Question

You have been asked for your advice in selecting a portfolio of assets and have been given the following data:

Expected return

Year Asset A Assest B Assest C

2019 12% 16% 12%

2020 14% 14% 14%

2021 16% 12% 16%

You have been told that you can create two portfolios—one consisting of assets A and B and the other consisting of assets A and C—by investing equal proportions (50%) in each of the two component assets.

a. What is the expected return for each asset over the 3-year period?

b. What is the standard deviation for each asset’s return?

c. What is the expected return for each of the two portfolios?

d. How would you characterize the correlations of returns of the two assets making up each of the two portfolios identified in part c?

e. What is the standard deviation for each portfolio?

f. Which portfolio do you recommend? Why?

(please show step by step really need help, thank you)

Answer 1

First we will prepare table containing all required data

C (A-A) (A-A)2 (B- B) (B-B2 (C-C) (C-C 12 14 16 42 AC Year 256 196 144 596 144 196 256 596 192 196 192 16 196 256 596 12 14 16 2019 196 256 596 12 42 2021

a. Expected Return

we will consider average return as expected return

$\bar{x}=\frac{\sum x}{n}$

Expected return of A

=42/3 =14

Expected return of B

=42/3 =14

Expected return of C

=42/3 =14

b. Standard Deviation

(a-T)

Standard deviation of A

3

$\sigma =1.63$

Standard deviation of B

3

$\sigma =1.63$

Standard deviation of C

3

$\sigma =1.63$

c. Expected return of portfolio

$R_{p}=w_{1}R_{1}+w_{2}R_{2}$

expected return of portfolio A & B

0.5 × 14 +0.5 × 14

$=14$

expected return of portfolio A & C

0.5 × 14 +0.5 × 14

$=14$

d. Correlation

n(Σ.ry) _ (Σ.r ) ( Σ y)

Correlation of portfolio A & B

$r_{AB}=\frac{3(580)-(42)(42)}{\sqrt{\left [ 3\times 596-(42)^{2} \right ]\left [ 3\times 596-(42)^{2} \right ]}}$

$r_{AB}=\frac{1740-1764}{\sqrt{\left [ 1788-1764 \right ]\left [ 1788-1764 \right ]}}$

$r_{AB}=\frac{-24}{\sqrt{\left [ 24 \right ]\left [ 24 \right ]}}$

AB-1

Correlation of portfolio A & C

$r_{AC}=\frac{3(596)-(42)(42)}{\sqrt{\left [ 3\times 596-(42)^{2} \right ]\left [ 3\times 596-(42)^{2} \right ]}}$

$r_{AC}=\frac{1788-1764}{\sqrt{\left [ 1788-1764 \right ]\left [ 1788-1764 \right ]}}$

$r_{AB}=\frac{24}{\sqrt{\left [ 24 \right ]\left [ 24 \right ]}}$

$r_{AB}=1$

e. Standard Deviation of Portfolio

$\sigma _{p}=\sqrt{w_{1}^{2}\sigma _{1}^{2}+w_{2}^{2}\sigma _{2}^{2}+2w_{1}w_{2}\sigma _{1}\sigma _{2}r_{p}}$

Standard deviation of Portfolio A & B

0.52 × 1.632 +0.52 × 1.632+2 × 0.5 × 0.5 × 1.63 × 1.63 × (-1)

V0.25 × 2.66+0.25 × 2.66-1.33

$=\sqrt{0.665+0.665-1.33}$

V1.33 1.33

$=0$

Standard deviation of Portfolio A & C

V0.52 × 1.632 +0.52 × 1.632+2 × 0.5 × 0.5 × 1.63 × 1.63 × 1

$=\sqrt{0.25\times 2.66+0.25\times 2.66+1.33}$

$=\sqrt{0.665+0.665+1.33}$

V1.33 1.33

$=1.63$

f. Recommended portfolio is A & B. Because standard deviation of portfolio is Zero, which means that portfolio has zero systematic risk.