Question

Assume you wish to evaluate the risk and return behaviors associated with various combinations of assets V and W under three assumed degrees of​ correlation: perfect​ positive, uncorrelated, and perfect negative. The following average return and risk values were calculated for these​ assets:

Asset	Average​ Return, r		Risk​ (Standard Deviation), s
V	7.9​%		4.6​%
W	12.7​%		9.7​%

a. If the returns of assets V and W are perfectly positively correlated​ (correlation coefficient = + 1​), describe the range of​ (1) return and​ (2) risk associated with all possible portfolio combinations.

b. If the returns of assets V and W are uncorrelated​ (correlation coefficient = 0​), describe the approximate range of ​(1​) return and​ (2) risk associated with all possible portfolio combinations.

c. If the returns of assets V and W are perfectly negatively correlated​ (correlation coefficient = - 1​), describe the range of​ (1) return and​ (2) risk associated with all possible portfolio combinations.

Answer 1

• Return of Portfolio (R_p)

It is weighted average of individual stock return in a portfolio. Two-Assets Portfolio Return formula provided below -

$R_{p} = W_{1}*R_{1}+W_{2}*R_{2}$

Where,

W = Weight of Stock

R = Return of Stock

• Standard Deviation(risk) of Portfolio ($\sigma _{p}$)

It measures the risk of Portfolio. Risk of two assets Portfolio formula provided below-

$\sigma _{p} = \sqrt{W_{1}^{2}*\sigma _{1}^{2}}+W_{2}^{2}*\sigma _{2}^{2}}+2*W_{1}*W_{2}*\sigma _{1}*\sigma _{2}*\rho _{12}}$

• Possible Portfolio Combination

It refers to portfolio on optimal portfolio frontier which means each portfolio provide higher return for higher level of risk. portfolio having low return for high risk should be rejected.

1.

Please refer to following spreadsheet and graph for Return and risk of Possible Portfolio combination having stock V and W perfectly positive correlated.

Formula Reference -

We can see in above calculation and graph, when V and W are perfectly positive correlated then all the combination A to Z are optimal which means there is always higher return for higher risk. combination highlighted in green are optimal.

2.

Please refer to following spreadsheet and graph for Return and risk of Possible Portfolio combination having stock V and W uncorrelated.

Formula Reference -

We can see in above calculation and graph, when V and W are uncorrelated then only combination C to Z are optimal .and combination A & B which are highlighted in red is not optimal which it has lower return for higher risk.

3.

Please refer to following spreadsheet and graph for Return and risk of Possible Portfolio combination having stock V and W perfectly negative correlated.

Formula Reference -

E47 Std. Deviation (s 0.046 0.097 Assets eturn( 0.079 0.127 Correlation (rvw Portfolio Std. Dev.(Sp) -SORT B472 SD$42A2+C47A2 SD$43 2+2 B47 C47 SD$42 SD$43 SC$44 -SORT(B48 2 SD$42 2+C48 2*SD$43 2+2 B48 C48 SD$42 SDS43 SC$44) 46 Comb Weight-v(Wv) Weight-(Ww) Portfolio return (Rp) SC$42 B47+SC$43*CA $C$42 B48+SC$43 C48 $C$42 B49+SC$43 C49 47 7 49 -1-B50 -1-B51 =1-B52 -1-B53 -1-B54 1-B55 -1-B56 -1-B57 50 D SC$42 B51+$C$43*C51 SC$42 B52+SC$43 C52 $C$42 B53 SC$43 C53 $C$ C54 -SORT B522 SD$42 2+C522 SD$43 2+2 B52 C52 SD$42 SD$43 SC$44 ORTB53"2" SDS42/24C53"2" SDS43"2+2"В53"c53"SDS42"SDS43"SC S44 -SQRT B54 2 SD$42 2+C54 2 SD$43 2+2 B54 C54 SD$42 SDS43*C$44 53 G 54 H S5 I $42"B54+SC$43 $42 B55+SCS43 SC$42 B56+$C$43*C56 SCS42 B57+SCS43 *C57 57 K

Combination highlighted in green D to Z are optimal and combination A,B&C highlighted in red are not optimal.