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QuestionAnalyze expected returns and risks for three investment alternatives using data from assets F, G, and H from 2013-2016. Calculate returns, standard deviations, and coefficients of variation. Recommend the best alternative.

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STEP 1

Assumptions1. The expected returns for assets F, G, and H are given for the years2013 to2016. . The three investment alternatives are investing100% in asset F,50% in asset F and50% in asset G, and50% in asset F and50% in asset H.
3. We assume that the returns are independent from year to year.

STEP 2

First, we need to calculate the expected return over the4-year period for each of the three alternatives. The expected return for an investment alternative is the sum of the annual returns for each asset in the alternative, divided by the number of years.
For alternative1, the expected return is the average of the returns of asset F over the4 years.
Expected returnalt1=14∑i=2012016Return,iExpected\, return_{alt1} = \frac{1}{4} \sum_{i=201}^{2016} Return_{,i}

STEP 3

Plug in the given values for the returns of asset F to calculate the expected return for alternative1.
Expected returnalt1=1(16%+17%+18%+19%)Expected\, return_{alt1} = \frac{1}{} (16\% +17\% +18\% +19\%)

STEP 4

Calculate the expected return for alternative1.
Expected returnalt1=14(16%+17%+18%+19%)=17.%Expected\, return_{alt1} = \frac{1}{4} (16\% +17\% +18\% +19\%) =17.\%

STEP 5

For alternative2, the expected return is the average of the returns of50% asset F and50% asset G over the4 years.
Expected returnalt2=14∑i=2013201(0.5×Return,i+0.5×ReturnG,i)Expected\, return_{alt2} = \frac{1}{4} \sum_{i=2013}^{201} (0.5 \times Return_{,i} +0.5 \times Return_{G,i})

STEP 6

Plug in the given values for the returns of asset F and asset G to calculate the expected return for alternative2.
Expected returnalt2=14[(0.5×16%+0.5×17%)+(0.5×17%+0.5×16%)+(0.5×18%+0.5×15%)+(0.5×19%+0.5×14%)]Expected\, return_{alt2} = \frac{1}{4} [(0.5 \times16\% +0.5 \times17\%) + (0.5 \times17\% +0.5 \times16\%) + (0.5 \times18\% +0.5 \times15\%) + (0.5 \times19\% +0.5 \times14\%)]

STEP 7

Calculate the expected return for alternative2.
Expected returnalt2=14[(0.5×16%+0.5×17%)+(0.5×17%+0.5×16%)+(0.5×18%+0.5×15%)+(0.5×19%+0.5×14%)]=16.5%Expected\, return_{alt2} = \frac{1}{4} [(0.5 \times16\% +0.5 \times17\%) + (0.5 \times17\% +0.5 \times16\%) + (0.5 \times18\% +0.5 \times15\%) + (0.5 \times19\% +0.5 \times14\%)] =16.5\%

STEP 8

For alternative3, the expected return is the average of the returns of50% asset F and50% asset H over the4 years.
Expected returnalt3=14∑i=20132016(0.5×Return,i+0.5×ReturnH,i)Expected\, return_{alt3} = \frac{1}{4} \sum_{i=2013}^{2016} (0.5 \times Return_{,i} +0.5 \times Return_{H,i})

STEP 9

Plug in the given values for the returns of asset F and asset H to calculate the expected return for alternative3.
Expected returnalt3=4[(.5×16%+.5×14%)+(.5×17%+.5×15%)+(.5×18%+.5×16%)+(.5×19%+.5×17%)]Expected\, return_{alt3} = \frac{}{4} [(.5 \times16\% +.5 \times14\%) + (.5 \times17\% +.5 \times15\%) + (.5 \times18\% +.5 \times16\%) + (.5 \times19\% +.5 \times17\%)]

STEP 10

Calculate the expected return for alternative3.
Expected returnalt3=4[(0.5×16%+0.5×14%)+(0.5×17%+0.5×15%)+(0.5×18%+0.5×16%)+(0.5×19%+0.5×17%)]=16.75%Expected\, return_{alt3} = \frac{}{4} [(0.5 \times16\% +0.5 \times14\%) + (0.5 \times17\% +0.5 \times15\%) + (0.5 \times18\% +0.5 \times16\%) + (0.5 \times19\% +0.5 \times17\%)] =16.75\%

STEP 11

Next, we need to calculate the standard deviation of returns over the4-year period for each of the three alternatives. The standard deviation is a measure of the variability of the returns. It is calculated as the square root of the variance, which is the average of the squared differences from the mean.
For alternative, the standard deviation is calculated as followsSD_{alt} = \sqrt{\frac{}{4} \sum_{i=2013}^{2016} (Return_{,i} - Expected\, return_{alt})^}

STEP 12

Plug in the given values for the returns of asset F and the expected return for alternative to calculate the standard deviation for alternative.
SDalt=4[(16%−17.5%)2+(17%−17.5%)2+(18%−17.5%)2+(19%−17.5%)2]SD_{alt} = \sqrt{\frac{}{4} [(16\% -17.5\%)^2 + (17\% -17.5\%)^2 + (18\% -17.5\%)^2 + (19\% -17.5\%)^2]}

STEP 13

Calculate the standard deviation for alternative.
SDalt=[(16%−17.5%)2+(17%−17.5%)2+(18%−17.5%)2+(19%−17.5%)2]=.12%SD_{alt} = \sqrt{\frac{}{} [(16\% -17.5\%)^2 + (17\% -17.5\%)^2 + (18\% -17.5\%)^2 + (19\% -17.5\%)^2]} =.12\%

