Multiple Objectives Optimization in Islamic Investment Portfolios A

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An empirical analysis of portfolio diversification and idiosyncratic risk in the

UK stock market

Tienyu Hwang

The Business School, Edinburgh Napier University, UK

Abstract

Capital market theory attempts to predict returns on securities based on the systematic risk of the

securities. An ideal portfolio should contain the least amount of idiosyncratic risk but over-

diversification unnecessarily lowers the systematic risk of a portfolio and under-diversification

unduly involves excessive risk. While small sized portfolios are exposed to high volatility of

returns due to bearing higher risk in these portfolios, indicating that there is a wide range of

dispersion in betas and returns, large sized portfolios yield the clustering effect as positive returns

diminish. Portfolio diversification increases with small sized portfolios and rapidly reduces as the

portfolio size increases above 12. Although idiosyncratic risk changes over different time horizons,

while the proportion of idiosyncratic risk relative to total risk decreases, a small proportion remains

in different sized portfolios, suggesting that this proportion of risk is an inherent part of

undiversifiable risk. The findings in this study show that as idiosyncratic risk is not fully

diversifiable and diversification provides diminishing marginal benefits as portfolio size increases,

this paper finds that 12 stocks in a portfolio is adequate to solve the issue of over-diversification

without limiting the benefits available. The findings in this paper reveal that there is a clustering

effect in large sized portfolios that creates a ceiling that prevents upside opportunities, squeezing

the range of returns. The evidence for small sized portfolios strengthens views on the advantages of

portfolio concentration, as implemented by Warren Buffett and Benjamin Graham (Cunningham,

2002; Graham, 2003).

Keywords: portfolio diversification, idiosyncratic risk, over-diversification, under-diversification

1. Introduction

Modern portfolio theory studies how to efficiently allocate funds across various risky assets to

maximise expected risk-adjusted return, commonly measured by the Sharpe ratio, which is the

excess return per unit of total risk measured by standard deviation. Since the 1950s, portfolio

analysis has been enthusiastically studied by academics and practitioners, especially after the

introduction of the safety-first criterion by Roy (1952) that selects a portfolio that can reduce the

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probability of a catastrophic outcome and a mean-variance efficient portfolio theory by Markowitz

(1952) that selects a portfolio with the highest expected return for a given level of risk. Although

the principle of portfolio theory asserts that investors should never put all their eggs in one basket,

this subject remains a matter of intense debate and controversy over the risk-reducing efficacy of

portfolio diversification. One of the fundamental problems of portfolio analysis lies in the

contention of the number of stocks required to minimise idiosyncratic risk or achieve a well-

diversified portfolio and the marginal benefit of risk reduction by adding an additional stock. In the

context of the Capital Asset Pricing Model (CAPM) by Sharpe (1964), the returns on securities are

compensated by the level of systematic risk. Since a well-diversified portfolio should contain a

large proportion of systematic risk relative to total risk, its portfolio return can simply be priced by

using the security market line. Moreover, a substantial body of literature commonly accepts that a

greater proportion of idiosyncratic risk relative to total risk positively affects portfolio returns (e.g.

Goyal and Santa-Claral, 2003; Bali, Cakici, Yan, and Zhang, 2005; Fletcher, 2007; Angelidis and

Tessaromatis, 2008; Bali and Cakici, 2008; Fu, 2009; Herskovic, Kelly, Lustig, and Van

Nieuwerburgh, 2016). Thus, contrary to the view that portfolio diversification can reduce

idiosyncratic risk, a portfolio with a large number of stocks means that it is less likely to earn an

abnormal return and hence over-diversification may create low returns for investors.

This paper examines the effect on idiosyncratic risk as the number of stocks in a portfolio

increases, using a set of stocks in the UK stock market. Surprisingly, there has been little research to

investigate the optimal number of stocks for a well-diversified portfolio in the UK stock market (e.g.

Poon, Taylor, and Ward, 1992) and whether the number of stocks in a portfolio is inversely related

to portfolio return. This paper contributes to the existing UK research by investigating the number of

stocks required in a well-diversified portfolio and also provides further evidence whether

idiosyncratic risk is relevant to portfolio diversification. Investigating the number of stocks in a

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portfolio provides a better understanding of increasing portfolio diversification and thus investors

may have more potential for gaining higher returns from portfolio diversification. For example,

DeMiguel, Garlappi, and Uppal (2009) assess various asset-allocation models to achieve high

returns and they suggest that the 1/N simple allocation rule should be used for allocating and

diversifying investors’ portfolios.

The variance-covariance principle is one of the basic underlying rules in determining portfolio

risk and describes portfolio risk1 that is calculated by the average variance of individual stocks and

the average covariance of individual stocks2. As the portfolio size becomes larger, the inherent

portfolio risk depends on the undiversifiable risk. It is generally accepted that adding stocks that are

not perfectly positively correlated with a portfolio reduces portfolio risk, but there is considerable

1 Bodie, Kane, and Marcus (2009, p195) and Reilly and Brown (2012, p201) state that when the portfolio size

becomes large the portfolio risk is close to the variance of the market portfolio. Since portfolio variance of return in n

stocks is 1 12 2 ( , )

nCov i j

p n n

where 2 is the average variance of return across all stocks in the market and ( , )Cov i j

is the average covariance between all pairs of two stocks, the minimum variance is attained for a very large number of

stocks (i.e. market portfolio) and close to the average covariance between two stocks in the market. Similarly, DeMigue,

Garlappi, and Uppal (2009) and Fletcher (2011) argue the efficiency of the 1/N portfolio strategy to create a well-

diversified portfolio.

2 The variance-covariance principle is able to distinguish between diversifiable risk and systematic risk. Since the

average variance of individual stocks is close to zero as the portfolio size becomes large, the variance of return on

individual stocks is considered to be diversifiable risk. However, the covariance between all stocks in a portfolio is close

to the average covariance as the portfolio size becomes large. The average covariance represents systematic risk that is

required to be carried by all stocks in the portfolio and cannot be diversified away. Accordingly, the systematic risk

represented by beta in the Capital Asset Pricing Model (CAPM) is measured by the covariance of the stock with the

market portfolio: 2/),( mmjj rrCov

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disagreement on the number of stocks to achieve a well-diversified portfolio. Several studies focus

on the relationship between portfolio size and risk minimisation. These studies have attempted to

use a small number of stocks to form a well-diversified portfolio. Evans and Archer (1968) find that

the maximum benefit of portfolio diversification is achieved by holding about 15 stocks. Jennings

(1971) concludes that a portfolio of 15 stocks can achieve efficient diversification. Evans and

Archer (1968) argue the economic benefits of holding more than 10 stocks in a portfolio as their

study shows a horizontal asymptote of the average standard deviation when portfolio size increases3.

Other studies also support small portfolio size documented in Tang (2004), and Brown, Gregoriou,

and Pascalau (2012). In contrast, several studies recommend that a larger portfolio size is required to

achieve diversification in Statman (1987), Newbould and Poon (1993), Beck, Perfect, and Peterson

(1996), Domian, Louton, and Racine (2007), Chance, Shynkevich, and Yang (2011), and Miffre,

Brooks, and Li (2013).

Tang (2004) follows Elton and Gruber's (1977) models, noting that 20 stocks can remove 95

percent of the diversifiable risk in a portfolio and the benefits of diversification on risk reduction

subsequently diminish as the portfolio size increases but Miffre, Brooks, and Li (2013) conclude the

opposite results. Elton and Gruber (1977) show that the major decline in portfolio risk occurs at a

small portfolio size between 10 and 20 stocks but they also suggest that equity managers should

consider holding a larger portfolio size beyond 15 stocks to protect against low probability risk of

3 Professional investors advocate small portfolio size as excessive portfolio diversification makes it difficult to

manage and monitor the performance of stocks but portfolio concentration reduces the free riding problem for stocks

without contributing value to a portfolio. For example, Warren Buffett argues a portfolio between 5 and 10 stocks and

Benjamin Graham asserts a portfolio between 10 and 30 (see Cunningham, 2002; Graham, 2003). Ivković, Sialm, and

Weisbenner (2008) find evidence that portfolio concentration outperforms excessive portfolio diversification.

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adverse events. However, Elton and Gruber (1977) point out that choosing few stocks may lead to

under-diversification, as small sized portfolios tend to have higher risk than large sized portfolios.

For example, Elton and Gruber (1977) show that a 15 stock portfolio has 32 per cent more risk when

compare to a 100 stock portfolio. Empirical studies by Solnik (1974) and Statman (1987) find that

portfolio risk is diminished as additional securities are added to that portfolio but the extent of

reducing risk by buying many different stocks is limited as in practice, many systematic risks, such

as inflation and economic growth, are prevailing to all firms. Statman (1987) has shown that the

optimal number of stocks in a portfolio is 30-40 stocks in a randomly selected portfolio.

Accordingly, Newbould and Poon (1993) and Domian, Louton, and Racine (2007) posit that

shortfall risk4 is likely to be alleviated through the use of a large portfolio size as a portfolio

concentrating on few stocks is poorly diversified. This thereby implies that holding the broadest

possible set of stocks can limit large unexpected losses. Yet, Campbell, Lettau, Malkiel, and Xu

(2001) find that the optimal portfolio size changes over time as a result of a declining trend in the

correlations among stock returns.

This paper investigates fundamental issues of asset pricing by examining the number of stocks

that are required for a well-diversified portfolio in the UK stock market. This paper utilises

computer simulation to generate 1,000 random portfolios with different time horizons to track and

compare the risk-reducing efficacy of portfolio diversification with different sets of stocks in a

portfolio. There are a number of studies in the UK that focus on the relationship between portfolio

size measured by market capitalisation and size effect (e.g. Levis, 1989; Dissanaike, 2002; Morelli,

2007; Hwang, Gao, and Owen, 2014). Hwang, Gao, and Owen (2014) construct efficient portfolios

with different market capitalisations and provide evidence of the size effect in the UK stock market.

