Capital Asset Pricing Model

The Kapitalgutpreismodell or pricing model for capital goods (abbreviation CAPM of English capital asset pricing model ) an equilibrium model (very restrictive) under assumptions explained the pricing of risky investments and enables important insights into the relationship between expected return and risk of securities. The Capital Asset Pricing Model (CAPM) was developed independently by William F. Sharpe , John Lintner and Jan Mossin in the 1960s and builds on the portfolio theory of Harry M. Markowitz . Although the CAPM is often criticized, it is a central component of modern capital market theory and forms the basis of many other models. The importance of the model is also expressed by the fact that Harry M. Markowitz and William F. Sharpe received the Alfred Nobel Memorial Prize for Economics ( Nobel Prize in Economics) in 1990 . Jan Mossin and John Lintner could not get the award because it is not awarded posthumously.

Assumptions of the CAPM

All investors:

1. try to maximize their economic benefit (the number of assets is given and fixed).
2. are rational and risk averse.
3. are broadly diversified across a range of investments.
4. are price takers , d. H. they cannot influence prices.
5. can lend and borrow unlimited amounts at the risk-free rate.
6. trade without transaction costs and taxes.
7. trade in securities that can be divided into any small packages (all assets are perfectly divisible and liquid).
8. have homogeneous expectations.
9. assume that all information is available to all investors at the same time.

Derivation of the fundamental equation of the CAPM

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When CAPM is assumed that investors behave as in the portfolio theory of Harry M. Markowitz has been described. Portfolio theory is based on two basic considerations. On the one hand, every investment decision is associated with risk (more precisely with the uncertainty about future returns ): Investors therefore evaluate every investment on the basis of its expected return and the risk involved in obtaining the return. In addition, portfolio theory takes account of the fact that investors invest in more than one investment, i.e. hold portfolios: the expected return and risk must therefore be measured in the portfolio context. The standard deviation (or, equivalently, the variance ) is regarded as the risk measure of an investment or portfolio .

To simplify the presentation, the portfolio should consist of two systems. The following then applies to the expected value and the variance of the returns of the portfolio:

${\displaystyle \mu =\operatorname {E} (R_{p})=\alpha \times \operatorname {E} (R_{1})+(1-\alpha )\times \operatorname {E} (R_{2})}$

and

${\displaystyle \sigma ^{2}=\operatorname {Var} (R_{P})=\alpha ^{2}\operatorname {Var} (R_{1})+(1-\alpha )^{2}\operatorname {Var} (R_{2})+2\alpha (1-\alpha )\operatorname {Cov} (R_{1},R_{2})=\alpha ^{2}\sigma _{1}^{2}+(1-\alpha )^{2}\sigma _{2}^{2}+2\alpha (1-\alpha )\rho _{1,2}\sigma _{1}\sigma _{2}}$,

Where:

• ${\displaystyle R_{p}}$ the return of the entire portfolio
• ${\displaystyle R_{i}}$ the return on investment i
• ${\displaystyle \alpha }$ the proportion of plant 1 in the portfolio
• ${\displaystyle \rho }$: the Bravais-Pearson correlation coefficient

Efficient edge

For (full correlation ), the total risk (measured by the standard deviation ) is an average of the risks of the individual investments weighted with the proportions . However, if the returns are not fully correlated (which of course they are not in reality either), the risk can be reduced by splitting them. In the adjacent figure, two systems are shown with their expected value and their variance. ${\displaystyle \rho =1}$ ${\displaystyle \sigma }$

For (incomplete correlation), diversification gives rise to new possibilities for the combination of expected return ( ) and risk ( ), all of which dominate the weighted average (connecting line - ), since they generate a higher return for the same risk or a lower risk for the same return or have both (lower risk and higher return). The less the returns are correlated, the more the risk can be eliminated. ${\displaystyle \rho <1}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$

As efficient edge ( English efficient frontier ) is then referred to the amount of non-dominated portfolio, the minimal risk can be achieved for a given for a given risk of the maximum yield and in return. In the μ - σ space the efficient edge (or the efficient limit) is a hyperbola (examples are shown in the diagram).