STEP 14

For alternative2, the standard deviation is calculated as followsSDalt2=4∑i=20132016[(0.×Return,i+0.×ReturnG,i−Expected returnalt2)2]SD_{alt2} = \sqrt{\frac{}{4} \sum_{i=2013}^{2016} [(0. \times Return_{,i} +0. \times Return_{G,i} - Expected\, return_{alt2})^2]}

STEP 15

Plug in the given values for the returns of asset F and asset G and the expected return for alternative2 to calculate the standard deviation for alternative2.
SDalt2=4[(0.5×%+0.5×17%−.5%)2+(0.5×17%+0.5×%−.5%)2+(0.5×18%+0.5×15%−.5%)2+(0.5×19%+0.5×14%−.5%)2]SD_{alt2} = \sqrt{\frac{}{4} [(0.5 \times\% +0.5 \times17\% -.5\%)^2 + (0.5 \times17\% +0.5 \times\% -.5\%)^2 + (0.5 \times18\% +0.5 \times15\% -.5\%)^2 + (0.5 \times19\% +0.5 \times14\% -.5\%)^2]}

STEP 16

Calculate the standard deviation for alternative2.
SDalt2=4[(0.5×16%+0.5×%−16.5%)2+(0.5×%+0.5×16%−16.5%)2+(0.5×18%+0.5×15%−16.5%)2+(0.5×19%+0.5×14%−16.5%)2]=.5%SD_{alt2} = \sqrt{\frac{}{4} [(0.5 \times16\% +0.5 \times\% -16.5\%)^2 + (0.5 \times\% +0.5 \times16\% -16.5\%)^2 + (0.5 \times18\% +0.5 \times15\% -16.5\%)^2 + (0.5 \times19\% +0.5 \times14\% -16.5\%)^2]} =.5\%

STEP 17

For alternative3, the standard deviation is calculated as followsSDalt3=4∑i=20132016[(0.5×Return,i+0.5×ReturnH,i−Expected returnalt3)2]SD_{alt3} = \sqrt{\frac{}{4} \sum_{i=2013}^{2016} [(0.5 \times Return_{,i} +0.5 \times Return_{H,i} - Expected\, return_{alt3})^2]}

STEP 18

Plug in the given values for the returns of asset F and asset H and the expected return for alternative3 to calculate the standard deviation for alternative3.
SDalt3=4[(0.5×16%+0.5×14%−16.75%)2+(0.5×17%+0.5×15%−16.75%)2+(0.5×18%+0.5×16%−16.75%)2+(0.5×%+0.5×17%−16.75%)2]SD_{alt3} = \sqrt{\frac{}{4} [(0.5 \times16\% +0.5 \times14\% -16.75\%)^2 + (0.5 \times17\% +0.5 \times15\% -16.75\%)^2 + (0.5 \times18\% +0.5 \times16\% -16.75\%)^2 + (0.5 \times\% +0.5 \times17\% -16.75\%)^2]}

STEP 19

Calculate the standard deviation for alternative3.
SDalt3=14[(.5×16%+.5×14%−16.75%)+(.5×17%+.5×15%−16.75%)+(.5×18%+.5×16%−16.75%)+(.5×19%+.5×17%−16.75%)]=1.71%SD_{alt3} = \sqrt{\frac{1}{4} [(.5 \times16\% +.5 \times14\% -16.75\%)^ + (.5 \times17\% +.5 \times15\% -16.75\%)^ + (.5 \times18\% +.5 \times16\% -16.75\%)^ + (.5 \times19\% +.5 \times17\% -16.75\%)^]} =1.71\%

STEP 20

Next, we need to calculate the coefficient of variation for each of the three alternatives. The coefficient of variation is a measure of risk per unit of return. It is calculated as the standard deviation divided by the expected return.
For alternative, the coefficient of variation is calculated as followsCValt=SDaltExpected returnaltCV_{alt} = \frac{SD_{alt}}{Expected\, return_{alt}}

STEP 21

Plug in the values for the standard deviation and the expected return for alternative1 to calculate the coefficient of variation for alternative1.
CValt1=1.12%17.5%CV_{alt1} = \frac{1.12\%}{17.5\%}

STEP 22

Calculate the coefficient of variation for alternative1.
CValt1=1.12%17.5%=0.064CV_{alt1} = \frac{1.12\%}{17.5\%} =0.064

STEP 23

For alternative, the coefficient of variation is calculated as followsCValt=SDaltExpected returnaltCV_{alt} = \frac{SD_{alt}}{Expected\, return_{alt}}

STEP 24

Plug in the values for the standard deviation and the expected return for alternative to calculate the coefficient of variation for alternative.
CValt=1.%16.%CV_{alt} = \frac{1.\%}{16.\%}

STEP 25

Calculate the coefficient of variation for alternative.
CValt=1.5%16.5%=0.091CV_{alt} = \frac{1.5\%}{16.5\%} =0.091

STEP 26

For alternative3, the coefficient of variation is calculated as followsCValt3=SDalt3Expected returnalt3CV_{alt3} = \frac{SD_{alt3}}{Expected\, return_{alt3}}

STEP 27

Plug in the values for the standard deviation and the expected return for alternative3 to calculate the coefficient of variation for alternative3.
CValt3=1.71%16.75%CV_{alt3} = \frac{1.71\%}{16.75\%}

STEP 28

Calculate the coefficient of variation for alternative3.
CValt3=1.71%16.75%=0.102CV_{alt3} = \frac{1.71\%}{16.75\%} =0.102

STEP 29

Based on the calculated coefficients of variation, we can see that alternative1 has the lowest coefficient of variation, which means it has the lowest risk per unit of return. Therefore, I would recommend alternative1.

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