4 Shortfall risk refers to the results that investors’ realised returns are less than their expected returns.

http://financial-dictionary.thefreedictionary.com/risk

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Levis (1989) develops five portfolios based on market size, dividend yield, P/E ratio, and share

prices5. Levis (1989) reports dividend yield, PE ratios, and firm size to be vital investment indicators.

However, this paper is different from these previous studies as the purpose of this research is to

determine the optimal number of stocks in a portfolio that can accurately reflects the risk inherent in

an undiversifiable portfolio. This paper not only investigates the effect of portfolio size on

idiosyncratic risk but also investigates whether the number of stocks in a portfolio is relevant for

maximising return.

In the introduction, the purpose of this research is highlighted, key research issues are

addressed and an in-depth review of key relevant literature in the area of portfolio diversification is

offered. The remainder of the paper is structured as follows. Section 2 describes the data in the UK

stock market and research methodology to form naive portfolios and estimate idiosyncratic risk in

portfolios. In this section the methodology of the formation of random portfolio selection is also

described, in addition to a discussion of different time horizons in forming different sizes of

portfolios. Section 3 analyses the results. The empirical results include the proportion of

idiosyncratic risk in a random portfolio and the relationship between beta and idiosyncratic risk.

Section 4 provides conclusions.

2. Data and research methodology

This section describes the data collected in testing a well-diversified portfolio and proposes the

methods to determine portfolio size and measure portfolio diversification. This paper collects

5 There is a standard sorting procedure forming portfolios that are simply sorting a set of stocks on beta, market

capitalisation, dividend yield, and PE ratios (Fama and French, 2004). This sorting procedure is commonly used in

empirical tests but it does not take into account the optimal number of stocks that need to be included in each portfolio.

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monthly returns for stocks listed on the FTSE All Share index from January 1985 to December

2011. While Poon, Taylor, and Ward (1992) utilise a pictorial analysis to examine portfolio

diversification in the UK market for the period 1955-1984, this paper extends the period of 1985 to

2011 and applies an empirical based approach to determine optimal portfolio size. I construct

different sizes of portfolios starting from one stock and adding one additional stock to build up to a

portfolio containing 204 stocks. 1,000 random portfolios are generated for each portfolio size to

calculate average portfolio return and variance. For example, the computer algorithms used to

generate the random portfolios are illustrated by using the example of 2 stocks in a portfolio. By

inputting a seed value the computer randomly generates two different numbers. These two numbers

are used to select two specified stocks and then form a two-stock portfolio in the first trial. Then

this paper uses the equal weight on each stock to calculate and obtain monthly returns on the

portfolio over the period under review and calculates the average return and the variance of returns

based on the set of monthly returns. Portfolio risk can be derived by equation (1) as discussed

below. Further, in the second trial a different random seed value is obtained and another set of

random numbers to select stocks in a portfolio is generated. The procedure to produce the average

return and the variance of returns is the same as in the previous step. Thus, 1,000 portfolios with 2

stocks are randomly generated and 1,000 average returns and 1,000 variances of returns are

obtained. Finally, the portfolio return is the average return based on 1,000 average returns and the

portfolio variance is the average variance based on 1,000 variances of returns. Furthermore, the

variance in variance is calculated based on 1,000 variances of returns derived from 1,000 trials.

Since the return sample of this paper is collected from the period of 1985 to 2011. I investigate

portfolio diversification using different time horizons, namely, one-year (the year 2011), two years

(the years 2010 to 2011), five years (the years 2007 to 2011), and the full period (the years 1985 to

2011). Different time horizons yield different returns and allow investors to evaluate the effect of

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the fluctuations of returns on portfolio diversification. The choice of a time horizon of returns may

affect the outcome of portfolio diversification. Previous studies investigating portfolio size and

portfolio diversification have commonly formed an equally weighted portfolio (EWP) with a set of

randomly selected stocks. For example, DeMiguel, Garlappi, and Uppal (2009) utilise 14 portfolio

diversification models and compare the performance of the optimal portfolio strategies to that of the

1/N strategy. This paper uses a naive 1/N model to estimate portfolio return and risk.

Portfolio risk with n stocks under the EWP method can be expressed as

1 12 2 ( , )n

Cov i jp n n

(1)

where

2

p portfolio risk in terms of the variance of returns in n stocks

2 is the average variance of return across all stocks in the market,

( , )Cov i j is the average covariance between all pairs of two stocks.

Since, the minimum variance of portfolio risk is attained for a very large number of stocks, for

a small number of stocks in a portfolio the first term of equation (1) does not approach zero and the

second term does not approach one. In such a case the number of stocks in a portfolio is determined

by both 21

n

and ),(1

jiCovn

n and the first term may be greater than the second term.

Total portfolio risk in equation (2) is broken down into systematic and idiosyncratic risk in the

single-index model.

2

idi

2

m

2

p

2

p σσβσ (2)

where 2

pσ is the variance of a portfolio return, 2

pβ is beta risk, 2

mσ is market risk, and 2

idiσ is

idiosyncratic risk.

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The expected return on a naive portfolio, pr , is calculated as

i

n

1i

ip rwr

(3)

where iw is the weighting of stock i for i=1,...,n; n is the number of stocks in a portfolio; ir is

the expected return on stock i. Equation (3) means that the expected return on the naive portfolio is

priced at total risk rather than systematic risk.

Further, since the expected return on the naive portfolio is proportional to total risk instead of

either idiosyncratic risk or systematic risk, equation (4) using regression analysis establishes

portfolio excess return or portfolio risk premium expressed by the proportion of the risk premium on

a unique part pα (idiosyncratic risk) and the proportion of the risk premium on the market portfolio

times pβ (systematic risk).

pfmppfp e)r-(rβαr-r (4)

where pe is a random error term.

3. Results and analysis

3.1. Portfolio diversification: data description and summary statistics

This paper uses monthly data from 1985 to 2011 and stock returns are divided into different time

periods. Table 1 exhibits the summary statistics for stocks and shows that there are large deviations

in average returns across different time horizons6 indicated by the difference between the minimum

6 One common method to eliminate outliers with extreme return values is to undertake a percentile winsorisation.

For example, a 90% winsorisation means removing the data in both tails of the distribution below the 5th percentile and

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and maximum monthly returns. While the 2011 monthly average stock return is 0.1087 percent, the

standard deviation of stock returns is 1.7653 percent and there is significant difference between the

minimum return of -5.8475 and the maximum return of 5.2912. Data for the average returns shows

that there are wide variations year by year which mirrors the annual volatility in the FTSE All

Share index. Extending the time period reduces these wide variations reflecting the weak-form of

the efficient market hypothesis (Poon, 1996; Yen and Lee, 2008). The interval between the

minimum and maximum average returns for other time periods provides evidence of the high risk

in the stock market as data in this paper shows that there is a wide range of losses and gains that

can occur. Accordingly, Roy (1952) stresses the importance of attempting to reduce the extreme

losses. Markowitz (1952) also confirms the necessity of forming a well-diversified portfolio on an

efficient frontier.

[Insert Table 1 here]

The CV value for average returns during the 2011 horizon is over 16, indicating that the

standard deviation of returns is about 16 times the average return. This is caused by the low returns

in that year. With the exception of 2007 where the CV is negative 10.1510, other CV statistics show

less variability.

By looking at the statistics for stock variances, the evidence supports the inconsistency of mean

variances across different time horizons. However, in comparison to the CV for average returns, the

CV for average variances is more stable inferring that risk is consistently present in the stock market,

above the 95th percentile. As this paper focuses on the effect of portfolio diversification on risk reduction, the

winsorisation method is not performed.

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showing that in general risk becomes more predictable over a longer period whereas return is much

more volatile. However, it is evident that the variability of returns does reduce in the long run.


The data presented in Table 2 exhibits the effect of portfolio diversification on portfolio risk by

randomly selecting a number of stocks in a naive portfolio over different time horizons. Table 2a is

based on a one-year time horizon for the year 2011. This paper applies the random seed to run 1000

times for each portfolio to calculate mean and variance. The number of stocks in a portfolio is

increased from one stock in a portfolio to a maximum of 204. Table 2a summarises these results.

Other time horizons in Tables 2b, 2c and 2d show similar reductions in the variance of portfolios.

This research follows the methods adopted by Elton and Gruber (1977) who consider up to 1,000

stocks in a portfolio and estimate variance and variance in variance in the portfolio. Their study

addresses total risk in portfolio but does not discuss idiosyncratic risk and systematic risk

individually. Their results show a comparable reduction in portfolio variance as the size of a

portfolio increases. The evidence from return data shows that as the size of a portfolio increases, the

variance reduces from a high of 61.82 down to 18.23. Significant reductions occur by expanding the

portfolio size; from one stock to three stocks as the variance falls from 61.82 to 32.04, a reduction of

48%; similarly from one stock to five stocks, the variance falls from 61.82 to 26.41, a reduction of

57%. Further expansion of portfolio size continues to show the variance reducing although declines

are less than 1% for each additional stock added in a portfolio for portfolio size of more than 11

stocks. For example, for 11 stocks in Table 2a the variance of the portfolio is 21.72; for 12 stocks

the variance of the portfolio is 21.38; a drop of only 0.85% in comparison to the variance for 11

stocks. Expanding the time horizon to include 2010-2011 as depicted in Table 2b, 2007 -2011 as

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depicted in Table 2c and the full period from 1985 in Table 2d, confirms consistent relationships in

the decline of the variance as portfolio size increases.

