# Is CAPM a good model

## The comparison of the Capital Asset Pricing Model and Arbitrage Pricing Theory as central valuation models of the capital market theory

### Table of Contents

List of figures

List of abbreviations and symbols

1 Introduction

2. CAPM

2.1 History and further development

2.2 Basics, premises and central model evaluation equation

2.2.1 Securities Market Line (WML) / Security Market Line (SML)

2.2.2 Beta factor (ß)

2.3 Empirical revision of the model using the single index model (SIM) ...

2.4 model review

3. APT

3.1 History and further development

3.2 Fundamentals, premises and central model evaluation equation

3.2.1 Securities market level (WME)

3.3 Empirical revision of the theory using the multi-index model (MIM)

3.4 Model review

4. CAPM and APT collation

5. Closing remarks and critical evaluation

bibliography

### List of figures

Figure 1 The modern portfolio theory Ralf Wahl www.ralf-wahl.de/Kapitalanlagen [Accessed June 7th, 2009] www.ralf-wahl.de/Kapitalanlagen.htm Original presentation

Figure 2 From the Efficient Frontier to the Capital Market Line Frankfurt-School Verlag, Portfolio Management Theory and Application Hendrik Garz, Stefan Günther, Cyrus Moriabadi 4th revised edition 2006 Page 45, Figure 15 and Page 59, Figure 24 Own illustration

Figure 3 Relationship between KML and WML Frankfurt-School Verlag, portfolio management theory and application Hendrik Garz, Stefan Günther, Cyrus Moriabadi 4th revised edition 2006 page 68, Figure 27 Own illustration

Figure 4 Securities market level with 2 risk factors Frankfurt-School Verlag, portfolio management theory and application Hendrik Garz, Stefan Günther, Cyrus Moriabadi 4th revised edition 2006 page 76, Figure 30 own illustration

### List of abbreviations and symbols

Figure not included in this excerpt

### Arrangement according to the German alphabet

Figure not included in this excerpt

### 1 Introduction

"Don't put all your eggs in one basket" has long been the guideline for many investment strategies. This apparently trivial finding has far-reaching effects on portfolio theory and models and the behavior of investors to this day. Harry M. Markowitz, Nobel laureate in economics and founder of modern portfolio theory, adapted and revolutionized this approach in his groundbreaking doctoral thesis "Portfolio Selection" from 1952.

The primary goal of every investor in asset allocation is, all in all, to achieve the highest possible return with the lowest possible risk under the basic premise of maximizing benefits. However, this proposition necessarily connotes the rationality of the decision-maker and establishes the relationship between the expected return on an investment option and its inherent risk. Markowitz summed up this complex, ambivalent relationship aptly: "I imagine my grief if the stock market shot up and I wasn't there - or if it crashed and I was completely there"^{1}.

In his opinion, the key to fine-tuning and optimizing the opportunity-risk structure lies in the optimal spread (diversification) and different weighting of the asset classes (asset allocation). “A good portfolio is more than a long list of stocks. It is a balanced unit that offers the investor both opportunity and protection amid a multitude of possible developments. Investors should therefore work towards an integrated portfolio that takes their individual needs into account. ''^{2}(see also Fig. 1).

In portfolio theory there is a nuance between systematic and unsystematic risk, as the environmental status of the overall risk. Due to the titled diversification, a reduction of the unsystematic risk can be empirically proven up to the complete elimination at the overall portfolio level. The non-diversifiable systematic risk as a residual quantity, secondary as market risk^{3} declared, is the starting point for the investor's expected return. As a core topic, Markowitz's theory devotes particular vigilance to this through the concept of the EF (Efficient Frontier) and the capital market line for the design of efficient portfolios.

The central capital market models CAPM (Capital Asset Pricing Model) and APT (Arbitrage Pricing Theory), as representatives of the explicit univariate and multivariate equilibrium models, are intended to demonstrate the price determination of efficient portfolios. There will be a detailed overview of the models, their assumptions, dissipations and merits. A comparison of the CAPM and the APT, along with a personal résumé, proclaims the conclusion of this work.

