Arbitrage Pricing Theory (APT)

January 2, 2023 by Youssef LOURAOUI

Arbitrage Pricing Theory (APT)

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the concept of arbitrage portfolio, a pillar concept in asset pricing theory.

This article is structured as follows: we present an introduction for the notion of arbitrage portfolio in the context of asset pricing, we present the assumptions and the mathematical foundation of the model and we then illustrate a practical example to complement this post.

Introduction

Arbitrage pricing theory (APT) is a method of explaining asset or portfolio returns that differs from the capital asset pricing model (CAPM). It was created in the 1970s by economist Stephen Ross. Because of its simpler assumptions, arbitrage pricing theory has risen in favor over the years. However, arbitrage pricing theory is far more difficult to apply in practice since it requires a large amount of data and complicated statistical analysis.The following points should be kept in mind when understanding this model:

Arbitrage is the technique of buying and selling the same item at two different prices at the same time for a risk-free profit.
Arbitrage pricing theory (APT) in financial economics assumes that market inefficiencies emerge from time to time but are prevented from occurring by the efforts of arbitrageurs who discover and instantly remove such opportunities as they appear.
APT is formalized through the use of a multi-factor formula that relates the linear relationship between the expected return on an asset and numerous macroeconomic variables.

The concept that mispriced assets can generate short-term, risk-free profit opportunities is inherent in the arbitrage pricing theory. APT varies from the more traditional CAPM in that it employs only one factor. The APT, like the CAPM, assumes that a factor model can accurately characterize the relationship between risk and return.

Assumptions of the APT model

Arbitrage pricing theory, unlike the capital asset pricing model, does not require that investors have efficient portfolios. However, the theory is guided by three underlying assumptions:

Systematic factors explain asset returns.
Diversification allows investors to create a portfolio of assets that eliminates specific risk.
There are no arbitrage opportunities among well-diversified investments. If arbitrage opportunities exist, they will be taken advantage of by investors.

To have a better grasp on the asset pricing theory behind this model, we can recall in the following part the foundation of the CAPM as a complementary explanation for this article.

Capital Asset Pricing Model (CAPM)

William Sharpe, John Lintner, and Jan Mossin separately developed a key capital market theory based on Markowitz’s work: the Capital Asset Pricing Model (CAPM). The CAPM was a huge evolutionary step forward in capital market equilibrium theory, since it enabled investors to appropriately value assets in terms of systematic risk, defined as the market risk which cannot be neutralized by the effect of diversification. In their derivation of the CAPM, Sharpe, Mossin and Litner made significant contributions to the concepts of the Efficient Frontier and Capital Market Line. The seminal contributions of Sharpe, Litner and Mossin would later earn them the Nobel Prize in Economics in 1990.

The CAPM is based on a set of market structure and investor hypotheses:

There are no intermediaries
There are no limits (short selling is possible)
Supply and demand are in balance
There are no transaction costs
An investor’s portfolio value is maximized by maximizing the mean associated with projected returns while reducing risk variance
Investors have simultaneous access to information in order to implement their investment plans
Investors are seen as “rational” and “risk averse”.

Under this framework, the expected return of a given asset is related to its risk measured by the beta and the market risk premium:

CAPM risk beta relation

Where :

E(r_i) represents the expected return of asset i
r_f the risk-free rate
β_i the measure of the risk of asset i
E(r_m) the expected return of the market
E(r_m)- r_f the market risk premium.

In this model, the beta (β) parameter is a key parameter and is defined as:

CAPM beta formula

Where:

Cov(r_i, r_m) represents the covariance of the return of asset i with the return of the market
σ²(r_m) is the variance of the return of the market.

The beta is a measure of how sensitive an asset is to market swings. This risk indicator aids investors in predicting the fluctuations of their asset in relation to the wider market. It compares the volatility of an asset to the systematic risk that exists in the market. The beta is a statistical term that denotes the slope of a line formed by a regression of data points comparing stock returns to market returns. It aids investors in understanding how the asset moves in relation to the market. According to Fama and French (2004), there are two ways to interpret the beta employed in the CAPM:

According to the CAPM formula, beta may be thought in mathematical terms as the slope of the regression between the asset return and the market return. Thus, beta quantifies the asset sensitivity to changes in the market return.
According to the beta formula, it may be understood as the risk that each dollar invested in an asset adds to the market portfolio. This is an economic explanation based on the observation that the market portfolio’s risk (measured by σ²(r_m)) is a weighted average of the covariance risks associated with the assets in the market portfolio, making beta a measure of the covariance risk associated with an asset in comparison to the variance of the market return.

Mathematical foundations

The APT can be described formally by the following equation

APT expected return formula

Where

E(r_p) represents the expected return of portfolio p
r_f the risk-free rate
β_k the sensitivity of the return on portfolio p to the k^th factor (f_k)
λ_k the risk premium for the k^th factor (f_k)
K the number of risk factors

Richard Roll and Stephen Ross found out that the APT can be sensible to the following factors:

Expectations on inflation
Industrial production (GDP)
Risk premiums
Term structure of interest rates

Furthermore, the researchers claim that an asset will have varied sensitivity to the elements indicated above, even if it has the same market factor as described by the CAPM.

Application

For this specific example, we want to understand the asset price behavior of two equity indexes (Nasdaq for the US and Nikkei for Japan) and assess their sensitivity to different macroeconomic factors. We extract a time series for Nasdaq equity index prices, Nikkei equity index prices, USD/CHY FX spot rate and US term structure of interest rate (10y-2y yield spread) from the FRED Economics website, a reliable source for macroeconomic data for the last two decades.

The first factor, which is the USD/CHY (US Dollar/Chinese Renminbi Yuan) exchange rate, is retained as the primary factor to explain portfolio return. Given China’s position as a major economic player and one of the most important markets for the US and Japanese corporations, analyzing the sensitivity of US and Japanese equity returns to changes in the USD/CHY Fx spot rate can help in understanding the market dynamics underlying the US and Japanese equity performance. For instance, Texas Instrument, which operates in the sector of electronics and semiconductors, and Nike both have significant ties to the Chinese market, with an overall exposure representing approximately 55% and 18%, respectively (Barrons, 2022). In the example of Japan, in 2017 the Japanese government invested 117 billion dollars in direct investment in northern China, one of the largest foreign investments in China. Similarly, large Japanese listed businesses get approximately 18% of their international revenues from the Chinese market (The Economist, 2019).

