**The Validity of Capital Asset Pricing Model (CAPM) in US Stock Market **

代写金融essay The illiquidity of these stocks weakens the scheme through which the CAPM works (arbitrage finally eliminates any abnormal returns)

**Summary 代写金融essay**

This research studies the validity of CAPM for US stock market from January 2^{nd} 1998 to November 13^{th} 2020.

The specific questions to be explored are:

- Is the systemic premium risk (the regression coefficient of Beta) linear, positive and statistically significant in explaining the expected stock returns?
- Is the rate of return for assets without systemic risk equals to the risk-free interest rate?

This research finds that CAPM beta is linear, positive and statistically significant in explaining the expected stock returns and that the rate of return for assets without systemic risk equals to the risk-free interest rate.

The results of this research implicates that the CAPM is valid for US stock market and on one can outperform the market.

**1 Introduction**

Since the Capital Asset Pricing Model (CAPM) was proposed in 1900s, there has been an ongoing debate about the model. As one core theory of modern finance, the validity of CAPM is being widely studied but there have been mixed results regarding the validity of the CAPM. If the CAPM is invalid and there exist great asset pricing deviations, various crises can occur in the financial market.

For example, investors may form unreliable evaluations about financial assets and make poor investment decisions; level of arbitrage activities may be too high, leading to improper allocation of social resources; enormous mis-priced financial products may even make it impossible to conduct normal financial transactions, inducing financial markets failure.

Thus, the study of validity of CAPM is of great importance in the sense that if CAPM is valid, then we can use it to make proper financial decisions. Instead, if CAPM is invalid and cannot be used to explain any emergence of asset pricing anomalies, then the CAPM needs to be modified so that there is as a “standard” in place for us to correctly evaluate financial products and understand the markets. The results of this study would be used for investment strategy recommendations for investors.

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**2 Literature Review 代写金融essay**

Capital pricing has always been a hot topic in financial area and a lot of endeavors have been done. The capital asset pricing idea can be traced back as early as in 1950s when Markowitz (1952) establishes the modern asset portfolio management theory. Then, Samuelson (1965) formulates the concept of effective market, Sharp (1964) and Litner (1965) and Mossin (1966) combine previous research results and propose the traditional capital asset pricing model (CAPM).

CAPM model is based on the premise of efficient markets. It has strict assumptions such as the frictionless market, same expectations among investors and so on. According to CAPM, there is positive and proportional relationship between returns and risks. CAPM has been considered one classic hypothetical model for asset pricing since its introduction.

The development of CAPM is inseparable from the efficient market hypothesis (EMH). Samuelson (1965) first proposes the concept of effective market to the world in 1965. Since then, after continuous exploration by researchers, it gradually improves and forms a theoretical system. Fama (2014) summarizes it as the efficient market hypothesis and points out that whether asset prices can effectively reflect all available information is the core issue of modern finance.

According to the EMH summarized by Fama (2014), the stock market can comprehensively, timely and accurately reflect the history, current and future information of stocks. According to the different degree of market's response to information, the market can be divided into weak effective market, semi-strong effective market and strong effective market.

In the weakly efficient market, the security price reflects all available historical information; in the semi-strong efficient market, the security price reflects all available historical and current public information; in the strong efficient market, the security price not only reflects all historical information and current public information, but also reflects the private information of investors.

__Early Empirical Examination of CAPM__

__Early Empirical Examination of CAPM__

There are three core points in the empirical test of CAPM based on cross-sectional regression or time series regression. First, the expected return of a security has a linear relationship with its systemic risk beta, and any other variables should have no explanatory power to the expected return; Second, the systemic risk premium (the cross-sectional regression coefficient of Beta) is positive, that is, the higher (lower) rate of return on high (low) risk assets; third, the rate of return for assets without systemic risk is equal to the risk-free interest rate, and the systemic risk premium is equal to the market average rate of return minus the risk-free rate of return.

Major work of empirical examination of the CAPM in the early period until the 1970s mainly includes Jensen, et al. (1972), Fama and MacBeth (1973). The researches conduct empirical tests on CAPM from a cross-sectional perspective. These studies provide strong support for the validity of CAPM theory, that is, the systemic risk of assets Beta has a significantly positive relationship with its expected rate of return.

However, there also exists slight imperfection: the CAPM shape predicted by the theory is too flat compared to the result of empirical test. That is to say, the intercept term obtained by the empirical test is larger than the average risk-free rate of return, while the expected rate of return of risky assets is lower than the theoretical rate of return.

In theory, the zero-beta CAPM derived by Black (1972) under the constraint of risk-free asset borrowing is more consistent with the empirical test results. The empirical results that CAPM is too flat have been further confirmed by time series testing. The main documents include Jensen and Black (1972) and Stambaugh (1982).