The question arises to what extent the risk can be eliminated by building a portfolio. The portfolio now consists of investments. It then has the risk: ${\displaystyle N}$

${\displaystyle \sigma _{p}^{2}=\sum \nolimits _{i=1}^{N}\sum \nolimits _{j=1}^{N}\alpha _{i}\alpha _{j}\sigma _{i,j}=\sum \nolimits _{i=1}^{N}\alpha _{i}^{2}\sigma _{i}^{2}+\sum \nolimits _{\begin{array}{c}i=1\\i\neq j\end{array}}^{N}\sum \nolimits _{j=1}^{N}\alpha _{i}\alpha _{j}\sigma _{i,j}}$

The naive diversification strategy is now that the portfolio is equally weighted, i.e. each investment is kept in proportion . For the risk then follows: ${\displaystyle \alpha _{i}=1/N}$

${\displaystyle \sigma _{p}^{2}={\frac {1}{N}}\sum \nolimits _{i=1}^{N}{\frac {1}{N}}\sigma _{i}^{2}+\sum \nolimits _{i=1}^{N}\sum \nolimits _{j=1}^{N}{\frac {1}{N}}\sigma _{i,j}}$

or.

${\displaystyle \sigma _{p}^{2}={\frac {1}{N}}{\overline {\sigma ^{2}}}\;+{\frac {N-1}{N}}{\overline {\sigma _{i,j}}}}$ .

Effects of diversification

The first summand is called company-specific risk. It turns out that the more investments are included in the portfolio, the company-specific risk (independent of the market) can be eliminated (the term converges to zero). As the number of investments increases, the risk converges against the average covariance of the portfolio.

Empirical studies regularly show that the average covariance is positive, meaning that the entire risk cannot be eliminated. This risk that remains after diversification is therefore also referred to as market risk (or systematic risk ). The company-specific risk is also called unsystematic, diversifiable risk . It has been empirically shown that with around 10 to 15 investments in a portfolio, the company-specific risk can hardly be significantly reduced.

Which combination is chosen depends on the respective risk preference of an investor. It is assumed that the investor behaves according to the Bernoulli principle , i. H. the target value, here the portfolio return , can be mapped in a (subjective) utility function , and the portfolio with the maximum expected utility is selected (max. ). ${\displaystyle R}$ ${\displaystyle u(R)}$ ${\displaystyle \operatorname {E} (u(R))}$

The Bernoulli principle is only a decision-making principle . It only becomes a decision rule when the utility function is precisely defined. It is assumed that the investor makes his investment decision exclusively on the basis of the parameters and . This requires that the utility function only depends on the first two moments of the return distribution. This can be justified with a quadratic utility function in relation to the rate of return or with a normal distribution of the rates of return. ${\displaystyle \mu }$ ${\displaystyle \sigma }$

A selection on the efficient edge also assumes that the marginal utility is positive for the rate of return and decreases as the rate of return increases. In the terminology of the risk benefit theory, there is then unsaturation and strict risk aversion . The second condition also implies that, ceteris paribus , the increase in the variance of the rate of return is not preferred . This means that the indifference curves in the μ-σ diagram are strictly monotonically increasing and convex from below (the further "northeast" the indifference curve is, the greater the benefit).

However, it is problematic that the preferences can hardly be determined and the formation of prices on the capital markets therefore cannot be determined due to the large number of different preferences. James Tobin has shown, however, that the selection of an optimal portfolio can be separated from individual preferences. If a risk-free investment ( ) is introduced into the analysis, the selection problem is significantly simplified. As the adjacent figure it can be seen all efficient portfolios are in μ-σ -diagram on through and lying straight line. The existence of a risk-free investment can be justified by the assumption of a perfect capital market , i.e. H. the investor can borrow and lend any sums at the same interest rate. ${\displaystyle R_{F}}$ ${\displaystyle R_{F}}$ ${\displaystyle M}$

Capital market line

The investor will then choose a combination of and , as he can then maximize his expected utility (he will in any case reach an indifference curve that lies further "northeast"). The structure of the risky portfolio is then independent of the investor's risk appetite. This property is known as individual separation ( Tobin separation ). ${\displaystyle R_{F}}$ ${\displaystyle M}$ ${\displaystyle M}$

The CAPM assumes that all investors behave as described in portfolio theory. If all investors have such homogeneous expectations, there are no taxes and transaction costs and none of them can influence market prices through actions, then all investors will hold a combination of and the same portfolio ( this is then called the market portfolio ). The CAPM expands the individual separation into a universal separation. Everyone holds the same portfolio , the structure of which is fixed. The figure opposite illustrates this relationship. ${\displaystyle R_{F}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

This division of the portfolio return into value and risk components independent of preferences enables the simple definition of a risk measure on the capital market and, based on this, the determination of an equilibrium price for one or more units of this measure. The straight line on which all optimal portfolios are located has the equation:

${\displaystyle \operatorname {E} (R_{p})=R_{F}+{\frac {\operatorname {E} (R_{M}-R_{F})}{\sigma _{m}}}*\sigma _{p}}$.

