Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1. In cell J1, calculate the sum of the asset weights.

J1: = SUM(D1:H1)

1.000

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1. In cell J1, calculate the sum of the asset weights.

J1: = SUM(D1:H1)

1.000

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1. In cell J1, calculate the sum of the asset weights.

J1: = SUM(D1:H1)

1.000

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

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A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

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14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

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12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

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12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

2. Set the values of the asset weights in cells B4:B8 equal to the values in cells D1:H1.

B4: = D1

0.200

B5: = E1

0.200

B6: = F1

0.200

B7: = G1

0.200

B8: = H1

0.200

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

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12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

3. Calculate the return contribution wiE(Ri ) of each of the assets in cells D10:H10.

D10: = D1*D2

0.016

Copy D10 to E10:H10.

0.021 0.024 0.015 0.017

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

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10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

3. Calculate the return contribution wiE(Ri ) of each of the assets in cells D10:H10.

D10: = D1*D2

0.016

Copy D10 to E10:H10.

0.021 0.024 0.015 0.017

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

3. Calculate the return contribution wiE(Ri ) of each of the assets in cells D10:H10.

D10: = D1*D2

0.016

Copy D10 to E10:H10.

0.021 0.024 0.015 0.017

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

3. Calculate the return contribution wiE(Ri ) of each of the assets in cells D10:H10.

D10: = D1*D2

0.016

Copy D10 to E10:H10.

0.021 0.024 0.015 0.017

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

3. Calculate the return contribution wiE(Ri ) of each of the assets in cells D10:H10.

D10: = D1*D2

0.016

Copy D10 to E10:H10.

0.021 0.024 0.015 0.017

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017

4. In cell J10, calculate the expected return of the portfolio as the sum of the return contributions.

J10: = SUM(D10:H10)

0.094

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017

4. In cell J10, calculate the expected return of the portfolio as the sum of the return contributions.

J10: = SUM(D10:H10)

0.094

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017

4. In cell J10, calculate the expected return of the portfolio as the sum of the return contributions.

J10: = SUM(D10:H10)

0.094

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

5. Calculate the risk contribution5∑

i=1

wiwjσij = wj

5∑i=1

wiσij of each of the assets in cells D11:H11.

D11: = D1*($B4*D4+$B5*D5+$B6*D6+$B7*D7+$B8*D8)

0.007

Copy D11 to E11:H11.

0.008 0.010 0.006 0.005

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

5. Calculate the risk contribution5∑

i=1

wiwjσij = wj

5∑i=1

wiσij of each of the assets in cells D11:H11.

D11: = D1*($B4*D4+$B5*D5+$B6*D6+$B7*D7+$B8*D8)

0.007

Copy D11 to E11:H11.

0.008 0.010 0.006 0.005

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

5. Calculate the risk contribution5∑

i=1

wiwjσij = wj

5∑i=1

wiσij of each of the assets in cells D11:H11.

D11: = D1*($B4*D4+$B5*D5+$B6*D6+$B7*D7+$B8*D8)

0.007

Copy D11 to E11:H11.

0.008 0.010 0.006 0.005

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

5. Calculate the risk contribution5∑

i=1

wiwjσij = wj

5∑i=1

wiσij of each of the assets in cells D11:H11.

D11: = D1*($B4*D4+$B5*D5+$B6*D6+$B7*D7+$B8*D8)

0.007

Copy D11 to E11:H11.

0.008 0.010 0.006 0.005

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

5. Calculate the risk contribution5∑

i=1

wiwjσij = wj

5∑i=1

wiσij of each of the assets in cells D11:H11.

D11: = D1*($B4*D4+$B5*D5+$B6*D6+$B7*D7+$B8*D8)

0.007

Copy D11 to E11:H11.

0.008 0.010 0.006 0.005

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005

6. In cell J11, calculate the return variance of the portfolio as the sum of the risk contributions.

J11: = SUM(D11:H11)

0.036

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005

6. In cell J11, calculate the return variance of the portfolio as the sum of the risk contributions.

J11: = SUM(D11:H11)

0.036

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005

6. In cell J11, calculate the return variance of the portfolio as the sum of the risk contributions.

J11: = SUM(D11:H11)

0.036

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

7. In cell J12, calculate the volatility of the portfolio as the square root of the return variance.

J12: = SQRT(J11)

0.191

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

7. In cell J12, calculate the volatility of the portfolio as the square root of the return variance.

J12: = SQRT(J11)

0.191

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

7. In cell J12, calculate the volatility of the portfolio as the square root of the return variance.

J12: = SQRT(J11)

0.191

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

8. In cell B14, calculate the Sharpe’s ratio of the portfolio.

B14: = (J10-B13)/J12

0.336

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

8. In cell B14, calculate the Sharpe’s ratio of the portfolio.

B14: = (J10-B13)/J12

0.336

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

8. In cell B14, calculate the Sharpe’s ratio of the portfolio.

B14: = (J10-B13)/J12

0.336

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11

Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1

Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative

Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.200 0.200 0.200 0.200 0.200

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.200

0.200

0.200

0.200

0.200

0.016 0.021 0.024 0.015 0.017 0.094

0.007 0.008 0.010 0.006 0.005 0.036

0.191

0.336

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

9. Apply Data/Analysis/Solver to identify the minimum variance portfolio with the non-negativity constraint.

Set Objective: $J$11 Objective: Min

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14

Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1

Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative

Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.231 0.096 0.115 0.197 0.361

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.231

0.096

0.115

0.197

0.361

0.019 0.010 0.014 0.015 0.031 0.089

0.008 0.003 0.004 0.006 0.012 0.033

0.181

0.324

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance

Asset expected return – Exercise 1

Exercise 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A B C D E F G H I J

Asset weight

Asset expected return

Covariance matrix

Portfolio expected return

Portfolio return variance

Portfolio volatility

Risk-free rate

Sharpe’s ratio

0.136 0.232 0.206 0.115 0.313

0.082 0.105 0.122 0.076 0.085

0.08370 0.03358 0.02030 0.01982 0.01132

0.03358 0.09189 0.03226 0.02484 0.02132

0.02030 0.03226 0.18625 0.00252 0.00866

0.01982 0.02484 0.00252 0.09669 0.01826

0.01132 0.02132 0.00866 0.01826 0.06541

0.030

1.000

0.136

0.232

0.206

0.115

0.313

0.011 0.024 0.025 0.009 0.027 0.096

0.004 0.010 0.011 0.003 0.010 0.037

0.192

0.343

10. Apply Data/Analysis/Solver to identify the tangent portfolio with the non-negativity constraint.

Set Objective: $B$14 Objective: Max

By Changing Variable Cells: $D$1:$H$1 Subject to the Constraints: $J$1 = 1

Make Unconstrained Variables Non-Negative Solve

Jukka Perttunen Fundamentals of finance