### 2. CAPM

### 2.1 History and further development

The CAPM was developed independently of each other by William Forsyth Sharpe (1964), John Virgil Lintner Jr. (1965, the rough basic idea of the elaboration can be attributed to Jack L. Treynor from 1961/1962) and Jan Mossin (1966) as a basic model, based on the portfolio theory of Harry M. Markowitz^{4}.

The abbreviation CAPM stands for Capital Asset Pricing Model and translates as “capital-asset-price model” or “price model for capital goods”.

The origin of the name of the model can be seen in its descriptive function, recursively Treynors Retrieval of a discounting method for long-term assets.

In the following years there was constant further development (the Swiss 3-factor model to be mentioned as an example)^{5}, the model bases were transferred to other areas (Intertemporales and Consumption CAPM, Tax-CAPM)^{6}.

### 2.2 Basics, premises and central model evaluation equation

The nucleus, as an evolution of Markowitz's PST, is the quantification and evaluation of individual value risks, i.e. the expected return on an investment in a perfectly capital market and fully diversified portfolio, depending on its inherent risk (objective and subjective uncertainty in the sense).

The univariate CAPM is a static equilibrium model and corresponds to the normative SIM in the universe of capital market models on a descriptive level.

As a result of its model premises and the elaboration of the linear interdependence of the singular expected value of an investment opportunity from an influencing variable (the risk), it can be classified under the category of one-factor models.

The basis of the concept is the EF including premises ^{7}^{8} (e.i. subsumed under the CAPM):

- All market participants are risk averse and hold efficient (market) portfolios, i.e. they act as portfolio optimizers and a priori rational investors in the sense of KMT

- Validity of the unsaturation and utility maximization principle

- homogeneous expectations regarding expected (WP) returns and covariance matrix

- Information efficiency in the sense of the term, investment decision without informational advantage

- Uniform, accessible investment universe without restrictions

- Frictionless markets (i.e. no taxes, no transaction costs)

- perfect capital market, (i.e. no market access restrictions, etc.)

- All of the investors act in the mean-variance framework

- Market equilibrium of the WP market (competitivity and spanning condition)

- Uniform market price of the risk reimbursed in terms of risk premiums

- No arbitrage opportunities / Non Arbitrage Condition / Law of one price

- Existence of a risk-free interest rate / "pure rate"

- Fixed amount of marketable securities that can be divided and traded at will

- univariate model / one-period model: singular influencing factor "the market"

The addition of the risk-free investment component by means of the risk-free interest rate to the efficient market portfolio induces a transformation from EF to KML.

Figure not included in this excerpt

The characteristics of the parameters and position of the market portfolio M are coherent for all investors as Homo Oeconomicus isotropic. These keep c.p. a corresponding linear combination of the risk-free investment and the tangential market portfolio. This allocation of the "risky portion" of the investable capital in the capital market equilibrium in a permanently identical market portfolio M, independent of the individual risk attitude (characteristics: neutrality / affinity / aversion), is called Tobin separation or two fund separation.

The linearity postulated at the beginning is expressed by the slope of the KML, which represents the disparity between the expected return (risk-adjusted cost of capital) of the market portfolio and the risk-free interest rate. The elongate and functional correlation between the market price of the risk / per risk unit and the expected return is signed as a securities market line^{9}.

#### 2.2.1 Securities Market Line (WML) / Security Market Line (SML)

In the (capital) market equilibrium, the WML differentiates between systematic risk that is relevant to valuation, remunerated as a relative contribution to the market risk on the market, and unsystematic risk that is not relevant to valuation and is not remunerated because it is diversifiable. Equal standard deviations are not mandatory. homogeneous earnings opportunities as a result of divergent shares in the intrinsic system. Portfolio risk.

Figure not included in this excerpt

Tangentialportfolio M by definition contains the spectrum of all tradable securities in the available investment universe pro rata of their published market value and market capitalization^{10}. The effect of full diversification is demonstrable; the variance of the M portfolio is entirely a matter of systemic risk. A WP can temporarily deviate from the postulated linear relationship due to market anomalies. WP above the WML are undervalued, with the same risk as WP on the WML, they have a higher expected return and vice versa. The supply / demand mechanism of the market leads this back to mean reversion, whereby the CAPM, as a static model, does not reflect such transitory states.

