Lecture 8: Capital Asset Pricing Model (CAPM) – Part 2

1. Derivation of the CAPM Formula

The Capital Asset Pricing Model (CAPM) formula is derived based on the principles of Modern Portfolio Theory (MPT) and the assumptions discussed in Part 1. Here’s a step-by-step explanation of how the formula is derived:

1. Market Portfolio and Risk-Free Asset :

   • The market portfolio (Rm​) represents the optimal risky portfolio that includes all investable assets, weighted by their market value.
   • Investors can combine the market portfolio with the risk-free asset (Rf​) to create portfolios along the Capital Market Line (CML) .

2. Expected Return of a Portfolio :
   The expected return of a portfolio combining the risk-free asset and the market portfolio is:

   
   
    

   Where wf​ is the weight of the risk-free asset.

3. Systematic Risk (Beta) :

   • The total risk of an asset consists of systematic risk (market risk) and unsystematic risk (specific to the asset).
   • Unsystematic risk can be diversified away, leaving only systematic risk relevant for pricing.
   • Systematic risk is measured by beta (βi​) , which quantifies the sensitivity of an asset’s returns to market returns:
   • ​
    

4. Linear Relationship Between Risk and Return :

   • The expected return of an asset is proportional to its systematic risk (βi​).
   • This relationship is expressed in the CAPM formula:
   • 
     Where:
   • E(Ri​): Expected return of asset $i$
   • Rf​: Risk-free rate
   • βi​: Beta coefficient of asset $i$
   • E(Rm​)−Rf​: Market risk premium

2. Security Market Line (SML)

The Security Market Line (SML) is a graphical representation of the CAPM formula. It shows the relationship between systematic risk ($\beta$) and expected return for individual assets or portfolios.

• Key Features of the SML :

  • The x-axis represents beta ($\beta$), measuring systematic risk.
  • The y-axis represents expected return ($E(R)$).
  • The SML starts at the risk-free rate (Rf​) on the y-axis and slopes upward, reflecting the market risk premium (E(Rm​)−Rf​).
   

• Interpretation of the SML :

  • Assets plotting on the SML are fairly priced, as their expected returns match their level of systematic risk.
  • Assets plotting above the SML are undervalued (offer higher returns for the same level of risk).
  • Assets plotting below the SML are overvalued (offer lower returns for the same level of risk).
   

• Equation of the SML :
  The SML equation is identical to the CAPM formula:

   

3. Applications of CAPM in Asset Valuation and Investment Decisions

The CAPM is widely used in finance for various purposes:

1. Asset Pricing :

   • CAPM helps determine whether an asset is fairly valued by comparing its expected return to the return predicted by the model.
   • If E(Ri​) > CAPM-predicted return, the asset is undervalued; if E(Ri​) < CAPM-predicted return, the asset is overvalued.
    

2. Cost of Equity :

   • Companies use CAPM to estimate the cost of equity (Re​) in discounted cash flow (DCF) models for valuing projects or businesses:Re​=Rf​+β⋅(E(Rm​)−Rf​)
    

3. Portfolio Management :

   • Investors use CAPM to construct efficient portfolios by selecting assets with appropriate levels of systematic risk.
    

4. Performance Evaluation :

   • CAPM serves as a benchmark for evaluating the performance of investment managers. Metrics like Jensen’s Alpha measure whether a portfolio outperforms or underperforms relative to its expected return based on CAPM.

4. Criticisms of the CAPM

Despite its widespread use, the CAPM has faced several criticisms due to its simplifying assumptions and empirical limitations:

1. Unrealistic Assumptions :

   • Assumes all investors have access to the same information and share homogeneous expectations, which is unrealistic.
   • Ignores taxes, transaction costs, and borrowing constraints.
    

2. Single-Factor Model :

   • CAPM assumes that systematic risk is the only factor driving returns, ignoring other factors such as company size, book-to-market ratios, or momentum.
    

3. Empirical Challenges :

   • Studies have shown that beta does not fully explain variations in asset returns. Other factors, such as firm size and value, also play a role (e.g., Fama-French Three-Factor Model).
    

4. Difficulty in Estimating Inputs :

   • Estimating inputs like the market risk premium (E(Rm​)−Rf​) and beta ($\beta$) can be challenging and subject to error.
    

5. Risk-Free Rate Limitations :

   • The assumption of a uniform risk-free rate may not hold in practice, especially in volatile or segmented markets.
    

6. Static Nature :

   • CAPM assumes a single-period framework, which may not reflect the dynamic nature of financial markets.

Key Takeaways

1. CAPM Formula : The CAPM formula links systematic risk ($\beta$) to expected return, providing a theoretical basis for asset pricing.
2. Security Market Line (SML) : The SML graphically represents the CAPM, showing the trade-off between systematic risk and expected return.
3. Applications : CAPM is used for asset valuation, cost of equity estimation, portfolio management, and performance evaluation.
4. Criticisms : CAPM faces criticism for its unrealistic assumptions, reliance on a single factor (beta), and empirical inconsistencies.

While CAPM remains a cornerstone of modern finance, practitioners often supplement it with multi-factor models and real-world adjustments to address its limitations. By understanding CAPM and its critiques, investors can make more informed decisions in complex financial markets.