Risk and Concentration of Portfolios on the Zimbabwe Stock Exchange after Currency Reform

Aims: The objectives of this study are to assess the level and impact of concentration of portfolios on the ZSE and to determine the number of stocks to be held in a concentrated portfolio to achieve effective risk reduction. Study Design: Portfolio Model. Place and Duration of Study: Zimbabwe, Department of Insurance and Actuarial Science and Department of Finance, between February 2013 and March 2013. Methodology: We analysed the level of concentration of portfolios held on the Zimbabwe Stock Exchange (ZSE). The market capitalization weights and the daily closing prices of 62 stocks in the industrial index for the four-year period form 19 February 2009 to 31 December 2012. Results: The Herfindahl-Hirschman Index and the Roll measure of concentration were employed to analyse the level of concentration of portfolios mimicking the industrial index and it was observed that portfolios held on the ZSE are highly un concentrated with an approximate measure of 14% under the HHI measure as at 31 December 2012. The daily returns over the period were calculated and used to estimate the risk of the portfolio. The findings indicates that stocks in the industrial index of the ZSE have relatively low correlation due to the small difference in risk between equally weighted portfolios with no Original Research Article British Journal of Economics, Management & Trade, 4(8): 1191-1202, 2014 1192 correlation and equally weighted portfolios with historical correlation. Conclusion: The empirical evidence highlights that an optimal portfolio size averaging 20 to 25 stocks of the Industrial Index stocks will have to be included in order to achieve effective risk reduction.


