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where ri = return of security i rf = return of the risk-free asset rm = return of the market i = sensitivity of security i relative to market movement m i = error term 2.4.2 The Security Market Line One of the key elements of modern portfolio theory is that, despite diversification, some risk still exists. The sensitivity of a specific position relative to the market is expressed by i (see Figure 2-4). The CAPM defines i as the relation of the systematic risk of a security i to the overall risk of the market.
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i =
covariance (rm , ri) systematic risk of title i = risk of market portfolios variance (rm)
i = cov (ri, rm) rm where ri = return of security i rf = return of the risk-free asset rm = return of the market i = sensitivity of security i relative to market movement m i = error term
(2.6)
The one-factor model is a strong simplification of the Markowitz model, and the CAPM is a theory. The main criticisms of the CAPM are as follows:
The market efficiency is not given in its strong form, as not all information is reflected in the market. This presents an opportunity for arbitrage profits, which can be generated as long as insider information is not available to the public. The normal distribution is a generalization, which distorts the results, especially for idiosyncratic risks.
The main message of market efficiency as it pertains to capital market models is that a market is considered efficient if all available data and information are reflected in the pricing and in the demand-and-supply relation.26 Fama distinguishes three types of market efficiency: weak, semistrong, and strong.27 In a study on Swiss equities, Zimmermann and Vock28 came to the conclusion that the test statistics (the standardized third and fourth moment as a measure for the skewness and kurtosis, the standardized span or studentized range, and the test from Kolmogorov-Smirnov) point to a leptokurtic return distribution (see Figure 2-5). The study concluded that the normal distribution has to be questioned from a statistical point of view. The deviations are empirically marginal. The leptokurtosis has been confirmed for U.S. equities in studies by Fama,29 Kon,30 Westerfield,31 and Wasserfallen and Zimmermann32 (see Figure 2-5). Zimmermann33 concluded that over a longer time horizon (1927 to 1987), the normal distribution fits the return distribution of Swiss equities.
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F I G U R E 2-5
Normal and Leptokurtic Distribution of Equity Returns.
Probability %
Leptokurtotic distribution
Normal distribution
Return
2.4.3 Modified Form of CAPM by Black, Jensen, and Scholes Black, Jensen, and Scholes conducted an empirical examination of the CAPM in 1972. They used a model without a risk-free interest rate, because the existence of a risk-free interest rate was controversial.34 In the model without a risk-free return, the security market line (SML) is no longer defined by the risk-free return and the market portfolio; instead, it is a multitude of combinations, as there is a multitude of zero-beta portfolios.35 The return that they were able to explain was significantly higher than the average risk-free return within the observation period. They concluded that the model is compatible with the standard form of the CAPM, but differentiates between borrowing and lending. The study supports the practical observation that borrowing is more expensive than lending money. Empirical studies support the development of the capital market line with two interest rates, one for borrowing and one for lending money. It is an important improvement, as it excludes the assumption that borrowing and lending are based on the same risk-free rate.36 Figure 2-6 is based on the following equations: E(ri) = rL + i [E(rm) rL] + i E(ri) = rB + i [E(rm) rB] + i (2.7) (2.8)
F I G U R E 2-6
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