The fundamental principle of the mean-variance portfolio theory is the idea that investors can assess the risk‐return trade-off of investment opportunities centered on the variances, expected returns, and correlations of those assets. Harry Markowitz (1952) is credited for this theory and according to Burton (1998:3), “Markowitz came along and there was light”. Prior to Harry Markowitz’s 1952 treatise on portfolio selection, there existed no theory on portfolio construction, the whole process was replete with just folklore and rules of thumb until Markowitz first made risk the cornerstone of portfolio management (Bernstein, 2007).
The mean variance theory forms the foundations of modern portfolio theory (MPT) which is a finance theory which aims at the maximization of expected return of a portfolio for a given level of portfolio risk, or minimize portfolio risk over a given level of portfolio expected return, by selecting varying proportions of a number of assets. Harry Markowitz advanced MPT in a 1952 paper (Markowitz, 1952) and a book in 1959 (Markowitz, 1959). According to Miller (1999), portfolio selection, as envisaged and published by Markowitz (1952), could be construed as the” big bang” of modern finance. Markowitz methodically developed what is now known as the risk-return trade-off in investment decision-making and mathematically deriving portfolios selection rule for the first time. Markowitz (1952) provided solutions to the questions of how to quantify risk, how to minimize risk and maximize returns and how capital should be allocated among different asset classes. This methodology formed the basis for all the subsequent theories regarding how risk can be quantified, how financial markets operate and how capital should be allocated by corporations (Bernstein, 1993).
Page | 18
• Description of the MPT
Underlying Assumptions (Elton and Gruber, 1997, Francis and Kim, 2013):
• All investors are risk averse (investors will choose less risk over more risk for any given level of expected return). Again, for a given level of risk, investors prefer more return to less return. It is however important to note that to some extent, all investors are risk averse even though investors may differ in their level of risk aversion.
• No taxes or transactions costs.
• Expected returns for all assets are known.
• For all assets, the variances and covariances of returns are known.
• To determine optimal portfolios, investors need to know only the variances, covariances and expected returns. Skewness, kurtosis, and all other elements of a distribution can be ignored by investors.
The fundamental takeaway of the MPT is that investors should not focus on individual assets’
risk and return in a portfolio but rather, each asset should be selected based on its variations in returns relative to variations in the returns of all the other securities in the portfolio (Francis and Kim, 2013). MPT again describes how combining a number of risky securities can still lead to a low risk for the portfolio, as long as their price variations of the assets in the portfolio are not perfectly positively correlated.
Figure 2.1: Effects of Portfolio Diversification
Source: Adopted from Meir Statman (1987), cited in Bodie, Kane and Marcus (2014: 207)
Page | 19 Figure 2.1 depicts the effect of portfolio diversification. The figure shows that portfolio risk actually reduces with diversification although the power of diversification is limited by the systematic risk.
The process of portfolio construction can be generalized to the situation of many risky assets and a risk-free security. This process can be broken down into three parts (Bodie et al., 2014).
First, there is the identification of the combination of risk and return available from the group of risky assets. Second in the process is identifying the portfolio weights that make up the steepest capital asset line (CAL)4 representing the optimal portfolio of risky assets. The final process involves choosing a complete portfolio through a combination of the optimal portfolio with the risk-free security.
The first step is to determine the risk–return opportunities available to the investor. The investor first ascertains the risk-return options available as given by the minimum-variance frontier of risky securities which is a graph of the least possible variance that can be achieved for a given level of portfolio expected return. With the input data for variances, covariances and expected returns available, the minimum-variance portfolio can be calculated for any desirable expected return. Figure 2.2 depicts the minimum variance frontier in a two‐asset scenario where, the only assets to be considered are Stocks A and B and the three‐asset scenario which includes Stock C. The benefits of diversification increase as the correlation between Stock C is not perfectly positive with either Stock A or Stock B.