The second factor, which is the 10y-2y yield spread, is linked to the shape of the yield curve. A yield curve that is inverted indicates that long-term interest rates are lower than short-term interest rates. The yield on an inverted yield curve decreases as the maturity date approaches. The inverted yield curve, also known as a negative yield curve, has historically been a reliable indicator of a recession. Analysts frequently condense yield curve signals to the difference between two maturities. According to the paper of Yu et al. (2017), there is a significant link between the effects of varying degrees of yield slope with the performance of US equities between 2006 and 2012. Regardless of market capitalization, the impact of the higher yield slope on stock prices was positive.

The APT applied to this example can be described formally by the following equation:

APT expected return formula example

Where

E(r_p) represents the expected return of portfolio p
r_f the risk-free rate
β_{p, Chinese FX} the sensitivity of the return on portfolio p to the USD/CHY FX spot rate
β_{p, US spread} the sensitivity of the return on portfolio p to the US term structure
λ_{Chinese FX} the risk premium for the FX risk factor
λ_{US spread} the risk premium for the interest rate risk factor

We run a first regression of the Nikkei Japanese equity index returns onto the macroeconomic variables retained in this analysis. We can highlight the following: Both factors are not statistically significant at a 10% significance level, indicating that the factors have poor predictive power in explaining Nikkei 225 returns over the last two decades. The model has a low R2, equivalent to 0.48%, which indicates that only 0.48% of the behavior of Nikkei performance can be attributed to the change in USD/CHY FX spot rate and US term structure of the yield curve (Table 1).

Table 1. Nikkei 225 equity index regression output.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 1 and 2 captures the linear relationship between the USD/CHY FX spot rate and the US term structure with respect to the Nikkei equity index.

Figure 1. Relationship between the USD/CHY FX spot rate with respect to the Nikkei 225 equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 2. Relationship between the US term structure with respect to the Nikkei 225 equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

We conduct a second regression of the Nasdaq US equity index returns on the retained macroeconomic variables. We may emphasize the following: Both factors are not statistically significant at a 10% significance level, indicating that they have a limited ability to predict Nasdaq returns during the past two decades. The model has a low R2 of 4.45%, indicating that only 4.45% of the performance of the Nasdaq can be attributable to the change in the USD/CHY FX spot rate and the US term structure of the yield curve (Table 2).

Table 2. Nasdaq equity index regression output.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 3 and 4 captures the linear relationship between the USD/CHY FX spot rate and the US term structure with respect to the Nasdaq equity index.

Figure 3. Relationship between the USD/CHY FX spot rate with respect to the Nasdaq equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 4. Relationship between the US term structure with respect to the Nasdaq equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Applying APT

We can create a portfolio with similar factor sensitivities as the Arbitrage Portfolio by combining the first two index portfolios (with a Nasdaq Index weight of 40% and a Nikkei Index weight of 60%). This is referred to as the Constructed Index Portfolio. The Arbitrage portfolio will have a full weighting on US equity index (100% Nasdaq equity index). The Constructed Index Portfolio has the same systematic factor betas as the Arbitrage Portfolio, but has a higher expected return (Table 3).

Table 3. Index, constructed and Arbitrage portfolio return and sensitivity table. img_SimTrade_portfolio_sensitivity
Source: computation by the author (Data: FRED Economics)

As a result, the Arbitrage portfolio is overvalued. We will then buy shares of the Constructed Index Portfolio and use the profits to sell shares of the Arbitrage Portfolio. Because every investor would sell an overvalued portfolio and purchase an undervalued portfolio, any arbitrage profit would be wiped out.

Excel file for the APT application

You can find below the Excel spreadsheet that complements the example above.

Why should I be interested in this post?

In the CAPM, the factor is the market factor representing the global uncertainty of the market. In the late 1970s, the portfolio management industry aimed to capture the market portfolio return, but as financial research advanced and certain significant contributions were made, this gave rise to other factor characteristics to capture some additional performance. Analyzing the historical contributions that underpins factor investing is fundamental in order to have a better understanding of the subject.

Useful resources

Academic research

Lintner, J. (1965) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1): 13-37.

Lintner, J. (1965) Security Prices, Risk and Maximal Gains from Diversification. The Journal of Finance 20(4): 587-615.

Roll, Richard & Ross, Stephen. (1995). The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning. Financial Analysts Journal 51, 122-131.

Ross, S. (1976) The arbitrage theory of capital asset pricing Journal of Economic Theory 13(3), 341-360.

Sharpe, W.F. (1963) A Simplified Model for Portfolio Analysis. Management Science 9(2): 277-293.

Sharpe, W.F. (1964) Capital Asset Prices: A theory of Market Equilibrium under Conditions of Risk. The Journal of Finance 19(3): 425-442.

Yu, G., P. Fuller, D. Didia (2013) The Impact of Yield Slope on Stock Performance Southwestern Economic Review 40(1): 1-10.

Business Analysis

Barrons (2022) Apple, Nike, and 6 Other Companies With Big Exposure to China.

The Economist (2019) Japan Inc has thrived in China of late.

Investopedia (2022) Arbitrage Pricing Theory: It’s Not Just Fancy Math.

Time series

FRED Economics (2022) Chinese Yuan Renminbi to U.S. Dollar Spot Exchange Rate (DEXCHUS).

FRED Economics (2022) 10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity (T10Y2Y).

FRED Economics (2022) NASDAQ Composite Index (NASDAQCOM).

FRED Economics (2022) Nikkei Stock Average, Nikkei 225 (NIKKEI225).

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth two-step regression method: the case of K risk factors

December 12, 2022 by Youssef LOURAOUI

Fama-MacBeth two-step regression method: the case of K risk factors

This article is structured as follows: we introduce the Fama-MacBeth two-step regression method. Then, we present the mathematical foundation that underpins their approach for K risk factors. We provide an illustration for the 3-factor mode developed by Fama and French (1993).

Introduction

Risk factors are frequently employed to explain asset returns in asset pricing theories. These risk factors may be macroeconomic (such as consumer inflation or unemployment) or microeconomic (such as firm size or various accounting and financial metrics of the firms). The Fama-MacBeth two-step regression approach found a practical way for measuring how correctly these risk factors explain asset or portfolio returns. The aim of the model is to determine the risk premium associated with the exposure to these risk factors.

The first step is to regress the return of every asset against one or more risk factors using a time-series approach. We obtain the return exposure to each factor called the “betas” or the “factor exposures” or the “factor loadings”.

The second step is to regress the returns of all assets against the asset betas obtained in Step 1 using a cross-section approach. We obtain the risk premium for each factor. Then, Fama and MacBeth assess the expected premium over time for a unit exposure to each risk factor by averaging these coefficients once for each element.

Mathematical foundations

We describe below the mathematical foundations for the Fama-MacBeth regression method for a K-factor application. In the analysis, we investigated the Fame-French three factor model in order to understand their significance as a fundamental driver of returns for investors under the Fama-MacBeth framework.

