4. Examples

Example 1

The risk-free rate of return is 3% and the market risk premium is 5%. A project your company is considering has a beta of 2. Based on this information, what is the appropriate risk-adjusted discount rate for the project's cash flows according to the CAPM?

 

The discount rate is 13%. To see this, we can write down the CAPM equation:

 

\(\mu_{project} = R_F + \beta_{project} \times MRP = 0.03 + 2 \times 0.05 = 0.13 = 13\% \)

 

According to the CAPM, the project's cash flows should therefore be capitalized with a discount rate of 13%. 

 

Example 2

A stock S has a volatility of 40% and exhibits a correlation of 0.75 with the market portfolio. The volatility of the market portfolio is 15% and its expected rate of return is 8%. The risk-free rate is 2%. Based on this information, what is the stock's expected rate of return?

  

First, we have to identify the systematic component of the stock's overall risk of 40%. Given a correlation of 0.75, that systematic risk is 30%. Consequently, the stock's beta is 2:

   

\(\beta_{S} = \frac{\sigma_S \times \rho_{MS}}{\sigma_M} = \frac{0.40 \times 0.74}{0.15} = 2.00 \)

  

Next, we have to extract the expected market risk premium. If the market portfolio's expected return is 8% and the risk-free return is 2%, the market risk premium is 6%:

  

\(MRP = \mu_M - R_F = 0.08 - 0.02 = 0.06 = 6\% \)

  

Now we are ready to plug the information in the CAPM equation:

   

\(\mu_{S} = R_F + \beta_{S} \times MRP = 0.02 + 2 \times 0.06 = 0.14 = 14\% \)

  

According to the CAPM, the stock's expected rate of return is 14%. 

 

We obtain the same beta, and consequently the same CAPM-implied expected rate of return, when we compute beta based on the stock's covariance with the market:

 

\( \sigma_{MS} = \sigma_S \times \sigma_M \times \rho_{MS} = 0.4 \times 0.15 \times 0.75 = 0.045 \)

 

\( \beta_S = \frac{\sigma_{MS}}{\sigma_S^2}=\frac{0.045}{0.15^2} = 2.00 \)

  

Example 3

With a beta of 1.5, a stock currently has an expected rate of return of 9.5%. The risk-free rate is 2%. Due to a change in the business of the firm, the beta of the stock increases to 3. Based on this information, what's the stock's new expected return if the other elements of the CAPM remain unaffected?

 

First, we can use the information about the current valuation (expected return of 7%, beta of 1.5) to extract the implied market risk premium. To do so, we can solve the CAPM equation for the market risk premium (MRP): 

 

\(\mu_{S}=R_F+\beta_{S}\times MRP \implies MRP = \frac{\mu_S-R_F}{\beta_S} = \frac{0.095-0.020}{1.5} = 0.05 = 5\% \)

 

The implied market risk premium is 5%. Now have all the necessary ingredients to compute the CAPM equation with the new beta:

   

\(\mu_{S}=R_F+\beta_{S}\times MRP= 0.02 + 3 \times 0.05 = 0.17 = 17\%\)

  

The stock's cost of capital increases from 9.5% to 17%.

 

Example 4

A stock has a beta of 1.75 and a correlation of 0.7 with the market. The market's volatility is 20%. Based on this information, what is the stock's volatility?

  

To find the answer, we can solve the beta equation for the stock's volatility (\(\sigma_S\)):

 

\( \beta_S=\frac{\sigma_S \times \rho_{MS}}{\sigma_M} \implies \sigma_S = \frac{\beta_S \times \sigma_M}{\rho_{MS}} = \frac{1.75 \times 0.20}{0.70} = 0.5 = 50\% \)

 

Based on the available information, the stock's volatility, therefore, is 50%.