Excerpt

## Table of Contents

1. Introduction

2. The CAPM

2.1 CAPM - Empirical Evidence

2.2 CAPM - Criticism

3. The three-factor-model

3.1 The three-factor model - Empirical Evidence

3.2 The three-factor model - Criticism

4. Other models

5. Summary of theory

6. Statistical analysis

7. Methodology

8. Results

9. Conclusion

References

Appendix

## 1. Introduction

Scientists use factor models to try to understand the relationship between risk and asset returns and to make estimations of the likely development of the returns in the future (Sharpe 2001, p.1). Today, two of the most renowned factor models^{[1]} to estimate expected returns of an asset or a firm^{[2]} are the Capital Asset Pricing Model (CAPM), introduced by Treynor (1962), Sharpe (1964), Lintner (1965) and Mossin (1966), and the three-factor model of Fama and French of 1992 (Bartholdy and Peare 2004, p.408). While the CAPM claims the existence of a positive linear relationship between the volatility/risk (market beta) and expected returns (Bali and Cakici 2004, p.57), Fama and French state that their three-factor model (3FM) has an improved performance in estimating returns as – so they claim – size and book-to-market equity have significant predictive power, too (Fama and French 1992, p.427).

## 2. The CAPM

The CAPM tries to explain the relationship between risk and return and predicts that the rate of return of a firm or an asset increases with risk, whereas beta is the risk indicator, the relative volatility (Bhatnagar and Ramlogan 2012, p.51). Risk refers to the non-diversifiable, economy-wide risk a company faces due to changes in macroeconomic conditions (such as the growth rate, unemployment, interest rates, inflation etc.; in contrary to company-specific risks).

CAPM considers only one risk factor: the overall market. According to the CAPM an investor is able to attain a higher point on the so-called “capital market line” - a higher rate of return -, only, if he is willing to incur additional risk (Sharpe 1964, p.425). Also, if one knows the beta of an investment, CAPM serves to calculate the theoretically appropriate rate of return.

The CAPM-formula in its simplest form^{[3]}:

**R = Rf + beta ( β ) x (RM – Rf)**

R: The expected return of a company / an asset

Rf: The rate of the risk-free investment

β: The company’s risk (volatility) compared to the beta of the overall market (=COV(R,RM) / VAR (RM))

RM: The expected market return

RM-Rf: The risk premium

One can see that the expected return increases with beta and therefore with increasing risk; higher average returns result from a higher beta and a higher risk premium (RM-Rf). A beta of 1 means the security or portfolio is as volatile and risky as the market, a lower but positive beta that the security or portfolio is less risky and volatile as the market. A beta of zero means the security or portfolio is completely unrelated to the market’s movements. Rf is a constant: The risk-free return that can be expected with a beta of zero.

### 2.1 CAPM - Empirical Evidence

As a scientific theory trying to picture reality, the CAPM has been tested empirically and has its supporters as well as its critics. The discussion whether the model holds or not reminds of the one about the Efficient-Market-Hypothesis (EMH): Although many have declared it dead, it is still alive – after more than 40 years (Ray, Savin and Tiwari 2008, p.721).

Several authors provided empirical evidence that supported the CAPM (or they could at least not refute it) and its claim that on average there is a positive trade-off between risk and return: In his famous paper “Capital market equilibrium with restricted borrowing” Black (1972, p.455) showed that while the positive relationship between risk and return exists, assets with low betas generate higher returns than predicted by the theory; Black, Jensen and Scholes found that “the beta factor seems to be an important determinant of security returns” (1972, p.83); a study of Fama and MacBeth (1973) provided evidence that the claimed risk-return relationship existed for the assets traded at the New York Stock Exchange.

### 2.2 CAPM - Criticism

Critique came from many sides. Roll (1977, p.130) wrote that while the CAPM holds in theory, it is nearly impossible to build a true market portfolio, which makes the theory not testable and that the benchmarks used are only proxies for the “real” portfolio. This might be the reason why the CAPM has never been falsified as its supporters claim.

Banz (1981, p.3) found in a study of firms listed on the New York Stock Exchange that “smaller firms have had higher risk adjusted returns, on average, than larger firms”. His discovery of this “size effect” that had “been in existence for at least forty years” weakened the CAPM. Other studies also identified different factors that influence average stock returns significantly: Rosenberg, Reid and Lanstein (1985) for market equity and book-to-market equity; Bhandari (1988) for leverage; Lakonishok, Shleifer and Vishny (1994) for book to market equity and the cashflow/price ratio.

