. Section 4 investigates the benefit and cost in diversification. Section 5 analyzes diversification in higher-moment world. Section 6 explores the efficient frontier in a risk tradeoff context. Section 7 makes the conclusion.
2. Constraints in portfolio diversification Mutual fund managers often make “sector” bets and concentrate their holdings in a few industries or stocks that expect to do well4. Over-diversification is probably the greatest enemy of portfolio outperformance, since the impact of a good idea is negligible. As a result, most aggressive growth funds will concentrate on a relatively
4
A previous written by the author titles “Can mutual fund managers outguess sectors?” finds that fund investment styles and sector exposures are closely related and different types of funds exhibit different sector timing abilities.
small set of stocks5. Some funds focus on types of investment strategies that lead to preferences for stocks with specific characteristics. For example, some fund managers are momentum/contrarian investors who try to follow/reverse trends in the stock market. It is obvious that the stocks held by these funds are likely to be highly correlated in the sense that they have moved together in the past and will continues so in the future. Hence, the portfolio holdings of these funds will be ideally highly un-diversified and could carry a great amount of idiosyncratic risk. If a fund investor is uncomfortable with these unusual risk exposures, he can eliminate such fund specific volatility by diversifying across mutual funds. An investor who implements diversification across funds faces with the same constraints as those he faces under diversification across stocks, which will be discussed as follow. 2.1. Cost/benefit consideration The principal that marginal costs should be compared to marginal benefits in determining the optimal levels of production or consumption is fundamental to economic theory. Hence, investors should perform a marginal cost/benefit analysis in order to determine an appropriate number of securities to hold. That is, diversification should be increased as long as the marginal benefits exceed the marginal costs. The benefits of diversification are in risk reduction. The costs are transaction costs. The usual argument for limited diversification is that marginal costs increase faster than marginal benefits as diversification increases6. Statman (1987) calculates the marginal benefits of diversification by comparing the expected return of a portfolio of say, 30 stocks, to the expected return of a 500-stock portfolio, levered so that its expected standard deviation is equal to the expected standard deviation of a 30-stock portfolio7. Generally speaking, the difference between the expected return of an n-stock portfolio, 5
For example, T. Rowe Price Spectrum Growth Fund holds only 8 stocks in its portfolio in March, 2003. 6
Mayshar (1979) develops a model that shows that it is optimal to limit diversification in the presence of transaction costs. 7
For example, Statman estimated at 0.52% the benefit of increasing diversification from 30 stocks to 500 stocks. An increase of diversification from 30 stocks to 500 stocks is worthwhile since the 0.52% benefit exceeds the 0.49% cost of the Vanguard Index 500 fund.
R, and the expected return of its corresponding levered m-stock portfolio, Rnm , is the benefit of increased diversification from n to m stocks, expressed in units of expected returns. ⎡ ⎤ ⎛σ ⎞ σ Bnm = Rnm − R = ⎢ R f + n EP ⎥ − R f + EP = ⎜⎜ n − 1⎟⎟ EP σm ⎣ ⎦ ⎝σm ⎠
[
]
(1)
Bnm is the benefit of increased diversification from n to m stocks, R f is the finance rate that investor can lend and borrow, σ n is the expected standard deviation of an n-stock portfolio, σ m is the expected standard deviation of an m-stock portfolio, and EP is the expected equity premium, which is the difference between R and R f . Given the above method for measuring the benefit of diversification across stocks, the benefit of diversification across mutual funds can be developed in the same way. The general investment product that provides investors diversification across funds is fund of funds (FOFs). The diversification benefit provided by FOFs does not come without a price. On average, a FOFs invests in 15 mutual funds and adds 0.63% in expenses, according to Morningstar (2003). 2.2. Higher moment risk preferences The probability distribution of the rate of return can be characterized by its moment. The reward for taking risks is measured by the first moment, which is the mean of the return distribution. Higher moments characterize the volatility risk and the asymmetry in payoffs. Investors’ risk preferences can be characterized by their preferences for the various moments of the distribution. The fundamental approximation theorem by Samuelson (1972) shows that when portfolios are revised often enough, and prices are continuous, the desirability of a portfolio can be measured by its mean and variance alone. The rate of return on well-diversified portfolios for holding periods that are not too long can be approximated by a normal distribution. Unfortunately, in reality the two assumptions underlying Samuelson’s theorem do not hold. Portfolio revisions