The mean return in Table 2a based on 1,000 randomly selected one-stock portfolios is 0.1219

which is close to the average return of 0.1087 on all stocks. The findings clearly show that standard

deviations are large in the small size portfolios where there is a wide range of gains and losses from

portfolio returns as posited by the maximum and minimum columns in Table 2a. Previous

researchers (Newbould and Poon, 1993; Campbell, Lettau, Malkiel, and Xu, 2001; Miffre, Brooks,

and Li, 2013) suggest that the best choice to mitigate risk is to form a portfolio with 30 to 50 stocks.

Although increasing portfolio size continues to reduce risk there is clearly a trade-off between

potential gains and losses. Investors may wish to accept lower risk for low return. Furthermore,

behavioural proponents observing behaviour of investors in their typical weighting function for

investment activity, describe an inverted S curve (e.g. Kahneman and Tversky, 1979). This suggests

that some investors are excessively optimistic and will overweight their portfolios with a small

number of stocks that could achieve high gains but have a low probability of occurring in reality

(Goetzmann and Kumar, 2008; Deaves, Lüders and Schröder, 2010). The findings in this research

support this view as shown in Table 2, as there is an inverse relationship between portfolio size and

risk represented by the range between the minimum return and maximum return7. For example,

Goetzmann and Kumar (2008) show that less diversified portfolios are more likely to have higher

volatility and higher skewness. Table 2a shows that in general small size portfolios are more likely

to have a high degree of negative skewness8. For example, the 3-stock portfolio has a negative

7 The Sharpe ratio is provided in the tables for comparison purposes but is not meaningful in large portfolios as

the risk is minimal.

8 Negative skewness exhibited in small size portfolios is relevant to portfolio diversification when assessing the

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skewness of 0.2296. The test statistic of the random sample skewness is

2.96471000/6/2296.0 t which gives a p-value of 0.0030 indicating that monthly portfolio

returns of 3 stocks are significantly skewed to the left at the 5% significance level. Similar findings

are presented in other time horizons in the results in Tables 2b, 2c, and 2d supporting the greater risk

that is present in small size portfolios. In addition, the test statistic of the random portfolios of

excess kurtosis of the return series in Table 2a is 7110.31000/24/5749.0 t which gives a p-

value of 0.0002 indicating that random returns in a 3-stock portfolio have long tails.

It should be noted that investors forming small size portfolios may choose stocks that could

result in large losses assuming that the extreme event is unlikely to occur and hence underweight or

even ignore the possibility of the extreme event. Accordingly, investors may suffer from illusion by

focusing on small size portfolios that are unlikely to generate high returns. For example, small size

portfolios were more likely to sustain large negative returns during 2008 with a minimum return of

negative 18.53% (maximum of 1.9%) as shown in Table 1. In reality, the gains and losses will be

randomly spread as supported by the efficient markets hypothesis (Fama, 1998; Poon, 1996; Yen

and Lee, 2008). Thus, over confidence as a result of forming a small size portfolio can even afflict

professional investors9 as they are attracted by the chance of making high returns as they try to

outperform the average returns.

likelihood of large losses. The negatively skewed distribution shows that very large losses are more likely as there is a

relatively higher probability of large losses in comparison to the symmetric distribution.

9 Professional investors who attempt to outperform the average returns from the market over time are unlikely to

consistently beat the market (e.g. Jensen, 1968; Cumby and Glen, 1990; Becker, Ferson, Myers, and Schill, 1999). In

contrast, Avramov and Wermers (2006), Coggin, Fabozzi, and Rahman (1993), Jiang, Yao, and Yu (2007) find evidence

of either risk-adjusted excess returns or positive market timing ability by pension funds or mutual funds.

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3.2. How many stocks in a well-diversified portfolio

While the findings of this research show that portfolio diversification reduces total risk, there is a

trade-off between minimising risk and maximising return. For example, as discussed previously,

the return for holding one stock in a portfolio over 1 year gains a maximum of 5.29% or loses a

maximum of 5.85%. However, the return for holding five stocks in a portfolio over 1 year has a

maximum gain of 2.37% or maximum loss of 2.59%. When the portfolio size reaches 10 stocks,

the return over one year narrows to a maximum gain of 1.91% or a maximum loss of 1.95%.

Moreover, in the large size portfolios the maximum percentage positive return is higher than the

minimum percentage negative return, indicating a greater possibility of a positive gain. In the

context of the maximum-minimum returns available based on 2011 returns, these findings suggest

that portfolio size with 15 stocks offers the possibility of a maximum gain of 1.86% which is

greater than the returns for portfolios consisting of more than 15 stocks. Further, portfolio returns

are volatile in the small size portfolios due to lack of diversification in these portfolios.

[Insert Figure 1 here]

In order to assess the number of stocks that could minimise the idiosyncratic risk in a

diversified portfolio, Figure 1 using graphical analysis10

exhibits the risk compared to the number of

stocks in a portfolio. The results are surprisingly consistent in terms of the optimal number of stocks

needed to effectively diversify idiosyncratic risk. Figures 1a to 1j consider annual results for years

2007 to 2011. For example, Figure 1e shows the total risk is decreasing as stocks increases from 1

10 Angelidis and Tessaromatis (2008) use a graphical analysis to investigate idiosyncratic volatility in the UK stock

market and Poon, Taylor, and Ward (1992) carry out a pictorial analysis to study portfolio diversification in the UK

stock market.

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stock to 204 stocks in a portfolio for 2009. By scrutinising the results for 2009, Figure 1f focuses on

the smaller sized portfolios from 1 to 13. As can be clearer seen the gradient of the concave curve

becomes flatter as the number of stocks increase. The relationship of risk versus the number of

stocks is apparently stable in all time horizon considered in this study. Idiosyncratic risk falls

substantially by increasing the number of stocks in a portfolio from 1 to 3. Further reductions in

idiosyncratic risk can be visualised by expanding the number of stocks up to 5 but only marginally

reductions are offered by expanding portfolio size up to 12. Beyond 12 stocks in a portfolio, the

reduction in idiosyncratic risk is negligible, as the line is close to horizontal, confirming the findings

by Evans and Archer (1968) who display a horizontal asymptote of the average standard deviation

when portfolio size becomes larger. Similar results are shown in Figure 3 that considers different

time horizons of 2 years, 5 years, 10 years, and full period under investigation as shown in Figures

3a, 3b, 3c, and 3d respectively. These results are also confirmed by the figures in Table 2 that depict

the reduction in risk is less than 1% beyond 11 stocks as discussed above. For ease of comparison,

Figures 1b, 1d, 1f, 1h, and 1j show the total risk for 3, 5, and 12 stocks highlighting the gradient of

the concave curve for those numbers. The rule of thumb based on these findings suggests that

significant idiosyncratic risk reduction occurs by increasing stocks in small sized portfolios but

further reductions of idiosyncratic risk is limited, suggesting that there is an inflection point that

diminishes risk reduction beyond that point as portfolio size increases. Although increasing the

number of stocks in a portfolio reduces total risk and enhances diversification, investors may choose

portfolios that are clearly over-diversified or under-diversified. Previous studies suggest that large

size portfolios can overcome under-diversification. In contrast, the results of this study indicate that

larger size portfolios are likely to be over-diversified in terms of the reduction in idiosyncratic risk.

Furthermore, Figure 1 shows that total risk varies from a minimum of 50 in the year 2007 to a

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maximum of 250 in the year 2009 over the five individual year periods. In addition, systematic risk

is highly positively correlated with total risk and reflects the changing total risk.


By comparing Figure 1 with Figure 2, the total risk and undiversifiable risk based on the

number of stocks in a portfolio in Figure 1, can be explained by the volatility of the FTSE All Share

index illustrated in Figure 2. For example in Figure 1, total risk measured by variance of percentage

returns, varies from a minimum of around 50 in the year 2007 to a maximum of about 250 in the

year 2009 and for a portfolio size of 12 stocks undiversifiable risk can be seen to vary from a

minimum of around 20 to a maximum of about 70 in the these two years. Viewing the volatility over

the period in Figure 2 from 2007 to 2011, the index data for both 2007 and 2011 show relatively

minor variations in the monthly index returns, reflecting that systematic risk proxied by the FTSE

All Share index, is relatively low in the UK stock market for those two years as shown in Figures 1b

and 1j. Furthermore, the index data for 2010 is comparatively more volatile in Figure 2, confirmed

by Figure 1d having a higher undiversifiable risk than Figures 1b and 1j. However, for 2008 and

2009 there are substantial variations in the FTSE All Share index with a significant downward trend

in 2008 and significant upward trend in 2009. These two years have high total risk and also high

undiversifiable risk as shown in Figures 1h and 1f.


Figure 3 illustrates the degree of the volatility of the number of stocks in portfolio using a

longer time horizons. Figure 3a represents total risk and undiversifiable risk using a 2-year returns.

When longer return periods are studied, the total risks and undiversifiable risks are seen to be

reduced. For example, Figure 3d covering the entire time horizon has total risk for the whole period

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of this study of 90 and the undiversifiable risk is levelled out around 32 for a portfolio size of 12

stocks.


Previously, the above results are based on aggregate analysis that investigates total risk and

systematic risk as portfolio size increases. Using 2011 return data, Figure 4 plots the returns of all

1,000 random portfolios to consider each individual portfolio, varying portfolio size. There is a large

dispersion in a one-stock portfolio that clearly shows that portfolio diversification is lacking, and the

percentage returns and the variance are in line with random walk theory. When adding one

additional stock to a portfolio as shown in Figure 4b, the degree of portfolio diversification is clearly

visible as seen by the clustering effect in the data. Undeniably, there is a positive association

between adding more stocks to a portfolio and reducing portfolio dispersion, as the 1,000 portfolios

bunch together, becoming more centred as stocks in a portfolio increase. For example in Figure 4f,

when the number of stocks in a portfolio is 12 stocks, all 1,000 random portfolios exhibit portfolio

concentration, indicating that portfolio diversification is reaching the maximum position, implying

that there is a limitation to portfolio diversification.