INTRODUCTION
Much of the focus of financial theory in relation to investment strategy for both corporates and individuals has focused on the importance of diversification as the major tool for portfolio optimisation and effective reduction of unsystematic risk in a portfolio of investments [1].Provides the first treatment of the principle of diversification which postulates that in order to lower the portfolio risk without sacrificing return, a portfolio with returns that are less than positively correlated can be created [2].Consider diversification as a strategy that is designed to reduce unique risk by spreading an investment portfolio across many investments, thus reducing variability of the returns of the portfolio. As the debate between the ideology of portfolio diversification and portfolio concentration rages on, several famous investors have adopted a concentrated position of investment strategy. Berkowits of Fairholme Capital was also quoted as saying, "The history of success, those who have succeeded well… they are focused on few activities" [3]. Berkowitz's ideology stems from the fact that one can only opt for diversification when one is not confident of their picks. Berkowitz believes that an investor needs only "a few good ideas" to do extremely well in his lifetime [3]. Other famous investors (Benjamin Graham, John Maynard Keynes, Bill Gates, John Rockefeller and Phillip Fisher) also followed this principle of investing more in one's best idea rather than investing in one's tenth best idea [4]. Concentration is the exact opposite of a diversification strategy.
Although no rigid definition of the concept of concentration has been documented in literature due to the little attention afforded to this concept, several scholars have sought to define it in different studies [5]. In their study of active Australian equity managers, define concentration as "the extent to which the portfolio weights held in stocks, industries and sectors deviate from the underlying index or market portfolio". The risk of a portfolio decreases as the number of different stocks in it increases [6]. Observe that the risk reduction effect diminishes rapidly as the number of stocks increases. However, they conclude that the benefits of diversification almost exhaust when a portfolio consists of approximately 10 stocks [7]. Refer to concentration as the "extent to which portfolio weights skew away from an equally weighted distribution of portfolio weights" [7]. further explain that if concentration is low, then the portfolio is more skewed towards equal weighting for all stocks in the portfolio whereas high concentration entails that most investors tend to concentrate most of their investments in a few stocks, for example 90% of investments held in 3 stocks in a stock market consisting of over 100 counters. [8] List several reasons why investors would hold a concentrated position as compared to a diversified portfolio. These include behavioural bias such as familiarity and overconfidence, and an investor's ability to monitor individual stocks. The investor identifies only those stocks that will provide higher expected abnormal returns after considering the cost of higher total risk. Behavioural bias leads an investor to a selection of familiar small subset of stocks, whose performance the investor is confident about (cognitive resource limitations). In addition, increasing the number of different investments may lessen one's ability to effectively monitor any of them.
Concentration of portfolio weights directly impacts on the specific risk of a portfolio. The specific risk of a portfolio is determined by the covariance of the underlying assets and the weighting structure of the portfolio [7]. Furthermore, [7] argue that although investors have no influence over the correlation structure among the assets, they can control their weighting structures, depending on their investment choices. In turn, the investor's choice in weighting structure affects the degree of concentration of the portfolio that investor holds. It has been widely suggested that concentration plays an important part in the structuring of portfolios in smaller markets such as the Johannesburg Stock Exchange (JSE) and the ZSE [7]. Most researches in finance were carried out for countries with developed stock markets [6]. Hence the need arises to analyse the risk and concentration of portfolios in emerging markets such as ZSE.
The debate over holding a concentrated stake or a diversified one has been raging on in recent years as most famous investors such as Warren Buffet have evidently strayed away from the Markowitz principle of diversification. Since concentration is less rigid, in terms of the definition and how many stocks constitute a concentrated portfolio, there is no universally acceptable definition of concentration that has been postulated [9]. As a benchmark and starting point to measuring concentration, investors often look at the number of stocks held in a portfolio [9]. However, in using such a measure for concentration, the Brandes Institute identifies a major pitfall in that the measure does not factor in the portfolio weights of the holding such that a portfolio comprising of 50 stocks may be equally-weighted (i.e. 2% invested in each stock) or the investor may hold 50% of his portfolio in one stock, while holding the remaining 50% in equal weights in the rest of the stock holding. This definition of concentration is, in many ways, similar to the percentage share approach [10]. The percentage share approach gauges the extent to which only a few firms account for a larger chunk of the industry's output [10]. This approach would, for example, present the percentage share of the market accounted for by the top 5, 10 or 20 companies within the industry [11].Employ Richard Roll's measure of concentration (RRC) in measuring portfolio concentration which was used initially for industrial concentration but has since been expanded to financial markets. Another measure, the Herfindahl-Hirschman Index (HHI), was first developed by economists Orris C. Herfindahl and Albert O. Hirschman to measure market competiton and the relative size of firms in comparison to the industry. Although this measure was initially developed for the industrial set-up, it has evolved over the years and hence its current adaptation in stock market concentration measures.
Although investors such as John Rockefeller and John Keynes have adopted the concentrated investment strategy, very few studies have been conducted to explore concentration in portfolios and its impact on risk. Most of these studies were conducted in smaller markets, compared to the New York Stock Exchange, as it has been documented that portfolios held in small markets such as the Johannesburg Stock Exchange tend to be highly concentrated in a few larger stocks [7,12].Study the different pension fund equity benchmarks on the JSE and observe that one of the many reasons that the All Share Index was not used as a benchmark by fund managers is due to the fact that it is highly concentrated in the larger stocks. Maritz noted that the All Share Index (ALSI) on the JSE was highly concentrated in larger (Resources) stocks and that the first share accounts for 16% of the index. The addition of the first eight shares gave a total of over 51% of the All Share Index on the JSE [7]. Also study the level of concentration of six indices on the JSE; namely ALSI Top-40, ALSI, ALSI Equally weighted, CAPI All Share, PENI All Share and the General Equity Average. They use the HHI and RRC. The HHI measure of the ALSI Top-40 is observed to be nearly twice as high as that of the General Equity Average, while the ALSI's HHI measure was approximately one and a half times higher than the average General Equity fund [7]. Submit that these differences depict the aversion that South African fund managers have to the high levels of concentration in the South African market. The level of expected total risk for each category was also consistent with the rankings based on the level of concentration.
The study is motivated by the absence of published work on portfolio concentration on the ZSE, along the lines of [7]. The question of market concentration in Zimbabwe is a tag of war between the conventional wisdom of diversification and market realities as dictated by competitive conditions in different sectors of the economy. Illiquid conditions in the Zimbabwe economy tend to favour market leaders in key sectors which are in turn expected to attract relatively more investor attention. As a result, we would expect the market to show significant signs of concentration in sectorial market leaders. On the other hand, conventional wisdom would dictate that investors should hold diversified portfolio in order to manage portfolio risk. It is against this background that this study is conducted to provide comprehensive measures of market concentration on the ZSE, and draw insight there from.
Two key questions of interest to investors are addressed in this paper. The first relates to whether the ZSE is concentrated. The second question pertains to the level of concentration in the market. Both questions are pertinent in that they provide indications of the level of risk aversion of the market, and to some extent the sensitivity of the market asymmetric shocks. Market concentration may also suggest possibility of cornering of the market, which should be a concern for regulators.
The paper extends the work of [7] to the Zimbabwe context, and makes a new contribution to the literature on the ZSE. Evidence on market concentration is interpreted in the context of findings on market efficiency on the ZSE by [13] to raise new questions regarding investor behaviour.

