Harry Markowitz (1952) laid the foundation of Modern Portfolio Theory (MPT) with the writing of his doctoral dissertation in statistics. According to Mangram (2013) Markowitz’s model introduced an original description of the effect of portfolio diversification by increasing the number of securities within a portfolio and the relationships of their covariances. For his pioneering work Markowitz shared a Nobel Prize with William Sharpe for his MPT contributions to the fields of economics and corporate finance. MPT is comprised of Markowitz’s portfolio selection theory and CAPM which was contributed by William Sharpe (1964). MPT is an investment framework for the selection and construction of investment portfolios based on the maximisation of expected returns of the portfolio and minimisation of the portfolios risk (Fabozzi, Gupta & Markowitz 2002).
The portfolio approach to investing plays a central role within the modern finance theory. One of the key implications and most quoted result from this theory is that increasing the size of one’s portfolio helps to reduce idiosyncratic risk, and it is thus possible to achieve superior risk-adjusted performance through the use of a portfolio technique to investing(Darly 2012). The selection of an optimal portfolio is problem of both theoretical and practical interest to both scholars and practitioners. Markowitz (1952) formulated the portfolio problem as a choice of the mean and variance of a portfolio of assets.
Markowitz’s portfolio theory has set up a clear quantitative framework for the selection of a portfolio, the process of portfolio selection has been summarised as an allocation of resources so as to tradeoff expected return and risk. By using statistical measurements of expected return and variance of return (variance being equated to risk), Markowitz characterised the benefits and risk connected with an investment (Oriakhi 2011).
Markowitz’s approach to investing formulates and solves a parametric quadratic program and has become a central decision model of many portfolio analytic and planning systems in constructing efficient frontiers, "efficient frontiers" can be viewed as the set of Pareto optimal (expected return, variance of return) combinations under conditions of uncertainty. The attractiveness of this simplistic unconstrained risk return model is that it is capable of being extended to capture market realisms such as cardinality constraints (a fixed number of assets) and transaction cost (fees associated with trading). This research will utilize GRG2 code non-linear solver to build an optimal portfolio.
Harry Markowitz (1952) laid the foundation of MPT with the writing of his doctoral dissertation in statistics. According to Mangram (2013) Markowitz’s model introduced a novel description of the impact of diversification on Portfolio risk for his work Markowitz shared a Nobel prize with William Sharpe for his MPT contributions to the fields of economics and corporate finance.
The aim of this research is to apply the Portfolio Optimisation Theory investment approach to the NSE. This theory postulates that it is possible to reduce portfolio risk through diversification.
2.2 Modern Portfolio Theory (MPT)
Harry Markowitz (1952) developed a theory of "portfolio choice," which allows investors to analyse risk relative to their expected return. For his work Markowitz, won a 1990 Nobel Memorial Prize in Economic Sciences which he shared with William Sharpe and Merton Miller.
Markowitz’s theory is what has developed to be referred as the Modern Portfolio Theory (MPT). MPT is a theory of investment in which an attempt is made to maximize portfolio expected return for a given amount of portfolio risk, or equally minimize risk for a given level of expected return, by carefully selecting the proportions of various assets to include in the portfolio (Rani 2012). According to Mangram (2013) MPT is comprised of Markowitz’s portfolio selection theory and CAPM which was contributed by William Sharpe (1964). MPT is an investment framework for selecting and constructing of investment portfolios based on the maximisation of expected returns and the minimisation risk of a portfolio (Fabozzi, 2002)
The Modern Portfolio Theory is an improvement of the traditional investment models which emphasized analysis of individual investment assets, and is a vital advance in the mathematical modelling of finance. The theory encourages asset diversification to prevaricate against market risk as well as risk that is unique to a specific company (idiosyncratic risk). In MPT risk can be measured using various mathematical models and can be reduced through diversification, whose major aim is to put together a weighted collection of assets which have lower risk factors than the individual asset.
