Question

You have a portfolio with a standard deviation of 22 % and an expected return of 16 %. You are considering adding one of the two shares in the table below. If after adding the shares you will have 20 % of your money in the new shares and 80 % of your money in your existing​ portfolio, which one should you​ add?

	Expected return	Standard deviation	Correlation with your​ portfolio's returns
Share A	13​%	26​%	0.4
Share B	13​%	16​%	0.6

Standard deviation of the portfolio with share A is

nothing​%.

​(Round to two decimal​ places.)

Standard deviation of the portfolio with share B is

nothing​%.

​(Round to two decimal​ places.)

Which share should you add and​ why?  ​(Select the best choice​ below.)

A.

Add Upper A since the portfolio is less risky when Upper A is added.Add A since the portfolio is less risky when A is added.

B.

Add Upper B because the portfolio is less risky when Upper B is added.

Answer 1

We have an old portfolio with a standard deviation of 22% and an expected return of 16%. We need to add stocks among A or B to the old portfolio. 20% of the money will be invested either in A or B and the remaining 80% will be the weight of the old portfolio.

We will chose the stocks for the which the standard deviation of the new portfolio is lesser.

Investment in old portfolio = W_o = 80%

Investment in A/B = W_A/B = 20%

Part 1 - Case 1: When A is added to the portfolio

Weight of the old portfolio = W_o = 80%, Standard Deviation of the old portfolio = σ_o = 22%

Weight of A = W_A = 20%, Standard deviation of A = σ_A = 26%

Correlation of A's return with old portfolio's returns = ρ_o,A = 0.4

Variance of the new portfolio (old portfolio + A) can be calculated using the below formula:

σ_P² =W_o²* σ²_o + W_A²* σ²_A + 2 W_o*W_A *ρ_o,A* σ_{o *} σ_A

σ_P² = 0.8²*(22%)² + 0.2²*(26%)² + 2*0.8*0.2*0.4*22%*26% = 0.030976+0.002704+0.0073216 = 0.0410016

therefore, standard deviation of the new portfolio (old portfolio + A) = σ_P = 0.0410016^1/2 = 0.202489 = 20.2489%

Answer -> standard deviation of the portfolio with share A = 20.25%

Part -2- Case 2: When B is added to the portfolio

Weight of the old portfolio = W_o = 80%, Standard Deviation of the old portfolio = σ_o = 22%

Weight of A = W_B = 20%, Standard deviation of A = σ_B = 16%

Correlation of A's return with old portfolio's returns = ρ_o,B = 0.6

Variance of the new portfolio (old portfolio + B) can be calculated using the below formula:

σ_P² =W_o²* σ²_o + W_B²* σ²_B + 2 W_o*W_B *ρ_o,B* σ_{o *} σ_B

σ_P² = 0.8²*(22%)² + 0.2²*(16%)² + 2*0.8*0.2*0.6*22%*16% = 0.030976+0.001024+0.0067584 = 0.0387584

therefore, standard deviation of the new portfolio (old portfolio + B) = σ_P = 0.0387584^1/2 = 0.196872 = 19.6872%

Answer -> standard deviation of the portfolio with share B = 19.69%

Part 3 - Since, the standard deviation of the new portfolio is less when B is added (19.69%) compared to the standard deviation when A is added (20.25%)

Answer-> Add B because the portfolio is less risky when B is added.