It is used as capital market line ( English capital market line ) refers. The slope of E (R _M -R _F ) / σ _M is referred to as the market price for the risk, because the expected market risk premium for a unit of the market risk ' represents. The price of an individual security depending on the risk can also be derived from the market price for a unit of risk. ${\displaystyle \sigma _{M}}$

A portfolio consists of the market portfolio and a security . The following then applies to the expected value of the return and the risk: ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle i}$

${\displaystyle \operatorname {E} (R_{p})=\alpha \operatorname {E} (R_{i})+(1-\alpha )\operatorname {E} (R_{M})}$

${\displaystyle \sigma _{p}={\sqrt {\alpha ^{2}\sigma _{i}^{2}+(1-\alpha )^{2}\sigma _{M}^{2}+2\alpha (1-\alpha )\rho _{1,2}\sigma _{i}\sigma _{M}}}}$.

The dependence of the expected value and the standard deviation of marginal changes in the share of the security in the portfolio can be determined by taking the first derivation according to: ${\displaystyle \alpha }$ ${\displaystyle i}$ ${\displaystyle \alpha }$

${\displaystyle {\frac {\partial \operatorname {E} (R_{p})}{\partial \alpha }}=\operatorname {E} (R_{i})-\operatorname {E} (R_{M})}$

${\displaystyle {\frac {\partial \sigma _{p}}{\partial \alpha }}={\frac {1}{2}}\left(\alpha ^{2}\sigma _{i}^{2}+(1-\alpha )^{2}\sigma _{M}^{2}+2\alpha (1-\alpha )\rho _{1,2}\sigma _{i}\sigma _{M}\right)^{-0.5}\left(2\alpha \sigma _{i}^{2}-2\sigma _{M}^{2}+2\alpha \sigma _{M}^{2}+2\sigma _{i,M}^{2}-4\alpha \sigma _{i,M}\right)}$.

In the market equilibrium , security i is represented in a certain proportion in the market portfolio . Changes in the proportion of this security cause an imbalance due to excess demand or supply. However, since there are no surpluses in the capital market equilibrium, α = 0 must be assumed for this situation . For the derivatives in equilibrium the following applies: ${\displaystyle \alpha _{i}}$ ${\displaystyle M}$

${\displaystyle {\frac {\partial \operatorname {E} (R_{p})}{\partial \alpha }}=\operatorname {E} (R_{i})-\operatorname {E} (R_{M})}$

${\displaystyle {\frac {\partial \sigma _{p}}{\partial \alpha }}={\frac {1}{2}}(\sigma _{M}^{2})^{-0.5}(-2\sigma _{M}^{2}+2\sigma _{i,M})={\frac {\sigma _{i,M}-\sigma _{M}^{2}}{\sigma _{M}}}}$.

For the marginal risk-return exchange ratio ( marginal rate of substitution between risk and expected return) in the market equilibrium then follows:

${\displaystyle {\frac {\partial \operatorname {E} (R_{p})/\partial \alpha }{\partial \sigma _{p}/\partial \alpha }}={\frac {\partial \operatorname {E} (R_{p})}{\partial \sigma _{p}}}={\frac {\operatorname {E} (R_{i})-\operatorname {E} (R_{M})}{(\sigma _{i,M}-\sigma _{M}^{2})/\sigma _{M}}}}$.

This marginal risk-return exchange ratio corresponds in the tangential point to both the slope of the efficient edge ( limit rate of transformation between risk and return) and the slope of the capital market line. Upon dissolution according to the expected return of the security i , the so-called results securities line ( English security market line ):

Stock line

${\displaystyle \operatorname {E} (R_{i})=R_{F}+\lbrack \operatorname {E} (R_{M})-R_{F}\rbrack {\frac {\sigma _{i,M}}{\sigma _{M}^{2}}}}$.

