Excerpt

## Content

List of abbreviations

List of symbols

1 Introduction

1.1 Introduction to asset pricing

1.2 Objective of this paper

2 The Capital Asset Pricing Model

2.1 Derivation of the CAPM

2.1.1 Firm-Specific Risk vs. Market Risk

2.1.2 The beta coefficient

2.1.3 The CAPM Equation

2.2 The Security Market Line (SML)

2.3 Assumptions of the CAPM

3 Problems of the CAPM

3.1 Unrealistic assumptions

3.2 Empirical Testing of CAPM

3.2.1 General Testing Problems

3.2.1.1 The Problem of ex ante Data

3.2.1.2 Roll’s critique

3.2.2 Results of empirical Tests

3.2.2.1 Controversy about Beta

3.2.2.2 Empirical support for other risk factors

3.2.3 Conclusion

4 New Developments

4.1 Neoclassical Models – Traditional Asset Pricing

4.1.1 ‘Smaller’ adjustments of the CAPM

4.1.1.1 Zero-Beta CAPM

4.1.1.2 Introducing taxes and transaction costs

4.1.1.3 International Capital Asset Pricing Model

4.1.1.4 Option pricing in CAPM context

4.1.2 Multi-factor Models

4.1.2.1 The Arbitrage Pricing Theory (APT)

4.1.2.2 Fama-French Three-Factor Model

4.1.3 Multi-period models

4.1.3.1 The intertemporal CAPM (ICAPM)

4.1.3.2 The Consumption-Based CAPM (CCAPM)

4.1.3.3 Production-based Asset Pricing Model

4.1.4 General Puzzles of Traditional Asset Pricing Models

4.1.4.1 Equity premium puzzle

4.1.4.2 Risk-Free Rate Puzzle

4.2 Behavioral Finance

4.2.1 Introduction

4.2.2 Evidence contradicting the efficient market

4.2.2.1 Royal-Dutch-Shell shares

4.2.2.2 IPO Palm

4.2.3 Pillars of Behavioral Finance

4.2.3.1 Psychology

4.2.3.2 Limits to arbitrage

4.2.3.3 Summary

4.2.4 Prospect-Theory Model

4.2.4.1 Key Elements

4.2.4.2 Assessment

4.2.5 Habit Formation Models

4.2.6 Models with heterogeneous Agents

4.2.7 Conclusion

4.3 Chaos, synergetic models and neural networks

5 Conclusion

Bibliography

Declaration of Academic Integrity

## List of abbreviations

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## List of symbols

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## 1 Introduction

### 1.1 Introduction to asset pricing

Asset pricing theory tries to explain why some assets pay higher average returns than others. Accordingly, the objective is to understand the prices or values of claims to uncertain payments. (Cochrane, 2005, p. XIII)

The central aspect is the risk-return tradeoff. It is rational that investors demand additional return for an asset incorporating more risk. This relationship can also be empirically examined when looking at the return development of different assets. For example, between 1926 and 1999, small U.S. stocks yielded average returns of almost 19%, while at the same time large stocks yielded 13% and US Treasury-Bills only about 4%. When looking at the risk of the assets, as measured by the standard deviation of the returns, the relationship becomes obvious: small stocks had a standard deviation of almost 40%, while large stocks and U.S. treasury-bills had 20 % and 3%, respectively. (Tuck School of Business, 2003, p. 2)

Problems arise, however, when one tries to determine the relevant risk factors and their expected compensation. The basis for this theory was already laid in the 1950s and 60s with the portfolio selection theory by Markowitz and the Capital Asset Pricing Model (CAPM) by Sharpe, for which he received a Nobel Prize in 1990. (Wilhelm, 2001, p. 15) The CAPM significantly shaped and changed financial management (Užík, 2004, p. VII). Today it is still widely used in practice and plays the centerpiece in the theoretical discussion of asset pricing, although it continues to be sharply criticized (Fama & French, 2004, p. 25). This leads to a variety of adaptations and further developments of the CAPM, but so far no model has been able to sufficiently persuade financial scientists and practitioners (Užík, 2004, p. VII).

As it might seem on first sight, asset pricing is not only solely important for financial investors, because in reverse this also means that companies have to meet the expected returns of their investors. This falls under the ‘Shareholder Value concept’, which has increased in significance over the past years and is being rigorously proclaimed by many investors. According to this concept, companies have to know the return expectations of the investors in order to include them in their capital costs for investment decisions. (Wallmeier, 1997, p. 1)

### 1.2 Objective of this paper

The objective of this paper is to give an overview of the most important movements of the complex area of asset pricing. This will be tried by logically structuring and building up the topic from its origins, the Capital Asset Pricing Model, and then over its main points of critique, in order to arrive at the different options developed by financial science that try to resolve those problematic aspects.