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed every day over the time period [0, T].
The risk factors denoted by F_k for k equal from 1 to K.

Step 1: time-series analysis of returns

For each asset i from 1 to N, we estimate the following linear regression model:

Fama-French time-series regression

From this model, we obtain the β_{i, Fk} which is the beta associated with the k^th risk factor.

Step 2: cross-sectional analysis of returns

For each period t from 1 to T, we estimate the following linear regression model:

Fama-French cross-sectional regression

Application: the Fama-French 3-Factor model

The Fama-French 3-factor model is an illustration of Fama-MacBeth two-step regression method in the case of K risk factors (K=3). The three factors are the market (MKT) factor, the small minus big (SMB) factor, and the high minus low (HML) factor. The SMB factor measures the difference in expected returns between small and big firms (in terms of market capitalization). The HML factor measures the difference in expected returns between value stocks and growth stock.

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed every day over the time period [0, T].
The risk factors denoted by F_MKT associated to the MKT risk factor, F_SMB associated to the MKT risk factor which measures the difference in expected returns between small and big firms (in terms of market capitalization) and F_HML associated to 𝐻𝑀𝐿 (“High Minus Low”) which measures the difference in expected returns between value stocks and growth stock

Step 1: time-series regression

img_SimTrade_Fama_French_time_series_regression

Step 2: cross-sectional regression

img_SimTrade_Fama_French_cross_sectional_regression

Figure 1 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the MKT factor.

Figure 1. Cross-sectional regression for the market factor.
Cross-section regression for the MKT factor Source: computation by the author.

Figure 2 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the SMB factor.

Figure 2. Cross-sectional regression for the SMB factor.
Cross-section regression for the SMB factor Source: computation by the author.

Figure 3 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the SMB factor.

Figure 3. Cross-sectional regression for the HML factor.
Cross-section regression for the HML factor Source: computation by the author.

Empirical study of the Fama-MacBeth regression

Fama-MacBeth seminal paper (1973) was based on an analysis of the market factor by assessing constructed portfolios of similar betas ranked by increasing values. This approach helped to overcome the shortcoming regarding the stability of the beta and correct for conditional heteroscedasticity derived from the computation of the betas for individual stocks. They performed a second time the cross-sectional regression of monthly portfolio returns based on equity betas to account for the dynamic nature of stock returns, which help to compute a robust standard error and assess if there is any heteroscedasticity in the regression. The conclusion of the seminal paper suggests that the beta is “dead”, in the sense that it cannot explain returns on its own (Fama and MacBeth, 1973).

Empirical study: Stock approach for a K-factor model

We collected a sample of 440 significant firms’ end-of-day stock prices in the US economy from January 3, 2012 to December 31, 2021. We calculated daily returns for each stock as well as the factor used in this analysis. We chose the S&P500 index to represent the market since it is an important worldwide stock benchmark that captures the US equities market.

Time-series regression

To assess the multi-factor regression, we used the Fama-MacBeth 3-factor model as the main factors assessed in this analysis. We regress the average returns for each stock against their factor betas. The first regression is statistically tested. This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2018. We use a t-statistic to explain the regression’s behavior. Because the p-value is in the rejection zone (less than the significance level of 5%), we can conclude that the factors can first explain an investor’s returns. However, as we will see later in the article, when we account for a second regression as proposed by Fama and MacBeth, the factors retained in this analysis are not capable of explaining the return on asset returns on its own. The stock approach produces statistically significant results in time-series regression at 10%, 5%, and even 1% significance levels. As shown in Table 1, the p-value is in the rejection range, indicating that the factors are statistically significant.

Table 1. Time-series regression t-statistic result.
Cross-section regression Source: computation by the author.

Cross-sectional regression

Over a second period from January 04, 2019, to December 31, 2021, we compute the dynamic regression of returns at each data point in time with respect to the betas computed in Step 1.

That being said, when the results are examined using cross-section regression, they are not statistically significant, as indicated by the p-value in Table 2. We are unable to reject the null hypothesis. The premium investors are evaluating cannot be explained solely by the factors assessed. This suggests that factors retained in the analysis fail to adequately explain the behavior of asset returns. These results are consistent with the Fama-MacBeth article (1973).

Table 2. Cross-section regression t-statistic result.
Source: computation by the author.

Excel file

You can find below the Excel spreadsheet that complements the explanations covered in this article.

Why should I be interested in this post?

Fama-MacBeth made a significant contribution to the field of econometrics. Their findings cleared the way for asset pricing theory to gain traction in academic literature. The Capital Asset Pricing Model (CAPM) is far too simplistic for a real-world scenario since the market factor is not the only source that drives returns; asset return is generated from a range of factors, each of which influences the overall return. This framework helps in capturing other sources of return.

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth regression method: the stock approach vs the portfolio approach

December 12, 2022 by Youssef LOURAOUI

Fama-MacBeth regression method: the stock approach vs the portfolio approach

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Fama-MacBeth regression method used to test asset pricing models and addresses the difference when applying the regression method on individual stocks or portfolios composed of stocks with similar betas.

This article is structured as follow: we introduce the Fama-MacBeth testing method. Then, we present the mathematical foundation that underpins their approach. We conduct an empirical analysis on both the stock and the portfolio approach. We conclude with a discussion on econometric issues.

Introduction

As a reminder, the Fama-MacBeth regression method is composed on two steps: step 1 with a time-series regression and step 2 with a cross-section regression.

The first step is to regress the return of every stock against one or more risk factors using a time-series regression. We obtain the return exposure to each factor called the “betas” or the “factor exposures” or the “factor loadings”.

The second step is to regress the returns of all stocks against the asset betas obtained in the first step using a cross-section regression for different periods. We obtain the risk premium for each factor used to test the asset pricing model.

The implementation of this method can be done with individual stocks or with portfolios of stocks as proposed by Fama and MacBeth (1973). Their argument is the better stability of the beta when considering portfolios. In this article we illustrate the difference with the two implementations.

Fama and MacBeth (1973) implemented with individual stocks

We downloaded a sample of daily prices of stocks composing the S&P500 index over the period from January 03, 2012, to December 31, 2021 (we selected the stocks present from the beginning to the end of the period reducing our sample from 500 to 440 stocks). We computed daily returns for each stock and for the market factor retained in this study. To represent the market, we chose the S&P500 index, an important global stock benchmark capturing the US equity market.

The procedure to derive the Fama-MacBeth regression using the stock approach can be achieved as follow:

Step 1: time-series regression

We compute the beta of the stocks with respect to the market factor for the period covered (time-series regression). We estimate the beta of each stock related to the S&P500 index. The beta is computed as the slope of the linear regression of the stock return on the market return (S&P500 index return). This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2018.