This is the fundamental equation of the CAPM . The verbal statement reads: The expected return on a risky capital investment i corresponds to the risk-free rate of return in the capital market equilibrium plus a risk premium that results from the market price for the risk multiplied by the risk level.

The risk level σ _iM / σ _M ² is referred to in the CAPM as beta β or beta factor . β _i only measures the contribution of the systematic risk of a security (= σ _{i, M} ) to the total risk of the portfolio (= σ _M ² ). If all investors hold very well diversified portfolios (which they are supposed to do: they hold the market portfolio), the unsystematic risk tends towards zero. Beta β is then the only relevant measure for the risk of a security. Unsystematic risk is not assessed. Using β = σ _iM / σ _M ² the CAPM takes the following form:

${\displaystyle \operatorname {E} (R_{i})=R_{F}+\beta _{i}\times \lbrack \operatorname {E} (R_{M})-R_{F}\rbrack }$.

Although the derivation is not trivial, you get a simple linear formula for the relationship between risk and return of individual investments. The simple formula and the catchy interpretation explain the widespread use of the model in practice.

Interpretation of the CAPM

Properties of the CAPM

The CAPM explains ex-ante in cross-section the yield structure of risky investments. The higher the systematic risk of an investment measured in terms of beta, the higher the return expectations investors will have. The natural anchor point is a beta of 1. According to CAPM, a beta of 1 results in the usual market return (return on the market portfolio). With a beta greater than 1, investors expect a higher return, and with a beta less than 1, a lower return. ${\displaystyle r_{M}}$

The CAPM does not specify how the betas are to be determined. They must be estimated using time series data. The beta results from a linear regression of the returns of the company to be valued on the returns of an efficient market portfolio. These time series estimates allow an additional interpretation of the betas. The estimated beta describes the extent to which the return on an investment replicates the return on the market portfolio. A beta factor of 1 means that the individual return develops proportionally to the market return. If the market return is e.g. B. 10% in one period, the individual return in this period should also be 10%. With a beta factor> 1, an investment should react disproportionately to changes in the market, ie the individual return fluctuates more than the market return. For example, with a beta of 1.5 and an increase (decrease) in the market index by 10%, the return on the relevant share should be 15% (−15%) over the same period.

Alternative representation

An alternative formulation of this return equation of the CAPM is the following " notation" ${\displaystyle \lambda }$

${\displaystyle \mu _{i}=r_{f}+\lambda \cdot \sigma _{i}\cdot \rho _{i,M}}$

with . ${\displaystyle \lambda ={\frac {\mu _{M}-r_{f}}{\sigma _{M}}}}$

In this formulation the beta factor is split up. It becomes clear that the expected return according to CAPM depends on the so-called Sharpe quotient , the “market price of the risk”. This is precisely the ratio of the market risk premium (MRP) to the scope of the market risk , i.e. the additional return per unit of risk. The scope of risk of the valuation object (uncertain return on the risky security) is expressed by the standard deviation of this return. The product expresses the scope of risk that the valuation subject has to bear (taking into account the risk diversification options). ${\displaystyle \lambda }$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \sigma _{i}\cdot \rho _{i,M}}$

Implementation of the model

For the practical implementation of the CAPM, three parameters must be estimated: the return on the risk-free investment ( ), the market return ( ) and the beta factor ( ). ${\displaystyle r_{f}}$ ${\displaystyle r_{M}}$ ${\displaystyle \beta }$

Determination of the risk-free investment

A risk-free investment is - in the sense of the theory - characterized by the fact that the return does not fluctuate ( volatility or standard deviation is zero) and there is no connection to other variables (covariances are zero). There is no such facility. An alternative must therefore be sought that comes as close as possible to this ideal. Bonds issued by the public sector with an excellent credit rating (“AAA”) are considered “quasi-secure” investments . The failure probabilities of these systems are extremely low (approx. 0.7% over a period of 10 years). The internal rate of return of these investments ( yields to maturity , yields to maturity , promised yields ) can therefore be considered to be almost risk-free.

In Germany, federal bonds with a five-year term or federal bonds with a term of 10–30 years are particularly suitable. With a normal interest rate structure , however, the yield to maturity of bonds increases with increasing maturity, so the maturity of the bond must be determined in order to accurately estimate the risk-free interest rate. The selection of the suitable term depends directly on the purpose of the discounting. With the help of discounting, future cash flows are compared with alternative investments on the capital market. For different maturities, however, different returns can be observed on the capital market with a non-flat interest rate structure. A suitable comparison therefore requires discounting with matching maturities, ie the comparative return in the denominator and the cash flows in the numerator should be considered with the same maturity.