In short, the early empirical evidence basically supports CAPM, at least supports the consistency between the zero-beta CAPM theory proposed by Jensen (1972) and the empirical results (Fama,1993).

__Anomalies Against the CAPM 代写金融essay__

__Anomalies Against the CAPM 代写金融essay__

In the late 1970s, more and more empirical evidence began to contradict the predictions of CAPM theory, and a large number of results no longer even support Black (1972)'s zero-beta CAPM (Fama, 1993). Empirical violations of CAPM are mainly in the two following aspects.

First, a series of market pricing anomalies based on the EMH and CAPM continue to emerge, including the earnings-to-price ratio anomaly, scale anomaly, leverage anomaly, value anomaly, momentum anomaly, liquidity anomaly, profitability anomaly, accrual anomalies, investment anomalies, trait volatility anomalies and so on. Harvey, et al. (2016) examine 316 market pricing anomalies recorded in 313 documents (250 published) and find that even if the t-test statistic is increased from 2 to 3, there are still nearly half of the pricing anomalies.

Hou, et al. (2020) systematically comb through the 452 types of pricing anomalies that have been recorded in academia, and find that even by setting more stringent test statistics thresholds and considering the impact of the small market capitalization company effect on the pricing anomalies, there still exist 158 asset pricing anomalies significantly. 代写金融essay

### Second, more and more empirical results have found that the systemic risk beta of assets in CAPM is not correlated with the expected rate of return.

On the basis of the systemic risk beta, Fama and French (1992) add company size, book-to-market ratio, financial leverage ratio, and price-to-earning ratio in to the model, use cross-sectional regression to test the explanatory power of the above five factors on the expected rate of return and find that after controlling for the company size factor, the stock beta is not related to the expected rate of return. This research has triggered a fierce debate on the validity of CAPM theory. Among these controversial studies, the competing discussions between Kothari, et al. (1995) and Fama and French (1996) are particularly eye-catching.

Kothari, et al. (1995) find two main points. First, when using individual stocks’ annual returns rather than monthly returns to estimate beta, the asset’s systemic risk has a significant risk premium. Second, the effect of book-to-market ratio found by Fama and French is mainly caused by the "survivor bias" of small market capitalization companies in the Compustat database, and only a weak effect of book-to-market ratio is found using the S&P Industrial Database. Therefore, Kothari, et al. (1995) provide further empirical support for the validity of CAPM.

### However, Fama and French (1996) put forward tit-for-tat research conclusions regarding the conclusions of Kothari et al. (1995). 代写金融essay

In view of whether the systemic risk premium is positive, Fama and French (1996) continue to use monthly returns to estimate the asset beta, and use the independent grouping method in terms of Beta and market-to-book ratio to control the influence of the company's market value on the relationship between beta and expected return.

However, Fama and French (1996) still haven't found the supporting evidence that risk premium is positive. Regarding whether the book-to-market rate effect is caused by the survivor bias of small market capitalization companies in the Compustat database, Fama and French (1996) cite the research of other scholars to refute, proving that the existence of the book-to-market ratio effect has nothing to do with the survivor bias of small market capitalization companies.

The problems encountered in the CAPM empirical test may be caused by the failure of the theory itself in then sense that its theoretical assumptions are inconsistent with the real world, or the test environment constructed cannot meet the requirements of the theory itself. For example, Roll (1977) believes that the core of testing CAPM is to find the market portfolio on efficient frontier, which includes not only securities assets, but also commodities, real estate, collectibles, human capital and so on. Therefore, in a strict sense, CAPM has never been truly tested (Roll, 1977). This argument is known as the famous "Roll's Critique".

However, Fama (2014) firmly believes that if the systemic risk beta cannot effectively and uniquely explain the difference between asset returns, then the fate of CAPM should end here.

However, due to data availability and/or technology issues, researchers in the above context are not able to employ a large enough dataset to perform better estimations. Meanwhile, as with the perfection of legislation and development of the security market, the financial market condition recently is getting closer to the environment assumptions under the CAPM. Thus, it is necessary to use a wide and updated dataset to re-examine the validity of CAPM. This is the inspiration of this research. 代写金融essay

**3 Rationale**

This research examines the validity of Capital Asset Pricing Model (CAPM) in US stock market during the period of January 2^{nd} 1998 to November 13^{th} 2020. The specific questions to be explored are:

- The systemic premium risk (the regression coefficient of Beta) is linear, positive and efficient in explaining the expected stock returns. This answer to this research question is essential to the validity of CAPM as is pointed out by Fama (2014): if the systemic risk beta cannot effectively and uniquely explain the difference between asset returns, then the fate of CAPM should end. If the estimated beta is positive and statistically significant, then this hypothesis is confirmed.
- The rate of return for assets without systemic risk is equal to the risk-free interest rate (the regression concept equals to zero). This answer to the previously mentioned pricing anomalies: if the rate of return for assets without systemic risk as not equal, then CAPM fails. If the estimated constant is statistically different from zero, then this hypothesis is disconfirmed.

