Modern Portfolio Theory and Capital Asset Pricing Model

Modern Portfolio Theory and Capital Asset Pricing Model Introduction The Capital Asset Pricing Model developed by William Sharpe has significant similarities with Harry Markowitz’s Portfolio theory. In fact, the later is rightly considered as the next logical step from the latter, with both based on similar foundations. There are also differences in how each model/theory is calculated, pertaining to risk considerations. This paper’s main objective is to identify these differences while highlighting the similarities as well to put things into perspective. The report will open with an overview of Markowitz’s portfolio theory and explain it further by means of describing the efficient frontier, the Capital Market Line, risk free asset and the Market Portfolio. The report will then switch its attention to the Capital Asset Pricing Model and explain it further through the Security Market Line. The report will then close by outlining the differences between the two with a view of answering the main objective. What will come through in this report is that Markowitz’s portfolio theory uses standard deviation as its risk measure and takes into account all risk in an efficient portfolio, while the Capital Asset Pricing Model uses the beta co-efficient to measure risk and takes into account both efficient and non-efficient portfolios – further more it measures the risks of individual assets within the portfolio. Modern Portfolio Theory Modern Portfolio Theory (MPT) was introduced by Harry Markowitz, way back in 1952. At a high level it proposes how rational investors use diversification to optimise their investment portfolios and give guidance on pricing risky assets. MPT assumes that investors are risk averse, i.e. given two assets A and B offering the same expected return, investors will opt for asset A if it is less risky. In effect, an investor who expects higher returns would need to accept more risk. The expected trade-off between risk and return depends on the individual’s level of risk aversion. The implication of this is a rational investor (a risk averse investor) will not invest in a portfolio if another one exists offering a better risk-return profile (Fabozzi Markowitz, 2002). For any given level of risk, investors will opt for portfolios with higher expected returns instead of those with lower returns. Another assumption under MPT is that investors are only interested in the expected return and the volatility of an investment, as measured by the mean and standard deviation respectively. Investors do not consider any other characteristics, for example, charges. In effect, based on the assumptions above, investors are concerned about efficient portfolios. To explain portfolio theory further, let us consider the formula for the expected return and risk of a portfolio under MPT. Suppose two assets A and B formed a portfolio in proportion (X) each, the expected return for that portfolio would be: R(p) = X(a)R(a) + X(b)R(b), where: R(p) = expected returns from portfolio R(a) = expected returns from asset A R(b) = expected returns from asset B The standard deviation or risk of that portfolio would be: SD(p) = √(XÂ ²aSDÂ ²a + XÂ ²bSDÂ ²b + 2XaXbRSDaSDb), where: SD(p) = standard deviation of expected returns of portfolio SDa = standard deviation of expected returns of asset A SDb = standard deviation of expected returns of asset B R = correlation coefficient between the expected returns of the two assets The efficient frontier Under MPT, Markowitz examined the efficient frontier curve. The efficient frontier curve gives a graphic presentation of a set of portfolios that offer the maximum rate of return for any given level of risk (McLaney, 2006). According to Markowitz, an efficient investor will opt for an optimum portfolio along the curve, based on their level of risk aversion and their perception of the risk and return relationship (Fabozzi Markowitz, 2002). Figure 1: Efficient Frontier Source: www.riskglossary.com The curve in the diagram above illustrates the efficient frontier. Portfolios on the curve are efficient – i.e. they offer maximum expected returns for any given level of risk and minimum risk for any given level of expected returns. The shaded region represents the acceptable level of investments when risk is compared against returns. For every point on the shaded region, there will be at least one portfolio that can be constructed and has a risk and return corresponding to that point (www.riskglossary.com) As aforementioned, each portfolio on the efficient frontier curve will have a higher rate of return for the same or lower level of risk or lower risk for an equal or better rate of return when compared with portfolios not on the frontier. It is important to note that the efficient frontier is really made up of portfolios rather than individual assets. This is because portfolios could be diversified, i.e. investors can hold assets which are imperfectly correlated (Fabozzi Markowitz, 2002). This will help to ensure that investors can reduce their risks associated with individual asses by holding other assets – a kind of set-off. The Capital Market Line The Capital Market Line (CML) is a set of risk return combinations that are available by combining the market portfolio with risk free borrowing and lending (www.lse.co.uk/financeglossary). The CML defines the relationship between risk and return for efficient portfolios of risky securities. It specifies the efficient set of portfolios can investor can obtain by combining the portfolio (which contains risk) with a risk free asset. The formula for CML is: E (r_c) = r(f) + SD(c)*[E(r_m)-r(f)]/SD(m) Where: E(r_c) = expected return on portfolio c R(f) = risk free rate SD(c ) = standard deviation of portfolio c E (r_m) = expected return on market portfolio SD(m) = standard deviation of market return The CML indicates that the expected return of an efficient portfolio is equal to the risk-free rate plus a risk premium. Both risk and return increase in a linearly along the CML. Figure 2: Capital Market Line Source: www.riskglossary.com In Figure 2 above, the CML is the line touching the efficient frontier curve. It passes through the risk free rate (assumed to be 5%). The point where the CML forms a tangent with the efficient frontier curve is the point called the super-efficient portfolio. The Risk free asset, Sharpe ratio and the Market Portfolio The risk free asset pays a risk free rate and has a zero variance in returns, e.g. government short-term securities. When combined with a portfolio of assets the change in return and risk is