Determination of the market return

Market return and stochastic properties of various indices (1998 = 100)

The CAPM is derived from the findings of portfolio theory. Accordingly, the market portfolio is a very broad portfolio in which there are no unsystematic risks. Investors only have to bear systematic risks when investing in the market portfolio. As a result, the market portfolio should consist of very different investments that are hardly correlated with one another, e.g. B. stocks, bonds, real estate, raw materials, foreign exchange, crypto currencies, etc. The construction of such a portfolio is hardly practicable. In practice, therefore, one restricts oneself to the asset class stocks and calculates market returns on the basis of readily available data from stock indices.

The stock index used should be - according to theory - at least a very broad index. An index that maps many regions and industries is to be preferred to indices that are limited to certain industries or regions. With over 1,600 stocks from 23 industrialized countries, the MSCI World is therefore better suited to calculating market returns than the DAX, which only shows a manageable 30 stocks from one industrialized country. When choosing a suitable stock index, one should make sure that it is a performance index . A performance index (total return index) not only reflects price developments, but also dividends and other income from investors (e.g. from subscription rights). Price indices , on the other hand, are calculated exclusively on the basis of the prices of the stocks contained in the index and therefore suppress a large part (approx. 30% of the performance of stocks is due to dividends) of the relevant return for investors. The MSCI World , the S&P 500 and the DAX are performance indices and meet these requirements.

Market returns can easily be calculated from the index changes. In performance indices, a compound interest effect builds up over time due to the reinvestment premise of dividends and other income. The index level at the beginning of the period under review can be interpreted as the originally invested capital (initial payment), the final level of the index then corresponds to the future value of the market portfolio generated with interest and compound interest. The market return can then be determined based on the well-known formula for calculating an internal rate of return:

${\displaystyle {\text{Index}}_{0}(1+r_{M})^{T}={\text{Index}}_{T}\;\Longleftrightarrow \;r_{M}={\sqrt[{T}]{\frac {{\text{Index}}_{T}}{{\text{Index}}_{0}}}}-1}$.

Example: The German share index (DAX) was introduced on July 1, 1988 and at the end of 1987 normalized to an index level of 1,000 points. This can be interpreted as if you had invested € 1,000 on January 1, 1988, for example, and with this € 1,000 at the end of 2019 (i.e. after 32 years) had generated a value of € 13,000 (with an index level of 13,000 points). This corresponds to an annual return of 8.35%.

The adjacent figure shows the determined market return for various indices and estimation periods. It becomes clear that the calculated market return depends significantly on the index used. The calculated return for the MSCI World (Gross Total Return Index) is regularly significantly higher than the DAX return. It is also clear that a portfolio of DAX companies is dominated by a portfolio of MSCI World companies: With a lower risk (measured in terms of volatility), even higher returns can be achieved with MSCI stocks. It therefore makes no sense for a rational investor to limit himself to a portfolio of large German DAX companies. The home bias is punished with a lower return and a higher risk.

Determination of beta factors

The beta factor of a listed company i results from the ratio of the covariance between the company's rate of return and the market rate of return to the variance of the rate of return of the market rate . The betas can be estimated on the basis of time series data with a simple linear regression (see beta factor # Determination of beta factors ). Alternatively, the beta can also be formulated using excess returns. The formulation with excess returns has the advantage that two hypotheses can be tested. If the CAPM is valid, should be and deviate significantly from zero. This can be checked with conventional hypothesis tests. ${\displaystyle r_{i}}$ ${\displaystyle r_{M}}$ ${\displaystyle \sigma _{M}^{2}}$ ${\displaystyle \alpha =0}$ ${\displaystyle \beta }$

Establishing the estimation period T is quite difficult. On the one hand, the estimation period should be chosen as long as possible in order to increase the quality of the estimation. On the other hand, the beta factors should represent the systematic risks of a company in the future - this speaks against the use of data reaching far into the past. Based on this consideration, an estimate of 5 years seems to be appropriate. However, there is no generally applicable rule - the arguments must be carefully weighed up in individual cases.

