2.2 Asset pricing models

  • Asset pricing is a fundamental concept in finance that focuses on determining the value of financial assets, which depends on various risk factors

  • Consequently, different models have been established, such as CAPM, ICAPM, APT, and FF\(3\), which differ in their assumptions

  • The first equilibrium model of asset pricing, known as Capital Asset Pricing Model (CAPM), was developed by Sharpe based on Markowitz’s modern portfolio theory

  • The CAPM says that in market equilibrium the risk premium (measured by excess return) on any asset is directly related to the magnitude of its non-diversifiable risk (beta coefficient)

CAPM is commonly estimated as a single factor time-series regression using the excess returns

\[\begin{equation} r_t-r_f=\alpha+\beta(r_m-r_f)+u_t \tag{2.8} \end{equation}\] \(r_t\) is a return on individual asset (firm specific)
\(r_f\) is a return on a risk-free asset (in practice a Treasury bill rate is used)
\(r_m\) is market return (well diversified portfolio or benchmark market index, like S\(\&\)P\(500\), is used as market proxy)
\(\alpha\) is an intercept (also called Jensen’s alpha)
\(\beta\) is systematic risk (also called non-diversifiable risk or covariance risk)
\(u_t\) is a normally distributed white noise (identically and independetly distributed random errors)

  • The empirical version (2.8) represents the unrestricted CAPM, while the originally proposed version imposes the restriction \(\alpha=0~\), and thus

\[\begin{equation} r_t=r_f+\beta(r_m-r_f)+u_t \tag{2.9} \end{equation}\]

  • Empirical version is used to test the theory, i.e. if the null hypothesis \(H_0:~\alpha=0\) is not rejected it means that the asset’s excess returns are fully explained by its exposure to market risk (beta)

  • Otherwise, positive or negative \(\alpha\) suggest positive or negative abnormal returns, indicating that asset outperforms or underperforms the market

  • CAMP estimation on daily basis requires transformation of daily observed \(1-\)month Treasury bill annualized rates into daily risk-free returns

  • If \(\beta \gt 1\) observed asset is more volatile than the market, if \(0 \leq \beta \leq 1\) the asset is less volatile (more stable) than the market, if \(\beta = 0\) the asset’s returns are uncorrelated with the market’s returns (similar to a risk-free asset), and if \(\beta \lt 0\) the asset moves in the opposite direction to the market (this type of asset can serve as a hedging instrument)

Example 8. According to empirical CAPM does Tesla outperforms or underperforms the market? Is Tesla riskier than the S\(\&\)P500? What is the the percentage of non-diversifiable (market specific) risk and the percentage of diversifiable (firm specific) risk?
Solution Copy the code lines below to the clipboard, paste them into an R Script file, and run them. 
library(quantmod)
library(modelsummary)

# Fetching data of Tesla stock and S&P500 index from Yahoo Finance source
getSymbols(c("TSLA","^GSPC"),
           src="yahoo", 
           from = as.Date("2021-01-01"),
           to = as.Date("2024-12-31"))
tesla.ret= dailyReturn(Cl(TSLA))
market.ret= dailyReturn(Cl(GSPC))
# Fetching 1-month T-bill annualized rates (observed daily) from FRED source
getSymbols("DGS1MO",
           src="FRED",
           from = as.Date("2021-01-01"),
           to = as.Date("2024-12-31"))

riskfree=DGS1MO[index(market.ret)] # keeping only matching dates for T-bills
riskfree=((1+(riskfree/100))^(1/252))-1 # transforming annualized rates to daily risk-free returns

# Estimating CAPM for Tesla using daily observations (few of them are missing with respect to risk-free return)
capm=lm((tesla.ret-riskfree)~(market.ret-riskfree)) # prinosi Tesle regresiraju se na prinose trzista

# Table summary of CAPM
modelsummary(list("CAPM (Tesla)"=capm),stars=TRUE,fmt=4)
  • Merton has introduced the extension of the CAPM well known as the Intertemporal Capital Asset Pricing Model (ICAPM).

Intertemporal Capital Asset Pricing Model (ICAPM):

Then notably Ross (1976) Arbitrage Pricing Theory (APT) multifactor asset pricing model progressed as the another CAPM substitutes. Macroeconomic variables represent multifactor in the Arbitrage Pricing Theory (APT) multifactor asset pricing model. The Arbitrage Pricing Theory (APT) multifactor asset pricing model is much flexible and complex as compared to the CAPM. The APT model of asset pricing measures the expected returns of assets as a linear function of different macroeconomic factors. Arbitrage Pricing Theory (APT):

  • Fama and French developed three factor asset pricing model expanding the Merton’s ICAPM Fama-French three-factor model (FF3):