entail transaction costs, meaning that rebalancing must necessarily be limited. Price continuation rules out certain phenomena such as the major stock price jumps that occur in response to takeover attempts. It also rules out such dramatic events as the 25% one-day decline of the stock market in October 1987. Therefore, the second moment (variance) alone is generally not an adequate measure of risk Even though small losses occur more likely than with a normal distribution investors prefer positively skewed distributions of their portfolio value. They do so because big losses are less probable than in a normally or a negatively skewed distribution8. As most investors are risk averse, they view the disutility of a loss as unevenly greater than the utility of a gain in the same proportion. Harvey and Siddique (2000) show that systematic skewness earns a risk premium and thus matters in the composition of portfolios. As investors desire positive skewness in portfolio returns, the risky asset will be held in higher proportion than predicted by a mean-variance framework due to its positive skewness. However, such positive skewness in portfolio returns do not come without a cost. Harvey and Siddique (2000) demonstrate that at any level of variance, there is a negative trade-off of mean return and skewness. That is, to get investors to hold low or negatively skewed portfolios, the expected return needs to be higher, to compensate investors for the increased probability of shortfall. Therefore, as the average risk-averse investor desires low downside risk and high upside potential, volatility or variance become inadequate as risk measures 9 . Skewness in combination with the first two moments is able to mirror the investor’s attitude towards both the upper and the lower part of the distribution. Skewness preferences becomes increasingly important as a decision making criterion when the investor has to choose among different risk levels that correspond to distinctly 8
For example, Cooley (1977) conducts an experiment to test the perception of risk on the part of institutional investors and finds that, among 56 institutional investors who were asked to rate distributions according to perceived risk, at least 29 associated the asymmetry of return distributions with risk. Particularly, the investors associated increases in risk with increases in negative skewness, indicating a preference for positive skewness. These results are intuitive since positively skewed distributions limit losses and provide the investor with downside protection. 9
Other risk measures such as Value-at-Risk or Lower Partial Moments focus on the lower end of the distribution and do not capture the degree of upside potential the investor desires to achieve.
differently skewed return distributions. Skewness increases both with the risk levels as well as with the time horizon of the investment due to the compounding effect over long time periods. 2.3. Diversification in a higher moment world Given the preference for positive skewness, it would be expected that investors would seek to construct portfolios that have this characteristic. However, this complicates the portfolio selection decision, because a desire on the part of investors to obtain positive skewness may not be compatible with the familiar method of constructing a diversified portfolio in order to reduce risk. As shown in Simkowitz and Beedles (1978), even though portfolio variance decrease as diversification occurs, skewness is rapidly reduced by diversification and becomes negative finally. Therefore, there exists a tradeoff between higher skewness and lower variances in portfolio diversification. Besides variance and skewness, there exist other risk measures that are of particular interests to certain investors. For example, some investors may be concerned more about the tails of the distribution (the kurtosis) or the lower tail of the distribution (VaR). VaR involves examining the extreme lower tail of the distribution of returns, in other words, it measures the worst expected loss. The assessment of such large losses is of great importance to investors, especially under tumultuous market environment. Since most asset returns exhibit non-normal distribution, I adjust the normal VaR formula for skewness and kurtosis analytically, by using the Comish-Fisher (1937) expansion as follows:
z CF = z C +
(
)
(
)
(
)
1 2 1 1 3 3 zC − 1 S + z C − 3z C K − 2 zC − 5zC S 2 6 24 36
(2)
where: z C =critical value for the probability (1- α ) with a standard normal distribution (-2.33 at 99%)
S=skewness K=excess kurtosis T=number of returns for the portfolio
μ =mean of portfolio return σ =standard deviation of the portfolio return Rt =return of the portfolio at time t and where the skewness and excess kurtosis of a distribution are defined as follows: 1 T ⎛R −μ⎞ S = ∑⎜ t ⎟ T t =1 ⎝ σ ⎠
3
1 T ⎛ Rt − μ ⎞ K = ∑⎜ ⎟ −3 T t =1 ⎝ σ ⎠
(3)
4
(4)
The modified VaR therefore comes to: ModifiedVaR = μ − z CF σ
(5)
If higher moments of returns play an important role in the portfolio decision process of investors, considering portfolio diversification in a three or higher moments world would result in a more precise picture of the portfolio distribution desired by the investor10. This paper specifically examines the relationship between diversification and the other information contained in the distribution beyond mean and variance, such as skewness, kurtosis, and VaR.