METHODOLOGY
The data was colleted from the ZSE website. The period of study is from 19 February 2009 to 31 December 2012. The daily stock prices and market capitalization quoted in United States dollars (USD) were collected. The study is limited to industrial index stocks and a total of 62 stocks are used in the analysis. The ZSE industrial index was assumed to be a good proxy for a market portfolio held by an investor attempting to effectively reduce exposure. The industrial index was used because it accounts for more than 90% of the total stocks listed on the ZSE. The methodology employed for the analysis was adopted from [7].

Measures of Concentration
We computed the market capitalisation weights for each stock as a ratio of the market capitalisation of each stock to the total market capitalisation (in USD) of the 62 stocks that constitute the Industrial index, that is:

Herfindahl-hirschman index
The HHI was used as the primary measure for concentration. The index was computed by squaring then summing the investment weights in a portfolio: Where i W is the investment weight in the i -th counter [7]. The index was adopted because it gives more weight to those counters that account for a larger proportion of the market share and at the same time, the index is mathematically tractable and simple to analyse. The HHI is also well known and has been used in other academic literature studying the ideology of concentration of portfolios. The HHI is interpreted based on [14]. An HHI below 0.15 refer to an un concentrated market. HHI between 0.15 and 0.25 imply a moderately concentrated market. A highly concentrated market is achieved by an HHI value above 0.25.

Roll's measure of concentration
This measure of concentration was calculated by: Where N is the number of counters in the portfolio [11]. The RRC has the desirable quality of measuring the deviation or departure of the portfolio under consideration from that of an equally weighted portfolio. Since it is a relative measure, interpretation of the results is very simple. It was however adopted as a second measure as it is less familiar than the HHI.

Calculation of Return
The return of the individual stocks was calculated using the formula: Where Pt is the stock price at time t and Pt 1 − is the stock price at time The model for calculation of return was used because it has both mathematical and modelling properties compared to using the simple return model [15]. This model is particularly suited for calculating return on individual stocks and not the return in portfolios.

Estimation of the Portfolio Risk
In attempting to determine the average number of stocks to be held in a concentrated portfolio for effective risk reduction, the risk of the simulated portfolios was calculated. We adopted the Markowitz definition of a risk as this was mathematically tractable and in line  [7]Analysed four scenarios with regard to the portfolio weighting structure. However, for the current research purposes, three different scenarios were considered. These variables under consideration in the three scenarios are namely; portfolio weighting structure (as a proxy for concentration in the portfolio), and correlation structure of the stocks. The three scenarios are given below:

Assessing How Various Levels of Concentration Affect the Resulting Risk of the Portfolio
Scenario 1: Equal weighting among the stocks in the portfolio, assuming no correlation between stocks. Scenario 2: Equal weighting among the stocks in the portfolio, assuming their historical correlation structure. Scenario 3: Market capitalisation weighting of stocks in the portfolio, assuming their historical correlation structure.
Scenario 1 is an oversimplified case that is not applicable under a real world scenario but rather is developed as a control and for comparison purposes. Scenario 2 is more realistic as the returns of stocks tend to correlate and is proxy of the famous Markowtiz portfolio diversification. This scenario is also indicative of studies conducted on large markets such as the New York Stock Exchange, and thus symbolises the adoption of international studies and assumptions on the local market, the ZSE. Scenario 3 is an adopted study of Markowitz diversification, adopted to fit the characteristics of the ZSE.
Since the methodology (as adopted by [7]) is based on 500 simulations for each portfolio size, we made use of both Microsoft Excel functions and VBA programming to achieve this. The advantages of using VBA programming include efficiency in automating repetitive tasks such as simulation, and it extends the range of functions offered by Microsoft Excel [17]. The methodology followed is summarised below: • k stocks are randomly selected from the universe of 62 stocks, starting from k=1.
• A portfolio of the k stocks is formed and depending on the scenario under consideration, the portfolio is based on either equal weights or market capitalisation weights. • The portfolio weights are rebased to sum to one.
• The risk (from Markowitz definition of risk) of the portfolio is computed.
• The process is repeated 500 times to obtain 500 estimates of the risk of the portfolio of size k. • The 500 estimates for portfolio size k is averaged and recorded.
• The portfolio size is increased by one stock to k+1 and the entire process repeated up to k=10 (We sampled these first 10 stocks because [6] found that maximum risk reduction effect by increasing the number of stocks is limited to 10 stocks in a portfolio). • Thereafter 5 stocks are added to the portfolio for sampling purposes and the exercise is completed when all 62 stocks have been sampled, i.e. k=62.

