CML and SML: efficiency versus pricing

EXERCISE 01

Draw both relationships, identify their axes and explain their scope.

A.What do the CML and SML describe?
B.Why does the tangency portfolio coincide with the market in CAPM equilibrium?
C.What is the risk–return trade-off and an efficient portfolio?
D.Can two portfolios with the same return, but 20% and 14% volatility, both be efficient?
E.Does adding a risk-free asset improve the opportunity set?

A.

\mathrm{CML}:\quad E[R_P]=r_f+\frac{E[R_M]-r_f}{\sigma_M}\sigma_P

\mathrm{SML}:\quad E[R_i]=r_f+\beta_i(E[R_M]-r_f)

Both vertical axes show expected return. CML uses total volatility on its horizontal axis and describes efficient combinations of cash and the tangency portfolio. SML uses beta and prices every asset and portfolio under CAPM, including inefficient portfolios.

CML in expected return–volatility space and SML in expected return–beta space, sharing the market portfolio

B.

With homogeneous expectations, a common investment universe and risk-free borrowing/lending, investors hold combinations of the same tangency portfolio and cash. Market clearing makes aggregate risky holdings equal the market portfolio. Therefore the market portfolio is mean–variance efficient under these assumptions.

C.

Investors require compensation for bearing priced risk. A portfolio is mean–variance efficient when no feasible portfolio offers a higher expected return at the same or lower volatility, or lower volatility at the same or higher expected return, with at least one strict improvement.

D.

No. The 14%-volatility portfolio dominates the 20%-volatility portfolio. This comparison alone does not prove that the lower-volatility one is efficient relative to every other opportunity.

E.

It permits combinations along capital-allocation lines and, at the best slope, the CML. On the efficient branch, these combine the risk-free asset and market portfolio; borrowing corresponds to a negative cash weight.

CML and SML: efficiency versus pricing

EXERCISE 02

The risk-free rate is 3%, market expected return 8% and market volatility 20%. Alfa invests 80% in the market and 20% risk-free. Lambda has expected return 7%, volatility 25% and market correlation 0.8.

A.Draw the CML and locate the portfolios.
B.Draw the SML, calculate betas and assess pricing.

A.

E[R_P]=0.03+\frac{0.08-0.03}{0.20}\sigma_P=0.03+0.25\sigma_P

E[R_\alpha]=0.20(0.03)+0.80(0.08)=7\%,\quad\sigma_\alpha=16\%

Alfa lies on the CML and is efficient. Lambda lies below it at (25%, 7%); it is feasible but inefficient as a standalone investment.

Side by side CML and SML for Alfa and Lambda

B.

E[R_i]=0.03+0.05\beta_i

\beta_\alpha=0.80,\quad\beta_\lambda=0.8\frac{0.25}{0.20}=1

Alfa is fairly priced on the SML. Lambda requires 8% but offers 7%, so it is below the SML and overpriced.

CML and SML: efficiency versus pricing

EXERCISE 03

The risk-free rate is 3.00%, market return 9.00% and market volatility 20.00%. Portfolio X invests 40.00% in the market and the remainder risk-free. Portfolio Y returns 4.00% with volatility 7.00%. Assume both are fairly priced under CAPM for the SML questions.

A.Draw the CML, calculate X and compare efficiency.
B.Draw the SML and calculate both betas.

A.

E[R_P]=R_f+\frac{E[R_M]-R_f}{\sigma_M}\sigma_P

E[R_X]=0.6(0.03)+0.4(0.09)=5.4000\%

\sigma_X=0.4(0.2)=8.0000\%

X is on the CML. At Y’s volatility, the CML return is 5.1000%, above Y’s return, so Y is inefficient. X and Y alone need not dominate one another; compare the full opportunity set. X and the market are both efficient; choosing between them depends on risk tolerance.

B.

E[R_i]=0.03+0.06\beta_i

\beta_X=0.4,\quad\beta_Y=\frac{0.04-0.03}{0.09-0.03}=0.166667

Both are on the SML by the fair-pricing assumption. An inefficient portfolio can still be fairly priced: the CML uses total volatility, whereas the SML uses systematic beta.

CML and SML: efficiency versus pricing

EXERCISE 04

The risk-free rate is 2.00%, market return 7.00% and market volatility 25.00%. Portfolio X invests 80.00% in the market and the remainder risk-free. Portfolio Y returns 6.00% with volatility 21.00%. Assume both are fairly priced under CAPM for the SML questions.

A.Draw the CML, calculate X and compare efficiency.
B.Draw the SML and calculate both betas.

A.

E[R_P]=R_f+\frac{E[R_M]-R_f}{\sigma_M}\sigma_P

E[R_X]=0.2(0.02)+0.8(0.07)=6.0000\%

\sigma_X=0.8(0.25)=20.0000\%

X is on the CML. At Y’s volatility, the CML return is 6.2000%, above Y’s return, so Y is inefficient. X dominates Y: equal or higher return and lower volatility. X and the market are both efficient; choosing between them depends on risk tolerance.

B.

E[R_i]=0.02+0.05\beta_i

\beta_X=0.8,\quad\beta_Y=\frac{0.06-0.02}{0.07-0.02}=0.800000

Both are on the SML by the fair-pricing assumption. An inefficient portfolio can still be fairly priced: the CML uses total volatility, whereas the SML uses systematic beta.

CML and SML: efficiency versus pricing

EXERCISE 05

The risk-free return is 2%. The market return is 7%, with volatility 20%. Alfa combines the market and risk-free asset and has volatility 25%. Waldo has expected return 6%, volatility 30% and beta 1.

A.Draw the CML and locate both investments.
B.Find Alfa’s composition and assess efficiency.
C.Draw the SML and assess pricing.

A.

E[R_P]=R_f+\frac{E[R_M]-R_f}{\sigma_M}\sigma_P

E[R_P]=0.02+0.25\sigma_P

Alfa is at (25%, 8.25%) on the CML. Waldo is at (30%, 6%), below the CML, whose return at 30% volatility is 9.5%.

CML and SML comparing the market, Alfa and Waldo

B.

w_M=\frac{0.25}{0.20}=1.25,\quad w_f=1-1.25=-0.25

Alfa invests 125% in the market, borrowing 25%, and is mean–variance efficient under the CAPM borrowing assumptions. Waldo is not efficient as a standalone portfolio.

C.

E[R_i]=0.02+0.05\beta_i

\beta_\alpha=1.25,\quad E[R_\alpha]=8.25\%

Alfa is on the SML. Waldo’s beta of 1 requires 7%; its forecast return is only 6%, an alpha of −1 percentage point. It is overpriced under CAPM.

CML and SML: efficiency versus pricing

EXERCISE 06

A.Must every asset lie on the CML in CAPM equilibrium? Can an inefficient portfolio still be correctly priced?

A.

\mathrm{CML}:E[R_P]=r_f+\frac{E[R_M]-r_f}{\sigma_M}\sigma_P,\quad\mathrm{SML}:E[R_i]=r_f+\beta_i(E[R_M]-r_f)

Only mean–variance-efficient combinations of the risk-free asset and market portfolio lie on the CML. Correctly priced individual securities and portfolios satisfy the SML even when they carry diversifiable risk and lie below the CML. Thus an inefficient portfolio can have a return fully consistent with CAPM. The SML is a pricing relation in beta–return space; it is not the efficient frontier in volatility–return space.