Scenario-based portfolio and CAPM analysis

EXERCISE 01

Scenario	Probability	Smart-Invest return
Recession	10%	−2%
Normal	50%	10%
Expansion	40%	30%

Smart-Invest has correlation 0.8 with a market whose volatility is 10%. JCF has expected return 10%, volatility 6% and beta 0.8. The risk-free rate is 2%. Form a portfolio with 50% Smart-Invest, 30% JCF and 20% risk-free; correlation between the two stocks is −0.5.

A.Calculate Smart-Invest’s mean, volatility and beta.
B.Calculate the portfolio moments and compare with JCF.

A.

E[R]=\sum_s p_sR_s,\quad\sigma=\sqrt{\sum_s p_s(R_s-E[R])^2}

E[R]=0.1(-0.02)+0.5(0.10)+0.4(0.30)=0.168

\sigma=\sqrt{0.1(-0.02-0.168)^2+0.5(0.10-0.168)^2+0.4(0.30-0.168)^2}=0.113208

\beta=0.8\frac{\sigma}{0.10}=0.905662

B.

E[R_P]=0.5(0.168)+0.3(0.10)+0.2(0.02)=11.8\%

\beta_P=0.5\beta_S+0.3(0.8)=0.692831

\sigma_P=\sqrt{0.5^2\sigma_S^2+0.3^2(0.06)^2+2(0.5)(0.3)\sigma_S(0.06)(-0.5)}

Portfolio volatility = 5.0091%.

The portfolio has a higher return, lower total volatility and lower beta than JCF. It dominates JCF on these stated metrics.

Scenario-based portfolio and CAPM analysis

EXERCISE 02

Scenario	Probability	Return A	Return B
Expansion	40%	14%	10%
Normal	40%	8%	4%
Recession	20%	−4%	0%

CAPM holds, both securities are fairly priced, risk-free return is 2% and market return 5%. Construct a portfolio of 50% A, 30% B and 20% risk-free.

A.Calculate means, volatilities, covariance and correlation.
B.Find security and portfolio betas, return and volatility.

A.

\mu_i=\sum_s p_sR_{is},\quad Cov(A,B)=\sum_s p_s(R_{As}-\mu_A)(R_{Bs}-\mu_B)

\mu_A=0.4(0.14)+0.4(0.08)+0.2(-0.04)=0.08

\mu_B=0.4(0.10)+0.4(0.04)=0.056

Scenario	A deviation	B deviation	Weighted covariance
Expansion	0.0600	0.0440	0.001056
Normal	-0.0000	-0.0160	0.000000
Recession	-0.1200	-0.0560	0.001344

\sigma_A^2=0.00432,\quad\sigma_B^2=0.001504,\quad Cov(A,B)=0.0024

\rho_{AB}=\frac{0.0024}{\sqrt{0.00432}\sqrt{0.001504}}=0.941554

Volatility A = 6.5727%; B = 3.8781%.

B.

\beta_A=\frac{0.08-0.02}{0.05-0.02}=2,\quad\beta_B=1.2

\beta_P=0.5(2)+0.3(1.2)=1.36,\quad E[R_P]=0.02+1.36(0.03)=6.08\%

\sigma_P=\sqrt{0.5^2(0.00432)+0.3^2(0.001504)+2(0.5)(0.3)(0.0024)}

4.3993%

Scenario-based portfolio and CAPM analysis

EXERCISE 03

State	Probability	Bititan return	Trititan return
1	30.000%	3.000%	6.000%
2	30.000%	6.000%	1.000%
3	40.000%	8.000%	1.500%

Both securities lie on the SML. The risk-free rate is 2%; Bititan has beta 0.7. Poker offers an expected return of 12% and has beta 1.4.

A.Compute means, variances, standard deviations, covariance and correlation.
B.Find the return and variance of a 40% Bititan / 60% Trititan portfolio.
C.Infer Trititan’s beta and assess Poker.

A.

Use A for Bititan and B for Trititan. Returns are expressed as decimals in the calculations.

E[R_A]=0.3(0.03)+0.3(0.06)+0.4(0.08)=0.059

E[R_B]=0.3(0.06)+0.3(0.01)+0.4(0.015)=0.027

State	p × (A − mean A)²	p × (B − mean B)²	p × (A − mean A)(B − mean B)
1	0.00025230	0.00032670	-0.00028710
2	0.00000030	0.00008670	-0.00000510
3	0.00017640	0.00005760	-0.00010080

Var(A)=0.0002523+0.0000003+0.0001764=0.000429

Var(B)=0.0003267+0.0000867+0.0000576=0.000471

\sigma_A=\sqrt{0.000429},\qquad\sigma_B=\sqrt{0.000471}

E[R]=\sum_s p_sR_s,\quad Var(R)=\sum_s p_s(R_s-E[R])^2

Cov(A,B)=\sum_s p_s(A_s-E[A])(B_s-E[B])

Measure	Bititan	Trititan
Mean	5.900%	2.700%
Variance	0.000429	0.000471
Standard deviation	2.071%	2.170%

Cov(A,B)=\sum_s p_s(A_s-E[A])(B_s-E[B])=-0.000393

\rho_{AB}=\frac{-0.000393}{\sqrt{0.000429\times 0.000471}}\approx -0.874286

Covariance = -0.000393; correlation = -0.874286.

B.

E[R_P]=0.4(0.059)+0.6(0.027)=0.0398

\sigma_P^2=0.4^2(0.000429)+0.6^2(0.000471)+2(0.4)(0.6)(-0.000393)=0.0000496

Return = 3.980%; variance = 0.0000496.

C.

MRP=\frac{0.059-0.02}{0.7}

E[R_M]=0.02+MRP\approx7.57143\%

\beta_T=\frac{0.027-0.02}{MRP}\approx0.12564

r_{Poker}=0.02+1.4(MRP)=9.8\%

\alpha_{Poker}=12\%-9.8\%=2.2\%

Market return = 7.571%; Trititan beta = 0.12564.

Poker requires 9.800%. Its offered 12% is above the SML: it is undervalued under the CAPM assumptions.