RESULTS AND DISCUSSION
The level of concentration of a portfolio created by mimicking the ZSE industrial index was calculated over the four-year period. The concentration of such a portfolio was evaluated as at the last trading day of March, June, September and December using the market capitalisation weights.
As noted from Fig. 1

Scenario 1: Equal Weights with No Correlation between Stocks
Fig. 2 below displays the graph for equally-weighted portfolios increasing in size from 1 stock in a portfolio to 10 stocks. Thereafter samples of portfolios after adding 5 stocks to the portfolios were taken and for each number of stocks the simulations were based on a sample of 500 samples which were averaged thereafter.
From Fig. 2 above, we observe that the average volatility (standard deviation) of single stocks in the Industrial Index on the ZSE is approximately 10% over the four year period. This gives an annualised volatility of 2.50% p.a. Moreover, if the 62 stocks constituting the Industrial index were equally weighted and uncorrelated, the portfolio risk could be reduced to below 2.00% (about 1.32%) by creating a portfolio constituting all 62 stocks as given by Fig. 2.

Scenario 2: Equally Weighted Portfolios Assuming Historical Correlation Structure
In the second scenario, we simulated portfolios assuming the stocks are equally weighted and also assuming the historical correlation structure of the stocks, estimated over the prior four-year period.
From Fig. 3, it is evident that introducing the historical correlation between stocks increases the risk of the portfolio. The risk increases from about 1.32% (for the 62 stock portfolio assuming no correlation between stocks) to 2.17% (for the 62 stock portfolio assuming the historical correlation between stocks). It can be noted that there is an increase of less than 1% in risk by the introduction of the correlation structure between stocks. The relatively small increase in risk between the two scenarios shows that stocks constituting the Industrial index on the ZSE have very low correlation between them. Therefore, by further identifying stocks that behave independently, the component of risk attributable to the correlation structure can be largely reduced.

Scenario 3: Market Capitalisation Weights-Historical Correlation Assumed
In scenario 3, we simulated portfolios assuming the market capitalisation weights of the portfolios rebased to sum up to 1. Stocks in the Industrial index were also assumed to exhibit their historical correlation structure.
As given in Fig. 4 above, the risk of a portfolio constituting 62 stocks on the ZSE with market capitalisation weights and the historical correlation structure is approximately 2%. As noted above, there is not a great difference among the risk of a 62 stock portfolio with equal weight and no correlation, equal weight and historical correlation; and one with market capitalisation weights and historical correlation. Therefore, although portfolios held on the ZSE are highly concentrated, the level of concentration does not affect the risk in the portfolio greatly. Therefore in order to effectively reduce the portfolio risk (market capitalisation and historical correlation structure) to a minimum possible, an investor has to hold more than 10 stocks in a portfolio of the ZSE Industrial Index up to a total of 62 stocks. The four scenarios presented provide a holding of an optimal portfolio size of between 20 to 25. This is consistent with [18] who recommend that a portfolio size of 20 to 30 approximate to a welldiversified portfolio in which unsystematic risk is reduced to a minimum.   show that portfolios mimicking the ZSE industrial index are highly unconcentrated with a concentration level of approximately 14% on the Herfindahl-Hirschman index. These results show that the portfolios held on the ZSE are generally highly unconcentrated. From the findings, it is noted that although the weighting structure of portfolios on the ZSE is rather skewed from that of equally weighted portfolios, the structure does not have a clear and direct impact on the risk of a portfolio. It was also noted that stocks on the ZSE have relatively low correlation due to the small difference between the risk of a portfolio consisting of 62 stocks on the ZSE with equal weighting and no correlation and that of a portfolio with 62 stocks with equal weighting and exhibiting the historical correlation structure. Most of the risk in the portfolio is diversified away in the first 10 stocks held. Further risk reduction can be achieved by holding more than 10 stocks in a portfolio up to the total of 62 stocks to achieve the lowest level of risk possible for a portfolio of the Industrial Index on the ZSE. An interesting point to note is that in order to achieve effective risk reduction, a portfolio must constitute an average optimal size of 20 to 25 stocks of the Industrial index on the ZSE. To our knowledge, this study is the first of its kind on the ZSE and can be expanded to investigate the relationship between concentration of a portfolio and the expected returns of that portfolio as well as factors that determine the level of concentration held by different investors, both institutional and individual.

Risk
We recommend that investors monitor concentration levels because higher concentration poses a higher risk in the event that the overall ZSE underperforms, they stand to lose out. Investors should also keep an optimal portfolio of an average size of 20 to 25 stocks to achieve an effective risk reduction. Pension funds are the major investors on the ZSE, as such they should come up with a risk appetite model that take cognisance of the concentration and such framework should come from the board of trustees who have the capacity to influence investment decisions on the stock market.