The Markowitz’s theory of portfolio selection is a normative theory Farbozzi et al (2002) "defines a normative theory as one that describes a standard or norm behaviour that investors should pursue in construction of a portfolio" equally Sharpe’s Asset Pricing Theory is regarded as a positive theory one which hypothesizes how investors actually behave as opposed to how they should behave. MPT is an investment decision approach that helps investors to classify, estimate, and control both the kind and the amount of expected return and risk; it is sometimes referred to as Portfolio Management Theory. The relationship and quantification between risk and return and the assumption that investors must be compensated for assuming risk is crucial to the portfolio theory.
The MPT formulates mathematically the concept of diversification in investing, with the endeavour of choosing a collection of investment assets that collectively have a lower risk than any individual asset. This is made possible by the fact that different types of assets often change in value in opposite ways. But diversification can lower risk even though assets' returns are not negatively correlated, even if they are positively correlated (Rani 2012).
MPT assumes that investors are rational and that markets are efficient. The fundamental concept and vital improvement upon traditional investment approaches of MPT is that assets in an investment portfolio should not be chosen individually, each on their own merits. Rather, it is very critical to regard as to how each asset changes price in comparison to how every other asset in the portfolio changes in price (Machuki 2014). Investing is a trade-off between risk and expected return. As a rule, assets with higher expected returns are riskier for a given amount of risk, the MPT shows how an investor can select a portfolio with the highest possible expected return. Or, for a given expected return, the MPT shows an investor how to select a portfolio with the lowest possible risk (Gruber 2011).
2.2.1 Concept of Expected Return & Risk
According to Gupta (2001) return and risk are the two most important features of any investment, it is imperative for an investor to know the sources of return and risk. The primary task of security analysis is to identify, evaluate/analyse the factors contributing to both return and risk. The results of this security analysis are the crucial components for portfolio construction, revision and evaluation as well as for setting long term investment strategies and policies.
2.2.1.1 Expected Return
Return is the basic motivating force and the principle reward for the investor in any investment process. Returns are classified into two, i) realized returns (this is the return which has already been earned) ii) expected returns (Ross et al (2002) define expected return as the return which the investor anticipates to earn over some future investment period). In order for an investor to predict expected return for an individual security or portfolio, the historical performance of returns is examined. When selecting a portfolio the calculation of expected return is the first step, expected return can also be referred to as the mean or average return and when calculating the return of a portfolio of securities the weighted average of expected individual returns is considered. Ross et al (2002)
2.2.1.2 Portfolio Risk
In investment analysis risk can be defined as " the unpredictability of future returns from investment" Elton et al(2011) has defined risk "as the possibility that the actual return may not be the same as the expected return" this implies that risk refers to the chance that the actual return from an investment will differ from an expected outcome. Risk can be viewed as a change of variation in return, investments having greater chances of variations are considered more risky than those with lesser chances of variations.
There is a difference between risk and uncertainty; risk has the possibility of the event either happening or not and this possibility can be quantified and measured. In a risk situation probabilities can be assigned to an event based on facts and figures available regarding the decision. Uncertainty on the other hand is a situation where the possibility of the event either happening or not happening cannot be measured and probabilities cannot be assigned. Portfolio risk therefore is the possibility that an investment portfolio may not achieve its objectives (Elton et al).
Many factors contribute to portfolio risk and while an investor can be able to minimise these factors he is not able to fully eliminate them. In investment risk can be classified into two:
Un-diversifiable risk also known as systematic risk or market risk and diversifiable risk also known as unsystematic or idiosyncratic risk. Modern portfolio theory postulates that expected return from a portfolio depends heavily on the level of risk on it.
2.2.1.3 Measurement of Risk
2.2.1.3.1 Variance
There are many ways to determine the volatility (risk) of any security; however the most common measure is variance. Variance is a measure of squared deviations of a securities return from its expected return; it is the average squared difference between the actual return and the average return. When we are talking about the portfolio variance measures the volatility of the group of securities a larger variance means that there is greater volatility.