linear. The Sharpe Ratio is a measure of the additional return to be obtained about a risk free rate for a given portfolio compared with its corresponding risk. On the efficient frontier the portfolio with the highest Sharpe Ratio is known as the market portfolio. The CML is the result of a comparison between the market portfolio and the risk free asset. The CML surpasses the efficient frontier with the exception of the point of tangency. The Capital Asset Pricing Model While the CML focuses on the risk and return relationship for efficient portfolios, it would be useful to consider the relationship between expected return and risk for individual assets or securities. The Capital Asset Pricing Model (CAPM) would be used for this. CAPM is an extension of Markowitz’s Portfolio Theory or MPT. It introduces the notions of systematic and specific risks. Let us define each: Systematic risk – this is the risk associated with holding the market portfolio of assets Individual assets are affected by market movements Specific risk – this risk is unique to an individual asset and represents that portion of an asset’s return which has no correlation with market movements. CAPM assumes the following (McLaney, 2006, 199): Investors are risk averse and maximise expected utility of wealth The capital market is not dominated by any individual investors Investors are interested in only two features of a security, its expected returns and its variance or standard deviation There exists a risk free rate at which all investors may borrow or lend without limit at the same rate There is an absence of dealing charges, taxes and other imperfections All investors have identical perceptions of each security This lends credence to the assertion that CAPM follows a natural progression from MPT. The assumptions are identical with the main difference being how risks are categorised and treated. This will be explored in detail in a later section. Under CAPM, the market place will compensate an investor for taking a systematic risk but not a specific risk. The rationale for this is that specific risks can be avoided or minimised through diversification. The formula for CAPM is as follows: r = Rf + Beta x (RM-RF), where: r = expected return on an asset Rf = rate of risk free investment RM = return rate of the appropriate asset class Beta is the relative risk contribution of an individual security to the overall market portfolio. It measures the security risk relative to the market portfolio and ignores the specific risk. The beta equation is as follows: Cov (i,M)/(SDm)Â ², where: Cov (i,M) = covariance between market portfolio and security i (SDm)Â ² = variance of the market’s return The betas for all assets are measured in relation to the market portfolio beta which is 1. In effect, if individual beta is greater than 1, then individual asset has a higher risk than the market risk. If individual beta equals 1, then individual asset risk and market risk are the same. If individual beta is less than 1, then the risk of that individual asset is less than the market risk. The value of beta provides an idea of the level or size of the change in an asset’s return when a corresponding change in the returns of an overall portfolio is experienced (McLaney, 2006). Beta has come under criticism from academics and investors who do not appreciate the value of beta as an appropriate risk measure. However, this is somewhat challenged by actual performance of the betas of portfolios and mutual funds. These are regarded as stable and can be used to predict future betas. Security Market Line CAPM can be applied by using the Security Market Line (SML). SML is a graphical representation showing the linear relationship between systematic risk and expected rates of return for individual assets. In the case of the SML, risk is measured by beta. It plots the expected returns on the y axis and the risk as denoted by beta on the x axis. In other words, the SML expresses the linear relationship between the expected returns on a risky asset and its covariance with market returns. Its formula is: Figure 3: CAPM and SML The line in the diagram above is the SML. Differences relating to MPT (CML) and CAPM (SML) To explain the differences, it is useful to consider the relationships between risk and return in the perspective of CML and SML. CML compares the relationship from an MPT perspective, while SML does from a CAPM perspective. The main difference pertaining to MPT’s relationship with CAPM is pertaining to risk. Under Portfolio theory, CML gives an indication of expected returns in comparison with risk. Here the risk is measured in terms of standard deviation of returns. The rationale for this is CML represents the trade-off for efficient portfolios, i.e. the risk is all systematic risk (McLaney, 2006). The SML on the other hand, indicates the risk/return trade-off, using beta as the measure of risk. In this case, only the systematic risk element of the individual asset is taken into consideration. The reason why CML shows no individual security’s risk profile is because all individual securities have an element of specific risk, implying that they are inefficient. CML only looks at efficient portfolios. The table below summarises the main differences between CML and SML Table 1: Tabular difference between CML and SML Summary As has been shown above, CAPM has been developed along the lines of Markowitz’s Portfolio theory. They both use expected returns and risk as the investor’s main determinant of their investment decisions. They both assume that investors are risk averse and do not consider anything else other than risk and returns. However, there are some subtle differences which will now be summarised below: Under Portfolio theory, the CML measures risk by standard deviation or total risk. The SML measures risk by beta or systematic risk under CAPM – it ignores specific risks The CML graph is interested in providing information on efficient portfolios only. The SML graph on the other hand provides insight into both efficient and non-efficient portfolio and securities REFERENCES AND BIBLIOGRAPHY Books Bodie, et al (2006) ‘Investments’ (7th edition), McGraw-Hill/Irwin, London Elton, E et al (2003) ‘Modern Portfolio Theory and Investment Analysis’, Wiley, London Fabozzi, F. Markowitz, H. (2002) ‘Theory and Practice of Investment Management’, Wiley, London McLaney, E. (2006) ‘Business Finance – Theory and Practice’ (7th edition), Prentice Hall, London O’neill, W.J. (2002) ‘How to Make Money in Stocks’, (3rd edition), McGraw-Hill, London Internet Sources www.lse.co.uk www.riskglossary.com www.wikipedia.com