However, it should be noted that the diversification effect is more pronounced as the portfolio

size increases from 1 through to 12 stocks, in comparison to increasing the portfolio size from 12 to

30 stocks. However, the bunching of the data also eliminates both extreme positive and negative

returns, reducing incentive for investors to earn abnormal returns, as supported by Keynes (1936)

who stated that the influence of speculation is immense. The findings based on Figure 4 strengthen

the results in Figure 1 and Figure 2, and the findings in this study attempt to support a small

portfolio size that a portfolio size of 12 stocks can achieve a large reduction in idiosyncratic risk. In

contrast with the findings of both Bloomfield, Leftwich, and Long (1977) and Newbould and Poon

18

(1993), who emphasise that a portfolio of more than 20 stocks is necessary to achieve the main

benefits of diversification. The findings in this study of portfolios with 12 stocks contradict these

findings of 20 stocks and are also contrary to the findings of Statman (1987) who argues that a

portfolio with more than 30 stocks is required. Other researchers also have alternative opinions, for

example Campbell, Lettau, Malkiel, and Xu (2001) found portfolios between 20 and 50 stocks

should be formed, and Domian, Louton, and Racine (2007) suggested more than 100 stocks are

needed. However, the findings in this study are consistent with the findings in Evans and Archer

(1958), Jennings (1971), Ivković, Sialm, and Weisbenner (2008), and Brown, Gregoriou, and

Pascalau (2012) whereas professional investors such as Warren Buffet and Benjamin Graham

suggest prefer portfolios of up to 10 stocks. Furthermore, as shown by Campbell, Lettau, Malkiel,

and Xu (2001) who assert that portfolio size changes over time as a result of the increasing in

idiosyncratic volatility, that the number of stocks required in a portfolio increases over time in order

to minimise diversifiable risk. In contrast, the findings in Figure 1 and Figure 3 that examine

optimal portfolio size year by year and period by period show weak evidence to support the view

(Campbell, Lettau, Malkiel, and Xu, 2001) of changing portfolio size over time in order to minimise

diversifiable risk.

3.3. Analysis of idiosyncratic risk

Table 3 uses the single-index model to break down total risk into systematic risk and idiosyncratic

risk. Although total risks vary in different time horizons, the proportion of systematic risk in

alternative portfolios in these time horizons show a relatively stable relationship. The findings of

this stable relationship differs from the findings of Campbell, Lettau, Malkiel, and Xu (2001).

However, although the findings from Table 3 show a consistent proportion of idiosyncratic risk

relative to total risk, the absolute idiosyncratic risk varies from period by period. Further

19

investigation into the UK stock market is needed to evaluate the impact of idiosyncratic risk on the

changing portfolio size. This study does not consider the impact of firms’ characteristics on

idiosyncratic risk and the effect on portfolio diversification, which an educated investor will take

into account.

While there is a positive relationship between portfolio size and systematic risk, there is an

inverse relationship between portfolio size and idiosyncratic risk. The empirical findings in this

study appear to support the conventional view on portfolio diversification, as the larger the portfolio

size the lower the idiosyncratic risk. However, the large reduction in portfolio diversification is

clearly evident by expanding from the smallest size of portfolios with 2 stocks to 20 stock portfolios.

Any further increase in portfolio size that can reduce idiosyncratic risk is marginal. Depending on

the investors’ risk attitude, the findings of this paper suggest that most idiosyncratic risk has been

eliminated in small sized portfolios.


3.4. Portfolio diversification and beta

If the CAPM is considered to be valid, the performance of optimal portfolio should be close to the

performance of the market index. Calculating beta for all time horizons and all portfolio sizes as

presented in Table 4, shows that beta is close to one due to the averaging effect of forming 1,000

random portfolios, representing the overall movement of the market index. However, on further

investigation on beta volatility using the standard deviation of betas on 1,000 random portfolios

displays an inverse relationship to portfolio size, exhibiting that increasing portfolio diversification

reduces the dispersion of betas as portfolio size becomes larger. It is interesting to note that

20

although the standard deviation of betas continues to decline as portfolio size increases, the rate of

the decline decays, with only marginal reductions beyond around 15 stocks.

The minimum and maximum betas are also presented in Table 4 to provide the full range of

systematic risk in each portfolio size in the data. It is evident that there is a consistent pattern of the

range of betas when compared to the standard deviation of betas. For example, Table 4a shows that

the ranges of the minimum and maximum betas reduce at similar rates to the standard deviation of

betas. Therefore, as portfolio size increases, volatility lessens. Hence, the reduction in beta volatility

in the portfolio provides a better predictability of portfolio betas. As a result, the benefit of portfolio

diversification results in the ability to diversify away a high proportion of idiosyncratic risk. More

controversially, large sized portfolios provide a better diversification effect but it should be noted

that small sized portfolios offer substantial reductions in volatility as the range of betas declines

rapidly as the number of stocks increases in small portfolios.


3.5. Application of stochastic processes to test the robustness of results

This paper uses different time horizons to form 1,000 random portfolios. The “stochastic process”

(indexed by random portfolio selection, not by time) is a collection of random portfolios in

different sizes in a particular time horizon. For example, in a one-year return horizon (Table 2a)

1,000 random portfolios are generated in each sized portfolio which consists of one stock, of two

stocks, of three stocks, etc., increasing the size of the portfolio up to a maximum of 204 stocks. Let

P represent portfolio return, for the return data there are different sized portfolios P1, P2, P3,..., P204,

where the subscript 1 denotes the average portfolio return with one stock from 1,000 random trials

and the subscript 204 denotes the average portfolio return with two hundred and four stocks from

21

1,000 random trials. Each of these P's is a random variable, apart from the maximum of P204. In

order to test whether a different random seed will affect the results in terms of the mean and

variance of different portfolio sizes, this paper naively applies for the unit root test, a test of

“stationarity or nonstationarity” of different portfolio sizes. Apart from Model 1 the estimated τ

values are larger than the critical values for different models. For example, for Model 2 the

estimated τ value is -4.4566 and the 10 percent critical value is -2.5742. In absolute terms the

former is greater than the latter. The conclusion is that the average returns on different sizes of

portfolios are “stationary”. The data in this paper appears to be a weakly stationary and the results

are not dependent on the choice of the initial random seed number. There is apparent evidence to

support this conclusion that is detailed in Appendix 1 which includes unit root tests on portfolio

returns based on different number of stocks in a portfolio in addition to the schematic diagrams that

clearly show that the average returns of these portfolios are stationary supported by the close

similarity of the diagrams presented.

4. Conclusions

Although there are a significant number of studies that endeavour to investigate the number of

stocks required for a well-diversified portfolio in order to minimise diversifiable risk, the results

from previous studies are inconsistent. This emphasises the great complexity of this popular topic

of portfolio diversification. This paper utilises return data to perform random portfolios with a

graphical analysis, uses the single-index model to differentiate idiosyncratic risk from total risk,

and maps individual portfolios in terms of risk and return. As random portfolios are drawn by the

set of random seeds, this paper naively applies time series terminology to test the stationarity of the

‘stochastic’ random portfolios, showing that random portfolios are weakly stationary and hence the

results in this study should not be dependent on the choice of the initial random seed number.

22

Both the aggregate analysis and individual stock analysis are used to assess the optimal number

of stocks in the UK stock market over the period 1985-2011. The results are able to clarify the issues

of over-diversification in addition to under-diversification. In agreement with the findings of

previous studies, this paper shows that there is a positive association between the increase in

portfolio size and the benefit of portfolio diversification. Further, this paper also concludes that the

variations in idiosyncratic risk in the UK stock market across time horizons are similar to the

findings of Campbell, Lettau, Malkiel, and Xu (2001). Their study asserts that idiosyncratic risk has

increased in the US stock returns over the 1962–1997. Furthermore, the results from the study by

Angelidis and Tessaromatis (2008) show that there are variations in idiosyncratic variances between

small and large stocks in the UK stock market over the period from 1980–2003. This paper suggests

that while forming large sized portfolios reduces total risk, the diversification effect for lessening

idiosyncratic risk becomes marginal as the portfolio size becomes larger than 12 stocks. However,

this paper also shows the likelihood of a high degree of skewness in small sized portfolio (under 5

stocks) indicating under-diversification is likely to cause large losses for investors with a relatively

high probability of occurring compared to the normal distribution. Investors who are overconfident

are likely to suffer from illusion as they narrow their confidence interval by forming small sized

portfolios. The findings in this research show an inverse relationship between portfolio size and the

difference between a maximum gain and maximum loss. In contrast, graphic analysis shows that

over-diversification occurs when portfolio size increases beyond 12 as the clustering effect become

centred. Furthermore, while over-diversification eliminates idiosyncratic risk, over-diversification

with the number of stocks greater than 12 in a portfolio is likely to diversify away the probability of

high and positive returns, as confirmed by the clustering effect. Thus, over-diversification restricts

risk taking and "speculating", as it limits returns by spreading investment by holding too many

stocks. Empirical evidence in this paper supports the view that the proportion of idiosyncratic risk to

23

total risk reduces with increasing portfolio size, but as portfolios become larger in size, idiosyncratic

risk exhibits diminishing marginal reductions. Overall, the results of this paper are similar to the

findings of other studies that support small sized portfolios. For example, Evans and Archer (1958)

and Jennings (1971), and Ivković, Sialm, and Weisbenner (2008) suggest concentrating on few

stocks will outperform holding a large number of stocks. This paper also echoes the professional

view, for example, by Warren Buffet and Benjamin Graham, who are in favour of investing in a

small number of enterprises.