Figure 2.2: How Increasing the Number of Assets Affect Portfolio Diversification
Source: Level II CFA Study Guide (2015: V5-137)
4 A graph depicting all feasible combinations of risk and accompanying return of a risk-free and risky asset.
Page | 20 The total risk, can be separated into systematic risk which is the covariance of the asset’s return with that of the return of the market portfolio and non-systematic risk. Systematic risk is the only relevant risk for decision making purposes. Figure 2.4 illustrates how diversification minimizes non-systematic risk for portfolios. The return variance of the portfolio is the total risk for the portfolio which is the systematic risk and the non-systematic risk. The horizontal axis illustrates the number of assets held in the portfolio.
Extensive diversification however cannot eliminate risk in situations where common elements of risk affect all firms. In Figure 2.3, the standard deviation of the portfolio falls with an increase in the number of securities, but does not be reduce to zero. Risk can be categorised as market risk or systematic risk and non-systematic risk. The market risk is the risk that lingers even after diversifying extensively, this risk is attributable to the market and is also is also called the systematic or non-diversifiable risk. On the other hand, certain risks can be eliminated through diversification these risks are firm-specific risks and are referred to as diversifiable risk or unsystematic risk. There are empirical studies that support this analysis (Bodie, Kane and Marcus, 2014).
Figure 2.3: The Elements of Portfolio Risk
Source: Bodie, Kane and Marcus (2014: 207)
As the number of asset held in the portfolio increases, the level of non-systematic risk is gradually eliminated or diversified away. This is supported by studies of different classes of
Page | 21 assets. A portfolio size of 20 randomly chosen assets for example, can completely remove any non-systematic risk and thereby leaving only the systematic risk (Drake and Fabozzi, 2010).
In summary, one can improve a portfolio’s risk‐return trade-off by increasing the number of investable securities, and for any given level of return, the minimum variance portfolio will depend on:
1. the individual assets’ expected returns, 2. the variance of each asset
3. the correlations among the returns of the asset in the portfolio, and 4. the number of assets in the portfolio.
Problems with the CAPM and Mean-Variance Theory
The CAPM and the Mean-Variance Theory are two pillars of modern finance, however, these models have been strongly criticized on both theoretically and empirically. The critique theoretically is that expected utility is fallacious, and some of the other assumptions underlying the models are invalid, and, paradoxical choices may be made if one adheres to the Mean- Variance Theory (Levy, 2010). Empirically, the models are further criticized because rates of returns of assets are not normally distributed and the CAPM has only mediocre explanatory power (Levy, 2010).
Allais (1953) shows that when using Expected Utility Theory in making decisions between pairs of alternatives, especially for those involving small probabilities, there may be some evidence of paradoxes (popularly known as the Allais paradox5) within Expected Utility Theory casting doubts on the validity of the Expected Utility Theory which among other assumptions, forms the foundation of the Mean-Variance Theory and CAPM. This paradox inspired the idea of using decision weights. Kahneman and Tversky (1979) suggest Prospect Theory as a substitute theory to the Expected Utility Theory. They posit that investors misrepresent probabilities, make choices based on variations in wealth, are prone to loss aversion and tend to maximize the anticipation of an S-shaped value function containing a risk-
5 Maurice Allais published a paper on a survey he had carry out with a hypothetical game. In the survey, subjects with in-depth knowledge of probability theory of probability and construed to behave rationally consistently violated the Expected Utility Theory. The game and its results are popularly referred to as the Allais Paradox.
Page | 22 seeking segment. Normative economic theories could be more appropriate for agents of artificial sort than for human agents, since Artificial Intelligence better adhere to idealized assumptions of rationality than people (Parkes and Wellman, 2015).
Levy, Giorgi and Hens (2012) however, argue that although the Cumulative Prospect Theory and Prospect Theory conflict with the Expected Utility Theory, and violates some of the underlying assumptions of the CAPM, the CAPM’s Security Market Line (SML) is intact in the Cumulative Prospect Theory framework and therefore, the CAPM is also intact in Cumulative Prospect Theory framework.
Baumol (1963), Leshno and Levy (2002) Levy, Leshno and Leibovich (2008), and Levy (2012) opine that the Mean Variance Theory is sufficient but it is not a necessary rule for investment decisions, and therefore not an optimal rule, precipitating an elimination of a segment, or some segments, of the efficient frontier away from the efficient set. The market portfolio consequently may be also removed from the efficient set, leading to a vague conclusion on the CAPM.