Due to the complexity of this subject and the limited scope of this paper, obviously it will not be possible to discuss each model or movement in depth. Coherently, the aim is to point out the main thoughts of each aspect discussed. For further information, especially concerning the deeper mathematical backgrounds and derivations of the models, the author would like to refer the reader to the books mentioned in this paper. Many of those works, finance journal publications and the literature on asset pricing in general, set their focus on different parts of this paper, which again underlines the complexity in terms of scientific scope and intellectual and mathematical intricacy of this topic.

## 2 The Capital Asset Pricing Model

As mentioned above, the Capital Asset Pricing Model (CAPM) laid the basis for modeling the risk-return relationship as it is considered “the basic theory that links risk and return for all assets.” (Gitman, 2006, p. 246)

The foundation of this model has to be seen in the portfolio choice model, especially as developed at the beginning of the 1950s by Harry Markowitz. Later, in the middle of the 60s, Sharpe, Lintner and Mossin adapted the basic idea of Markowitz by generalizing the individual decision problem of a single investor to all capital market participants. This step led to the CAPM and other asset pricing models (Wilhelm, 2001, p. 66-67). Accordingly, the CAPM builds upon the model of portfolio choice. Due to the limited scope of this paper the portfolio selection model can not be discussed in detail and is assumed to be known.^{[1]}

### 2.1 Derivation of the CAPM

The initial development of the Capital Asset Pricing Model is generally attributed to William F. Sharpe^{[2]} based on his article in the ‘Journal of Finance’ from 1964 about Capital Asset Prices (Gitman, 2006, p. 246).

In this article he summarizes that:

*“In equilibrium, capital asset prices have adjusted so that the investor, if he follows rational procedures (primarily diversification), is able to attain any desired point along a capital market line. He may obtain a higher expected rate of return on his holdings only by incurring additional risk. In effect, the market presents him with two prices: the price of time, or the pure interest rate […] and the price of risk, the additional expected return per unit of risk borne […].”* (Sharpe, 1964, p. 425)

Accordingly, there are two important aspects of the CAPM. First of all, the investor is compensated for delaying consumption over the planning horizon with the *price of time*. Secondly, the investor is rewarded with the *risk premium* for taking on the risk associated with the investment. Obviously, the latter is the more complex issue. In the following it will be analyzed on what this risk premium depends.

#### 2.1.1 Firm-Specific Risk vs. Market Risk

Sharpe speaks of diversification. By diversifying the portfolio one is able to eliminate firm-specific risks. But assets also contain another type of risk which is called the ‘market risk’. It is attributed to factors that affect all firms and thus cannot be eliminated through diversification. Therefore it is also called ‘nondiversifiable risk’. (Gitman, 2006, p. 247)

Accordingly, the only part of risk that a rational, diversified investor has to focus on is the market risk. Furthermore, in an efficient market only the nondiversifiable risk is compensated. (Weston, Besley & Brigham, 1996, p. 204)

#### 2.1.2 The beta coefficient

But if one will always be stuck with the market risk and the firm specific risk will not play an important role in a diversified portfolio, does that mean that all assets are equally risky? No, because not all assets react the same way on changes in the market. Thus, there are assets that might only be slightly affected by changes in the market and others that depend to a high degree on the market development and will, therefore, react strongly to a changing market. (Weston et al., 1996, p, 201)

This leads us to the central aspect of the CAPM, namely the *beta coefficient*. It is a measure of nondiversifiable risk, so it measures the “degree of an asset’s return in response to a change in the market return.” (Gitman, 2006, p. 247) Although the beta should be forward-looking and compared to the whole market return, in practice it is usually based on historical returns and a common stock index as a proxy for the market return (Gitman, 2006, p. 247).