24

Table 1: Statistics for stocks in different time horizons

Statistics for stock return Statistics for stock variance

Mean Standard

deviation

CV Median Minimum Maximum Mean Variance in

variance

CV Median Minimum Maximum

2011 return horizon

0.1087 1.7653 16.2343 0.0698 -5.8475 5.2912 61.9171 3545.5246 0.9616 41.3946 4.4429 457.2800

2010 return horizon

1.6524 2.4221 1.4658 1.3917 -4.2811 11.4983 74.1698 7999.9464 1.2059 49.5738 3.3110 895.9998

2009 return horizon

4.4828 3.8890 0.8675 3.5417 -1.3558 27.2403 253.0805 184593.7946 1.6976 119.364 5.8517 3723.285

2008 return horizon

-4.6593 3.2492 -0.6973 -4.0452 -18.5340 1.9111 152.96 29233.6340 1.1177 108.8392 7.5096 1726.6289

2007 return horizon

-0.2031 2.0622 -10.1510 -0.0322 -6.5517 6.5224 51.5346 1502.0690 0.7520 43.1553 2.0566 257.3370

2010-2011return horizon

0.8805 1.5806 1.7949 0.7160 -4.4450 7.4009 67.7765 4531.3148 0.9931 46.7633 3.7206 649.286

2007-2011 return horizon

0.2763 1.0348 3.7450 0.2432 -3.3091 4.4098 126.4686 19628.3086 1.1077 82.4359 8.8837 1031.7960


0.7525 0.7672 1.0195 0.6428 -1.3809 5.3335 98.6519 8039.1300 0.9088 71.2237 6.9076 717.0260


0.9802 0.3748 0.3824 0.9057 -0.3440 2.4557 90.2121 3109.7068 0.6181 78.0184 22.4070 333.9427

Notes: Statistics for stock return: Mean is the average return on all stocks and is calculated by the sum of average returns of all stocks divided by the total number of stocks.

Standard Deviation is the standard deviation based on average returns of all stocks in this study. CV is the coefficient of variation calculated by standard deviation of returns

divided by the average return. Median is the middle of the average return on all stocks. Minimum is the minimum of the average return based on all stocks. Maximum is the

maximum of the average return based on all stocks.

Statistics for stock variance: Mean is the average variance on all stocks and is calculated by the sum of average variances of all stocks divided by the total number of stocks.

Variance in variance is the variance of variances of all stocks. CV is the coefficient of variation calculated by standard deviation of variances divided by the average variance.

Median is the middle of the average variance on all stocks. Minimum is the minimum of the average variance based on all stocks. Maximum is the maximum of the average

variance based on all stocks.

25

Table 2a: Effect of portfolio diversification on portfolio risk: 1000 random portfolios based on one-year return horizon

Number of stocks Variance Variance in variance Mean Standard

deviation

Median Skewness Excess

Kurtosis

JB Minimum

Maximum

EPR Sharpe

1 61.8284 3392.1359 0.1219 1.7833 0.0698 -0.1147 1.4327 87.7189 -5.8476 5.2912 0.0819 0.0459

2 40.3395 815.7199 0.1101 1.2253 0.0882 -0.0574 0.7910 26.6172 -4.6414 4.0685 0.0700 0.0571

3 32.0392 303.6643 0.0105 1.0522 0.0698 -0.2296 0.5749 22.5596 -4.2676 3.5640 -0.0295 -0.0281

4 28.4523 173.9904 0.0317 0.9019 0.0477 -0.2434 0.4340 17.7250 -3.4925 2.6105 -0.0083 -0.0092

5 26.4079 128.9479 0.0514 0.8025 0.0649 -0.0667 0.0932 1.1038 -2.5845 2.3695 0.0114 0.0142

6 25.0066 100.0101 0.0625 0.7244 0.0653 -0.0799 0.1709 2.2797 -2.2443 2.4219 0.0224 0.0310

7 24.0682 80.4597 0.0633 0.6658 0.0663 -0.1271 0.3103 6.7065 -2.5473 2.1103 0.0233 0.0350

8 23.2926 65.8064 0.0608 0.6231 0.0848 -0.2236 0.5631 21.5425 -2.4139 2.1199 0.0208 0.0333

9 22.7124 56.0347 0.0657 0.5840 0.0689 -0.2108 0.5026 17.9333 -2.2309 1.9343 0.0256 0.0439

10 22.2626 51.1379 0.0672 0.5551 0.0766 -0.1793 0.7002 25.7916 -1.9456 1.9065 0.0272 0.0490

11 21.7210 44.1853 0.0702 0.5245 0.0857 -0.1450 0.5354 15.4447 -1.8228 1.6794 0.0301 0.0574

12 21.3820 39.2335 0.0724 0.5016 0.0842 -0.1528 0.4358 11.8058 -1.7309 1.5728 0.0323 0.0645

13 21.1250 34.6503 0.0786 0.4799 0.0784 -0.0954 0.3989 8.1492 -1.5978 1.5886 0.0385 0.0803

14 20.9429 31.3037 0.0828 0.4637 0.0807 -0.0817 0.4781 10.6347 -1.4985 1.8531 0.0427 0.0921

15 20.7585 29.8070 0.0836 0.4513 0.0799 -0.0081 0.4925 10.1186 -1.4253 1.8619 0.0436 0.0966

16 20.6053 26.5187 0.0851 0.4405 0.0940 -0.0189 0.3314 4.6361 -1.4159 1.7865 0.0451 0.1023

17 20.4398 24.5935 0.0916 0.4290 0.1014 -0.0392 0.3089 4.2323 -1.3719 1.7453 0.0515 0.1201

18 20.3744 23.7120 0.0918 0.4135 0.1071 -0.0888 0.2728 4.4136 -1.3397 1.3425 0.0518 0.1251

19 20.2411 22.6051 0.0945 0.4005 0.0980 -0.0595 0.2167 2.5469 -1.3193 1.3588 0.0544 0.1359

20 20.1425 21.3057 0.0935 0.3862 0.0970 -0.0639 0.1981 2.3168 -1.3237 1.3472 0.0534 0.1384

25 19.7293 16.8326 0.0965 0.3381 0.1009 -0.1425 0.3191 7.6291 -1.1316 1.1683 0.0565 0.1670

30 19.3621 13.4032 0.1037 0.3053 0.1150 -0.1572 0.2611 6.9593 -1.0714 1.0378 0.0637 0.2087

50 18.7821 6.7193 0.1084 0.2187 0.1207 -0.1815 0.1856 6.9255 -0.6295 0.7351 0.0684 0.3127

100 18.4028 2.2006 0.1032 0.1283 0.1106 -0.2486 -0.0217 10.3172 -0.3526 0.4626 0.0631 0.4921

150 18.3042 0.7710 0.1092 0.0752 0.1074 -0.0506 -0.0474 0.5205 -0.1218 0.3167 0.0692 0.9201

200 18.2301 0.0427 0.1096 0.0180 0.1095 0.1003 0.0409 1.7477 0.0564 0.1637 0.0696 3.8572

Notes: Number of stocks is a random choice of stocks in a portfolio; Variance is the average variance of 1000 variances of random portfolios. Variance in variance is the

variance of variances of 1,000 random portfolios. Mean is the average return on 1,000 random portfolios. Standard deviation is the standard deviation based on mean returns

of 1,000 random portfolios. Median is the middle of the average portfolio return and it is based on a sorted list of 1,000 portfolio returns. Skewness measures the extent of

asymmetry of portfolio returns. Excess kurtosis is defined as kurtosis minus 3. The excess kurtosis of 1,000 random portfolio returns that are normally distributed is zero. JB

is the Jarque-Bera test statistic for the normal distribution of portfolio returns. Minimum is the minimum of 1,000 random portfolio returns. Maximum is the maximum of

1,000 random portfolio returns. EPR is portfolio excess return calculated as average portfolio return minus the risk free rate. Sharpe is the Sharpe ratio measuring the

portfolio excess return per unit of total risk.

26

Table 2b: Effect of portfolio diversification on portfolio risk: 1000 random portfolios based on two-year return horizon


deviation


Kurtosis

JB Minimum

Maximum

EPR Sharpe

1 68.1700 4416.7982 0.8768 1.5531 0.7203 0.3536 2.7489 335.6998 -4.4451 7.4010 0.8361 0.5384

2 44.5370 919.7335 0.8743 1.1163 0.7964 0.3113 1.3620 93.4408 -3.5142 5.3010 0.8335 0.7467

3 36.3032 346.2235 0.8120 0.9315 0.8084 -0.0752 0.6011 15.9998 -3.0901 3.6755 0.7713 0.8280

4 32.1780 198.7330 0.8237 0.8067 0.8248 -0.0601 0.6700 19.3062 -2.3128 4.0945 0.7829 0.9705

5 29.8672 144.3926 0.8230 0.6975 0.8148 -0.0426 0.2950 3.9295 -1.5844 3.4727 0.7822 1.1215

6 28.3269 111.8867 0.8316 0.6373 0.8286 -0.0012 0.1752 1.2790 -1.1692 2.9035 0.7909 1.2411

7 27.3121 89.2916 0.8373 0.5767 0.8353 -0.0081 0.2214 2.0529 -1.0365 2.7586 0.7966 1.3812

8 26.3588 73.1326 0.8218 0.5351 0.8168 -0.0274 -0.0092 0.1285 -0.8415 2.4172 0.7811 1.4598

9 25.7252 63.4673 0.8297 0.5093 0.8419 -0.1512 0.0542 3.9352 -1.1080 2.2720 0.7890 1.5492

