 # The Capital Asset Pricing Model (CAPM), the Fama-French Model, and the Pastor-Stambaugh Model

## The Capital Asset Pricing Model (CAPM)

According to CAPM, investors evaluate the risk of assets based on the systematic risk they contribute to their total portfolio. The expected return on an asset is calculated as:

$$\text{Required return on share }i=\text{Current expected risk-free return}+\beta_{1}\times (\text{Equity risk premium})$$

$$\text{E}(\text{R}_{i})=\text{R}_{F}+\text{B}_{i}[\text{E}(\text{R}_{\text{M}}-\text{R}_{\text{F}})]$$

The asset’s beta measures its systematic risk, which is the sensitivity of its return to the returns on the market portfolio.

#### Example: Applying CAPM

Consider the following information:

$$\small{\begin{array}{l|r}\text{Current risk-free rate} & 5.0\% \\ \hline\text{Equity market return} & 10\% \\ \hline\text{Market beta} & 1.10\\\end{array}}$$

The required rate of return is closest to:

#### Solution

\begin{align*}\text{Required return}&=\text{R}_{F}+\text{B}_{i}[\text{E}(\text{R}_{\text{M}})-\text{R}_{\text{F}})]\\&=5\%+1.10\%\times[10\%-5\%]\\&=10.5\%\end{align*}

## Multifactor Models

As opposed to the CAPM, which is a single factor model, multifactor models consider multiple factors when estimating the required return. Arbitrage price theory (APT) models estimate required returns based on a set of risk premia. It is expressed as:

$$\text{r}=\text{R}_{\text{F}}+(\text{Risk Premium})_{1}+(\text{Risk Premium})_{2}+…+(\text{Risk Premium})_{k}$$

Where:

$$(\text{Risk premium})_{i}=(\text{Factor sensitivity})_{i}\times(\text{Factor risk premium})_{i}$$

Factor sensitivity is the asset’s sensitivity to a particular factor holding all other factors constant.

The factor risk premium for factor $$i$$ is the expected return above the risk-free rate accruing to an asset with unit sensitivity to factor $$i$$ and zero sensitivity to all other factors.

### The Fama-French Model

The Fama-French model attempts to account for the higher return on small-cap stocks, than that predicted by the CAPM. This model estimates the required return as:

$$\text{r}_{i}=\text{R}_{\text{F}}+\beta_{i}^{mkt}\text{RMRF}+\beta_{i}^{size}\text{SMB}+\beta_{i}^{value}\text{HML}$$

Where:

$$\beta_{mkt}=$$ Market beta.

$$\beta_{size}=$$ Size beta.

$$\beta_{value}=$$ Value beta.

The Fama-French model considers three factors:

1. RMRF: The equity risk premium is calculated as the difference between the return on a value-weighted market index and the risk-free rate.
2. SMB (Small Minus Big): This factor accounts for differences in company market capitalization. It is estimated as the difference between the average return on three small-cap portfolios and the average return on three large-cap portfolios. The smaller the company the greater the required return. It is usually positive for small-cap stocks and negative for large-cap stocks.
3. HML (High Minus Low): It is estimated as the difference between the average return on two high book-to-market (HBM) portfolios, representing value bias, and the average return on two low book-to-market (LBM) portfolios, representing a growth bias. It is usually positive for stocks with high book-to-market ratios and negative for stocks with low book-to-market ratios.

#### Example: Fama-French Model

Consider the following information:

$$\small{\begin{array}{l|r}\text{Risk-free rate} & 5.00\% \\ \hline\text{Equity risk premium} & 7.00\% \\ \hline\text{Beta} & 1.22 \\\hline\text{Size premium} & 4.20\% \\ \hline\text{Size beta} & 0.14 \\ \hline\text{Value premium} & 5.80\% \\ \hline\text{Value beta} & 0.36\\\end{array}}$$

The required return based on the Fama French Model is closest to:

#### Solution

\begin{align*}\text{r}_{i}&=\text{R}_{\text{F}}+\beta_{i}^{mkt}\text{RMRF}+\beta_{i}^{size}\text{SMB}+\beta_{i}^{value}\text{HML}\\&=5\%+1.22\times7\%+0.14\times4.20\%+0.36\times5.80\%\\&=16.22\%\end{align*}

### The Pastor-Stambaugh Model

Investors require a return premium for assets that are relatively illiquid, i.e., assets that cannot be quickly sold in quantity without high transaction costs. The Pastor-Stambaugh model (PSM) adds to the FFM a compensation for the degree of liquidity of an equity investment. It is estimated as:

$$\text{r}_{i}=\text{R}_{\text{F}}+\beta_{i}^{mkt}\text{RMRF}+\beta_{i}^{size}\text{SMB}+\beta_{i}^{value}\text{HML}+\beta_{i}^{liq}\text{LIQ}$$

Where:

$$\beta_{i}^{liq}=$$ Liquidity beta.

Liquidity refers to the ease and potential price impact of the sale of an equity interest into the market.

#### Example: Pastor-Stambaugh Model

Consider the following information about a common stock issue

$$\small{\begin{array}{l|r}\text{Risk-free rate} & 5.00\% \\ \hline\text{Equity risk premium} & 7.00\% \\ \hline\text{Beta} & 1.22 \\ \hline\text{Size premium} & 4.20\% \\ \hline\text{Size beta} & 0.14 \\ \hline\text{Value premium} & 5.80\% \\ \hline\text{Value beta} & 0.36\\ \hline\text{Liquidity premium} & 5\% \\ \hline\text{Liquidity beta} & -0.15 \\\end{array}}$$

Determine the cost of capital using the Pastor-Stambaugh model.