#### 2.1.3 The CAPM Equation

The following is the CAPM equation and calculates the required return on asset i:

illustration not visible in this excerpt

Where Abbildung in dieser Leseprobe nicht enthalten= expected return on the market portfolio

Accordingly, the market risk premium is the additional return over the risk-free rate needed to compensate investors for taking on the average amount of risk associated with holding the market portfolio of assets. Thus, it depends on the average degree of risk aversion of investors and is calculated by the difference between the return on the market portfolio and the risk free rate of return. (Weston et al., 1996, p. 207)

If an asset has the same response as the market, its beta must be 1. On the other hand, if it responds stronger than the market, its Abbildung in dieser Leseprobe nicht enthaltenis higher than 1, leading to a higher risk premium. Obviously, for a less responsive asset the beta is smaller than 1 and it therefore has a lower risk premium. (Gitman, 2006, p. 249)

### 2.2 The Security Market Line (SML)

The Security Market Line graphically depicts the risk-return trade-off. Accordingly, it is the visualization of the CAPM and graphs the required return of an asset dependant on each level of beta (and thus its market risk). The interception of the SML with the y axis (*b* = 0) is at the risk-free rate of return. (Gitman, 2006, p. 252)

illustration not visible in this excerpt

The Security Market Line (SML) (Weston et al., 1996, p. 208)

Since the risk-free rate is the investor’s compensation for the ‘price of time’, in other words the delay of consumption over the period, this rate will change with a change in inflationary expectations. Obviously, with growing inflationary expectation the compensation for delaying consumption must also be higher. Thus, inflation shifts the SML curve upwards. (Weston et al., 1996, pp. 209/210)

As discussed, the market risk premium (depicted by the slope of the SML) depends on the risk aversion of investors. Increasing risk aversion will therefore make the SML steeper and vice versa. (Gitman, 2006, pp. 254/255)

### 2.3 Assumptions of the CAPM

Since the preconditions of the CAPM will be of great importance in the following course of this paper, one has to be aware of their rigidity in order to assess the limitations of the models and to discuss new developments in asset pricing theory. The key assumptions of the CAPM were first stated by Michael C. Jensen in 1972 and are as follows: (Brigham & Gapenski, 1996, p. 68 and Jensen, 1972, pp. 358/359)

1. All investors think in terms of a single period.

2. Investors act rational and choose their portfolio solely based on the expected return and its standard deviation over that period, which means that returns have to be normally distributed (Wilhelm, 2001, p. 69).

3. All investors can borrow or lend an unlimited amount of money at a given (and for all equal) risk-free rate of interest.

4. All investors have homogeneous expectations, meaning that they identically estimate expected returns, standard deviations and correlations of returns among all assets.

5. Homogeneous expectations require that all investors have constant and free access to all required information regarding the investment decision. Furthermore, this information has to be analyzed and evaluated equally by all. (Hug, 1993, pp. 131/132)

6. All assets are perfectly divisible and are perfectly marketable at the going price.

7. There are no transaction costs, taxes and restrictions on short sales of any asset.

8. The market is not constricted by any institution (Weber, 1990, p. 72).

9. Investors assume that their own acting will not affect prices (= price takers).

10. The quantities of all assets are given and fixed.

11. Investors are risk averse (Hug, 1993, p. 130).

## 3 Problems of the CAPM

The decisive question is to what extent the CAPM is able to fulfill its high-aimed objective, namely to explain the risk – return relationship of assets. The title of this paper already indicates that the CAPM has its limitations. Actually, most empirical tests of the CAPM have even had trouble to explain the past, let alone to predict the future (Weber, 2006, pp. 83-86). In those tests, empirical research even noticed a few regularities in the divergences (Užík, 2004, p.47). These empirical problems might be the result of theoretical failings, namely many simplifying assumptions (Fama & French, 2004, p. 25). These will be discussed in the next paragraph. Afterwards, problematic aspects of testing the CAPM and a few of the unexplained phenomena will be analyzed.

### 3.1 Unrealistic assumptions

Already when the assumptions of the CAPM were stated in the upper part of the paper, the reader must have been surprised by their rigorousness and lack of reality. Here, only the most important assumptions are to be discussed.

One of them is obviously the single period time horizon of the model. This means that investors are only concerned with the wealth their portfolio produces at the end of the current period. Investors in the real world have the intention of securing their lifetime consumption level by the means of investing. Making optimal investment decisions by considering returns over the next period only (single period model), is just achievable under further assumptions. (Armitage, 2005, p. 52)

Another problem is that the model should only be based on forward-looking data, e.g. the expected rate of return and the expected beta. Obviously, these cannot be estimated with precision and are therefore often historically based. (Brigham & Gapenski, 1996, p. 85)

Other assumptions that do not comply with reality are the lack of free and instantly available information (information market efficiency) and the exclusion of taxes and transaction costs (Hug, 1993, pp. 151-162). Furthermore, in reality a risk-free asset does not exist. Even government bonds, which play this role in the practical usage of the CAPM actually contain risk as well. (Užík, 2004, p. 53)

It is also quite unrealistic that all investors have homogeneous expectations and that they all act rationally, based on the expected return and the standard deviation (Shefrin, 2005, pp. 1-5).