3. Data and methodology The following analysis is performed on 485 mutual funds over the period April, 1997-
10
Indeed, Gaivoronski and Pflug (2000) conclude that investors who are concerned with VaR will not achieve comparable results using a portfolio selection methodology that relies on another risk measure, such as variance.
July, 2002 and sub-periods April, 1997 – December, 1999 and January, 2000 – July 2002. Monthly total return index data for virtually all equity mutual funds (unit trusts) that existed during any given quarter between April, 1997 and July, 2002 (inclusive) were extracted from Datastream. The definition of return index is expressed as: RI t = RI t − k ×
Pt + Dt Pt − k
(6)
RI t , return index on day t RI t − k , return index on day t-k Pt , the closing bid price on ex-date t Pt − k , the closing bid price on date t-k Dt , the dividend payment associated with ex-date t Hence, the monthly returns for each fund can be expressed as: Rt = RI t / RI t − k − 1
(7)
A similar expression is used for returns on the market portfolio proxy the S&P 500 Index. The 485 sample funds are drawn from the universe of 2,400 funds identified in this database over the sample period. The fund category is identified by examining the fund name and more than 20 such categories are identified. The main U.S. mutual fund categories include bond fund, equity fund, international funds, global funds and sector funds. In this study, I focus on actively managed equity funds and investigate mainly 5 types of equity funds, including aggressive growth, growth, growth & income, value, and balanced funds. Funds within the aggressive growth and growth sector invest in stocks that provide capital growth. Funds within the value sector invest in stocks that have a good earnings track record and provide income. Funds within the growth & income sector invest in stocks that provide moderate income with decent growth prospects. Balanced funds invest in multiple assets in the market in an
attempt to time the market. For aggressive growth and growth funds, I allocate small-cap growth funds to the aggressive growth sector while including large-cap growth funds in the growth sector, as small-cap companies on average exhibit a higher growth rate than large-cap companies. Among these five types of mutual funds, I randomly select an arbitrary number of funds in each category. Table 1 summarizes the risk characteristics of the sample funds and subgroup of sample funds. There are several points worth a few lines of comments. First, except the aggressive growth funds, all other types of funds exhibit negative monthly return during the sample period. The aggressive growth funds also have the highest volatility compared to other funds. Second, all types of funds exhibit negative skewness (-0.47) and some sorts of leptokurtic distribution (1.13)11. Indeed, among 485 sample funds, only 50 funds have positive skewness while the majority of those funds’ returns are negatively skewed. Also, the in-group variation of kurtosis is very high, which is caused by a couple of high kurtosis funds. This suggests a potential problem of under-estimating the risk faced by fund managers. Third, as we proceed along the spectrum of mutual funds from the conservative types of funds to the aggressive types of funds, we see a monotonic increased pattern, with the aggressive growth funds exhibiting the highest VaR while the balanced funds exhibiting the lowest VaR, at both the 99% and 95% confidence levels. I form 5000 random portfolios with N funds (N=1, 2,…, 15), calculate the moments across time for each given portfolio and compute the average for each size N. The assumption of equally weighted rebalancing typical in this type of study is invoked. Consider the case where N=3. Three mutual funds are selected at random without replacement from the sample of 485. A portfolio is formed with equal weight in each of the three funds and the portfolio return is computed for each year from 1997 to 2002. Using these estimates, the portfolio risk characteristics are computed across time employing the above formulas, including: the mean, the variance, the skewness, 11
It is measured by the kurtosis, which is based on the size of a distribution's tails. Distributions with relatively large tails are called "leptokurtic"; those with small tails are called "platykurtic." A distribution with the same kurtosis as the normal distribution is called "mesokurtic."