10 25.2197 56.5587 0.8333 0.4809 0.8333 -0.1502 0.0623 3.9201 -0.9709 2.4419 0.7926 1.6481

11 24.6796 49.6061 0.8348 0.4628 0.8344 -0.1205 0.1030 2.8623 -0.8365 2.2808 0.7941 1.7159

12 24.3208 43.7909 0.8393 0.4426 0.8419 -0.1380 0.1569 4.1999 -0.6727 2.2523 0.7986 1.8043

13 24.0065 38.8680 0.8430 0.4241 0.8417 -0.1499 0.1091 4.2428 -0.5831 2.1489 0.8023 1.8919

14 23.7490 34.3722 0.8467 0.4082 0.8491 -0.0921 0.1375 2.2020 -0.5051 2.2545 0.8059 1.9744

15 23.5258 32.0009 0.8509 0.3890 0.8438 -0.0483 0.1705 1.6004 -0.4341 2.2766 0.8101 2.0824

16 23.3466 28.7512 0.8538 0.3773 0.8420 -0.0345 0.0479 0.2940 -0.3362 2.2337 0.8131 2.1549

17 23.1423 26.8552 0.8571 0.3663 0.8543 -0.0203 0.0602 0.2198 -0.2031 2.2765 0.8164 2.2284

18 23.0620 25.3297 0.8590 0.3604 0.8550 -0.0490 0.1576 1.4359 -0.4387 2.0215 0.8183 2.2702

19 22.9215 23.6708 0.8613 0.3500 0.8603 -0.0460 0.1577 1.3894 -0.3898 2.0754 0.8205 2.3442

20 22.8097 22.4993 0.8628 0.3347 0.8586 -0.0588 0.1119 1.0973 -0.3310 2.0529 0.8220 2.4557

25 22.3399 17.2376 0.8661 0.2920 0.8522 0.0117 -0.2434 2.4917 -0.0601 1.6658 0.8254 2.8266

30 22.0394 13.7053 0.8697 0.2685 0.8708 -0.0283 -0.0376 0.1919 -0.0843 1.7974 0.8290 3.0868

50 21.3588 7.4050 0.8735 0.1935 0.8683 0.1118 -0.1020 2.5174 0.2783 1.5095 0.8328 4.3033

100 20.9063 2.2710 0.8757 0.1146 0.8741 -0.0014 -0.1364 0.7757 0.4907 1.1779 0.8349 7.2883

150 20.7531 0.8186 0.8805 0.0684 0.8803 -0.0445 -0.1187 0.9171 0.6416 1.0803 0.8397 12.2737

200 20.6996 0.0487 0.8815 0.0160 0.8815 -0.1901 0.8393 35.3730 0.8073 0.9369 0.8407 52.4023








27

Table 2c: Effect of portfolio diversification on portfolio risk: 1000 random portfolios based on five-year return horizon Number of stocks Variance Variance in variance Mean Standard

deviation


Kurtosis

JB Minimum

Maximum

EPR Sharpe

1 126.8666 19236.2089 0.2685 1.0170 0.2291 0.2415 2.1510 202.5062 -3.3091 4.4099 0.0807 0.0794

2 83.9984 4538.2407 0.2724 0.7249 0.2408 0.3247 1.5824 121.9092 -2.2737 3.5902 0.0847 0.1168

3 69.0238 1671.8746 0.2185 0.5905 0.1879 0.2053 0.4860 16.8633 -1.9511 2.4630 0.0308 0.0522

4 61.6389 955.3929 0.2299 0.5215 0.2061 0.2670 0.8139 39.4902 -1.4163 2.6111 0.0422 0.0809

5 57.7933 715.6697 0.2341 0.4602 0.2187 0.1701 0.4556 13.4737 -1.1879 2.2110 0.0464 0.1007

6 55.0801 568.7432 0.2414 0.4174 0.2207 0.2317 0.3698 14.6433 -1.1110 1.9711 0.0537 0.1286

7 53.2426 436.5324 0.2452 0.3786 0.2386 0.1424 0.3755 9.2541 -0.9979 1.6744 0.0575 0.1519

8 51.5221 346.1996 0.2398 0.3452 0.2288 0.0443 0.0566 0.4599 -0.9456 1.4210 0.0521 0.1509

9 50.4649 289.7037 0.2454 0.3244 0.2446 -0.0397 0.2776 3.4748 -0.9674 1.2939 0.0577 0.1777

10 49.4922 246.9991 0.2480 0.3040 0.2415 0.0107 0.0178 0.0321 -0.8095 1.1696 0.0603 0.1983

11 48.5532 224.3431 0.2513 0.2955 0.2489 0.0753 0.1254 1.5999 -0.7070 1.3449 0.0636 0.2152

12 47.8907 198.6388 0.2532 0.2821 0.2444 0.0606 0.0497 0.7147 -0.6172 1.1492 0.0655 0.2321

13 47.3399 175.4790 0.2550 0.2715 0.2482 -0.0160 -0.0791 0.3031 -0.6047 1.1325 0.0673 0.2478

14 46.9020 156.7220 0.2571 0.2630 0.2503 0.0313 -0.1078 0.6478 -0.5536 1.0652 0.0694 0.2639

15 46.5437 144.1840 0.2595 0.2499 0.2494 -0.0287 -0.0369 0.1940 -0.5703 0.9519 0.0717 0.2870

16 46.2981 128.4034 0.2615 0.2434 0.2545 -0.0590 0.0107 0.5857 -0.5560 0.9233 0.0738 0.3030

17 45.9136 119.3669 0.2625 0.2347 0.2557 -0.0197 -0.0740 0.2930 -0.5568 0.9562 0.0748 0.3186

18 45.8335 113.8670 0.2641 0.2317 0.2635 -0.0023 0.0678 0.1924 -0.6362 1.0519 0.0764 0.3297

19 45.5832 104.8814 0.2654 0.2243 0.2674 -0.0119 0.1529 0.9977 -0.5914 1.0355 0.0777 0.3463

20 45.3664 98.6782 0.2653 0.2161 0.2656 -0.0484 0.0244 0.4157 -0.5634 0.9836 0.0776 0.3591

25 44.5117 76.0590 0.2686 0.1879 0.2610 0.0644 -0.1196 1.2864 -0.4151 0.8504 0.0809 0.4304

30 43.9837 58.7345 0.2683 0.1750 0.2735 0.0405 0.0385 0.3356 -0.3817 0.8830 0.0806 0.4604

50 42.8860 30.3012 0.2728 0.1261 0.2718 0.0769 -0.1768 2.2874 -0.0736 0.6635 0.0851 0.6744

100 42.0463 9.3934 0.2738 0.0758 0.2767 -0.0028 -0.0739 0.2288 0.0564 0.5160 0.0860 1.1346

150 41.8081 3.3965 0.2753 0.0446 0.2748 -0.0313 -0.3333 4.7923 0.1441 0.3972 0.0875 1.9613

200 41.6501 0.2283 0.2765 0.0102 0.2767 -0.1132 0.3746 7.9819 0.2397 0.3065 0.0887 8.6767








28

Table 2d: Effect of portfolio diversification on portfolio risk: 1000 random portfolios based on full-period return horizon


deviation


Kurtosis

JB Minimum

Maximum

EPR Sharpe

1 89.9280 2915.9728 0.9795 0.3692 0.9016 0.8835 2.3531 360.8019 -0.3441 2.4557 0.4582 1.2412

2 58.2494 581.6203 0.9789 0.2591 0.9490 0.3797 0.6522 41.7511 0.0526 1.9279 0.4577 1.7665

3 47.2808 222.0097 0.9620 0.2013 0.9524 0.2539 0.2063 12.5209 0.2387 1.6580 0.4408 2.1891

4 41.9663 125.8172 0.9651 0.1785 0.9522 0.3357 0.4725 28.0885 0.3429 1.6515 0.4439 2.4862

5 38.9911 89.6748 0.9685 0.1637 0.9597 0.2344 0.2085 10.9683 0.4475 1.5003 0.4472 2.7325

6 36.9147 67.9891 0.9697 0.1494 0.9604 0.1688 0.0398 4.8142 0.5094 1.4241 0.4485 3.0008

7 35.5009 53.1583 0.9727 0.1375 0.9636 0.1832 -0.0972 5.9903 0.5571 1.4209 0.4514 3.2843

8 34.2997 43.8738 0.9732 0.1274 0.9675 0.0903 -0.1152 1.9120 0.5661 1.3579 0.4519 3.5468

9 33.4955 37.0774 0.9756 0.1196 0.9706 0.0307 -0.0510 0.2656 0.5710 1.3111 0.4543 3.7973

10 32.8077 31.7593 0.9763 0.1142 0.9755 0.0238 -0.1124 0.6210 0.5799 1.3134 0.4551 3.9855

11 32.2361 28.2530 0.9774 0.1099 0.9743 0.0583 0.0019 0.5669 0.5667 1.3168 0.4562 4.1502

12 31.7538 24.3505 0.9775 0.1050 0.9740 0.0520 -0.1299 1.1535 0.5921 1.2873 0.4563 4.3441

13 31.3189 20.8305 0.9766 0.1015 0.9730 0.1053 -0.0084 1.8501 0.6455 1.3283 0.4554 4.4887

14 30.9965 18.9234 0.9768 0.0974 0.9731 0.1550 -0.1134 4.5406 0.6695 1.3317 0.4555 4.6785

15 30.7227 17.6998 0.9775 0.0934 0.9746 0.1621 -0.1163 4.9438 0.6852 1.3002 0.4562 4.8846

16 30.4934 15.7600 0.9774 0.0891 0.9767 0.1563 -0.0012 4.0712 0.6982 1.3105 0.4561 5.1198