Step 2: cross-sectional regression

Over a second period from January 04, 2019, to December 31, 2021, we compute the dynamic regression of returns at each data point in time with respect to the betas computed in Step 1.

With this procedure, we obtain a risk premium that would represent the relationship between the stock returns at each data point in time with their respective beta for the sample analyzed.

Test the statistical significance of the results obtained from the regression

Results in the time-series regression using the stock approach are statistically significant. As shown in Table 1, the p-value is in the rejection area, which implies that the factor that the market factor can be considered as a driver of return.

Table 1. Time-series regression t-statistic result.
img_SimTrade_Fama_MacBeth_cross_sectional_regression_stat_result Source: computation by the author.

However, when analyzed in the cross-section regression, the results are not statistically significant anymore. As shown in Table 2, the p-value is not in the rejection area. We cannot reject the null hypothesis (H0: non significance of the market factor). Market factor alone cannot explain the premium investors are considering.
This means that the market factor fails to explain properly the behavior of asset returns, which undermines the validity of the CAPM framework. These results are in line with the Fama-MacBeth paper (1973).

Table 2. Cross-section regression t-statistic result.
img_SimTrade_Fama_MacBeth_cross_sectional_regression_stat_result Source: computation by the author.

You can find below the Excel spreadsheet that complements the explanations covered in this part of the article (implementation of the Fama and MacBeth (1973) method with individual stocks).

Fama and MacBeth (1973) implemented with portfolios of stocks

The procedure to derive the Fama-MacBeth regression using the portfolio approach can be achieved as follow:

Step 1: time-series regression

We first compute the beta of the stocks with respect to the market factor for the period covered (time-series regression). We estimate the beta of each stock related to the S&P500 index. The beta is computed as the slope of the linear regression of the stock return on the market return (S&P500 index return). This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2015. We build twenty portfolios based on stock betas ranked in ascending order. The betas of the portfolios are then estimated again on a subperiod from January 04, 2016, to December 31, 2018.

It is challenging to maintain beta stability over time. Fama-MacBeth aimed to remedy this shortcoming through its novel technique. However, some issues must be addressed. When betas are calculated using a monthly time series, the statistical noise in the time series is significantly reduced in comparison to shorter time frames (i.e., daily observation). When portfolio betas are constructed, the coefficient becomes considerably more stable than when individual betas are evaluated. This is due to the diversification impact that a portfolio can produce, which considerably reduces the amount of specific risk.

Step 2: cross-sectional regression

Over a second period from January 03, 2019, to December 31, 2021, we compute the dynamic regression of portfolio returns at each data point in time with respect to the betas computed in Step 1.

With this procedure, we obtain a risk premium that would represent the relationship between the portfolio returns at each data point in time with their respective beta for the sample analyzed.

Test the statistical significance of the results obtained from the regression

Results in the cross-section regression using the portfolio approach are not statistically significant. As captured in Table 3, the p-value is not in the rejection area, which implies that the factor is statistically insignificant and that the market factor cannot be considered as a driver of return.

Table 3. Cross-section regression with portfolio approach t-statistic result.
img_SimTrade_Fama_MacBeth_Portfolio_cross_sectional_regression_stat_result Source: computation by the author.

You can find below the Excel spreadsheet that complements the explanations covered in this part of the article (implementation of the Fama and MacBeth (1973) method with portfolios of stocks).

Econometric issues

Errors in data measurement

Because regression uses a sample instead of the entire population, a certain margin of error must be accounted for since the authors derive estimated betas for the sample.

Asset return heteroscedasticity

In econometrics, heteroscedasticity is an important concern since it results in unequal residual variance. This indicates that a time series exhibiting some heteroscedasticity has a non-constant variance, which renders forecasting ineffective because the time series will not revert to its long-run mean.

Asset return autocorrelation

Standard errors in Fama-MacBeth regressions are solely corrected for cross-sectional correlation. This method does not fix the standard errors for time-series autocorrelation. This is typically not a concern for stock trading, as daily and weekly holding periods have modest time-series autocorrelation, whereas autocorrelation is larger over long horizons. This suggests that Fama-MacBeth regression may not be applicable in many corporate finance contexts where project holding durations are typically lengthy.

Why should I be interested in this post?

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth regression method: Analysis of the market factor

December 12, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Fama-MacBeth two-step regression method used to test asset pricing models. The seminal paper by Fama and MacBeth (1973) was based on an investigation of the market factor by evaluating portfolios of stocks with similar betas. In this article I will elaborate on the methodology and assess the statistical significance of the market factor as a fundamental driver of return.

This article is structured as follow: we introduce the Fama-MacBeth testing method used in asset pricing. Then, we present the mathematical foundation that underpins their approach. I then apply the Fama-MacBeth to recent US stock market data. Finally, I expose the limitations of their approach and conclude to discuss the generalization of the original study to other risk factors.

Introduction

The two-step regression method proposed by Fama-MacBeth was originally used in asset pricing to test the Capital Asset Pricing Model (CAPM). In this model, there is only one risk factor determining the variability of returns: the market factor.

The first step is to regress the return of every asset against the risk factor using a time-series approach. We obtain the return exposure to the factor called the “beta” or the “factor exposure” or the “factor loading”.

The second step is to regress the returns of all assets against the asset betas obtained in Step 1 during a given time period using a cross-section approach. We obtain the risk premium associated with the market factor. Then, Fama and MacBeth (1973) assess the expected premium over time for a unit exposure to the risk factor by averaging these coefficients once for each element.

Mathematical foundations

We describe below the mathematical foundations for the Fama-MacBeth two-step regression method.

Step 1: time-series analysis of returns

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed over the time period [0, T].
The risk factor denoted by F for the market factor impacting the asset returns.

For each asset i (for i varying from 1 to N) we estimate the following time-series linear regression model:

Fama MacBeth time-series regression

From this model, we obtain the following coefficients: α_i and β_i which are specific to asset i.

Figure 1 represents for a given asset (Apple stocks) the regression of its return with respect to the S&P500 index return (representing the market factor in the CAPM). The slope of the regression line corresponds to the beta of the regression equation.

Figure 1. Step 1: time-series regression for a given asset (Apple stock and the S&P500 index).
Source: computation by the author.

Step 2: cross-sectional analysis of returns

For each period t (from t equal 1 to T), we estimate the following cross-section linear regression model:

Fama MacBeth cross-section regression

Figure 2 plots for a given period the cross-sectional returns and betas for a given point in time.

Figure 2 represents for a given period the regression of the return of all individual assets with respect to their estimated individual market beta.

Figure 2. Step 2: cross-section regression for a given time-period.

Source: computation by the author.

We average the gamma obtained for each data point. This is the way the Fama-MacBeth method is used to test asset pricing models.