the kurtosis, the 99% and 95% confidence level VaR. This process is repeated and averages of the 5000 risk measures are calculated.
4. Benefits and costs in diversification The traditional argument on the benefits of diversification is that increased portfolio diversification tends to decrease portfolio standard deviation. The relationship between standard deviation and the number of funds in a portfolio is investigated and the results are presented in Table 2. It implies that 16% of a portfolio standard deviation is eliminated as diversification increases from 1 to 5 funds. Adding 5 more funds eliminates as additional 2 percent of the standard deviation. Increasing the number of funds to 15 eliminates only an additional 1 percent of the standard deviation. However, in order to determine the desired level of diversification in a fund portfolio, one need consider the costs and benefits associated with increased diversification. Recall that the benefits of increased diversification can be estimated by comparing a portfolio of n funds to a portfolio with a large number of funds, m. I set m to be 485, the total number of funds in the sample. Consider the case where the expected equity premium is 8.79%, the mean realized equity premium during 1926-2001 according to Statman (2002)12. Thus, benefits, in terms of expected returns, of increasing the number of funds in various portfolios to 485 are presented in Table 3. The expected annual benefit of increasing diversification from 1 fund to the 485 funds is 2.27% while the expected annual benefit when diversification increases from 5 to 485 funds is 0.51%. According to Morningstar, the average fund of funds adds 0.63% in expenses. These additional expenses can be viewed as the cost associated with the increased diversification in portfolio consisting of mutual funds. Therefore, the expected annual cost of an increase in diversification from 1 or 5 funds to 485 is
12
Today there is little agreement that it is a fair estimate of the expected equity premium. Fama and French (2001) estimated the expected equity premium based on P/E ratios and dividend yield. The average of the two is 3.44%.
0.63%13. It turns out that the optimal level of diversification is 4-5 funds when the equity premium is 8.79%. The benefit of increasing diversification from 4 to 485 is 0.70, a little bit higher than the 0.63% cost of replacing a 4-fund portfolio with the 485-fund portfolio. The benefit of increasing diversification from 5 to 485 is 0.51%, a little bit lower than the 0.63% cost of replacing a 5-fund portfolio with the 485-fund portfolio. (See Table 3 and Figure 1) The benefits of diversification are smaller when the equity premium is smaller. When the equity premium declines from 8.79% to 3.44%14, the expected annual benefit of increasing diversification from 1 fund to the 485 funds is 0.89% while the expected annual benefit when diversification increases from 2 to 485 funds is 0.52%. Compared to the cost of 0.63%, diversification across funds seems unattractive.
5. Diversification in a higher moment world The average moments are tabulated in Panel A, Table 4 for portfolio compositions ranging from 1 to 15. Figure 2 graphically show how the risk measures vary with the number of funds in the portfolio. Mean return is very stable across all values of N. However, an investor can significantly reduce the dispersion of returns by increasing the number of funds in portfolio. As shown in Table 2, it is possible to eliminate more than 18%15 of the total risk with a portfolio of 15 funds. More than 85%16 and 95% of the unsystematic risk is eliminated with 5-fund and 10-fund portfolios, respectively. Portfolio skewness is negative everywhere and generally decreases (more left skew)
13
Indeed, this estimated cost is quite reasonable. For example, Barclays Global Investment S&P 500 fund is an index fund with an expense ratio of 0.15%, while Barclays Global Investment AA fund is a fund of funds with an expense ratio of 0.75%, which has 517 funds in its portfolio in November, 2002. The difference in expense is about 0.6%. Both funds belong to the same fund family—Wells Fargo.