Grinold (1993, p.28) wrote that what makes beta controversial is primarily “its role in the Capital Asset Pricing Model, which says expected residual returns should be zero”. He also said that beta serves well as a measure of risk and co-movement for portfolios; the notion, however, that expected excess returns move proportionally to beta was doubtful (Grinold 1993, p.33).

Fama and French are among the most dedicated and influential critics of the CAPM. They claim that the “empirical record of the model is poor” and that most applications of the model are invalid (2004, p.25-26). Fama and French tested the theory for cross-sectional stock data of small and large cap stocks and found that CAPM failed to explain stock price developments while estimations using firm size and book-to-market value were more reliable. Their own 3FM uses therefore market return, firm size and the book-to-market ratio to predict returns (Fama and French 1992, p.450-452).

The study of Michailidis et al. (2006) of the Greek stock market refuted the CAPM’s assumptions. Their findings do not support the assumption that higher risk (beta) leads to higher returns.

Jagannathan and McGrattan (1995, p.7) summarize that there is empirical support in favour of the CAPM in the long run, but “for some sample periods, the relation between average return and beta is completely flat”. They also write that the additional factors firm size and book-to-market equity - suggested by Fama and French – do better in explaining the cross-sectional variation in average asset returns.

## 3. The three-factor-model

The 3FM of Fama and French of 1992 is an extension of the CAPM. It tries to fix the claimed weaknesses of the CAPM theory by including the factors size and book-to-market-ratio that Fama and French state have significant influence on firm returns. It takes into account the phenomenon that small cap stocks outperform the market on a regular basis and that value stocks do so, too. A Value stock tends to trade below its intrinsic value relative to its fundamentals (sales, earnings, dividends etc.). A low price-to-book ratio is considered an indicator for a value stock (Fama and French 1998, p.1975).

The Fama and French equation (1995) is an extension of the CAPM:

**R = Rf + beta (β) x (RM – Rf) + S x Size + H x P/B**

R: The expected return of a company / an asset

Rf: The rate of the risk-free investment

β: The company’s risk (volatility) compared to the beta of the overall market

RM: The expected market return

RM-Rf: The risk premium

S: Size-beta, which is positive for a small size that is considered to have a positive influence on the average return.

Size: Total assets

H: P/B-beta, which is positive for a small P/B ratio that is considered to be adding to the adding to the average return.

P/B: Price-to-book ratio

According to Fama and French a higher percentage of the average returns can be explained by their 3FM than by the CAPM – although they admit that the “three-factor risk-return equation is, however, just a model. It surely does not explain return on all securities and portfolios” (Fama and French 1996a, pp.81-82).

For the purpose of this report and because of the available data provided, the original Fama and French formula was slightly simplified (original formula: E(rA) = r(f) + βA(E(rm) - rf) + s x SMB + h x HML). The factor SMB (small minus big) was replaced by size, the value premium factor HML (high minus low) by the price-to-book ratio P/B.

### 3.1 The three-factor model - Empirical Evidence

In several studies, published in 1993, 1995, 1996 and 1998, Fama and French provided more evidence for and additional insights into their own theory and its quality in real world applications. Chan, Hamao and Lakonishok (1991) found evidence for a strong relationship between low P/B-ratios and average return in Japanese securities. Drew, Tony and Veeraragavan (2005) for the Shanghai Stock Exchange as well as Connor and Sehgal (2001) found evidence for a significant impact of size and book-equity-to-market-equity on average returns for the Indian market.

### 3.2 The three-factor model - Criticism

Fama and French’s criticism of the CAPM was itself criticized by others. Several studies challenged their findings. Kothari, Shanken, and Sloan (1995) argue that they depend on the interpretation of the reader. They also suggest that Fama and French’s findings were likely to be influenced by survivorship bias^{[4]}. Amihud, Christensen, and Mendelson (1992) and Black (1993) argue that the data provided by Fama and French to invalidate the CAPM is too noisy, meaning meaningless.

**[...]**

^{[1]} Factor models are also used in capital budgeting, project evaluation, and provide for example a theoretical justification for passive investing.

^{[2]} What is of interest in this report is the modelling of the returns of firms. However, as the model’s most frequent application is the estimation of securities’ returns, many studies concern this kind of use of the model.

^{[3]} Or: R - Rf = beta (β) x (RM – Rf)

^{[4]} The tendency of mutual fund companies to drop mutual funds that have been unsuccessful in the past. This results in an overestimation of past returns of mutual funds.

- Quote paper
- Master of Arts UZH Stefan Heini (Author), 2012, Factor models on explaining firm’s returns in a credit risk context, Munich, GRIN Verlag, https://www.grin.com/document/272034

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