**4 Data, Method and Analysis 代写金融essay**

__Data__

The data of stock price is discovered through Google Dataset Search and accessed through Kaggle (https://www.kaggle.com/tsaustin/us-historical-stock-prices-with-earnings-data). It contains daily data over 20 years (from January 2^{nd} 1998 to November 13^{th} 2020) on 7786 stock prices listed on NASDAQ, NYSE & AMEX.

The yield on three-month treasury bill is used as a proxy for the risk-free rate. The data of daily yield of three-month treasury bill is retrieved from FRED (https://fred.stlouisfed.org/series/DGS3MO).

The market return is calculated as the arithmetic return of NASDAQ Composite, Amex Composite and NYSE composite. The ideal way of to compute market return would be calculating the value-weighted return of all the 7786 stocks. However, since the dataset has no information on market cap and all the three indices are value-weighted, this research calculates the arithmetic return of the tree indices alternatively. The index values of all the tree indices are obtained from Yahoo Finance.

__Method__

__Method__

The model to be tested is the CAPM model: 代写金融essay

where the is the expected return for stock i; is the risk-free returns; is the Beta for stock i; is the excess return of the market.

The corresponding equation to be estimated is:

(1)

To answer the three research questions.

- The systemic premium risk (the regression coefficient of Beta) is linear, positive and statistically significant in explaining the expected stock returns (confirms my hypothesis). The hypothesis to be tested is:

.

- The rate of return for assets without systemic risk is equal to the risk-free interest rate (confirms my hypothesis). The hypothesis to be tested is:

.

__Analysis 代写金融essay__

__Analysis 代写金融essay__

To analyze hypothesis 1, the excess return of each stock (the return of individual stock minus the risk-free return) is regressed against the excess return of the market (the return of market minus the risk return). Then, the t-statistic for individual beta is computed. Next, a right-tailed t-test is carried out for each beta and the corresponding p-value is computed. A smaller p-value indicates rejection of the null hypothesis and the acceptance of the alternative hypothesis, which confirms my hypothesis: beta is linear, positive and statistically significant in explaining the expected stock returns. The testing results for hypothesis 1 is summarized in the table 1 below.

[Table 1 Here]

Based on Table 1, we find out that the CAPM is successful in about 90% cases in the sense that the CAPM is positive and significant in explaining the expected return of individual stocks with a significance level of 5% . For the failed cases, this research finds that most of the failures are due to the less liquidity of the stocks or extremely limited number of observations. For example, for the failing cases, there exist negative betas or zero betas. However, most of cases are stocks that are thinly traded.

#### For example, the most failing 10 cases with the greatest negative betas are for stock markers: 代写金融essay

BSMU, IBTA, IBTE, IBTD, IBTF, IBTG, IBTJ, IBTK, IBTH and IBTI. The average trading volume for these stocks reported by Yahoo Finance is barely above 1,000,000 and some even get an averaged trading volume below 1000. In contrast, regular traded stocks can have millions of trading volumes. For example, AAPL (Apple) has an average volume above 1000,000,000 as is reported by Yahoo Finance as of April 18^{th}, 2021. Moreover, for the failing cases, 76.1% of them has observations less than 1000.

The illiquidity of these stocks weakens the scheme through which the CAPM works (arbitrage finally eliminates any abnormal returns) and the extremely limited sample size for these stocks can lead to giant estimation biases. Thus, this research concludes that the CAPM is positive and significant in explaining the expected return of individual stocks. 代写金融essay

Similarly, to analyze hypothesis 2, the excess return of each stock (the return of individual stock minus the risk-free return) is regressed against the excess return of the market (the return of market minus the risk return). Then, the t-statistic for individual constant in equation (1) is computed. Next, a two-tailed t-test is carried out for each constant and the corresponding p-value is computed.

A smaller p-value indicates rejection of the null hypothesis and the acceptance of the alternative hypothesis, which disconfirms my hypothesis: the rate of return for assets without systemic risk (the regression concept) is equal to the risk-free interest rate. The testing results for hypothesis 2 is summarized in the table 1 below.

#### [Table 2 Here]

Based on Table 2, we find out that the CAPM is successful in over 99% cases in the sense that the rate of return for assets without systemic risk equals to the risk-free interest rate. For the failed cases, there exists positive alpha. In this part, this research similarly finds that most of the failures are due to the less liquidity of the stocks or extremely limited number of observations.