14
Suggested by Fama and French (2001).
15
18% = (0.0667-0.0542)/0.0542.
16
77% = (0.0667-0.0560)/(0.0667-0.0542)
with diversification. One exception is the case when the diversification level increases from 1 fund to 2 funds. The skewness coefficient indeed increases (less left skew), which suggests that a naive diversification strategy with two funds can provide benefits to investors, since less negative skewness is preferable to investors. Beyond that level, the skewness of the diversified portfolio generally decreases as more funds come in. As presented in Panel B, Table 4, about 33% and 73% of diversifiable skewness is eliminated with 5-fund and 10-fund portfolios, respectively. Therefore, for investors who prefer positive skewness, diversification with a higher level is not a good strategy. The reported kurtosis has a monotonic pattern. Generally, it decreases as the degree of diversification increases. The high standard deviations in the small-sized portfolio are caused by a couple of high kurtosis stocks and the fact that, in some samples, the kurtosis coefficient does not exist. The last two columns of Panel A, Table 4 show that, at both the 95% and 99% confidence levels, the VaR is decreasing as the degree of diversification increases. Looking through Panel B, Table 4, we find that as the portfolio size increases from 1 to 5, the loss decreases by 22% at the 99% level and by about 14% at the 95% level. Therefore, the results show that diversification is desirable for investors who worry about extreme losses. In general, diversification across mutual funds tends to decrease skewness, kurtosis, and VaR. If fund investors prefer lower standard deviation, higher skewness, lower kurtosis, and lower VaR, there is a tradeoff between the desirable reduction of unsystematic return dispersion, kurtosis, VaR and an undesirable decrease in the return skewness with mutual fund diversification. Diversification across mutual funds at a moderate level, say 5 funds, may be suitable for most fund investors. With 5-fund portfolio, the decrease in skewness is moderate while it captures most of the reduction in standard deviation, kurtosis, and VaR.
6. Risk tradeoff and efficient frontier The above analysis suggests that there exit tradeoffs between different risk measures when diversifying across mutual funds. It leads us to the following interesting question: how these tradeoffs affect the portfolio construction process of investors? Investors’ risk preferences can be characterized by their preferences for the various moments of the distribution. Generally, a moment preference function for an investor can be developed as follows: E (U ) = α 1 mean − α 2 var iance + α 3 skewness − α 4 kurtosis + ...
(8)
All the even moments represent the likelihood of extreme values. The odd moments represent measures of asymmetry. Thus, we can characterize the risk aversion of any investor by the above preference scheme that the investor assigns to the various moments of the distribution. In terms of asymmetric payoff structure, variance or standard deviation become inadequate as risk measures, as the average risk-averse investor desires low downside risk and high upside potential. Portfolio analysis would then be based on a tradeoff among expected means, standard deviation and skewness of returns rather than on a tradeoff of means and standard deviation only. Indeed, Golec and Tamarkin (1998) prove that the loss in utility by reducing skewness can be completely offset by the gain in utility due to the decrease in variance. It must be emphasized that both the lower and the upper end of the distribution matter for the preference structure, which is captured by the skewness coefficient. VaR, by contrast, only focuses on the lower end of the distribution and do not capture the degree of upside potential the investor desires to achieve. If investors are so conservative, they will be more averse to the risk of losses on the downside than of gains on the upside. For these investors, they may trade off lower skewness for lower VaR. Figure 3 provides the efficient frontiers related to the tradeoff between various moments of the return distribution. These graphs are formed based on the previous portfolio simulation, which clearly support the above discussions.