Empirical study of the Fama-MacBeth regression

The seminal paper by Fama and MacBeth (1973) was based on an analysis of the market factor by assessing constructed portfolios of similar betas ranked by increasing values. This approach helped to overcome the shortcoming regarding the stability of the beta and correct for conditional heteroscedasticity derived from the computation of the betas for individual stocks. They performed a second time the cross-sectional regression of monthly portfolio returns based on equity betas to account for the dynamic nature of stock returns, which help to compute a robust standard error and assess if there is any heteroscedasticity in the regression. The conclusion of the seminal paper suggests that the beta is “dead”, in the sense that it cannot explain returns on its own (Fama and MacBeth, 1973).

Empirical study: Stock approach

We downloaded a sample of end-of-month stock prices of large firms in the US economy over the period from March 31, 2016, to March 31, 2022. We computed monthly returns. To represent the market, we chose the S&P500 index.

We then applied the Fama-MacBeth two-step regression method to test the market factor (CAPM).

Figure 3 depicts the computation of average returns and the betas and stock in the analysis.

Figure 3. Computation of average returns and betas of the stocks.
Source: computation by the author.

Figure 4 represents the first step of the Fama-MacBeth regression. We regress the average returns for each stock with their respective betas.

Figure 4. Step 1 of the regression: Time-series analysis of returns
Source: computation by the author.

The initial regression is statistically evaluated. To describe the behavior of the regression, we employ a t-statistic. Since the p-value is in the rejection area (less than the significance limit of 5 percent), we can deduce that the market factor can at first explain the returns of an investor. However, as we are going deal in the later in the article, when we account for a second regression as formulated by Fama and MacBeth (1973), the market factor is not capable of explaining on its own the return of asset returns.

Figure 5 represents Step 2 of the Fama-MacBeth regression, where we perform for a given data point a regression of all individual stock returns with their respective estimated market beta.

Figure 5. Step 2: cross-sectional analysis of return.
Source: computation by the author.

Figure 6 represents the hypothesis testing for the cross-sectional regression. From the results obtained, we can clearly see that the p-value is not in the rejection area (at a 5% significance level), hence we cannot reject the null hypothesis. This means that the market factor fails to explain properly the behavior of asset returns, which undermines the validity of the CAPM framework. These results are in line with Fama-MacBeth (1973).

Figure 6. Hypothesis testing of the cross-sectional regression.
Source: computation by the author.

Excel file for the Fama-MacBeth two-step regression method

You can find below the Excel spreadsheet that complements the explanations covered in this article to implement the Fama-MacBeth two-step regression method.

Limitations of the Fama-McBeth approach

Selection of the market index

For the CAPM to be valid, we need to determine if the market portfolio is in the Markowitz efficient curve. According to Roll (1977), the market portfolio is not observable because it cannot capture all the asset classes (human capital, art, and real estate among others). He then believes that the returns cannot be captured effectively and hence makes the market portfolio, not a reliable factor in determining its efficiency.

Furthermore, the coefficients estimated in the time-series regressions are sensitive to the market index chosen for the study. These shortcomings must be taken into account when assessing CAPM studies.

Stability of the coefficients

The beta of individual assets are not stable over time. Fama and MacBeth attempted to address this shortcoming by implementing an innovative approach.

When betas are computed using a monthly time-series, the statistical noise of the time series is considerably reduced as opposed to shorter time frames (i.e., daily observation).

Using portfolio betas makes the coefficient much more stable than using individual betas. This is due to the diversification effect that a portfolio can achieve, reducing considerably the amount of specific risk.

Conclusion

Why should I be interested in this post?

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

Business Analysis

NEDL. 2022. Fama-MacBeth regression explained: calculating risk premia (Excel). [online] Available at: [Accessed 29 May 2022].

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth two-step regression method

January 30, 2024July 2, 2022 by Youssef LOURAOUI

In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Fama-MacBeth two-step regression method used to test asset pricing models.

This article is structured as follow: we introduce the Fama-MacBeth testing method. Then, we present the mathematical foundation that underpins their approach. We conclude with a practical case study followed by a discussion on econometric issues.

Introduction

Risk factors are frequently employed to explain asset returns in asset pricing theories. These risk factors may be macroeconomic (such as consumer inflation or unemployment) or microeconomic (such as firm size or various accounting and financial metrics of the firms). The Fama-MacBeth two-step regression approach a practical way for measuring how correctly these risk factors explain asset or portfolio returns. The aim of the model is to determine the risk premium associated with the exposure to these risk factors.

Mathematical foundations

We describe below the mathematical foundations for the Fama-MacBeth two-step regression method.

Step 1: time-series analysis of returns

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed over the time period [0, T].
The risk factors denoted by F₁ for the market factor influencing the asset returns.

For each asset i from 1 to N, we estimate the following parameters:

From this model, we obtain a series of coefficients: α_i which is the risk premium for asset i and the β_{i, F₁ associated to the market risk factor.}

Figure 1 represents for a given asset the regression of a return with respect to the market factor (as in the CAPM). The slope of the regression line corresponds to the market beta of the regression.

Figure 1 Time-series regression.
Source : computation by the author.

The econometric issues (estimation bias, heteroscedasticity, and autocorrelation) related to the model are discussed in more details in the econometric limitation section.

Step 2: cross-sectional analysis of returns

For each period t from 1 to T, we estimate the following linear regression model:

Figure 2 plots for a given period the cross-sectional returns and betas for a given point in time.

Figure 2 represents for a given period the regression of the return of all individual assets with respect to their estimated individual market beta.

Figure 2. Cross-sectional regression.
Time-series regression Source: computation by the author.

We average the gamma obtained for each data point. This is the way the Fama-MacBeth method is used to test asset pricing models.

Empirical study by Fama and MacBeth (1973)

Fama-MacBeth performed a second time the cross-sectional regression of monthly stock returns on the equity betas computed on the initial workings to account for the dynamic nature of stock returns, which help to compute a robust standard error to gauge the level of error and assess if there is any heteroscedasticity in the regression. The conclusion of the seminal paper suggests that the beta is “dead”, in the sense that it cannot explain returns on its own (Fama and MacBeth, 1973).

New empirical study

We then applied the Fama-MacBeth two-step regression method to test the market factor (CAPM).

Figure 3 depicts the computation of average returns and the betas and stock in the analysis.

Figure 3. Computation of average returns and betas of the stocks.
Source: computation by the author.

Figure 4 represents the first step of the Fama-MacBeth regression. We regress the average returns for each stock with their respective betas.

Figure 4. Step 1 of the regression: Time-series analysis of returns
Source: computation by the author.