The average trading volume for these stocks reported by Yahoo Finance is barely above 1,000,000 and they have observations no more than 2671. The illiquidity of these stocks weakens the scheme through which the CAPM works and the extremely limited sample size for these stocks can lead to giant estimation biases. Thus, this research concludes that the rate of return for assets without systemic risk equals to the risk-free interest rate.

**5 Significance and Conclusion 代写金融essay**

This research is of great importance in the sense that it once again finds supportive evidence for the validity of CAPM for US stocks using over 20 years of daily price data. The valid CAPM suggests that the expected return of individual stock is positively and significantly correlated with its CAPM beta. Thus, investors can safely make security selection decisions based on their risk-return preferences and the relationship predicted by CAPM. Meanwhile, the validity of CAPM indicates that in the long run, no one can beat the market because in the long run the rate of return for assets without systemic risk equals to the risk-free interest rate (hypothesis 2) and on one can continuously obtain positive abnormal returns.

Put it another way, the EMH holds and all information has been reflected in the market price so technical analysis is useless. As a result, the best investment suggestions for investors is that instead of trying to outperform the market, one better strategy is to invest in an market index as in the long run it would provide higher returns. This strategy is especially true if we consider other costs (such as transaction fees) associated with frequent buying and selling under active trading strategies.

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**6 References 代写金融essay**

Fama, E. F. (2014). Two pillars of asset pricing. *American Economic Review*, *104*(6), 1467-85.

Markowitz, H. (1952) Portfolio Selection. *The Journal of Finance*, Vol. 7, No. 1, pp. 77-91. March. 1952.

Prices, C. A. (1964). A theory of Market Equilibrium under Conditions of Risk. *Journal of Finance*, *19*(3), 425-444.

Lintner, J. (1965). Security prices, risk, and maximal gains from diversification. *The journal of finance*, *20*(4), 587-615.

Mossin, J. (1966). Equilibrium in a capital asset market. *Econometrica: Journal of the econometric society*, 768-783.

Samuelson, P. A. (1965). Proof That Properly Anticipated Prices Fluctuate Randomly. *Management Review*, *6*(2).

Jensen, M. C., Black, F., & Scholes, M. S. (1972). The capital asset pricing model: Some empirical tests.

Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. *Journal of political economy*, *81*(3), 607-636.

#### Black, F. (1972). Capital market equilibrium with restricted borrowing. *The Journal of business*, *45*(3), 444-455.

Stambaugh, R. F. (1982). On the exclusion of assets from tests of the two-parameter model: A sensitivity analysis. *Journal of financial economics*, *10*(3), 237-268.

Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. *Journal of financial economics*, *33*(1), 3-56.

Harvey, C. R., Liu, Y., & Zhu, H. (2016). … and the cross-section of expected returns. *The Review of Financial Studies*, *29*(1), 5-68.

Hou, K., Xue, C., & Zhang, L. (2020). Replicating anomalies. *The Review of Financial Studies*, *33*(5), 2019-2133. 代写金融essay

Fama, E. F., & French, K. R. (1992). The cross‐section of expected stock returns. *the Journal of Finance*, *47*(2), 427-465.

Kothari, S. P., Shanken, J., & Sloan, R. G. (1995). Another look at the cross‐section of expected stock returns. *The journal of finance*, *50*(1), 185-224.

Fama, E. F., & French, K. R. (1996). The CAPM is wanted, dead or alive. *The Journal of Finance*, *51*(5), 1947-1958.

Roll, R. (1977). A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory. *Journal of financial economics*, *4*(2), 129-176.

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**Appendix**

**Appendix 1 Model Output**

Due to the large size of the regression output (7786 Beta and Alpha estimations), output is not displayed here but is stored in myest.dta file. Please refer to the file for model output.

**Appendix 2 - Table 1 Examination Results of Hypothesis 1**

Significance Level | 5% | 10% |

Number of results accepting /rejecting | 6976 | 7086 |

Number of results rejecting /accepting | 810 | 700 |

Total | 7786 | 7786 |

Percentage of results accepting /rejecting | 89.6% | 91.09% |

Table 1 Examination Results of Hypothesis 1

**Appendix 2 - Table 2 Examination Results of Hypothesis 2 代写金融essay**

Significance Level | 5% | 10% |

Number of results accepting /rejecting | 7781 | 7761 |

Number of results rejecting /accepting | 5 | 25 |

Total | 7786 | 7786 |

Percentage of results accepting /rejecting | 99.94% | 99.68% |

Table 2 Examine Results of Hypothesis 2 代写金融essay

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