7. Conclusion Diversification across mutual funds should be considered in a benefit/cost context. A diversification level of 5-fund is the break-event point where the marginal benefit of diversification equals the marginal cost. I also analyze the change in the risk characteristics of a portfolio of mutual funds as the number of funds increases. Mutual fund investors can substantially reduce unsystematic risk by diversifying. A portfolio of 15 funds can reduce the return dispersion by more than 18%. Fund investors face a tradeoff between desirable reduction in standard deviation, kurtosis and VaR and undesirable reduction in skewness. With 5-fund portfolio, approximately 85% of the reduction in the portfolio standard deviation and more than 70% of the reduction in kurtosis, VaR at the 99% level and 95% level. However, only a moderate decrease in skewness (about 33%) with 5-fund portfolio. The above findings have several important implications. First, when implementing a naive diversification strategy with mutual funds, investors should consider the relevant cost and benefit associated with the diversification. For moderate level of diversification across mutual funds, the benefit of increased diversification may just cover the associated cost. However, for higher level of diversification, the benefit of increased diversification could fall short of the associated cost. Second, for a fund investor, the optimal diversification level for a portfolio consisting of funds is determined by the tradeoff between gain and loss in the investor’s utility, which are generated from the changes in risk characteristics of the portfolio with increased diversification. The 5-fund portfolio suggested here is just a general thumb of rule and it does not intend to suit all tastes. Different investors may come at different level of diversification. Third, it should be noted that unlike most stock fund managers that hold relatively undiversified portfolios of stocks, most FOFs have higher diversification level of more than 5 funds in their portfolio. These funds may add value to investors by providing optimal risk exposures management or timing that goes beyond diversification service. The former can be viewed as active investing
while the latter can be viewed as passive investing. However, if these funds fail to add value by active investing, over-diversification can be a potential enemy to fund performance.
Table 1 Type
Number
Risk Statistics Summary of the Sample Funds for 1997-2002
Return
Volatility
Skewness
Kurtosis
VaR-99%
VaR-95%
Average
Stdev
Average
Stdev
Average
Stdev
Average
Stdev
Average
Stdev
Average
Stdev
ALL FUNDS
485
-0.12%
0.82%
6.71%
2.74%
-0.4676
57.42%
1.1293
281.69%
11.02%
5.13%
8.78%
3.93%
AGGRESSIVE GROWTH
73
0.14%
0.51%
9.63%
2.01%
-0.1227
41.76%
0.9800
161.65%
17.26%
5.55%
13.03%
2.76%
GROWTH
193
-0.23%
1.18%
7.47%
2.96%
-0.4439
47.82%
0.7682
197.86%
11.81%
4.88%
9.71%
4.42%
GROWTH & INCOME
50
-0.10%
0.33%
5.17%
0.70%
-0.4868
49.36%
0.8997
279.66%
8.26%
1.29%
6.79%
0.94%
VALUE
95
-0.16%
0.41%
5.72%
0.88%
-0.5682
61.07%
1.4750
363.84%
9.39%
2.14%
7.34%
1.17%
BALANCED
69
-0.03%
0.31%
3.97%
1.06%
-0.7460
75.92%
1.9814
408.89%
6.47%
1.96%
5.10%
1.35%
Note: The column “Average” states the mean value of the risk statistic within groups of funds. The column “Stdev” states the variation of the risk statistic within groups of funds.