17 30.2436 14.9026 0.9775 0.0867 0.9761 0.1745 -0.1199 5.6722 0.7261 1.2799 0.4563 5.2632

18 30.0904 14.1071 0.9775 0.0840 0.9770 0.1133 -0.1308 2.8532 0.7217 1.2791 0.4563 5.4341

19 29.9059 12.8566 0.9777 0.0808 0.9787 0.0983 -0.0586 1.7540 0.7251 1.2794 0.4564 5.6495

20 29.7500 12.1437 0.9777 0.0782 0.9774 0.0637 -0.0374 0.7349 0.7095 1.2536 0.4565 5.8351

25 29.1016 9.2065 0.9780 0.0689 0.9774 0.0762 -0.1805 2.3242 0.7840 1.1932 0.4568 6.6255

30 28.6784 6.8781 0.9803 0.0623 0.9802 0.0517 -0.0370 0.5023 0.7604 1.1946 0.4590 7.3629

50 27.8957 3.7940 0.9815 0.0475 0.9801 0.1719 0.0735 5.1518 0.8203 1.1478 0.4602 9.6819

100 27.2588 1.1837 0.9805 0.0270 0.9783 0.0258 0.0025 0.1116 0.8810 1.0667 0.4593 17.0203

150 27.0646 0.4370 0.9803 0.0157 0.9799 -0.0026 -0.1270 0.6734 0.9358 1.0294 0.4590 29.2760

200 26.9801 0.0269 0.9802 0.0036 0.9805 -0.4372 0.6642 50.2351 0.9629 0.9922 0.4590 126.2945








29

Table 3a: Idiosyncratic risk for one-year time horizon

Number of stocks Total risk Systematic risk Idiosyncratic risk % systematic % idiosyncratic

1 61.8184 22.8755 38.9429 0.4028 0.5972

2 40.3299 19.9341 20.3958 0.4933 0.5067

3 32.0302 18.0465 13.9837 0.5496 0.4504

4 28.4432 17.4874 10.9558 0.6029 0.3971

5 26.3987 17.2166 9.1821 0.6427 0.3573

6 24.9974 17.0872 7.9101 0.6721 0.3279

7 24.0589 16.9417 7.1172 0.6958 0.3042

8 23.2835 16.8111 6.4723 0.7153 0.2847

9 22.7032 16.7501 5.9531 0.7310 0.2690

10 22.2534 16.7129 5.5405 0.7444 0.2556

11 21.7119 16.6097 5.1022 0.7598 0.2402

12 21.3729 16.5584 4.8145 0.7707 0.2293

13 21.1158 16.5329 4.5830 0.7791 0.2209

14 20.9337 16.5505 4.3832 0.7865 0.2135

15 20.7492 16.5600 4.1892 0.7939 0.2061

16 20.5960 16.5658 4.0302 0.8001 0.1999

17 20.4306 16.5386 3.8920 0.8054 0.1946

18 20.3652 16.5718 3.7934 0.8097 0.1903

19 20.2320 16.5661 3.6658 0.8147 0.1853

20 20.1334 16.5702 3.5632 0.8190 0.1810

25 19.7201 16.5372 3.1829 0.8356 0.1644

30 19.3527 16.4431 2.9095 0.8473 0.1527

50 18.7726 16.3484 2.4242 0.8698 0.1302

100 18.3933 16.3500 2.0433 0.8886 0.1114

150 18.2948 16.3860 1.9087 0.8955 0.1045

200 18.2208 16.3858 1.8350 0.8993 0.1007

Total risk is the sum of systematic risk and idiosyncratic risk. Systematic risk is calculated using the formula 2

m

2

pσβ . Beta is

obtained by regressing portfolio excess return against market risk premium. The variance of market returns is equal to

k

1i

2

mm,i )r(r1-k

1 where k is the number of months, mr is market return, mr is average market return under investigation.

Idiosyncratic risk is the variance of regression residuals and it is equal to

k

1i

2

i )residual(residual1-k

1. % systematic is the

mean systematic risk in each portfolio from 1,000 random portfolios. % idiosyncratic is the mean idiosyncratic risk in each

portfolio from 1,000 random portfolios.