Frantz &Payne (2009) asserts that "when many securities are held together in a portfolio, assets decreasing in value are often offset by portfolio assets increasing in value thereby minimising risk. Therefore, the total variance of a portfolio of assets is always lower than a simple weighted average of the individual asset variances." According to Frantz ibid increasing the number of portfolio assets improves its efficient frontier.
2.2.1.3.2Standard Deviation
Standard deviation is a very common measure of volatility (risk). The portfolio selection model assumes that investors make their investment decisions based on expected returns and risk. Standard deviation as a measure of return is calculated by squaring the variance i.e. it’s the squared deviations (variability) of securities.
2.2.1.3.3 Covariance of Return
When measuring the relationship between returns for one stock and returns on another it’s necessary to measure their covariance or correlation. Covariance is the expected value of the product of two deviations; it’s a measure of how returns on assets move together (Elton et al 2011). When the positive and negative deviations occur at similar times the covariance is a large positive number, if they have the positive and negative deviations at diverse times then the covariance is negative, if the positive and negative deviations are unrelated it tends to be zero(ibid).
2.2.1.3.4Correlation Coefficient of Returns
Correlation measures the relationship between variables "MPT tries to analyse the interrelationship between different investments, it uses statistical measures such as correlation to quantify the diversification effect on portfolio performance." (Mangram 2013) correlation coefficient is the covariance of two assets or more divided by the product of the standard deviation of each asset. "Dividing the correlation by the product of the standard deviations does not change the properties of the covariance it simply scales it to have values between -1 and +1(Elton et al 2011)
2.2.2 Diversification
Diversification concept is a cornerstone concept of the MPT, diversification concept relates to a well-known adage "don’t put all your eggs in one basket." If the basket is dropped all the eggs are broken; if placed in more than one basket the risk that all the eggs will be broken is dramatically reduced. An investor can achieve diversification by investing in different stocks, asset classes’ i.e. bonds, real estate and commodities such as gold or oil. Diversification effect refers to the relationship between correlations and portfolios. When correlations between assets are imperfect these effect results. Markowitz argued that diversification cannot eliminate all risk because investors are affronted with two types of risk: systematic and unsystematic risk.
According to many analysts unsystematic risk is the type of risk which can be reduced or eliminated because the basis for these types of risk are events which are unique to a particular company or industry or even country. Systemic risk on the other hand cannot be diversified because be diversified because it comes from external factors which affect all or majority of the companies. Analysts have noted that while a diversified portfolio can improve returns and significantly reduce unsystematic risk, no amount of diversification can eliminate all risk because there are simply too many variables. Furthermore, no amount of diversification can do away with or reduce systematic risk which affects all or most companies and markets at the same time
2.2.3 Assumptions of the Modern Portfolio Theory
There are many assumptions about markets and investors in the modern portfolio theory. Some of these assumptions are explicit in MPT equations; for example the use of Normal distributions to model returns. Others are implicit, such as the neglect of transaction fees and taxes. None of these assumptions are entirely true, and each of them compromises the theory to a little degree: Efficient Market Hypothesis, the chief among the MPT’s assumptions is the efficient market hypothesis (EMH). This hypothesis claims that financial markets are "information ally efficient". In other words, an investor cannot consistently achieve returns in excess of average market returns on a risk-adjusted basis, given the information available at the time that the investment is made. There are three major adaptations of the EMH hypothesis: "weak", "semi-strong", and "strong". The weak EMH states that prices of traded assets (for example, stocks, bonds, or property) already show all the past publicly available information. The semi-strong EMH asserts that prices reflect all publicly available information and that prices change to reflect any new public information. The strong EMH additionally claims that prices instantly reflect even hidden or "insider" information. Research has shown some evidence for and against the weak and semi-strong EMHs, while the evidence against the strong EMH is very powerful. Extensive researches done by analysts have revealed signs of inefficiency in financial