Two-asset portfolios: weights, return and volatility

EXERCISE 01

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%.

Two-asset portfolios: weights, return and volatility

EXERCISE 02

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.

Two-asset portfolios: weights, return and volatility

EXERCISE 03

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.

Two-asset portfolios: weights, return and volatility

EXERCISE 04

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%.

Two-asset portfolios: weights, return and volatility

EXERCISE 05

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%

Two-asset portfolios: weights, return and volatility

EXERCISE 06

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%

Two-asset portfolios: weights, return and volatility

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 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%

Two-asset portfolios: weights, return and volatility

EXERCISE 08

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%