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Assume you wish to evaluate the risk and return behaviors associated with various combinations of assets...

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.

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  • Return of Portfolio (Rp)

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}* ho _{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.

P14 C. (i IC 2 1 Assets Return(r) Std. Deviation (s) 7.90% 12.70% 4.60% 9.70% Risk-Return 14.00% 13.00% 12.00% 11.00% E 10.00% 9,00% 8.00% Correlation (rvw) 12.70% 8 Comb. Weight-V(Wv)Weight-(Ww) Portfolio return (Rp) Portfolio Std. Dev.(Sp 1.74% 11.26% 7.90% 8.38% 8.86% 34% 9.82% 4.60% 5.11% 5.62% 10.78% 1096 10 B 11 С 12 D 13 E 34% 70% 0% 7.15% 15 G 16H 17 18 J 19 K 40% 30% 60% 6.00% 8.17% 8.68% 1.74 12.228 5.00% 10% 096 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Std. Deviation(Risk) 9.00% 10.00% 11.00% 100%

Formula Reference -

E9 Std. Deviation (s) 0.046 0.097 Assets 0.079 0.127 Correlation (rvw) Weight-V(Wv Weight-(Ww) ortfolio return (Rp) Portfolio Std. Dev.(Spl SC$4 B10+SC$5 C10 SCS4 B11+CSS C11 SCSA B12+S SCSA B13+SC$5 C13 10 B 11 С 12 D 13 E 14 F 15 G 16 H 17 I 18 J 19 K 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1-B10 1-B11 1-812 -1-B13 1-B14 -1-B15 =1-B16 1-B17 1-B18 1-B19

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.

P14 21 Assets Return Std. Deviation (s) Risk-Return 23 24 25 26 27 Comb. Weight-V(W) Weight-(Ww Portfolio return (Rp Portfolio Std. Dev.(Sp 28 29 30 С 31 D 32 E 33 F 34 G 35 H 36 I 37 J 38 К 7.90% 12.70% 4.60% 970% 14.00% Correlation (rw) 12.70% 22% 74% 7.90% 8.38% 8.86% 34% 9.82% 4.60% 4.25% 4.16% 4.34% 4.76% 5.37% 6.10% 6.93% 81% 0 78% 10.00% | 82% 9.34% 8.86% 8.38% 7.90% 30% 11.25% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 10% std. deviation(Risk) 100%

Formula Reference -

E28 21 2 Std. Deviation 0.046 0.097 0.079 24 25 26 27 Comb 28 29 30 с 31 D 32 E 33 F 34 G 35 H 36 I 37 38 K 0.127 Correlation (rvw Weight-V(Wv) Weight-(Ww) Portfolio return (R SC$23 B28+$C$24*C28 $C$23 B29+SC$24 C29 SC$23 B30+SC$24*C30 -SC$23в 31+SCS24C31 $C$23 B32+SC$24 C32 SC$23 833+SC$24 C33 SC$23 B34+SC$24 C34 SC$23 B35+SC$24 SC$23 B36+$C$24*C36 SC$23 B37+$C$24 C37 $C$23 B38+SC$24 C38 Portfolio Std. Dev.(Sp) SORT(828a2 SDS2342 SORT(B29 2 $D$23 2+C29 2 SD$24 2+2 B29 C29 SD$23 SD$24 $C$25) SQRT(B30 2 $D$23 2+C302 SD$24 2+2 B30 C30 $D$23*D$24 $C$25 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1-B30 1-B31 1-B32 1-B33 1-B34 1-B35 1-B36 -1-B37 1-B38 QRT(B31 2SDS232+C312SDS242+2В 31°C31.SDS23° SDS24.SCS25) ORT (B3242 $D$2312+C3242 5D$24A2+2 B32 C32 SDS23 SD$24 SC$25 SQRT(B33 2 $D$23 2+C332 SD$24 2+2 B33 C33 SD$23*SD$24 $C$25 SQRT(B342 SDS23 2+C34n2 SD$2412+2 B34 C34 SDS23 SDS24 SC SORT B35 2 SDS2342+03542 SD$2442+2 B35 C35SDS23 SD$24 SC$25 SQRT(B36 2 $D$23 2+C362 SD$24 2+2 B36 C36 $D$23*SD$24 $C$25

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.

P14 39 Assets Return(r Std. Deviation (s Risk-Return 42 43 4.60% 9.70% 14.00% 13.00% 12.00% 11.00% E 10.00% 9.00% 8.00% 7.00% 6.00% 5.00% 12.70% Correlation (rvw 45 46 Comb. Weight-V(Wv) Weight-(Ww) Portfolio return (Rp) Portfolio Std. Dev.(Sp) 47 7.90% 8.38% 8.86% 4.60% 3.17% 1.74% 0.31% 1.12% 2.55% 3.98% 5.41% 6.84% 827% 9,70% 80% 3496|1 50 D 51 E 52 F 53 G 54 H 55I 56 J 57 K 9.82% 10130% 10.78% 11.26% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 10% Std. devation (risk) 100% 12.70%

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

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

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