General uses

• Stock valuation
• Performance analysis
• Assessment of investment projects, company valuation
• Portfolio management
• Investment decision that maximizes market value

Critical appreciation of the CAPM

The strict premises of the CAPM may seem unrealistic at first glance. However, many of the assumptions can be relaxed without questioning the fundamental statements of the CAPM. In the 1970s and 80s in particular, some of the original model assumptions were replaced by more realistic ones. This shows that the core statement of the security line model continues to apply even under less strict assumptions . Various observations (anomalies) that are incompatible with the CAPM are documented in numerous empirical studies. These include the value effect, the small business effect , the momentum effect and the January effect . See also the Fama-French three-factor model . However, William F. Sharpe already stated in 1964 that a theory should not be tested in terms of the closeness to reality of its premises , but rather in terms of the acceptability of its implications. The CAPM not only provides the best-known explanation for the exchange relationship ( trade-off ) between return and risk. B. an important tool in measuring the performance of investment funds .

Particularly when valuing unlisted companies, restrictions on the applicability of the capital goods price model (CAPM) must be observed when determining the cost of capital (or risk discounts).

1. Homogeneity of expectations and planning consistency: How should the individual level of information (e.g. with regard to risks) be taken into account when determining (subjective) decision-making values?
2. Diversification: How should non-diversified (idiosyncratic) risks flow into the cost of capital and valuation if the valuer does not have a perfectly diversified portfolio and possibly cannot realize it?
3. Risk measure and restrictions: What are the consequences if, as an alternative to the beta factor or the standard deviation of the CAPM, other risk measures are used for the valuation, because in an imperfect capital market (a) there are financing restrictions on the part of the creditors and / or (b) the evaluator does Scope of downside risks, e.g. B. want to limit the likelihood of insolvency (safety first)?

The CAPM eludes an empirical review because the market portfolio of all risky assets cannot be reconstructed, criticizes Roll. Because of this, sub-portfolios are used. Tests of these sub-portfolios only provide information about the risk efficiency of these sub-portfolios. Moreover, the CAPM cannot do justice to the claim to explain the stock exchange prices in reality, since a state of equilibrium can hardly be postulated for real capital markets.

Another problem for the empirical verification of the CAPM is that it is sometimes used as a prediction model. However, tests for the risk efficiency of a portfolio are only carried out on the basis of actual stock market prices from the past and usually do not take into account the expectations of investors. Further problems with the empirical verification are the individual behavior of the investors, their influence on the stock exchange prices, structural changes of the portfolio and data gaps. Data is not actually available for all examined values ​​and time periods, so that certain assumptions have to be made for missing data.

Empirical studies on CAPM show in the vast majority "unexpected", i.e. H. Influences on stock returns that cannot be explained by beta, so-called "anomalies". The study by Banz (1981) showed the size effect . The study by Basu (1977) finds that stocks with a low valuation level (P / E) can expect above-average returns that cannot be explained by the beta of the CAPM.

Based on an empirical study in 1992, Eugene Fama and Kenneth French developed the three-factor model in 1993 as a more prognostic alternative to the CAPM. It includes both the price / book value ratio (“value factor”) and the size of the company (stock market value) as explanatory factors for the stock returns. These results are confirmed for the German stock market. Carhart's (1997) derived four-factor model takes into account the momentum factor uncovered in many empirical studies as a further explanatory variable for the stock return. Jegadeesh and Titman (1993 and 2011) again demonstrate a pronounced (risk-adjusted) outperformance of momentum investment strategies. Stocks with the highest return in the last three to twelve months draw a significantly above-average return in the following three to six months.

In 2015, Fama and French presented a five-factor model. The 5 factors are: (1) market risk, (2) company size , (3) value , (4) profitability and (5) investment patterns. This model explains between 71% and 94% of the variance in returns between two diversified US portfolios. The five-factor model therefore has a higher explanatory power than the three-factor model with regard to the mentioned factor portfolios.

Walkshäusl (2012) shows the existence of a significantly negative risk-return relationship for the stock market and thus questions a central implication of the CAPM: more risk leads to a higher expected return. It even shows that stocks with lower volatility also have a very low beta and at the same time a very clearly positive alpha, while the low-return stocks with high volatility have a beta factor greater than one and negative alpha.

