Market and Portfolio Beta

EXERCISE 01

Correlation with the market is 0.6; asset volatility is 20.000%, market volatility 16.000%, risk-free return 4.000% and market risk premium 8.000%.

A.Calculate beta and the CAPM required return.

A.

\beta_i=\rho_{iM}\frac{\sigma_i}{\sigma_M},\quad E[R_i]=r_f+\beta_i(E[R_M]-r_f)

\beta_i=0.6\frac{0.2}{0.16}\approx 0.75

E[R_i]=0.04+(0.75)(0.08)\approx 0.1

Beta = 0.75000; required return = 10.000%.

Market and Portfolio Beta

EXERCISE 02

A.What does beta measure? Is it proportional to the correlation with market returns?

A.

\beta_i=\frac{Cov(R_i,R_M)}{Var(R_M)}=\rho_{iM}\frac{\sigma_i}{\sigma_M}

Beta measures the sensitivity of a security’s return to the market return, or its contribution to systematic risk. It equals correlation multiplied by the ratio of security volatility to market volatility. It is proportional to correlation only with those volatilities held fixed; a high-volatility stock need not have a high beta.

Market and Portfolio Beta

EXERCISE 03

Risk-free rate = 4.000%; market risk premium = 6.000%.

Asset	Weight	Beta
1	0.60000	2.16
2	0.40000	0.69

A.Calculate portfolio beta and expected return.

A.

\beta_P=\sum_iw_i\beta_i,\quad E[R_P]=r_f+\beta_PMRP

\beta_P=(0.6)(2.16)+(0.4)(0.69)\approx 1.572

E[R_P]=0.04+(1.572)(0.06)\approx 0.13432

Beta = 1.57200; expected return = 13.432%.

Market and Portfolio Beta

EXERCISE 04

Your portfolio beta is 1.18. Weights are 15% risk-free, 30% stock A and 55% stock B. Stock A has the same systematic risk as the market.

A.Calculate beta of B.

A.

\beta_P=\sum_iw_i\beta_i

1.18=0.15(0)+0.30(1)+0.55\beta_B

\beta_B=\frac{1.18-0.30}{0.55}

1.60.

Market and Portfolio Beta

EXERCISE 05

The S&P 500 has expected return 16% and volatility 25%. Microsoft volatility is stated as 20% and covariance with the S&P 500 as 0.10. The risk-free rate is 6%. Intel’s expected return is half Microsoft’s. Portfolio weights are 0.25 Microsoft, 0.10 Intel, 0.75 S&P 500, −0.20 GM (beta 0.80), and 0.10 risk-free.

A.Check whether the inputs are jointly feasible.
B.Compute the formal CAPM results using the covariance as supplied.

A.

|Cov(i,M)|\leq\sigma_i\sigma_M=0.20(0.25)=0.05

The stated covariance 0.10 violates this bound.

There is no joint return distribution with all the stated statistics.

B.

\beta_{MS}=0.10/0.25^2=1.6,\quad E[R_{MS}]=22\%

\beta_{Intel}=(0.11-0.06)/0.10=0.5

\beta_P=0.25(1.6)+0.10(0.5)+0.75(1)-0.20(0.8)+0.10(0)=1.04

E[R_P]=0.06+1.04(0.16-0.06)=16.4\%

Portfolio beta = 1.04; formal expected return = 16.4%.

These are algebraic results from the supplied covariance; they do not remove the inconsistency identified above.