Corporate finance, Module 7, CAPM, covariance, practice problems

(The attached PDF file has better formatting.)

** Exercise 7.1: Beta

The risk-free rate equals the market risk premium, and the covariance of a stocks returns with the overall market return equals the variance of the overall market returns. Because of changes in operations, the covariance of the stocks returns with the overall market returns decreases 10% but the variance of the market return does not change. 

A.    What is the stocks beta before the change in operations?

B.    What is the stocks expected return as a function of the risk-free rate before the change in operations?

C.    What is the stocks beta after the change in operations?

D.    What is the stocks expected return as a function of the risk-free rate after the change in operations?

E.    What is the percentage change in the stocks expected return from the change in opearations?

Part A: The CAPM beta is the covariance of the stocks return with the market return divided by the variance of the market return. In this exercise, the covariance of a stocks returns with the overall market return equals the variance of the overall market returns, so the CAPM beta is one.

Part B: If the risk-free rate is R, the market risk premium is also R. Before the change in operations, the stocks expected return is R + 1.00 R = 2R.

Part C: If the covariance of the stocks return with the market return decreases 10% and the variance of the market return does not change, the CAPM beta decreases 10%.

Part D: After the change in operations, the stocks expected return is R + 0.90 R = 1.90R.

Part E: The percentage change in the expected return is 1.90 / 2.00 1 = 5.00%.

** Exercise 7.2: Beta

The standard deviation of overall market returns is 41%. The correlation of an oil companys stocks returns with the overall market return is 59%, and the standard deviation of the oil stocks returns is 71.5%. 

Because of war in Iran, the standard deviation of the oil companys stocks return increases 25%. The variance of the market return and the correlation of the stocks return with the market return do not change.

A.    What is the formula for the CAPM beta in terms of standard deviations and correlations?

B.    If the standard deviation of the stocks return increases P%, what is the change in its CAPM beta?

Part A: The CAPM beta is the covariance of the stocks return with the market return divided by the variance of the market return. The correlation is the covariance divided by the product of the standard deviations of the two variables. The standard deviation is the square root of the variance.

The CAPM beta () = covariance(r_s, r_m) / σ²(r_m) = [ ρ(r_s, r_m) σ(r_s) σ(r_m) ] / σ²(r_m) = [ ρ(r_s, r_m) σ(r_s) ] / σ(r_m)

Part B: If all else remains the same, the CAPM is proportional to the standard deviation of the stocks return.  If the standard deviation increases 25%, the increases 25%. The figures in the exercise (41%, 59%, 71.5%) do not affect the solution.

** Exercise 7.3: Betas

The CAPM beta of stock W is 1.200, the CAPM beta of stock Y is 0.800, and the risk-free rate equals the market risk premium.

A.    What is the relation of the covariance of Stock Ws return with the market return to the covariance of Stock Ys return with the market return?

B.    What is the relation of the expected return on Stock W to the expected return on Stock Y?

C.    What is the relation of the excess of the expected return for stock W over the risk-free rate (r_w r_f) to the excess of the expected return for stock Y over the risk-free rate (r_y r_f)?

Part A: The CAPM beta is the covariance of returns for the stock with the overall market returns divided by the variance of the overall market returns. The beta of stock W is 50% higher than the beta of stock Y, so the covariance is 50% higher.

Part B: Let the risk-free rate be R, so the market risk premium is also R.

        The expected return on Stock W is R + 1.2 R = 2.2 R.

        The expected return on Stock Y is R + 0.8 R = 1.8 R.

The ratio of the expected returns is 2.2R / 1.8R = 1.222

Part C: The risk-free rate reflects inflation, the time value of money, and the effect of taxes on investment income. The CAPM beta measures the additional return required for risky common stock over risk-free bonds.

From the CAPM equation:

        r_Y r_f = _Y market risk premium

        r_W r_f = _w market risk premium

➾ (r_W r_f) / (r_Y r_f) = _w / _y = 1.200 / 0.800 = 1.5 = 50% greater.

** Exercise 7.4: CAPM Beta and correlation

The variance of the overall market return is 25%, the variance of returns on stock ABC is 49%, and the stocks beta is 0.840.