Ballwieser sees the CAPM as “anything but” empirically confirmed and refers to a corresponding statement by Kruschwitz , p. 227: “Against the background of the numerous and quite contradicting tests, the conclusion must be drawn that the CAPM today is only slightly empirical Finds support. The illustration also showed that no 'true test' of the CAPM is known yet. "

See also

• Cost of capital
• Risk diversification
• Fama-French three factor model
• Carhart 4-factor model
• Alternative factor models
• Consumption Capital Asset Pricing Model (CCAPM)
• Duplication principle

literature

The original essays can be found at:

• Harry M. Markowitz: Portfolio Selection. In: Journal of Finance, Volume 7, 1952, pp. 77-91.
• William F. Sharpe: Capital asset prices: A theory of market equilibrium under conditions of risk , In: Journal of Finance, Volume 19, 1964, pp. 425-444.
• John Lintner: Security prices, risk and maximal gains from diversification , In: Journal of Finance 20, 1965, 587-615
• Jan Mossin: Equilibrium in a capital asset market , In: Econometrica, Volume 35, 1965, pp. 768-783.

The CAPM is the subject of numerous books on finance. So you can find clear derivations z. B. at

• Richard Brealey, Steward C. Myers, Franklin Allen: Principles of Corporate Finance. 12th edition, McGraw-Hill 2016, ISBN 978-1-259-25333-1 .
• David Hillier, Stephen A. Ross, Randolph W. Westerfield: Corporate Finance , 2nd Edition. McGraw-Hill 2013, ISBN 978-0-07-713914-8 .
• Glen Arnold, Deborah Lewis: Corporate Financial Management , 6th Edition, Harlow et al. a. 2019, ISBN 978-1-292-14044-5 .

The critical examination of the CAPM and special aspects are dealt with in the following publications:

• MM Carhart: On Persistence in Mutual Fund Performance , Journal of Finance 52 (1), 1997, pp. 57-82.
• M. Dempsey: The Capital Asset Pricing Model (CAPM): The History of a Failed Revolutionary Idea in Finance? , in: ABACUS, Volume 49, Issue Supplement S1, 1997, pp. 7-23.
• H. Dirrigl: Company valuation for the purpose of tax assessment in the field of tension between individualization and capital market theory - a current problem against the background of the inheritance tax reform (also a contribution to the commemorative publication for Franz W. Wagner on his 65th birthday) (PDF; 1.9 MB). In: arqus-Working Paper No. 68, 2009. Online at franz-w-wagner.de.
• D. Ernst, W. Gleißner: How problematic are the restrictive assumptions of the CAPM for the company valuation ?, in: Der Betrieb, Heft 49, 2012. S. 2761–2764.
• EF Fama: Risk-Adjusted Discount Rates and Capital Budgeting under Uncertainty , in: Journal of Financial Economics, 5/1977, pp. 3–24.
• EF Fama, KR French: Common risk factors in the returns on stocks and bonds, in: Journal of Financial Economics, Vol. 47, 1993. pp. 3-56.
• EF Fama, KR French: Dissecting Anomalies, in: Journal of Finance , volume 63, issue 4, August 2008, pp. 1653-1678.
• W. Gleißner: Uncertainty, Risk and Enterprise Value , in: K. Petersen, C. Zwirner, G. Brösel (Ed.): Handbook Company Valuation , Bundesanzeiger Verlag, 2012. ISBN 978-3-89817-917-1
• P. Fernandez: Are calculated betas worth for anything? , IESE Business School, University of Navarra, February 17, 2004, pp. 1-34.
• W. Gleißner, M. Wolfrum: Equity costs and the valuation of unlisted companies: relevance of degree of diversification and degree of risk , in: FINANZ BETRIEB, 9/2008, pp. 602–614.
• M. Hagemeister, A. Kempf: CAPM and expected returns: An investigation based on the expectations of market participants , in: DBW, 2/2010, pp. 145–164.
• Hanauer, M./Kaserer, C./Rapp, MS: Risk factors and multifactor models for the German stock market , in: Betriebswirtschaftliche Forschung & Praxis, 65, No. 5, 2013, pp. 469–492.
• T. Hering: Financial enterprise valuation, Deutscher Universitätsverlag , Wiesbaden 1999.
• N. Jegadeesh, S. Titman: Momentum , August 29, 2011, working papers series.
• R. Roll: A critique of the asset pricing theory s tests , Journal of Financial Economics 4, 1977. pp. 129-176.
• Peter Seppelfricke: Company evaluations: methods, overviews and facts for practitioners , 2020. ISBN 978-3-7910-4734-8
• William Sharpe: Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk , 1964, in: Journal of Finance, pages 425-442
• K. Spremann: Valuation: Basics of modern company valuation , Oldenbourg Wissenschaftsverlag, 2004.
• C. Walkshäusl: Fundamental Risks and Stock Returns - Here too, less risk leads to better performance in: CORPORATE FINANCE biz, 3/2013, pp. 119–123.