Table 2
Average Mean and Standard Deviation across 5,000 Samples (Sampling from the 485 Funds, 1997-2002) Average
Ratio of portfolio standard
Proportions of
standard
deviation to standard
unsystematic risks
deviation
deviation of a single fund
eliminated
-0.12%
0.0667
100.00%
0.00%
2
-0.12%
0.0609
91.35%
46.34%
3
-0.12%
0.0586
87.90%
64.80%
4
-0.13%
0.0572
85.74%
76.40%
5
-0.12%
0.0560
84.04%
85.47%
6
-0.12%
0.0560
83.95%
85.98%
7
-0.12%
0.0556
83.40%
88.95%
8
-0.12%
0.0549
82.37%
94.47%
9
-0.12%
0.0549
82.31%
94.77%
10
-0.11%
0.0548
82.17%
95.49%
11
-0.12%
0.0548
82.19%
95.41%
12
-0.13%
0.0543
81.48%
99.23%
13
-0.12%
0.0544
81.59%
98.62%
14
-0.12%
0.0542
81.33%
100.00%
15
-0.12%
0.0542
81.33%
100.00%
Portfolio
Average
size
mean
1
Table 3 Number of Funds in Portfolio
Benefits and Costs Associated with Diversification
Benefits of
Excess of the
Benefits of
Excess of the
diversification
benefit of
diversification
benefit of
(equity risk
diversification over
(equity risk
diversification over
premium =
the additional cost
premium =
the additional cost
8.79%)
0.63%
3.44%)
0.63%
1
2.27%
1.64%
0.89%
0.26%
2
1.32%
0.69%
0.52%
-0.11%
3
0.94%
0.31%
0.37%
-0.26%
4
0.70%
0.07%
0.27%
-0.36%
5
0.51%
-0.12%
0.20%
-0.43%
6
0.50%
-0.13%
0.20%
-0.43%
7
0.44%
-0.19%
0.17%
-0.46%
8
0.32%
-0.31%
0.13%
-0.50%
9
0.32%
-0.31%
0.12%
-0.51%
10
0.30%
-0.33%
0.12%
-0.51%
11
0.30%
-0.33%
0.12%
-0.51%
12
0.23%
-0.40%
0.09%
-0.54%
13
0.24%
-0.39%
0.09%
-0.54%
14
0.21%
-0.42%
0.08%
-0.55%
15
0.21%
-0.42%
0.08%
-0.55%
Table 4
Average Distributional Characteristics and Their Standard Deviations across 5,000 Samples (Sampling from the 485 Funds, 1997-2002) Panel A
Risk Statistics
Portfolio
Average
size
skewness
1
-0.4698
57.00%
1.0412
279.35%
10.96%
5.12%
8.72%
3.93%
2
-0.3906
33.34%
0.4561
108.94%
9.61%
3.10%
8.08%
2.58%
3
-0.3997
27.70%
0.2973
76.54%
9.08%
2.47%
7.80%
2.07%
4
-0.4107
23.15%
0.1966
59.78%
8.74%
1.94%
7.61%
1.60%
5
-0.4141
22.69%
0.1800
55.38%
8.55%
1.88%
7.46%
1.54%
6
-0.4237
21.55%
0.1464
50.90%
8.50%
1.69%
7.46%
1.37%
7
-0.4356
19.99%
0.1439
47.57%
8.43%
1.58%
7.41%
1.26%
8
-0.4325
18.70%
0.1163
42.62%
8.31%
1.44%
7.33%
1.15%
9
-0.4403
17.18%
0.1147
40.87%
8.30%
1.39%
7.32%
1.09%
10
-0.4427
17.71%
0.1221
40.12%
8.30%
1.37%
7.31%
1.05%
11
-0.4468
15.37%
0.0936
36.55%
8.26%
1.25%
7.32%
0.97%
12
-0.4536
14.28%
0.0904
34.85%
8.17%
1.11%
7.24%
0.90%
13
-0.4549
14.06%
0.0897
34.20%
8.19%
1.15%
7.26%
0.91%
14
-0.4622
12.21%
0.0807
31.65%
8.14%
1.03%
7.23%
0.84%
15
-0.4616
11.82%
0.0721
30.99%
8.13%
0.99%
7.23%
0.81%
(std)
Average kurtosis
Panel B
(std)
Average VAR (99%)
(std)
Average VAR (95%)
Reduction in Risk Characteristics
Proportion of
Proportion of
Proportion of
diversifiable
diversifiable
diversifiable
skewness
kurtosis
VaR (99%)