markets. Critics have blamed the belief in rational markets for much of the financial crises that occur in financial markets: the second assumption is that asset returns are (jointly) normally distributed random variables, despite this assumption, evidence from frequent many observations show that returns in equity and other markets are not normally distributed. Research has evidenced that large swings (3 to 6 standard deviations from the mean) occur in the market far more frequently than the normal distribution assumption would predict. While the model can also be justified by assuming any return distribution which is jointly elliptical, all the joint elliptical distributions are symmetrical whereas asset returns empirically are not: another assumption is that correlations between assets are fixed and constant forever. Correlations of stocks depend on relationships between the underlying assets that consist the portfolio, and change when these relationships change. In reality during financial crisis, all assets tend to become positively correlated, because they all move (down) together. That is saying, the MPT fails to function when investors are most in need of shielding from risk: another assumption is that all investors aim to maximize economic utility, the investors aim is to maximize their economic utility in order to make as much money as possible, regardless of any other considerations. This is one of the central assumptions of the efficient market hypothesis, upon which the MPT is built: the MPT also assumes that investors have an precise conception of possible returns, the probability beliefs of investors match the true distribution of returns. A different possibility is that of investors' expectations being biased, therefore causing market prices to be information ally inefficient: another assumption of the theory is that there are no taxes or transaction costs, in reality real financial products are subject both to taxes and transaction costs, and taking these into account will alter the composition of the optimum portfolio: the theory also assumes that all investors are price takers and that their actions do not influence prices of securities. In reality, large sales or purchases of individual assets can shift market prices for that asset and others (using cross-elasticity of demand). An investor may not even be able to assemble the theoretically optimal portfolio if the market moves too much while they are buying the required securities: Another assumption is that all securities can be divided into pieces of any size in reality, fractional shares cannot usually be bought or sold, and some assets have minimum order sizes. The above assumptions can be relaxed with more complicated versions of the MPT that take into account a more sophisticated model of the world (for example one with non-normal distributions, taxes and transaction costs) but all mathematical models of finance still rely on many impracticable assumptions as stated above.
2.3 Portfolio Optimisation Theory
Portfolio theory postulates how a risk averse investor can optimally allocate his investments between different assets. Since the seminal work of Markowitz (1952), many studies have been published using his Mean variance optimization (MVO) as a quantitative tool which allows an investor to make allocations in investments by considering the trade-off between risk and return. These mathematical models of investor attitudes and asset return dynamics aid in the portfolio selection process.
Mathematical methods of optimization have been successfully used in financial portfolio management. The Mean-Variance method propagated by Harry Markowitz is one of the most widely used techniques of portfolio selection. In this model the risk of the portfolio is modelled by the quadratic variance of its daily returns. The Mean-Variance method seeks to minimize this quadratic objective function subject to constraints concerning the permissible structure of the portfolio and a requirement that the expected returns exceeds a pre-specified level. This leads to a constrained quadratic optimization problem which has been extensively studied in the mathematical literature. Efficient computational algorithms have been developed and coded to solve even very large scale quadratic optimization problems (Engels 2004).
The key ingredients of the Mean-Variance portfolio model are the covariance matrix and the mean vector of the daily returns of the securities in the portfolio. Portfolio theory makes an assumption that these quantities are known. But in reality these values have to be estimated from observed market data. In order to have good quality statistical estimates for the large covariance and mean return structures one needs a very large sample of historical returns data. But because markets keep on changing over the time, only a limited set of recent historical data is relevant for the purposes of planning for future investments. The estimated covariance matrix and the mean return vectors are the key inputs of the optimization program (Denteh 2004).