Web links

Commons : Portfolio Theory and CAPM  - Collection of Images, Videos and Audio Files

• The CAPM

Individual evidence

1. ^ William F. Sharpe: A Theory of Market Equilibrium under Conditions of Risk . tape 19 , no. 3 . The Journal of Finance, p. 425-444 .
2. ↑ John Lintner: SECURITY PRICES, RISK, AND MAXIMUM GAINS FROM DIVERSIFICATION * . In: The Journal of Finance . tape 20 , no. 4 , December 1965, ISSN  0022-1082 , p. 587-615 .
3. ^ Jan Mossin: Equilibrium in a Capital Asset Market . In: Econometrica . tape 34 , no. 4 , October 1966, ISSN  0012-9682 , p. 768 , doi : 10.2307 / 1910098 .
4. ^ Harry Markowitz: Portfolio Selection . In: The Journal of Finance . tape 7 , no. 1 , March 1952, ISSN  0022-1082 , p. 77 , doi : 10.2307 / 2975974 .
5. ^ Arnold, Glen .: Corporate financial management . 4th ed. Pearson Financial Times / Prentice Hall, Harlow, Eng. 2008, ISBN 978-0-273-71041-7 , pp. 354 .
6. ↑ Franziska Ziemer: The beta factor in science . In: The beta factor . Springer Fachmedien Wiesbaden, Wiesbaden 2017, ISBN 978-3-658-20244-6 , p. 139-333 .
7. ↑ Gleißner, W .: Uncertainty, Risk and Enterprise Value, in: Petersen, K. / Zwirner, C. / Brösel, G. (Ed.), Handbook Company Valuation, Bundesanzeiger Verlag, 2013, SS 691–721. (PDF; 2.5 MB)
8. ↑ Gleißner, W. (2011): Risk analysis and replication for company valuation and value-oriented corporate management, in: WiSt, 7/2011, pp. 345–352.
9. ↑ cf. z. B. Kerins / Smith, JK / Smith, R., 2004, pp. 385-405
10. ^ Richard Roll, A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory , in: Journal of Financial Economics 4 (2/1977), pages 129-176
11. ↑ http://www.cfr-cologne.de/download/workingpaper/cfr-07-01.pdf CAPM and expected returns: A study based on the expectations of market participants
12. ↑ See also Haugen, R. 2004. The new finance. New York: Pearson Education; Ulschmid, C. (1994) Empirical Validation of Capital Market Models, Frankfurt am Main and Hagemeister / Kempf, DBW 2010, pp. 145–164.
13. ↑ cf. Hagemeister / Kempf, 2010
14. ^ Matthias X. Hanauer, C. Kaserer, Marc S. Rapp: Risk factors and multifactor models for the German stock market . In: Business Research & Practice . tape 65 , no. 5 , 2013, p. 469-492 ( ssrn.com ).
15. ↑ see e.g. B. Jegadeesh / Titman (1993 and 2011)
16. ^ Eugene F. Fama, Kenneth R. French: A five-factor asset pricing model . In: Journal of Financial Economics . tape 116 , no. 1 , April 2015, p. 1–22 , doi : 10.1016 / j.jfineco.2014.10.010 ( elsevier.com [accessed July 9, 2020]).
17. ↑ Walkshäusl, C. (2012): The volatility anomaly on the German stock market: Less risk leads to better performance, in: Corporate Finance biz, 02/2012, p. 84.
18. ↑ Ballwieser, W. (2008): Betriebswirtschaftliche (capital market theoretical) requirements for company valuation, in: WPg, Volume 61, special issue 2008, pp. 102-108
19. ↑ Carhart, MM (1997): On Persistence in Mutual Fund Performance, in: Journal of Finance, 52 (1), pp. 57-82.