Lecture 4

                                           [INS:   :INS]

1.      In the following table there are three markets with only two securities. Calculate the
weights for minimal risk portfolio on each market. What will be the expected return and risk of
every portfolio? Which of this three markets will be the best investment?

                                              Market

                                             Security

                                               r[i]

                                               risk

                                  [INS: Correlation[A] :INS] [,B]

                                                 I

                                                 A

                                               0,22

                                               0,30

                                               0,15

                                                 B

                                               0,31

                                               0,32

                                                II

                                                 A

                                               0,26

                                               0,29

                                               -0,06

                                                 B

                                               0,34

                                               0,33

                                                III

                                                 A

                                               0,18

                                               0,20

                                               0,09

                                                 B

                                               0,41

                                               0,38


2.      For investment opportunities in three companies calculate the minimum variance portfolio.
In the next step calculate the optimal portfolio with return of 15 %. For every portfolio calculate
the return and risk.


                                             Company 1

                                             Company 2

                                             Company 3


                                            Correlation

                                                 m

                                                0,8

                                                0,3

                                                0,6


                                              s[1,2]

                                               -0,1

                                                 s

                                                1,2

                                                0,8

                                                1,1


                                              s[1,3]

                                               -0,5






                                              s[2,3]

                                                0,3



3.      In the following table you find covariance matrix and the vector of returns. What will be
the optimum portfolio (weights) with minimum risk and then calculate the weights with return of 5
%. For all versions calculate return and risk.


                                              Sec[1]

                                              Sec[ 2]

                                              Sec[ 3]

                                              Sec[ 4]

                                              Sec[ 5]

                                              Sec[ 6]

                                              Sec[ 7]

                                             r[i] (%)

Sec[ 1]

                                               80,5

                                               82,7

                                               85,3

                                               85,1

                                               123,9

                                                22

                                                3,5

                                                1,9

Sec[ 2]

                                               82,7

                                               184,7

                                               131,5

                                               69,4

                                               49,5

                                                58

                                               -9,9

                                                6,1

Sec[ 3]

                                               85,3

                                               131,5

                                               374,2

                                               384,5

                                               366,5

                                               103,8

                                               343,5

                                                2,9

Sec[ 4]

                                               85,1

                                               69,4

                                               384,5

                                               684,8

                                               599,1

                                               51,6

                                               502,7

                                                 4

Sec[ 5]

                                               123,9

                                               49,5

                                               366,5

                                               599,1

                                               871,4

                                               -21,2

                                               520,4

                                                5,7

Sec[ 6]

                                                22

                                                58

                                               103,8

                                               51,6

                                               -21,2

                                               89,7

                                               74,4

                                                3,4

Sec[ 7]

                                                3,5

                                               -9,9

                                               343,5

                                               502,7

                                               520,4

                                               74,4

                                               574,6

                                                4,9


4.      Consider an investment opportunity with risky asset and risk free asset. The return of risk
free asset will be 3,5 %.

Risky portfolio

                                                                A

                                                                                                   B

                                                                                                       C

                                                                                                           D

               6,2%

                                                                                                  4%

                                                                                                     7,5%

                                                                                                         8,4%

               14,5%

                                                                                                 9,7%

                                                                                                      17%

                                                                                                          20%


Weights by different portfolios:


                                                1.

                                                2.

                                                3.

                                                4.

                                                5.

                                               0,20

                                               0,40

                                               0,50

                                               0,60

                                               0,80

                                             Portfolio

                                               0,80

                                               0,60

                                               0,50

                                               0,40

                                               0,20