2.3.1. Efficient Frontier
The efficient frontier is one of the concepts of MPT it is a curve that shows all efficient portfolios in a risk-return framework. An efficient portfolio is defined as the portfolio that maximizes the expected return for a given amount of risk (standard deviation), or the portfolios that minimises the risk subject to a given expected return (Engels 2011). Portfolios lying on the efficient frontier represent the best possible combination for expected return and portfolio risk (Elton 2011). One of the major assumptions of MPT is that an investor is risk averse and because an investor is risk averse he will always invest in an efficient portfolio. One of the major implications of the Markowitz’s efficient frontier theory is its presumption of the benefits of diversification. The efficient frontier theory implies that rational investors seek out portfolios that generate the largest possible returns with the least amount of risk.
2.3.1.1 Minimum Variance Portfolio
According to Engels (2011) when an investor desires to invest in a portfolio with the least amount of risk and he does not care about his expected return he will always invest in an efficient frontier, and will therefore chose the portfolio on the efficient frontier with the minimum standard deviation at this point also the variance is minimal that is why it’s called the minimum variance portfolio. This minimum variance portfolio can be calculated by minimising the variance subject to the necessary constraint that an investor can only invest the amount of capital he has. This is called the budget constraint. The minimization problem is
Min{θ^T ∑▒〖θ ↓ 1 ̅^T 〗 θ=C_0 }
2.3.1.2Sharpe Portfolio
If an investor has other preferences other than taking the least possible amount of risk, and thereby investing in the minimum variance portfolio, another option open to him is to invest in the portfolio with the maximum Sharpe ratio. The Sharpe ratio is defined as the return-risk ratio Sharpe ratio=mean/(standard deviation)
It represents the expected return per unit of risk, therefore, the portfolio with the maximum Sharpe ratio gives the highest expected return per unit of risk, and is thus the most risk efficient portfolio.
2.3.1.3 Optimal Portfolio
We have looked at two portfolios that an investor can prefer to invest in, the minimum variance portfolio if he desires minimum amount of risk and the tangency portfolio if his objective is to maximise the portfolio’s Sharpe ratio. However, Markowitz’s portfolio selection theory assumes a different kind of preference for the investor. It postulates that an investor’s goal is to maximise his utility function. Utility is a function of expected return, variance and a new parameter γ. γ is called the parameter of absolute risk aversion, which is a measure of investors risk averseness. It is different for each investor and even for the same investor it can change through time. The greater the γ the more risk averse an investor is. The parameter of risk aversion is assumed to be positive because MPT assumes that all investors are risk averse. Therefore, the optimal portfolio is a combination of the minimum variance portfolio and the Sharpe portfolio, when a proportion of the investor’s capital is invested in the minimum variance portfolio and the other proportion in the tangency portfolio
2.4 Empirical Review
2.4.1 Return, Risk, Size and Portfolio Optimisation
There have been several academic papers seeking to give an answer to the question of how many stocks constitute a diversified portfolio. A study on diversification in the Malaysian Stock Market conducted by Zulkifli et al., (2010), after examining the Malaysian stock market concluded that 15 stocks are enough to diversify away a satisfiable amount of diversifiable risk. They further argued that well diversified portfolios contain weakly related assets, where the degree of correlation between them is low and therefore, the revenue is low.
Statman (2004), Benjelloun and Siddiqi (2006) using equally weighted portfolios estimated the size of a well-diversified portfolio, they contrasted the benefits of diversification with its costs. They concluded that as portfolio size increases risk decreases but management costs also increase. Both papers’ argument were built around the idea that it is worth increasing portfolio size as long as marginal benefit of increased diversification exceeds its marginal cost. They both reached the conclusion that a diversified portfolio consists of hundreds of stocks.
According to the Barclays Global Investors (2006) correlation can be used to measure the magnitude of the relationship between any two investments. Ragunathan and Mitchell (1997) analysed correlation between national stock market returns as a basis for portfolio risk reduction which shows that portfolios with negative or a low positive correlation would help in reducing risks
A study conducted by Kitur et al (2015) at the NSE concludes that diversification results in risk reduction benefits and that an optimal portfolio size for an investor in the NSE is between 18 to 22 securities. They found out that portfolio risk decreased as the number of securities in the portfolio increased.