eliminated
eliminated
eliminated
1
N/A
0.00%
0.00%
0.00%
2
0.00%
60.38%
47.61%
42.92%
3
12.86%
76.76%
66.31%
61.81%
4
28.38%
87.15%
78.41%
74.94%
5
33.11%
88.87%
85.06%
84.43%
6
46.68%
92.34%
87.01%
84.67%
7
63.44%
92.59%
89.41%
88.06%
8
59.06%
95.44%
93.64%
93.61%
9
70.06%
95.60%
93.94%
94.00%
10
73.34%
94.84%
94.14%
94.55%
11
79.09%
97.79%
95.29%
94.47%
12
88.73%
98.11%
98.69%
99.45%
13
90.54%
98.18%
97.97%
98.20%
14
99.76%
99.12%
99.66%
99.97%
15
100.00%
100.00%
100.00%
100.00%
Portfolio size
(std)
Proportion of diversifiable VaR (95%) eliminated
Figure 1
Optimal Level of Diversification Based on Tradeoff of Benefit and Cost
Cost of diversification (0.63%)
2.50%
2.00%
1.50%
Benefit of diversification when the equity premium is 3.44%. The break-even portfolio contains 1-2 funds
1.00%
0.50%
Benefit of diversification when the equity premium is 8.79%. The break-even portfolio contains 4-5
0.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 2
Changes in Risk Statistics with Increased Diversification
Panel A
Standard Deviation
0.0680 0.0660
Stdev
0.0640 0.0620 0.0600 0.0580 0.0560 0.0540 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 2
Changes in Risk Statistics with Increased Diversification (Continued) Panel B:
Skewness
0.0000 -0.0500 -0.1000
-0.2000 -0.2500 -0.3000 -0.3500 -0.4000 -0.4500 -0.5000 1
2
3
4
5
6
7
Panel C:
8
9
10
11
12
13
14
Kurtosis
1.2500 1.0500 0.8500 Kurtosis
Skewness
-0.1500
0.6500 0.4500 0.2500 0.0500 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
15
Figure 2
Changes in Risk Statistics with Increased Diversification (Continued) Panel D:
VaR (at the 99% Confidence)
11.50% 11.00%
10.00% 9.50% 9.00% 8.50% 8.00% 7.50% 1
2
3
4
5
Panel E:
6
7
8
9
10
11
12
13
13
14
14
VaR (at the 95% Confidence)
8.80% 8.60% 8.40% VAR(95%)
VAR (99%)
10.50%
8.20% 8.00% 7.80% 7.60% 7.40% 7.20% 1
2
3
4
5
6
7
8
9
10
11
12
15
15
Figure 3
Efficient Frontiers Related to the Tradeoff Between Various Moments of the Return Distribution
Panel A:
Standard Deviation and Return
0.2
0.15
Return
0.1
0.05
0 -0.04
0.01
0.06
-0.05 Standard Deviation
Panel B:
Skewness and Return 0.05 0.04 0.03 0.02
Return
0.01
-3.12
-2.62
-2.12
-1.62
-1.12
0 -0.62 -0.12 -0.01 -0.02 -0.03
Skewness
-0.04 -0.05
0.38
Panel C:
Kurtosis and Return
0.05 0.04 0.03 0.02
Return
0.01 0 -2
-1
0
1
2
3
4
5
6
7
8
-0.01 -0.02 -0.03
Kurtosis
-0.04 -0.05
Panel D:
VaR (99%) and Return
0.04 0.03 0.02
Return
0.01 0 -0.05
0
0.05
0.1
0.15
-0.01 -0.02 -0.03 -0.04
VaR (99%)
0.2
0.25
0.3
9
10
Panel E:
Variance and Skewness
3 2 1 0 -0.04
-0.03
-0.02
-0.01
Skewness
-0.05
0
0.01
0.02
0.03
0.04
0.05
-1 -2 -3 -4
Variance -5 -6
Panel F:
VaR and Skewness
3 2 1 0
Skewness
-0.2
0
0.2
0.4
-1 -2 -3 -4 -5 -6
VaR
0.6
0.8
1
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