### 3.2 Empirical Testing of CAPM

Obviously, models will always be a simplification of the real world in order to create a processible model. Pierre-Yves Moix states that “Models have proven to be very successful in the field of engineering since they can adequately capture the relationships in the physical world. Economic models on the other hand involve a formalization of human (economic) behavior, which is definitely, and fortunately, too rich to be fully described in quantitative terms.” (Moix, 2001, p. 5) Accordingly, the important part is that the models are good approximations of reality. This can only be decided by empirical tests. The following part will discuss problems involved with testing the CAPM and show results of some empirical tests.

#### 3.2.1 General Testing Problems

##### 3.2.1.1 The Problem of ex ante Data

An important problem when testing the CAPM is that the model should be based on *expected*, forward-looking data. Actually though, it is based entirely on historical data. The expected return is an example and “there is no reason to believe that realized rates of return over the past holding periods are necessarily equal to the expected rates of return.” (Brigham & Gapenski, 1996, p. 84) The same holds true for the beta values. Accordingly, one has to be careful with the result of the tests. (Brigham & Gapenski, 1996, p. 84 and Q3, pp. 133/134)

##### 3.2.1.2 Roll’s critique

Roll argued in 1977 that the CAPM cannot be tested. It can neither be proven wrong nor true. (Armitage, 2005, p. 51) The general problem is that the CAPM is based on the market portfolio. Alongside stocks, the market portfolio actually contains real estate and other risky assets, even human capital. Human capital actually seems quite important since, for example, it is worth about 2/3 of the US-GDP. (Jahnke, 2006, pp. 45/46)

Obviously, it is impossible to hold the market portfolio. Thus, a proxy to the market portfolio has to be found. Roll argues that ex post, the possibility is given to create an efficient portfolio. But this proxy portfolio does not necessarily have to represent the real market portfolio. (Weber, 2006, p. 84) On the other hand, if a portfolio that ex post does not indicate a relation between the mean return and the beta, all one can infer for sure is that the market-proxy portfolio is not efficient ex post. (Armitage, 2005, pp. 51/52)

Thus, nobody so far has neither been able to definitively deny nor prove the CAPM, because it has never ‘really’ been tested (Fama & French, 2004, p. 41).

#### 3.2.2 Results of empirical Tests

Roll’s critique did not stop science from trying to test the CAPM. This paragraph is dedicated to the results that were found, independent from Roll’s critique, and to their conclusions.

##### 3.2.2.1 Controversy about Beta

In 1980 Wallace wrote an article about the CAPM entitled “Is Beta Dead?” (Weber, 2006, p. 84). In an article in 1992, Fama and French came to a similar conclusion, stating that their research did not show that there is any necessary relationship between the average stock return and its betas (Weber, 2006, p. 85).

But others found a correlation and in turn criticized the assumptions made by Fama and French and the database they used (Weber, 2006, p. 85). One of the most important and often cited studies that support the CAPM was carried out by Black, Jensen and Scholes in 1972. The study came to the conclusion that a positive relationship between the beta and the average return exists. Unfortunately, the slope of the security market line was too flat to explain the empirical values. Furthermore, the empirical analysis indicated a risk-free rate that was higher than the actual one. This conclusion was supported by many other studies, e.g. Blume and Friend (1973) and Fama and MacBeth (1973). (Fama & French, 2004, p. 32)

Accordingly, the CAPM approach of trying to explaining all risk with the market risk represented by *β*, seems not to be the perfect choice. A natural reaction to this is the search for other factors that influence risk. A few of the factors that science came up with will be discussed in the next paragraph.

**[...]**

^{[1]} For information on portfolio selection and risk and return fundamentals please refer to standard finance textbooks, e.g. Gitman (2006), pp. 224 ff.

^{[2]} Almost at the same time the model was also independently developed by John Lintner (1965) and Jan Mossin (1966) (Užík, 2004, p. 27). But the main credit is given to Sharpe (Gitman, 2006, p. 246).

- Quote paper
- Manuel Kürschner (Author), 2008, Limitations of the Capital Asset Pricing Model (CAPM), Munich, GRIN Verlag, https://www.grin.com/document/92947

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