Gupta, et al (2001), using the random approach found out that the size of a well-diversified portfolio for the borrowing investor is 30 while that for the lending investor is 50. Suquaier and Ziyud (2011), using equally weighted portfolios, conclude that most of the diversification benefits can be obtained by holding 15-16 stocks. They recommend that diversification is important for all equity investors. Diversification should be increased as long as the marginal benefits and risk reduction exceed marginal costs and transaction costs.
According to Byrne and Lee (2000) the growth of the size of a portfolio has a direct impact on its risk. Studies prove that risk of naïve portfolio strongly reduces when including 20 to 40 assets. Further increase of the number of assets makes the reduction of risk negligible. Tang (2004) examined naïve (equally weight) diversification and analytically showed that for an infinite population of stocks, a portfolio size of 20 is required to eliminate 95% of the diversifiable risk on average. However, an addition of 80 stocks (the size of 100) is required to eliminate an extra 4 %( 99% total) of diversifiable risk. Solniki (2007), after examining the U.S stock market, indicates that the sufficient number of stocks in a portfolio in the U.S. is 20. The results of these scientists’ previous findings are far different, after performing; the six years weekly return analysis in eight different countries.
Using mutual funds quarterly returns O’Neil (1997) investigates portfolio diversification. He finds that when time series standard deviation is used as a measure of risk, one fund is all that is needed to achieve diversification. However, when O’Neil uses standard deviation of terminal wealth, a measure of cross sectional risk, he finds that a large number of funds are needed to achieve diversification.
Campbell, Lettau, Malkiel and Xu (2001) traced changes in the volatility of individual stocks, industries and the overall market from 1962 to 1997. Campbell et al. come to the following conclusions: the volatility of individual stocks has increased over time and the correlation among stocks returns has decreased over time, the volatility of the market and most of the industries have not changed, and the number of stocks necessary to achieve diversification has increased. They concluded that 20 stocks are necessary to achieve a large percentage of diversification benefits.
Gilmore and McManus (2002) empirically tested the correlation between stock securities, they concluded that because of a low short term correlation between the three central European markets and the US, benefits from international diversification were available to the US investors; they also showed that stock returns in emerging countries were positively correlated.
K.M. Kiani(2011) in his study on emerging markets and using value at risk (VaR) as a measure of risk concluded that portfolio diversification across markets, industry and country reduces an investors exposure to risk. However, because of the repeated episodes of financial crises and the global financial turmoil a number of measures can be taken to enhance investors’ confidence for investing in the emerging markets particularly with reference to the market risk together with the currency and local market risks that need to be priced in the returns from the emerging markets
Daryl and Shawn (2012) also used alternative weighting strategies, to investigate the benefits of diversification. While not mentioning the number of stocks needed in order to have an optimum portfolio, they concluded that the absolute benefits of diversification for a market capitalisation weighted portfolio are smaller than the absolute benefits of diversification for an equally weighted portfolio but investors should not just look at absolute benefits but achievable benefits as well.
Evans and Archer (2010), in a study conducted in the US market, suggested that as few as 20 securities are adequate to have a well-diversified portfolio. They further concluded that for a randomly selected and equally weighted portfolio there is little risk reduction to be obtained from expanding a portfolio beyond 10 to 15 securities. Elton and Gruber (1997) studied and discussed the previous literature and developed an exact expression formula for determining the effect of diversification on risk. By using this approximation they found that total risk goes down at lower rate as more securities are added. They recommend that 15 stocks would appear to be significant for good diversification benefits.
A study by Boscaljion et al (2005), suggested that a randomly selected portfolio of 30 stocks or less selected from industry leaders and equally weighted stocks could provide the same level of diversification as the S&P 500 Index. This study analysed the return of 500 stock portfolio traded in NYSE and S&P index. The study concluded that a well-diversified portfolio must contain at least 30 stocks.