The initial regression is statistically evaluated. To describe the behaviour of the regression, we employ a t-statistic. Since the p-value is in the rejection area (less than the significance limit of 5 percent), we can deduce that the market factor can at first explain the returns of an investor. However, as we are going deal in the later in the article, when we account for a second regression as formulated by Fama and MacBeth, the market factor is not capable of explaining on its own the return of asset returns.

Figure 5 represents Step 2 of the Fama-MacBeth regression, where we perform for a given data point a regression of all individual stock returns with their respective estimated market beta.

Figure 5. Step 2: cross-sectional analysis of return.
Source : computation by the author.

Figure 6 represents the hypothesis testing for the cross-sectional regression. From the results obtained, we can clearly see that the p-value is not in the rejection area (at a 5% significance level), hence we cannot reject the null hypothesis. This means that the market factor fails to explain properly the behaviour of asset returns, which undermines the validity of the CAPM framework. These results are in line with the Fama-MacBeth paper (1973).

Figure 6. Hypothesis testing of the cross-sectional regression.
Source: computation by the author.

Excel file for the Fama-MacBeth two-step regression method

You can find below the Excel spreadsheet that complements the explanations covered in this article to apply the Fama-MacBeth two-step regression method.

Econometric issues

Errors in data measurement

Because regression uses a sample instead of the entire population, a certain margin of error must be accounted for since the authors derive estimated betas for the sample.

Asset return heteroscedasticity

Asset return autocorrelation

Limitation of CAPM

Roll: selection of the appropriate market index

Furthermore, the coefficients are sensitive to the market index chosen for the study. These shortcomings must be taken into account when assessing other CAPM studies.

Fama-MacBeth: Stability of the coefficients

The stability of the beta across time is difficult. Fama-MacBeth attempted to address this shortcoming by implementing its innovative approach. However, some points need to be addressed:

When betas are computed using a monthly time series, the statistical noise of the time series is considerably reduced as opposed to shorter time frames (i.e., daily observation).

Constructing portfolio betas makes the coefficient much more stable than when assessing individual betas. This is due to the diversification effect that a portfolio can achieve, reducing considerably the amount of specific risk.

Why should I be interested in this post?

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

Business Analysis

NEDL. 2022. Fama-MacBeth regression explained: calculating risk premia (Excel). [online] Available at: [Accessed 29 May 2022].

About the author

The article was written in May 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Systematic risk and specific risk

November 30, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the systematic risk and specific risk of financial assets, two fundamental concepts in asset pricing models and investment management theories more generally.

This article is structured as follows: we introduce the concept of systematic and specific risk. We then explain the mathematical foundation of this concept. We finish with an insight that sheds light on the relationship between diversification and risk reduction.

Portfolio Theory and Risk

Markowitz (1952) and Sharpe (1964) developed a framework on risk based on their significant work in portfolio theory and capital market theory. All rational profit-maximizing investors seek to possess a diversified portfolio of risky assets, and they borrow or lend to get to a risk level that is compatible with their risk preferences under a set of assumptions. They demonstrated that the key risk measure for an individual asset is its covariance with the market portfolio under these circumstances (the beta).

The fraction of an individual asset’s total variance attributable to the variability of the total market portfolio is referred to as systematic risk, which is assessed by the asset’s covariance with the market portfolio. In the article systematic risk, we develop the economic sources of systematic risk: interest rate risk, inflation risk, exchange rate risk, geopolitical risk, and natural risk.

Additionally, due to the asset’s unique characteristics, an individual asset exhibits variance that is unrelated to the market portfolio (the asset’s non-market variance). Specific risk is the term for non-market variance, and it is often seen as minor because it can be eliminated in a large diversified portfolio. In the article specific risk, we develop the economic sources of specific risk: business risk and financial risk.

Mathematical foundations

Following the Capital Asset Pricing Model (CAPM), the return on asset i, denoted by R_i can be decomposed as

img_SimTrade_return_decomposition

Where:

R_i the return of asset i
E(R_i) the expected return of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the expected return of the market
R_M – E(R_M) the market factor
ε_i the specific part of the return of asset i

The three components of the decomposition are the expected return, the market factor and an idiosyncratic component related to asset only. As the expected return is known over the period, there are only two sources of risk: systematic risk (related to the market factor) and specific risk (related to the idiosyncratic component).

The beta of the asset with the market is computed as:

Beta

Where:

σ_i,m : the covariance of the asset return with the market return
σ_m² : the variance of market return

Total risk can be deconstructed into two main blocks:

Total risk formula

The total risk of the asset measured by the variance of asset returns can be computed as:

Decomposition of total risk

Where:

β_i² * σ_m² = systematic risk
σ_εi² = specific risk

In this decomposition of the total variance, the first component corresponds to the systematic risk and the second component to the specific risk.

Effect of diversification on portfolio risk

Diversification’s objective is to reduce the portfolio’s standard deviation. This assumes an imperfect correlation between securities. Ideally, as investors add securities, the portfolio’s average covariance decreases. How many securities must be included to create a portfolio that is completely diversified? To determine the answer, investors must observe what happens as the portfolio’s sample size increases by adding securities with some positive correlation. Figure 1 illustrates the effect of diversification on portfolio risk, more precisely on total risk and its two components (systematic risk and specific risk).

Figure 1. Effect of diversification on portfolio risk

Source: Computations from the author.

The critical point is that by adding stocks that are not perfectly correlated with those already held, investors can reduce the portfolio’s overall standard deviation, which will eventually equal that of the market portfolio. At that point, investors eliminated all specific risk but retained market or systematic risk. There is no way to completely eliminate the volatility and uncertainty associated with macroeconomic factors that affect all risky assets. Additionally, investors can reduce systematic risk by diversifying globally rather than just within the United States, as some systematic risk factors in the United States market (for example, US monetary policy) are not perfectly correlated with systematic risk variables in other countries such as Germany and Japan. As a result, global diversification eventually reduces risk to a global systematic risk level.

You can download below two Excel files which illustrate the effect of diversification on portfolio risk.

The first Excel file deals with the case of independent assets with the same profile (risk and expected return).

Figure 2 depicts the risk reduction of total risk in as we increase the number of assets in the portfolio. We manage to reduce half of the overall portfolio volatility by adding five assets to the portfolio. However, the decrease becomes more and more marginal as we add more assets.

Figure 2. Risk reduction of the portfolio. img_SimTrade_systematic_specific_risk_1 Source: Computations from the author.

Figure 3 depicts the overall risk reduction of a portfolio. The benefit of diversification are more evident when we add the first 5 assets in the portfolio. As depicted in Figure 2, the diversification starts to fade at a certain point as we keep adding more assets in the portfolio. It can be seen in this figure how the specific risk is considerably reduced as we add more assets because of the effect of diversification. Systematic risk (market risk) is more constant and doesn’t change drastically as we diversify the portfolio. Overall, we can clearly see that diversification helps decrease the total risk of a portfolio considerably.

