Security Market Line

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%.

Security Market Line

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%.

Security Market Line

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.

Security Market Line

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.

Security Market Line

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.

Security Market Line

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.

Security Market Line

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.

Security Market Line

EXERCISE 08

Gamma has a forecast expected return of 20.00% and beta 1.5. The risk-free rate is 4% and the market expected return is 12%.

A.Assess pricing under CAPM.

A.

r_i=r_f+\beta_i(E[R_M]-r_f),\quad\alpha_i=E[R_i]-r_i

r_i=0.04+1.5(0.12-0.04)=0.16

\alpha=0.2-0.16

Alpha = 4.00%. Above the SML; undervalued relative to the forecast.

Security Market Line

EXERCISE 09

Gamma has a forecast expected return of 13.00% and beta 1.5. The risk-free rate is 4% and the market expected return is 12%.

A.Assess pricing under CAPM.

A.

r_i=r_f+\beta_i(E[R_M]-r_f),\quad\alpha_i=E[R_i]-r_i

r_i=0.04+1.5(0.12-0.04)=0.16

\alpha=0.13-0.16

Alpha = -3.00%. Below the SML; overvalued relative to the forecast.

Security Market Line

EXERCISE 10

Gamma has a forecast expected return of 15.00% and beta 1.5. The risk-free rate is 4% and the market expected return is 12%.

A.Assess pricing under CAPM.

A.

r_i=r_f+\beta_i(E[R_M]-r_f),\quad\alpha_i=E[R_i]-r_i

r_i=0.04+1.5(0.12-0.04)=0.16

\alpha=0.15-0.16

Alpha = -1.00%. Below the SML; overvalued relative to the forecast.

Security Market Line

EXERCISE 11

Gamma has a forecast expected return of 18.00% and beta 1.5. The risk-free rate is 4% and the market expected return is 12%.

A.Assess pricing under CAPM.

A.

r_i=r_f+\beta_i(E[R_M]-r_f),\quad\alpha_i=E[R_i]-r_i

r_i=0.04+1.5(0.12-0.04)=0.16

\alpha=0.18-0.16

Alpha = 2.00%. Above the SML; undervalued relative to the forecast.

Security Market Line

EXERCISE 12

The market expected return is 8.5%, and the risk-free rate is 1.5%.

Portfolio	Expected return	Beta
A	2.9%	0.2
B	8.5%	1
K	5.0%	2

A.Which portfolios lie on the SML?

A.

r_i=r_f+\beta_i(E[R_M]-r_f)

r_A=0.015+0.2(0.085-0.015)=0.029

r_B=0.015+1(0.07)=0.085,\quad r_K=0.015+2(0.07)=0.155

A and B lie on the SML; K lies below it.

Security Market Line

EXERCISE 13

Asset A has expected return 14% and beta 1.1 and lies on the SML. The risk-free rate is 5.2%. Asset B has beta 1.4 and expected return 18%.

A.Is B correctly priced?

A.

MRP=(0.14-0.052)/1.1=8\%

r_B^{CAPM}=5.2\%+1.4(8\%)=16.4\%

B offers 18%: above the SML and undervalued.

Security Market Line

EXERCISE 14

Correlation with the market is 0.8; asset volatility is 10.000%, market volatility 5.000%, risk-free return 3.000% and market risk premium 6.000%.

A.Calculate beta and the CAPM required return.
B.Assess another security with the same beta and an expected return of 15.000%.

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=0.8\frac{0.1}{0.05}\approx 1.6

E[R_i]=0.03+(1.6)(0.06)\approx 0.126

Beta = 1.60000; required return = 12.600%.

B.

\alpha=0.15-0.126\approx 0.024

It lies above the SML and is undervalued, assuming the inputs and CAPM are correct.

Security Market Line

EXERCISE 15

Correlation with the market is 0.6; asset volatility is 30.000%, market volatility 15.000%, risk-free return 3.000% and market risk premium 5.000%.

A.Calculate beta and the CAPM required return.
B.Assess another security with the same beta and an expected return of 8.000%.

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{0.3}{0.15}\approx 1.2

E[R_i]=0.03+(1.2)(0.05)\approx 0.09

Beta = 1.20000; required return = 9.000%.

B.

\alpha=0.08-0.09\approx -0.01

It lies below the SML and is overvalued, assuming the inputs and CAPM are correct.

Security Market Line

EXERCISE 16

Correlation with the market is 0.8; asset volatility is 25.000%, market volatility 15.000%, risk-free return 4.000% and market risk premium 6.000%.

A.Calculate beta and the CAPM required return.
B.Assess another security with the same beta and an expected return of 11.000%.

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=0.8\frac{0.25}{0.15}\approx 1.333333

E[R_i]=0.04+(1.333333)(0.06)\approx 0.12

Beta = 1.33333; required return = 12.000%.

B.

\alpha=0.11-0.12\approx -0.01

It lies below the SML and is overvalued, assuming the inputs and CAPM are correct.