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The portfolio return is the weighted return of the individual positions, and the portfolio risk is the weighted risk of all individual assets and the covariance between those assets: Rp = =
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(2.2)
The covariance can be expressed as a correlation term as follows: ij = ij i j The risk-adjusted portfolio return is: rp rf p (2.4) (2.3)
The efficient frontier is an outcome of Markowitz s theory, a borderline of all portfolios with optimal risk return relations (Figure 2-3). His ap-
Market Risk
F I G U R E 2-3
Efficient Frontier Curve and Capital Market Line.
Return
Efficient frontier
pi Ca
a lm
t rke D
e lin
rf C A
Risk p
proach was developed further by Tobin.21 Tobin improved the correlation between the assets and the risk aversion by including a risk-free position. Through combining the portfolios on the efficient frontier of Markowitz and a risk-free position, Sharpe further developed the conceptual modeling of market risks and introduced the capital market line as the tangent from the risk-free asset to the efficient frontier. 2.4.1 The Capital Asset Pricing Model
The complexity of Markowitz s portfolio model and some generalization of assumptions led to further developments. The capital asset pricing model (CAPM) was developed by Sharpe,22 Lintner,23 and Mossin,24 and later was enhanced by Black. It is a logical extension of the ideas behind modern portfolio theory as first outlined by Markowitz. Because the Markowitz approach makes no statement about the pricing of equities, the CAPM offers a statement on the relevant investment risks and the risk return relation under the condition that the markets are in equilibrium. The CAPM is an equilibrium model for the capital market.
The CAPM is based on the following nine assumptions:25
Utility maximization. Investors try to maximize their own utilities; they are risk-averse. Decision basis. Investors make their decisions only on the basis of risk and return. Expectations. Investors have homogeneous expectations regarding return and risk (variance and covariance) of the assets. One-period time horizon. Investors have identical time horizons of one period. Information efficiency. Information is free and simultaneously available to all market participants. Risk-free asset. Investors can borrow or invest in an unlimited amount of risk-free assets. Markets without friction. No taxes, transaction fees, restrictions on short positions or other market restrictions exist. Capital market equilibrium. The sum of all instruments is given and in possession of the investors. All instruments are marketable, and the assets are divisible to any degree. Supply and demand are not influenced by anything other than price. Distribution. The CAPM, like the Markowitz approach, is based on the normal distribution of returns or a quadratic utility function.
All combinations are on the line between a risk-free investment and the uncertain investment of the efficient frontier. The part between rf and D is called the capital market line (CML) and contains only one efficient portfolio, which is at the tangential point between the efficient frontier and the capital market line (see Figure 2-3). It is not enough to know the return distribution (variance) of a position; the return must be viewed relative to the market and risk components. The CAPM assumes that a certain portion of the risk of a position is a reflection of the overall market risk, which is carried by all positions in the market and thus cannot be diversified. This part of the risk is defined as systematic risk, which cannot be eliminated through diversification. This risk premium is defined as the market risk premium. In contrast, the specific risk (or unsystematic risk) cannot be explained by market events and has its origins in position-specific factors (e.g., management errors and competitive disadvantages). This component can be diversified and is not rewarded by a premium. The expected return of a specific stock is calculated as follows: E (ri) = rf + i [E(rm) rf] + i (2.5)
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