3 Theoretical Background
3.2 Financial Theory
3.2.1 Modern Portfolio Theory
In 1952, Markowitz developed a portfolio construction theory in which investors would be compensated with higher returns for bearing higher risk. Markowitz proved through his research that the correlation between the assets affects the portfolio variance, and thereby the investor could lower his risk by diversification of the assets. This theory is known as the Mean-Variance Analysis and is central in Capital Asset Pricing Theory (Guerard, 2010).
Capital Asset Pricing Theory
The capital asset pricing theory describes the relationship between risk and expected returns for assets, particularly equities. Further, the capital allocation line is widely used throughout finance when optimizing the capital allocation in investment portfolios. The capital allocation combines the risk and return of the risky assets with a risk-free asset to reduce risk. The capital allocation line is expressed as:
𝐸(𝑟𝑐) = 𝑟𝑓+𝐸(𝑟𝑝) − 𝑟𝑓
𝜎𝑝 𝜎𝑐 (3.1)
where 𝑟𝑓 is the risk-free rate of return and 𝐸(𝑟𝑝) is the expected return on the risky portfolio.
The model also incorporates risk through the standard deviations of the complete portfolio, 𝜎𝑐, and the risky portfolio, 𝜎𝑝. The last term is the Sharpe ratio multiplied by the standard deviation of the complete portfolio. The Sharpe ratio is a proper measurement of portfolio performance, as it measures the performance relative to risk. The higher the Sharpe ratio,
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the higher expected return corresponding to any level of volatility (Corporate Finance Institute, 2018).
In simple portfolio theory, an investor has the ability to invest in two different assets at time 𝑡. The two assets have a risk and return from 𝑡 to 𝑡𝑡+1 and are noted as assets 𝑟𝑒 and 𝑟𝑏. According to the simple portfolio theory, the expected return on the investment portfolio is expressed as the weighted average of the expected return of the two assets:
𝐸(𝑟𝑝) = 𝑤𝑒∗ 𝐸(𝑟𝑒) + 𝑤𝑏∗ 𝐸(𝑟𝑏) (3.2)
The corresponding standard deviation is shown in expression (3.3). The square root of three fractions estimates the standard deviation. The first two, is the weighted average of the variance, and the last fraction incorporates the covariance:
𝜎𝑝= √[𝑤𝑒2∗ 𝜎𝑒2+ 𝑤𝑏2∗ 𝜎𝑏2+ 2 ∗ 𝑤𝑒𝑤𝑏𝐶𝑜𝑣(𝑟𝑒, 𝑟𝑏)] (3.3)
where the portfolio weights are the composition of a particular holding in equities (𝑤𝑒) or bonds (𝑤𝑏). The last fraction incorporates the covariance between the two assets 𝑜𝑣(𝑟𝑒, 𝑟𝑏).3
Diversification
The covariance is a measure of the joint variability between two random variables. The higher the values of one variable mainly correspond to the higher values with the other variable if the covariance is positive. If the covariance is negative, the higher values of one variable will correspond to lower values with the other variable, i.e., opposite behavior. The covariance is expressed as:
3 Bodie, Kane and Marcus (2014)
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𝐶𝑜𝑣(𝑟𝑒, 𝑟𝑏) = 𝜌𝑒,𝑏∗ 𝜎𝑒∗ 𝜎𝑏 (3.4)
where 𝑝𝑒,𝑏 is the correlation between assets. The correlation must fall within the range of 1 to 1. A coefficient of 1 indicates the assets vary perfectly together, while a correlation of -1 denotes that the two assets vary the complete opposite of each other (Bodie, Kane &
Marcus, 2014). More assets in the portfolio results in diversification benefits as additional assets decrease non-systematic risk. The non-systematic risk is a measure of firm-specific risk that is associated with a single firm. An example of firm-specific risk is a sudden strike by the employees, or a new governmental regulation affecting a particular group of firms.
In other words, diversification in asset allocation is essential and reduces an investor’s overall risk (Wiafe, Basu & Chen, 2015).4
Figure 3-1: Illustration of diversification benefits from non-systematic risk.
Figure 3-1 illustrates the principles of how diversification reduces the overall risk in asset allocation. When the correlation coefficient equals to 1, the portfolio will not benefit from diversification, as can be seen from the straight line. In this scenario, the standard deviation is only a weighted average between the two assets. However, as the correlation coefficient decreases from 1 and approaches -1, the portfolio benefits from diversification. As seen, the lower the correlation coefficient reduces the risk for a given level of return. In the event of a correlation of -1, the investment portfolio will have the least amount of risk for a given level of return.
4 Bodie, Kane and Marcus (2014) 2%
3%
4%
5%
6%
7%
0% 5% 10% 15% 20% 25%
EXPECTED RETURN
STANDARD DEVIATION
Correlation: 1 Correlation: 0,5 Correlation: 0 Correlation: -1
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The Efficient Frontier
The modern portfolio theory relies on the claim that investors choose portfolios that generates the lowest amount of risk for a defined rate of return. In other words, investors seek out portfolios on the efficient frontier. Portfolios that lies below the efficient frontier are suboptimal, as they do not provide enough return for the defined level of risk. Figure 3-2 illustrates the linking between the suboptimal portfolios and the efficient frontier. As can be seen, the portfolios that cluster below or to the right of the efficient frontier are suboptimal because they have a higher level of risk given the return of the investment (Bodie, Kane & Marcus, 2014).
Figure 3-2: Illustration of the efficient frontier. The striped gray figure expresses the efficient frontier, and the dark red dots denote suboptimal portfolios.
Figure 3-2 shows the efficient frontier. However, there are not one efficient frontier in the security market. The efficient frontier can change to meet the need and characteristics of different of portfolio managers and investors. For example, if an investor may require a minimum return on the investment portfolio or may rule out investments in ethically or politically undesirable industries (InvestingAnswers, 2018).
1%
3%
5%
7%
9%
0% 5% 10% 15% 20%
EXPECTED RETURN
STANDARD DEVIATION
Efficient frontier Suboptimal portfolios
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