Figure 3. Risk decomposition of the portfolio. img_SimTrade_systematic_specific_risk_2 Source: Computations from the author.

The second Excel file deals with the case of dependent assets with the different characteristics (expected return, volatility, and market beta).

Academic research

A series of studies examined the average standard deviation for a variety of portfolios of randomly chosen stocks with varying sample sizes. Evans and Archer (1968) and Tole (1982) calculated the standard deviation for portfolios up to a maximum of twenty stocks. The results indicated that the majority of the benefits of diversification were obtained relatively quickly, with approximately 90% of the maximum benefit of diversification being obtained from portfolios of 12 to 18 stocks. Figure 1 illustrates this effect graphically.

This finding has been modified in two subsequent studies. Statman (1987) examined the trade-off between diversification benefits and the additional transaction costs associated with portfolio expansion. He concluded that a portfolio that is sufficiently diversified should contain at least 30–40 stocks. Campbell, Lettau, Malkiel, and Xu (2001) demonstrated that as the idiosyncratic component of an individual stock’s total risk (specific risk) has increased in recent years, it now requires a portfolio to contain more stocks to achieve the same level of diversification. For example, they demonstrated that the level of diversification possible in the 1960s with only 20 stocks would require approximately 50 stocks by the late 1990s (Reilly and Brown, 2012).

Figure 4. Effect of diversification on portfolio risk Source: Computation from the author.

You can download below the Excel file which illustrates the effect of diversification on portfolio risk with real assets (Apple, Microsoft, Amazon, etc.). The effect of diversification on the total risk of the portfolio is already significant with the addition of few stocks.

We can appreciate the decomposition of total risk in the below figure with real asset. We can appreciate how asset with low beta had the lowest systematic out of the sample analyzed (i.e. Pfizer). For the whole sample, specific risk is a major concern which makes the major component of risk of each stock. This can be mitigated by holding a well-diversified portfolio that can mitigate this component of risk. Figure 5 depicts the decomposition of total risk for assets (Apple, Microsoft, Amazon, Goldman Sachs and Pfizer).

Figure 5. Decomposition of total risk Source: Computation from the author.

You can download below the Excel file which deconstructs the risk of assets (Apple, Microsoft, Amazon, Goldman Sachs, and Pfizer).

Why should I be interested in this post?

If you’re an investor, understanding the source of risk is essential in order to build balanced portfolios that can withstand market corrections and downturns.

If you are a business school or university undergraduate or graduate student, this content will help you in broadening your knowledge of finance.

Useful resources

Academic research

Campbell, J.Y., Lettau, M., Malkiel, B.G. and Xu, Y. 2001. Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk. The Journal of Finance, 56: 1-43.

Evans, J.L., Archer, S.H. 1968. Diversification and the Reduction of Dispersion: An Empirical Analysis. The Journal of Finance, 23(5): 761–767.

Markowitz, H. 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica, 34(4): 768-783.

Reilly, R.K., Brown C.K. 2012. Investment Analysis & Portfolio Management, Tenth Edition. 239-245.

Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. Management Science, 9(2): 277-293.

Sharpe, W.F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3): 425-442.

Statman, M. 1987. How Many Stocks Make a Diversified Portfolio?. The Journal of Financial and Quantitative Analysis, 22(3), 353–363.

Tole T.M. 1982. You can’t diversify without diversifying. The Journal of Portfolio Management. Jan 1982, 8 (2) 5-11.

About the author

The article was written in November 2021 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Security Market Line (SML)

November 20, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Security Market Line (SML), a key concept in asset pricing derived from the Capital Asset Pricing Model (CAPM).

This article is structured as follows: we first introduce the concept of Security Market Line (SML). We then present the mathematical foundations of the SML. We finish by presenting an investment strategy that can be implemented relying on the SML.

Security Market Line

The SML reflects the risk-return combinations accessible in the capital market at any given time for all risky assets. Investors would choose investments based on their risk appetites; some would only consider low-risk investments, while others would welcome high-risk investments. The SML is derived from the Capital Asset Pricing Model (CAPM), which describes the trade-off between risk and expected return for efficient portfolios.

The expected relationship between risk and return is depicted in Figure 1. It demonstrates that as perceived risk increases, investors’ required rates of return increase.

Figure 1. Security Market Line.

Source: Computation by the author.

Under the CAPM framework, all investors will choose a position on the capital market line by borrowing or lending at the risk-free rate, since this maximizes the return for a given level of risk. Whereas the CML indicates the rates of return of a specific portfolio, the SML represents the risk and return of the market at a given point in time and indicates the expected returns of individual assets. Also, while the measure of risk in the CML is the standard deviation of returns (total risk), the measure of risk in the SML is the systematic risk, or beta. Figure 2 depicts the SML line combined with four different assets. Asset A and B are above the SML line, which implies that they are overvalued. Asset C and D are below the SML which implies that they are undervalued. From Figure 2, we can implement an investment strategy by going long if the asset or portfolio lies under the SML and going short if the asset or portfolio is greater than the SML.

Figure 2. Security Market Line with a plot of different assets.

Source: Computation by the author.

Mathematical foundation

The SML plots an individual security’s expected rate of return against systematic, undiversifiable risk. The risk associated with an individual risky security is determined by the volatility of the security’s return, not by the market portfolio’s return. Individual risky securities bear a proportional share of the systematic risk. The only risk that an investor should be compensated for is systematic risk, which cannot be neutralized through diversification. This risk is quantified using the beta, which refers to a security’s sensitivity to market fluctuations. The slope of the SML is equal to the market risk premium and reflects the risk-reward trade-off at a particular point in time. We can define the line of the SML as:

Mathematically, we can deconstruct the SML as:

SML_formula

Where

E(R_i) represents the expected return of asset i
R_f is the risk-free interest rate
β_i measures the systematic risk of asset i
E(R_M) represents the expected return of the market
E[R_M – R_f] represents the market risk premium.

Beta and the market factor

William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) independently developed the Capital Asset Pricing Model (CAPM). The CAPM was a significant evolutionary step forward in capital market equilibrium theory because it allowed investors to value assets correctly in terms of risk. The CAPM makes a distinction between two forms of risk: systematic and specific risk. Systematic risk refers to the risk posed by the market’s basic structure, its participants, and all non-diversifiable elements such as monetary policy, political events, and natural disasters. By contrast, specific risk refers to the risk inherent in a particular asset and so is diversifiable. As a result, the CAPM solely captures systematic risk via the beta measure, with the market’s beta equal to one, lower-risk assets having a beta less than one, and higher-risk assets having a beta larger than one.