30

Table 3b: Idiosyncratic risk for two-year time horizon


1 68.1656 24.7273 43.4383 0.3986 0.6014

2 44.5325 21.7870 22.7455 0.4930 0.5070

3 36.2989 20.5631 15.7358 0.5583 0.4417

4 32.1736 19.8583 12.3153 0.6112 0.3888

5 29.8627 19.6346 10.2281 0.6513 0.3487

6 28.3223 19.4381 8.8843 0.6795 0.3205

7 27.3075 19.3070 8.0005 0.7011 0.2989

8 26.3543 19.1547 7.1995 0.7216 0.2784

9 25.7206 19.0710 6.6497 0.7358 0.2642

10 25.2151 18.9894 6.2257 0.7484 0.2516

11 24.6750 18.9023 5.7727 0.7619 0.2381

12 24.3162 18.8498 5.4664 0.7714 0.2286

13 24.0020 18.7835 5.2184 0.7791 0.2209

14 23.7445 18.7377 5.0068 0.7855 0.2145

15 23.5212 18.7111 4.8102 0.7920 0.2080

16 23.3421 18.6988 4.6433 0.7975 0.2025

17 23.1377 18.6358 4.5019 0.8021 0.1979

18 23.0574 18.6628 4.3946 0.8064 0.1936

19 22.9170 18.6476 4.2694 0.8105 0.1895

20 22.8052 18.6452 4.1600 0.8145 0.1855

25 22.3353 18.5861 3.7492 0.8297 0.1703

30 22.0347 18.5673 3.4674 0.8407 0.1593

50 21.3541 18.4023 2.9518 0.8608 0.1392

100 20.9016 18.3553 2.5463 0.8780 0.1220

150 20.7485 18.3552 2.3933 0.8846 0.1154

200 20.6950 18.3800 2.3150 0.8881 0.1119


m

2

pσβ . Beta is


k

1i

2

mm,i )r(r1-k



k

1i

2





31

Table 3c: Idiosyncratic risk for five-year time horizon


1 126.9684 44.2118 82.7565 0.3912 0.6088

2 84.1389 39.5389 44.6000 0.4910 0.5090

3 69.1820 36.9346 32.2473 0.5467 0.4533

4 61.8055 36.1237 25.6818 0.5976 0.4024

5 57.9657 36.0708 21.8949 0.6339 0.3661

6 55.2551 35.8613 19.3937 0.6592 0.3408

7 53.4198 35.7178 17.7020 0.6774 0.3226

8 51.6982 35.5242 16.1740 0.6947 0.3053

9 50.6430 35.4408 15.2022 0.7062 0.2938

10 49.6719 35.3170 14.3549 0.7170 0.2830

11 48.7327 35.1811 13.5515 0.7275 0.2725

12 48.0716 35.0776 12.9939 0.7351 0.2649

13 47.5217 35.0131 12.5086 0.7416 0.2584

14 47.0845 34.9860 12.0985 0.7472 0.2528

15 46.7267 34.9589 11.7678 0.7523 0.2477

16 46.4814 34.9819 11.4995 0.7566 0.2434

17 46.0981 34.8757 11.2224 0.7603 0.2397

18 46.0188 34.9697 11.0491 0.7635 0.2365

19 45.7692 34.9464 10.8228 0.7668 0.2332

20 45.5527 34.9367 10.6160 0.7700 0.2300

25 44.6995 34.8520 9.8475 0.7821 0.2179

30 44.1751 34.8078 9.3673 0.7900 0.2100

50 43.0782 34.7019 8.3762 0.8067 0.1933

100 42.2390 34.6208 7.6183 0.8201 0.1799

150 42.0014 34.6208 7.3806 0.8244 0.1756

200 41.8435 34.6112 7.2323 0.8272 0.1728


m

2

pσβ . Beta is


k

1i

2

mm,i )r(r1-k



k

1i

2





32

Table 3d: Idiosyncratic risk for full-period time horizon


1 90.0950 24.4589 65.6361 0.3293 0.6707

2 58.4092 23.5169 34.8924 0.4276 0.5724

3 47.4327 22.7676 24.6651 0.4923 0.5077

4 42.1199 22.6467 19.4732 0.5472 0.4528

5 39.1464 22.6633 16.4831 0.5863 0.4137

6 37.0712 22.6480 14.4232 0.6164 0.3836

7 35.6586 22.6392 13.0194 0.6392 0.3608

8 34.4553 22.6077 11.8476 0.6595 0.3405

9 33.6518 22.6272 11.0245 0.6751 0.3249

10 32.9641 22.6207 10.3434 0.6888 0.3112

11 32.3926 22.6290 9.7635 0.7010 0.2990

12 31.9107 22.6157 9.2950 0.7109 0.2891

13 31.4765 22.5974 8.8791 0.7198 0.2802

14 31.1548 22.6123 8.5426 0.7274 0.2726

15 30.8811 22.6164 8.2647 0.7338 0.2662

16 30.6523 22.6351 8.0173 0.7397 0.2603

17 30.4027 22.6067 7.7960 0.7448 0.2552

18 30.2503 22.6284 7.6219 0.7492 0.2508

19 30.0660 22.6287 7.4373 0.7537 0.2463

20 29.9101 22.6323 7.2778 0.7577 0.2423

25 29.2622 22.6223 6.6400 0.7739 0.2261

30 28.8405 22.5927 6.2478 0.7840 0.2160

50 28.0587 22.6032 5.4555 0.8060 0.1940

100 27.4212 22.5928 4.8284 0.8241 0.1759

150 27.2275 22.5969 4.6306 0.8300 0.1700

200 27.1431 22.6269 4.5162 0.8336 0.1664


m

2

pσβ . Beta is


k

1i

2

mm,i )r(r1-k



k

1i

2





33

Table 4a: Beta and portfolio diversification for one-year time horizon

Number of stocks Beta Standard deviation Minimum Maximum

1 1.0288 0.6411 -0.2882 2.9321

2 1.0291 0.4704 -0.1789 2.4909

3 1.0052 0.3856 0.0038 2.2189

4 1.0093 0.3233 0.0260 2.0464

5 1.0131 0.2816 0.1103 2.1621

6 1.0146 0.2606 0.1748 1.8082

7 1.0161 0.2360 0.2911 1.7685

8 1.0160 0.2177 0.3032 1.6655

9 1.0167 0.2051 0.3833 1.5859

10 1.0173 0.1963 0.4344 1.5945

11 1.0165 0.1828 0.5182 1.5794

12 1.0168 0.1721 0.5101 1.5675

13 1.0173 0.1638 0.5247 1.5531

14 1.0187 0.1583 0.5352 1.5396

15 1.0195 0.1552 0.5804 1.5247

16 1.0207 0.1488 0.6363 1.4511

17 1.0204 0.1444 0.6274 1.4335

18 1.0218 0.1419 0.6308 1.4385

19 1.0220 0.1393 0.6671 1.4265

20 1.0226 0.1360 0.6804 1.4656

25 1.0234 0.1215 0.6899 1.4695

30 1.0218 0.1090 0.7115 1.4153

50 1.0216 0.0785 0.7457 1.2792

100 1.0237 0.0451 0.8685 1.1861

150 1.0255 0.0269 0.9371 1.1113

200 1.0258 0.0064 1.0040 1.0446

Beta is the average beta based on 1,000 random portfolios. Standard deviation is the standard deviation of betas in portfolios.

Minimum is the minimum beta in portfolios. Maximum is the maximum of beta in portfolios.

34

Table 4b: Beta and portfolio diversification for two-year time horizon


1 0.9276 0.5276 -0.4770 2.6278

2 0.9294 0.3733 0.0040 2.3680

3 0.9250 0.3020 0.1430 1.8946

4 0.9222 0.2527 0.2170 1.8134

5 0.9238 0.2250 0.2690 1.8792

6 0.9232 0.2068 0.3422 1.6802

7 0.9231 0.1919 0.3251 1.5678

8 0.9221 0.1777 0.3904 1.5152

9 0.9215 0.1699 0.3918 1.4301

10 0.9211 0.1607 0.3826 1.3950

11 0.9205 0.1514 0.4778 1.3700

12 0.9204 0.1439 0.5185 1.3476

13 0.9199 0.1368 0.5525 1.3685

14 0.9195 0.1311 0.5557 1.3317

15 0.9194 0.1272 0.5344 1.3088

16 0.9197 0.1229 0.5247 1.2754

17 0.9186 0.1191 0.5761 1.2456

18 0.9197 0.1157 0.5856 1.2435

19 0.9197 0.1130 0.6030 1.2694

20 0.9199 0.1103 0.6072 1.3369

25 0.9199 0.0976 0.6546 1.2072

30 0.9204 0.0878 0.6758 1.2221

50 0.9181 0.0654 0.7198 1.1654

100 0.9186 0.0361 0.8058 1.0306

150 0.9190 0.0219 0.8527 0.9839

200 0.9199 0.0054 0.8987 0.9379



35

Table 4c: Beta and portfolio diversification for five-year time horizon


1 1.1071 0.5635 -0.1530 3.5405

2 1.1033 0.4031 0.0775 3.2507

3 1.0926 0.3085 0.2417 2.5085

4 1.0923 0.2595 0.4206 2.0149

5 1.0967 0.2366 0.5650 1.9252

6 1.0971 0.2188 0.6123 1.8794

7 1.0980 0.2019 0.5859 1.8891

8 1.0976 0.1867 0.6143 1.8412

9 1.0980 0.1765 0.6960 1.7286

10 1.0977 0.1657 0.6768 1.6727

11 1.0965 0.1593 0.7435 1.7074

12 1.0959 0.1521 0.7433 1.6397

13 1.0958 0.1449 0.7556 1.6309

14 1.0961 0.1397 0.6907 1.5929

15 1.0963 0.1346 0.7004 1.6767

16 1.0974 0.1285 0.7474 1.6139

17 1.0961 0.1249 0.7527 1.5992

18 1.0979 0.1222 0.7549 1.5454

19 1.0980 0.1177 0.8025 1.4938

20 1.0981 0.1153 0.7914 1.4700

25 1.0980 0.1028 0.8270 1.5010

30 1.0983 0.0910 0.8807 1.4624

50 1.0983 0.0674 0.8764 1.3579

100 1.0985 0.0379 0.9693 1.2144

150 1.0989 0.0229 1.0323 1.1690

200 1.0990 0.0060 1.0745 1.1130



36

Table 4d: Beta and portfolio diversification for full-period time horizon


1 0.9979 0.2776 0.3279 2.1496

2 0.9962 0.1978 0.3976 1.8342

3 0.9867 0.1584 0.4077 1.5754

4 0.9877 0.1335 0.4854 1.3757

5 0.9898 0.1201 0.6366 1.3953

6 0.9904 0.1117 0.6244 1.3578

7 0.9910 0.1043 0.6663 1.3643

8 0.9909 0.0987 0.7040 1.2924

9 0.9918 0.0935 0.7155 1.2741

10 0.9922 0.0875 0.7337 1.2365

11 0.9928 0.0829 0.7500 1.2126

12 0.9929 0.0783 0.7667 1.2197

13 0.9928 0.0736 0.7707 1.2009

14 0.9933 0.0712 0.7648 1.2055

15 0.9936 0.0691 0.7890 1.1893

16 0.9942 0.0663 0.7971 1.1942

17 0.9937 0.0646 0.8217 1.2250

18 0.9943 0.0629 0.8017 1.2167

19 0.9944 0.0604 0.8223 1.1992

20 0.9946 0.0592 0.8129 1.1923

25 0.9948 0.0521 0.7972 1.1800

30 0.9944 0.0458 0.8532 1.1588

50 0.9951 0.0341 0.9018 1.1649

100 0.9953 0.0194 0.9403 1.0696

150 0.9955 0.0120 0.9564 1.0341

200 0.9962 0.0029 0.9843 1.0041



37

38

39

40

41

42

Appendix 1: Unit root tests on portfolio returns based on different number of stocks in a portfolio

One-year return horizon

α0 α1 δ γ R2 D-W

Model 1 - - -0.0032

(-0.6438)

- 0.0020 2.03

Model 2 0.0181***

(4.4075)

- -0.1751***

(-4.4566)

- 0.0899 1.87

Model 3 0.0232***

(5.4253)

0.0001***

(3.3879)

-0.2577***

(-5.6755)

- 0.1393 1.82

Model 4 0.0251***

(5.4937)

0.0001***

(3.4177)

-0.2773***

(-5.6719)

0.1072

(1.5479)

0.1402 1.15

Two-year return horizon


Model 1 - - 0.0000

(0.0169)

- 0.0000 2.26

Model 2 0.0677***

(2.8347)

- -0.0775***

(-2.8342)

- 0.0384 2.18

Model 3 0.1397***

(4.3397)

0.0000***

(3.2469)

-0.1629***

(-4.3449)

- 0.0865 2.10

Model 4 0.1323***

(3.9080)

0.0000***

(2.9900)

-0.1542***

(-3.9120)

-0.0628

(-0.8934)

0.0889 1.12

Five-year return horizon


Model 1 - - 0.0000

(0.0325)

- 0.0000 2.43

Model 2 0.0333***

(3.6194)

- -0.1226***

(-3.6168)

- 0.0611 2.29

Model 3 0.0531***

(4.7519)

0.0000***

(3.0035)

-0.2023***

(-4.7563)

- 0.1016 2.20

Model 4 0.0471***

(4.0183)

0.0000***

(2.7395)

-0.1794***

(-4.0280)

-0.1313*

(-1.8734)

0.1210 0.88

All-year return horizon


Model 1 - - 0.0000

(0.0335)

- 0.0000 2.11

Model 2 0.1505***

(4.0893)

- -0.1536***

(-4.0892)

- 0.0768 1.96

Model 3 0.1723***

(4.3936)

0.0000

(1.5684)

-0.1761***

(-4.3935)

- 0.0880 1.94

Model 4 0.1772***

(4.2950)

0.0000

(1.5523)

-0.1811***

(-4.2946)

0.0309

(0.4345)

0.0884 1.10

The dependent variable in each model is ΔPn. The variableΔPn=Pn-Pn-1 is called the first difference of the series Pn. R2 is the

coefficient of determination indicating the percentage of the variation in the dependent variable that is explained by the regression

equation. D-W is the Durbin-Watson Statistic and values lower than 2 suggest that the residuals may be serially correlated. *, **,

and *** indicates that the null hypothesis is rejected with the following the levels of significance: 10 percent, 5 percent, and 1

percent respectively. t-values are reported in parentheses.

Model 1: ΔPn=δPn-1+νn

Model 2: ΔPn=α0+ δPn-1+νn

Model 3: ΔPn=α0+α1t+δPn-1+νn

Model 4: ΔPn=α0+α1t+δPn-1+γΔPn-1+νn

The critical 1, 5, and 10 percent τ values for Model 1 are -2.5757, -1.9412, and -1.6164, respectively. The critical 1, 5, and 10

percent τ values for Model 2 are -3.4637, -2.8757, and -2.5742, respectively. The critical 1, 5, and 10 percent τ values for Model 3

are -4.0061, -3.4329, and -3.1400, respectively.

43

44

45

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