Portfolio Risk and Return

EXERCISE 01

Stock	Expected return	Shares	Price
A	8%	520	25
B	15%	300	48
C	6%	250	26

A.Calculate expected portfolio return.

A.

E[R_P]=\frac{\sum_iN_iP_iE[R_i]}{\sum_iN_iP_i}

V_P=520(25)+300(48)+250(26)=33900

E[R_P]=\frac{520(25)(0.08)+300(48)(0.15)+250(26)(0.06)}{33900}

10.59%

Portfolio Risk and Return

EXERCISE 02

Use portfolio weights 65% in RRR and 35% in WWW.

Asset	Expected return	Volatility	Weight
A	7.000%	16.000%	0.65000
B	10.000%	20.000%	0.35000

Correlation = 0.22.

A.Calculate portfolio expected return and volatility.

A.

E[R_P]=w_AE[R_A]+w_BE[R_B]

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B

E[R_P]=0.65(0.07)+(0.35)(0.1)\approx 0.0805

\sigma_P^2=(0.65)^2(0.16)^2+(0.35)^2(0.2)^2+2(0.65)(0.35)(0.22)(0.16)(0.2)

\sigma_P=\sqrt{0.0189192}\approx 0.137547

Expected return = 8.050%; volatility = 13.755%.

Portfolio Risk and Return

EXERCISE 03

Asset	Expected return	Volatility	Weight
A	10.000%	18.000%	0.60000
B	15.000%	25.000%	0.40000

Correlation = 0.40.

A.Calculate portfolio expected return and volatility.
B.Can these assets eliminate all portfolio volatility?

A.

E[R_P]=w_AE[R_A]+w_BE[R_B]

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B

E[R_P]=0.6(0.1)+(0.4)(0.15)\approx 0.12

\sigma_P^2=(0.6)^2(0.18)^2+(0.4)^2(0.25)^2+2(0.6)(0.4)(0.4)(0.18)(0.25)

\sigma_P=\sqrt{0.030304}\approx 0.17408

Expected return = 12.000%; volatility = 17.408%.

B.

No, because their correlation is 0.40, strictly between −1 and +1.

Portfolio Risk and Return

EXERCISE 04

Invest equally in Target and Gamma. An alternative uncorrelated security offers a 16% return and 30% volatility.

Asset	Expected return	Volatility	Weight
A	20.000%	40.000%	0.50000
B	12.000%	30.000%	0.50000

Correlation = 0.00.

A.Calculate portfolio expected return and volatility.
B.Is holding the alternative security alone preferable?

A.

E[R_P]=w_AE[R_A]+w_BE[R_B]

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B

E[R_P]=0.5(0.2)+(0.5)(0.12)\approx 0.16

\sigma_P^2=(0.5)^2(0.4)^2+(0.5)^2(0.3)^2+2(0.5)(0.5)(0)(0.4)(0.3)

\sigma_P=\sqrt{0.0625}\approx 0.25

Expected return = 16.000%; volatility = 25.000%.

B.

No. The two-asset portfolio has the same 16% expected return and only 25% volatility, so it dominates the alternative held alone. Its lack of correlation may still make it useful as an additional portfolio constituent.

Portfolio Risk and Return

EXERCISE 05

Initially hold 100 Macrosoft shares at $80 and 300 Intelligence shares at $40. Use the stated Macrosoft volatility of 18%.

Asset	Expected return	Volatility	Weight
A	15.000%	18.000%	0.40000
B	20.000%	20.000%	0.60000

Correlation = 0.38.

A.Calculate portfolio expected return and volatility.
B.Recalculate after selling 200 Intelligence shares.

A.

E[R_P]=w_AE[R_A]+w_BE[R_B]

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B

E[R_P]=0.4(0.15)+(0.6)(0.2)\approx 0.18

\sigma_P^2=(0.4)^2(0.18)^2+(0.6)^2(0.2)^2+2(0.4)(0.6)(0.38)(0.18)(0.2)

\sigma_P=\sqrt{0.0261504}\approx 0.161711

Expected return = 18.000%; volatility = 16.171%.

B.

For the remaining two-stock portfolio, excluding the sale proceeds:

w_A=\frac{100(80)}{100(80)+100(40)}=\frac23,\qquad w_B=\frac13

E[R_P]=\frac23(0.15)+\frac13(0.20)\approx16.667\%

\sigma_P=\sqrt{(\frac23)^2(0.18)^2+(\frac13)^2(0.20)^2+2(\frac23)(\frac13)(0.38)(0.18)(0.20)}

New weights: 2/3 and 1/3. Expected return = 16.667%; volatility = 15.787%.

Portfolio Risk and Return

EXERCISE 06

The portfolio contains 250 shares of A at €24 and 200 shares of B at €50. Their return standard deviations are 30.00% and 45.00%; correlation is 0.25.

A.Calculate weights and portfolio standard deviation.

A.

w_i=\frac{N_iP_i}{\sum_jN_jP_j}

w_A=\frac{250(24)}{250(24)+200(50)}=0.375,\quad w_B=0.625

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho_{AB}\sigma_A\sigma_B

\sigma_P^2=0.375^2(0.3)^2+0.625^2(0.45)^2+2(0.375)(0.625)(0.25)(0.3)(0.45)=0.10757812

\sigma_P=\sqrt{0.10757812}

32.80%

Portfolio Risk and Return

EXERCISE 07

The portfolio contains 250 shares of A at €24 and 200 shares of B at €50. Their return standard deviations are 20.00% and 35.00%; correlation is 0.5.

A.Calculate weights and portfolio standard deviation.

A.

w_i=\frac{N_iP_i}{\sum_jN_jP_j}

w_A=\frac{250(24)}{250(24)+200(50)}=0.375,\quad w_B=0.625

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho_{AB}\sigma_A\sigma_B

\sigma_P^2=0.375^2(0.2)^2+0.625^2(0.35)^2+2(0.375)(0.625)(0.5)(0.2)(0.35)=0.06988281

\sigma_P=\sqrt{0.06988281}

26.44%

Portfolio Risk and Return

EXERCISE 08

The portfolio contains 250 shares of A at €24 and 200 shares of B at €50. Their return standard deviations are 20.00% and 18.00%; correlation is 0.7.

A.Calculate weights and portfolio standard deviation.

A.

w_i=\frac{N_iP_i}{\sum_jN_jP_j}

w_A=\frac{250(24)}{250(24)+200(50)}=0.375,\quad w_B=0.625

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho_{AB}\sigma_A\sigma_B

\sigma_P^2=0.375^2(0.2)^2+0.625^2(0.18)^2+2(0.375)(0.625)(0.7)(0.2)(0.18)=0.03009375

\sigma_P=\sqrt{0.03009375}

17.35%

Portfolio Risk and Return

EXERCISE 09

The portfolio contains 250 shares of A at €24 and 200 shares of B at €50. Their return standard deviations are 25.00% and 15.00%; correlation is 0.8.

A.Calculate weights and portfolio standard deviation.

A.

w_i=\frac{N_iP_i}{\sum_jN_jP_j}

w_A=\frac{250(24)}{250(24)+200(50)}=0.375,\quad w_B=0.625

\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho_{AB}\sigma_A\sigma_B

\sigma_P^2=0.375^2(0.25)^2+0.625^2(0.15)^2+2(0.375)(0.625)(0.8)(0.25)(0.15)=0.03164062

\sigma_P=\sqrt{0.03164062}

17.79%

Portfolio Risk and Return

EXERCISE 10

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.

Portfolio Risk and Return

EXERCISE 11

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%

Portfolio Risk and Return

EXERCISE 12

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.