In the late 1970s, the portfolio management industry sought to replicate the market portfolio return, but as financial research advanced and significant contributions were made, it enabled the development of additional factor characteristics to capture additional performance. This resulted in the development of what is now known as factor investing.

Estimation of the Security Market Line

You can download an Excel file with data to estimate the Security Market Line.

Why should I be interested in this post?

The security market line is frequently used by portfolio managers and investors to determine the suitability of an investment product for inclusion in a portfolio. The SML is useful for determining whether a security’s expected return is favourable in comparison to its level of risk. The SML is frequently used to compare two similar securities that offer approximately the same rate of return to determine which one has the lowest inherent market risk in relation to the expected rate of return. Additionally, the SML can be used to compare securities of comparable risk to determine which one offers the highest expected return for that level of risk.

If you are a business school or university undergraduate or graduate student, this content will help you in broadening your knowledge of finance.

Useful resources

Academic research

Drake, P. and Fabozzi, F., 2010. The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management. John Wiley and Sons Edition.

Lintner, J. 1965a. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1): 13-37.

Lintner, J. 1965b. 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, 34(4): 768-783.

Reilly, R. K., Brown C. K., 2012. Investment Analysis & Portfolio Management, Tenth Edition.

Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. Management Science, 9(2): 277-293.

Sharpe, W.F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3): 425-442.

About the author

The article was written in November 2021 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Capital Asset Pricing Model (CAPM)

September 6, 2021 by Jayati WALIA

Capital Asset Pricing Model (CAPM)

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the Capital Asset Pricing Model (CAPM).

Introduction

The Capital Asset Pricing Model (CAPM) is a widely used metrics for the financial analysis of the performance of stocks. It shows the relationship between the expected return and the systematic risk of investing in an asset. The idea behind the model is that the higher the risk in an investment in securities, the higher the returns an investor should expect on his/her investments.

The Capital Asset Pricing Model was developed by financial economists William Sharpe, John Lintner, Jack Treynor and Jan Mossin independently in the 1960s. The CAPM is essentially built on the concepts of the Modern Portfolio Theory (MPT), especially the mean-variance analysis model by Harry Markowitz (1952).

CAPM is very often used in the finance industry to calculate the cost of equity or expected returns from a security which is essentially the discount rate. It is an important tool to compute the Weighted Average Cost of Capital (WACC). The discount rate is then used to ascertain the Present Value (PV) and Net Present Value (NPV) of any business or financial investment.

CAPM formula

The main result of the CAPM is a simple mathematical formula that links the expected return of an asset to its risk measured by the beta of the asset:

CAPM risk beta relation

Where:

E(r_i) represents the expected return of asset i
r_f the risk-free rate
β_i the measure of the risk of asset i
E(r_m) the expected return of the market
E(r_m)- r_f the market risk premium.

The risk premium for asset i is equal to β_i(E(r_m)- r_f), that is the beta of asset i, β_i, multiplied by the risk premium for the market, E(r_m)- r_f.

The formula shows that investors demand a return higher than the risk-free rate for taking higher risk. The equity risk premium is the component that reflects the excess return investors require on their investment.

Let us discuss the components of the Capital Asset Pricing Model individually:

Expected return of the asset: E(r_i)

The expected return of the asset is essentially the minimum return that the investor should demand when investing his/her money in the asset. It can also be considered as the discount rate the investor can utilize to ascertain the value of the asset.

Risk-free interest rate: r_f

The risk-free interest rate is usually taken as the yield on debt issued by the government (the 3-month Treasury bills and the 10-year Treasury bonds in the US) as they are the safest investments. As government bonds have very rare chances of default, their interest rates are considered risk-free.

Beta: β

The beta is a measure of the systematic or the non-diversifiable risk of an asset. This essentially means the sensitivity of an asset price compared to the overall market. The market beta is equal to 1. A beta greater than 1 for an asset signifies that the asset is riskier compared to the overall market, and a beta of less than 1 signifies that the asset is less risky compared to the overall market.

The beta is calculated by using the equation:

CAPM beta formula

Where:

Cov(r_i, r_m) represents the covariance of the return of asset i with the return of the market
σ²(r_m) the variance of the return of the market.

The beta of an asset is defined as the ratio of the covariance between the asset return and the market return, and the variance of the market return.

The covariance is a measure of correlation between two random variables. In practice, the covariance is calculated using historical data for the asset return and the market return.

The variance is a measure of the dispersion of returns. The standard deviation, equal to the square root of the variance, is a measure of the volatility in the market returns over time.

Expected market return

The expected market return is usually computed using historical data of the market. The market is usually represented by a stock index to which the stock belongs to.

For example, for calculating the expected return on APPLE stock, we usually consider the S&P 500 index. Historically, the expected return for the S&P 500 index is around 9%.

Assumptions in Capital Asset Pricing Model

The CAPM considers the following assumptions which forms the basis for the model:

Investors are risk averse and rational – In the CAPM, all investors are assumed to be risk averse. They diversify their portfolio which neutralizes the non-systematic or the diversifiable risk. So, in the end only the systematic or the market risk is considered to calculate the expected returns on the security.
Efficient markets – The markets are assumed to be efficient, thus all investors have equal access to the same information. Also, all the assets are considered to be liquid, and an individual investor cannot influence the future prices of an asset.
No transaction costs – The CAPM assumes that there are no transaction costs, taxes, and restrictions on borrowing or lending activities.
Risk premium – The CAPM model assumes that investors require higher premium for more risk they take (risk aversion).

Example

As an example, lest us consider an investor who wants to calculate the expected return on an investment in APPLE stock. Let’s see how the CAPM can be used in this case.

The risk-free interest rate is taken to be the current yield on 10-year US Treasury bonds. Let us assume that its value is 3%.

The S&P 500 index has an expected return of 9%.

The beta on APPLE stock is 1.25.

The expected return on APPLE stock is equal to 3% + 1.25*(9% – 3%) = 10.50%

Useful resources

Acadedmic articles

Lintner, J. (1965) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets The Review of Economics and Statistics 47(1) 13-37.

Markowitz, H. (1952) Portfolio Selection The Journal of Finance 7(1) 77-91.

Mossin, J. (1966) Equilibrium in a Capital Asset Market Econometrica 34(4) 768-783.

Merton, R.C. (1973) An Intertemporal Capital Asset Pricing Model Econometrica 41(5) 867-887.

Sharpe, W.F. (1964) Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk The Journal of Finance 19(3) 425-442.

Business sources

Mullins, D.W. Jr (1982) Does the Capital Asset Pricing Model Work? Harvard Business Review.

About the author

The article was written in September 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

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