#### Solution

\begin{align*}\text{r}_{i}&=\text{R}_{\text{F}}+\beta_{i}^{mkt}\text{RMRF}+\beta_{i}^{size}\text{SMB}+\beta_{i}^{value}\text{HML}+\beta_{i}^{liq}\text{LIQ}\\&=5\%+1.22\times7\%+0.14\times4.20\%+0.36\times5.80\%-0.15\times5\%\\&=15.47\%\end{align*}

### Macroeconomic and Statistical Multifactor Models

Macroeconomic factors and statistical analyses have also been used to model required returns. Macroeconomic factor models are economic variables that affect the expected future cash flows of companies and/or the discount rate that is appropriate to determine their present values. On the other hand, statistical factor models are applied to historical returns to determine portfolios of securities that explain those returns.

An example of the macroeconomic factor models is the five-factor BIRR model. The model is expressed as:

\begin{align*} \text{r}_{i}&=\text{T-bill rate}\\&+(\text{Sensitivity to confidence risk}\times\text{Confidence risk})\\&+(\text{Sensitivity to time horizon risk} \times \text{Time horizon risk})\\&+(\text{Sensitivity to inflation risk}\times\text{Inflation risk})\\&+(\text{Sensitivity to business cycle risk}\times\text{Business cycle risk})\\&+(\text{Sensitivity to market timing risk}\times\text{Market timing risk}) \end{align*}

The factor definitions of the BIRR model are as outlined below:

1. Confidence risk: The unexpected change in the difference between the return on risky corporate bonds and government bonds of the same maturity. Investors are willing to accept a smaller reward for bearing the added risk of corporate bonds.
2. Time horizon risk: The unexpected change in the difference between the return on 20-year government bonds and 30-day T-bills. It reflects investors’ willingness to invest for the long term.
3. Inflation risk: The unexpected change in the inflation rate. Return decline with positive surprises in inflation. thedentallounge.com
4. Business cycle risk: The unexpected change in the level of real business activity. A positive surprise indicates that the expected growth rate of the economy has increased.
5. Market timing risk: The portion of the total return of an equity market is not explained by the first four factors. Almost all stocks have positive sensitivity to this factor.

#### Example: Applying a Multifactor Model

Consider the following information:

• Confidence risk = 3.0%
• Time horizon risk = 4.0%
• Inflation risk = 5.5%
• Business cycle risk = 3.0%
• Market timing risk = 5.0%

The sensitivities for the stock are 0.2, -0.3, 1.5, 0.1 and 0.3, respectively. Applying a risk-free rate of 2%, the required rate of return from using a multifactor approach is closest to:

#### Solution

\begin{align*} \text{Required return}&=0.02+(0.2\times3\%)+(-0.3\times4\%)\\&+(1.5\times5.5\%)+(0.1\times3\%)+(0.3\times5\%)\\&=11.45\% \end{align*}

## Build-Up Method Estimates of the Required Return on Equity

The buildup method estimates the required return on an equity investment as the sum of the risk-free rate and a set of risk premia. It is usually used to value closely held companies.

$$\text{r}_{i}=\text{Risk-free rate}+\text{Equity risk premium}±\text{One or more premia (discounts)}$$

### Build-Up Approaches for Private Business Valuation

Analysts use a valuation approach that relies on building up the required rate of return by adding a set of premia to the risk-free rate. The premia include the equity risk premium and one or more additional premia based on factors such as size and perceived company-specific risks.

$$\text{r}_{i}=\text{Risk-free rate}+\text{Equity risk premium}+\text{Size premium}_{i}+\text{Specific-company premium}_{i}$$

• Equity risk premium: This is often estimated with reference to equity indexes of publicly traded companies.
• Size premium: This is a premium related to the excess returns of small stocks over large stocks reflecting an incremental return for small size. The level of the size premium is assumed to be inversely related to the size of the company being valued.
• Specific-company premium: This risk premium includes a premium for unsystematic risk of the subject company under the assumption that such risk related to a privately held company may be less easily diversified away.

Two additional issues should be considered when estimating the required return for private companies:

1. The relative values of controlling versus minority interests in share values.
2. The effect on the share value of the lack of marketability for a small equity.

## Bond Yield Plus Risk Premium

For companies with publicly traded debt, the bond yield plus risk premium method can be used to estimate the cost of equity:

$$\text{BYPRP cost of equity}=\text{YTM on the company’s long-term debt}+\text{Risk premium}$$

The YTM on the company’s long-term debt includes:

1. The real interest rate and a premium for expected inflation.

#### Example: Applying the Bond Yield Plus Risk Premium Method

Consider a company that has issued a 20-year bond with a YTM of 5%. A risk premium of 4% is applied to account for the risk associated with the company’s equity. The company’s cost of equity using the bond yield plus risk premium approach is closest to:

#### Solution

\begin{align*}\text{BYPRP cost of equity}&=\text{YTM on the company’s long-term debt}+\text{Risk premium}\\&=5\%+4\%\\&=9\%\end{align*}

## Question

The factor that most likely differentiates the Pastor-Stambaugh model from the Fama-French model is:

1. Liquidity.
2. Size.
3. Value.

#### Solution

The liquidity beta is the risk premium that is added to the Fama-French model when calculating The Pastor-Stambaugh model to account for a relatively illiquid asset.

B and C are incorrect. The size and value betas are risk premiums that are both considered when using the Pastor-Stambaugh model and Fama-French model.

LOS 21 (c) Estimate the required return on an equity investment using the capital asset pricing model, the Fama–French model, the Pastor–Stambaugh model, macroeconomic multifactor models, and the build-up method (e.g., bond yield plus risk premium).

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