Inferring risk-free and market returns

EXERCISE 01

A correctly priced stock has expected return 9.65%, volatility 25% and beta 1.1. The risk-free return is 3%.

A.Infer the market expected return.

A.

E[R_M]=r_f+\frac{E[R_i]-r_f}{\beta_i}

E[R_M]=0.03+\frac{0.0965-0.03}{1.1}

9.05%

The supplied volatility is not needed. The result from the stated inputs is 9.04545%, rather than exactly 9%.

Inferring risk-free and market returns

EXERCISE 02

CAPM holds. Stock A has expected return 3.5% and beta 0.5; B has expected return 8% and beta 2.

A.Find the market risk premium, risk-free rate and market return.

A.

E[R_i]=r_f+\beta_iMRP

MRP=\frac{0.08-0.035}{2-0.5}=0.03

r_f=0.035-0.5(0.03)=0.02

E[R_M]=0.02+0.03

Risk-free rate 2%; market expected return 5%.

Inferring risk-free and market returns

EXERCISE 03

A has beta 1.5, variance 0.1225 and expected return 11%. B has beta 0.7, variance 0.0324 and expected return 7.8%. The market has variance 0.04 and expected return 9%.

A.Infer the risk-free rate and market premium.
B.Find correlations with the market.
C.Find a mixture of A and B returning 9.5%, and its beta.

A.

MRP=\frac{0.11-0.078}{1.5-0.7}=4\%,\quad r_f=5\%

B.

\rho_{iM}=\beta_i\sigma_M/\sigma_i

\rho_{AM}=\frac{1.5\sqrt{0.04}}{\sqrt{0.1225}}\approx0.85714

\rho_{BM}=\frac{0.7\sqrt{0.04}}{\sqrt{0.0324}}\approx0.77778

Correlation A–market = 0.85714; B–market = 0.77778.

C.

w_A=\frac{0.095-0.078}{0.11-0.078}=0.53125

w_B=1-w_A=0.46875

\beta_P=0.53125(1.5)+0.46875(0.7)=1.125

Weight A = 53.125%; weight B = 46.875%; beta = 1.125.

Inferring risk-free and market returns

EXERCISE 04

A has expected return 10%, volatility 20% and beta 1.2. B has expected return 15%, volatility 25% and beta 1.5. Both are assumed to obey the same CAPM.

A.Find return and beta of a 60% A / 40% B portfolio.
B.What is the volatility of a fully invested mixture that earns exactly B’s return?
C.Infer the risk-free rate and market return.

A.

E[R_P]=0.6(0.10)+0.4(0.15)=12\%

\beta_P=0.6(1.2)+0.4(1.5)=1.32

Expected return = 12%; beta = 1.32.

B.

0.10w+0.15(1-w)=0.15\Rightarrow w=0

\sigma_P=\sqrt{0^2(0.20)^2+1^2(0.25)^2}=25\%

Since A and B have distinct expected returns, the weights must be 0% A and 100% B. Volatility is 25%; there is no diversification in this allocation.

C.

MRP=\frac{0.15-0.10}{1.5-1.2}=16.6667\%

r_f=0.10-1.2(MRP)=-10\%

E[R_M]=r_f+MRP=-10\%+16.6667\%\approx6.6667\%

Market return = 6.6667%.

The unusually negative risk-free rate is implied by these particular inputs; it must not be replaced silently with a positive number.

Inferring risk-free and market returns

EXERCISE 05

Alfa has market covariance 0.23 and expected return 9.75%; Gamma has covariance 0.41 and expected return 14.25%. Market variance is 0.20. Both lie on the SML.

A.Recover the risk-free rate and market return.
B.Calculate a 30% Alfa / 40% Gamma / 30% market portfolio.
C.Replace the market position with Lambda, which has beta 1 and expected return 7%. Which portfolio is preferable?

A.

\beta_A=0.23/0.20=1.15,\qquad\beta_G=0.41/0.20=2.05

r_f=0.0975-1.15(0.05)=4\%

E[R_M]=4\%+5\%=9\%

\beta_A=1.15,\quad\beta_G=2.05,\quad MRP=\frac{14.25\%-9.75\%}{2.05-1.15}=5\%

Risk-free rate = 4%; market return = 9%.

B.

\beta_P=0.3(1.15)+0.4(2.05)+0.3(1)=1.465

E[R_P]=0.3(0.0975)+0.4(0.1425)+0.3(0.09)=11.325\%

Beta = 1.465; expected return = 11.325%.

C.

\beta_{new}=0.3(1.15)+0.4(2.05)+0.3(1)=1.465

E[R_{new}]=0.3(0.0975)+0.4(0.1425)+0.3(0.07)=10.725\%

\Delta E[R]=0.3(9\%-7\%)=0.6\%

Beta remains 1.465 but expected return falls to 10.725%. Lambda is below the SML. Under CAPM pricing, the first portfolio offers 0.6 percentage points more expected return at the same systematic risk.

Inferring risk-free and market returns

EXERCISE 06

CAPM holds. Market variance is 0.40. Alfa has market covariance 0.25 and expected return 9%; Gamma has covariance 0.50 and return 12%. Portfolio P holds 25% Alfa, 25% Gamma and 50% market. Portfolio Q replaces that market allocation with Lambda, whose beta is 1 and forecast return 7%.

A.Infer the risk-free and market returns.
B.Find P’s return and beta.
C.Compare Q with P.

A.

\beta_A=0.25/0.40=0.625,\quad\beta_G=1.25

0.09=R_f+0.625\pi_M,\quad0.12=R_f+1.25\pi_M

\pi_M=\frac{0.12-0.09}{1.25-0.625}

4.8%; risk-free return = 6%; market return = 10.8%.

B.

E[R_P]=0.25(0.09)+0.25(0.12)+0.50(0.108)=10.65\%

Beta = 0.25(0.625) + 0.25(1.25) + 0.50 = 0.96875.

C.

E[R_Q]=0.25(0.09)+0.25(0.12)+0.50(0.07)=8.75\%

Q has the same beta, 0.96875, but a return 1.90 percentage points lower. Lambda’s forecast is inconsistent with CAPM fair pricing: beta 1 requires 10.8%. P offers the better systematic-risk-adjusted return; total-volatility dominance cannot be inferred without additional covariances.

Inferring risk-free and market returns

EXERCISE 07

Write the CAPM as E[Rᵢ] = a + βᵢ E[Rₘ].

A.Find a for beta 1, 2 and 0. What does a = 0 imply?

A.

a=(1-\beta_i)r_f

Beta 1 gives a = 0; beta 2 gives a = −rƒ; beta 0 gives a = rƒ. If rƒ is nonzero, a = 0 implies beta 1. If rƒ = 0, the intercept is zero for every beta.