Such temporary divergences are specified by the symbol a (alpha)^{11}. If there is a positive deviation, the portfolio is more efficient than the market, in the reciprocal case it is inefficient. The KML is an economic instrument for pricing efficient portfolios, the limit represents the evaluation and statement on microeconomic risks.

#### 2.2.2 Beta factor (ß)

The BETA factor represents a relative risk measure (without a unit), which relates the average covariance risk of an investment to the average risk of variance of the overall market. As a sensitivity indicator, it measures the relative contribution of an investment, i.e. the non-diversifiable systematic risk, in relation to the overall portfolio risk at the overall portfolio level^{12}.

Figure not included in this excerpt

A BETA> 1 indicates an above-average system. Risk, <1 antonym.

At the overall portfolio level, it is the arithmetic average of the individual value betas:

Figure not included in this excerpt

A high beta factor of a WP implies a high expected return and is reciprocal. The formal-logical representation of the mathematical determination of the optimum as an equation of return^{13}shows that the expected return of a risk-bearing investment object in market equilibrium is composed of the interest rate of financially risk-free investments and a risk premium. The risk premium is the product of the market price for the risk (slope of the WML) and the risk amount of the investment object under consideration (measured by ß). Analogous to the one-period planning, the calculated return can be converted into an equilibrium rate.

### 2.3 Empirical revision of the model using the single index model (SIM)

The relevance of the model is based on the simplicity and plausibility of the premises and input parameters. The problem of being related to the past is ambivalent, since expected values cannot be determined unconditionally based on ex-ante data.

The parameter approximation is implemented using the market or single index model (SIM)^{14}according to Sharpe using a linear regression model:

Figure not included in this excerpt

Beta embodies the slope of the regression degrees, which adequately approximates the pairwise observations of the market return and the WP in the scatter diagram. The condition of this approach is that the returns of the market portfolio flow into the market model equation as input factors. Since this in detail cannot be observed or replicated, an equivalent surrogate must be applied.

The assumption of uncorrelated regression individuals has far-reaching implications for the informative value, although this is only given for the market portfolio M (complete correlation of all securities contained with the index). This condition of empirical uncorrelation of the residual individuals is also applied to the surrogate.

The diagonal of the variance / covariance matrix describes the division of the variance of the surrogate portfolio into a systematic and unsystematic part^{15}:

Figure not included in this excerpt

If the selected index meets the requirement of uncorrelatedness, the pairwise covariances no longer have to be predicted in pairs since:

Figure not included in this excerpt

The result of the optimization is an efficiency line identical to that of the Markowitzian approach. If an assumption is not adhered to, this solution concept establishes an approximate solution. For practical use, a decision must be made between time / cost savings and the accuracy of the result depending on the disturbance variable.

**[...]**

^{1}See www.Spiegel.de, article “Why most investors are stupid” [accessed on June 8, 2009]

^{2}See Markowitz, Harry M. (1959), Portfolio Selection: Efficient Diversification of Investments

^{3}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p.66

^{4}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p.14

^{5}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p.13 f

^{6}See the dictionary of business economics, 6th edition 2007, pp. 878 ff

^{7}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), pp. 14 ff, 55 ff

^{8}See Frankfurt-School Verlag, Portfolio Management, 4th edition 2006, p. 67

^{9}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p.16, 50 f

^{10}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p. 55 f

^{11}See Portfolio Management Handbook, Uhlenbruch Verlag (2002), p. 466 ff

^{12}See Handbook Portfoliomanagement, Uhlenbruch Verlag (2002), p.255 f

^{13}See Frankfurt-School Verlag, Portfolio Management, 4th edition 2006, p. 68

^{14}See Portfolio Management Handbook, Uhlenbruch Verlag (2002), pages 12, 52, 311

^{15}See Frankfurt-School Verlag, Portfolio Management, 4th edition 2006, p. 72

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