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Concept Of Risk And Measurement Techniques

  Concept of Risk and Risk Measurement Techniques

Illustration 107

Suppose an investment in offshore oil exploration, there are two feasible results, the success of the project capitulating a payoff of $45 per share with a possibility 0.35 and the failure capitulating a payoff $5 per share with a possibility of 0.65. What is the anticipated value of investment per share.

Also determine what would be the investment per share in case of 5 probabilities as given below.


Cash Flows
Value in Million $ - Xi

Possibilities
Pi

25

0.4

35

0.5

45

0.6

55

0.5

65

0.4

Solution

  1. The anticipated value of investment per share

Formula to ascertain the anticipated value per share in case of two possibilities is,
            __
E (X) or X =          P1X1 + P2X2

                =          0.35 * 45 + 0.65 * 5

               =          15.75   +          3.25

               =          19

  1. Formula to ascertain 2 probable outcomes is

  E (X) or X =          P1X1 + P2X2 + …. PnXn
                              n
                  =          Σ          PiXi
                              t=1

Therefore,

                  =          (0.4 * 25) + (0.5 * 35) + (0.6 * 45) + (0.5 * 55) + (0.4 * 65)

                  =          10        +          17.5     +          27        +          27.5     +          26

                  =          108

Or
     


Cash Flows
Value in Million $ - Xi

Possibilities
Pi

Anticipated Value
PiXi

25

0.4

10

35

0.5

17.5

45

0.6

27

55

0.5

27.5

65

0.4

26

 

Σ PiXi = 108

                                               __
                                    E(X) or X        =          108

Illustration 108

Determine the standard deviation for the following probable outcomes.

Cash Flow

Probability

15

0.15

25

0.25

35

0.35

45

0.45

55

0.55

Solution

Cash Flow
Outcome - X

Probability
Pi

       _
X – X

_
(X – X)^ 2

        _
(X – X)^2 Pi

15

0.15

- 20

400

60

25

0.25

- 10

100

25

35

0.35

0

0

0

45

0.45

10

100

4.5

55

0.55

20

400

11

 _
 X = 35

 

0

 

100.5

                                    Σ = √ 100.5     =          10.02

Illustration 109

Determine the standard deviation and Coefficient Variation for the following.

Project A

Cash Flow

Probability

50

0.50

60

0.75

70

0.90

80

0.75

90

0.50

Project B

Cash Flow

Probability

40

0.75

55

0.85

70

0.95

85

0.85

100

0.75

  1. Standard Deviation and Variance for the Project A is as follows:

Cash Flow

Probability

       _
X – X

  _
(X – X)^ 2

         _
(X – X)^2 Pi

50

0.50

- 20

400

200

60

0.75

- 10

100

75

70

0.90

0

0

0

80

0.75

10

100

75

90

0.50

20

400

200

_
X = 70

     

550

                                                      n                      __
Standard Deviation σ       =          Σ          √ (X1 – X1)^ 2. P1
                                                      t=1

                                          =          √550                =          23.45

Co-efficient Variance       =          VA                   =          σ A      =          23.45
                                                                                          R A                   550

                                                                              =          0.04

  1. Standard Deviation and Variance for the Project B is as follows:

Cash Flow

Probability

       _
X – X

  _
(X – X)^ 2

          _
(X – X)^2 Pi

40

0.75

- 30

900

675

55

0.85

- 15

225

191.25

70

0.95

0

0

0

85

0.85

15

225

191.25

100

0.75

30

900

675

_
X = 70

     

1766.25

                                                      n                      __
Standard Deviation σ       =          Σ          √ (X1 – X1)^ 2. P1
                                                      t=1

                                          =          √1766.25         =          42.02

Co-efficient Variance       =          VA                   =          σ A      =          42.02
                                                                                          R A                 1766.25

                                                                              =          0.02

Therefore, the standard deviation as well as the variance is greater for Project B. This entails the relative risk of project B is comparatively lesser than that of project A.

Thus, as the anticipated value of project B is greater and its relative risk too is greater, the manager will decide to invest in this project.

Illustration 110

The table presents the probability allocation and anticipated value on investments. Determine the less risk project among the two.

State of Nature

Probability Pi

Outcome Monetary Return Xi

 

Investment in Project C

 

Inflation Null

0.25

100

Reasonable Inflation

0.40

175

Huge Inflation

0.30

300

 

Investment in Project D

 

Inflation Null

0.25

150

Reasonable Inflation

0.40

200

Huge Inflation

0.30

250

Investment Project C

Anticipated Value E(X)    =          P1X1 + P2X2 + P3X3

                                          =          0.25 * 100 + 0.40 * 175 + 0.30 * 300

                                          =          25        +          70        +          90

                                          =          185

S D σ         =          √ P1 (X1 – E(X)) ^2 + P2 (X2 – E(X)) ^ 2 + P3 (X3 – E(X)) ^2

                  =          √ 0.25 (100 –185) ^2 + 0.40 (175–185) ^2 + 0.30 (300 –185) ^2

                  =          √ 0.25 (85) ^2 + 0.40 (-10) ^2 + 0.30 (115) ^2

                  =          √ 0.25 (7225) + 0.40 (100) + 0.30 (13325)

                  =          √ 1806.25 + 40 + 3997.5

                  =          √ 5843.75        =          76.44

Investment Project D

P1X1 + P2X2 + P3X3

                                          =          0.25 * 150 + 0.40 * 200 + 0.30 * 250

                                          =          37.5     +          80        +          75

                                          =          192.5

S D σ         =          √ P1 (X1 – E(X)) ^2 + P2 (X2 – E(X)) ^ 2 + P3 (X3 – E(X)) ^2

                  =          √0.25(150–192.5)^2+0.40(200–192.5)^2 + 0.30 (250–192.5)^2

                  =          √ 0.25 (42.5) ^2 + 0.40 (-7.5) ^2 + 0.30 (57.5) ^2

                  =          √ 0.25 (1806.25) + 0.40 (56.25) + 0.30 (3306.25)

                  =          √ 451.56 + 22.5 + 991.875

                  =          √ 1466 =          38.28

  1. It is unambiguous from the above that anticipated return from investment Project C and D are slightly different, that is 185 and 192.5.
  1. However the investment Project C integrates double the risk as compared to investment D that is 76.44 and 38.28 correspondingly.
  1. Thus, the manager would choose Project D which has lesser risk comparatively.
  1. It is thus unambiguous that decision making involving risk is based not only on the anticipated value but also on the degree of risk incorporated and in the person’s approach toward risk.
  1. It has to be noted that the scrutiny of decision making by a person integrating risk and uncertainty is helpful not only for choosing the investments but also scrutinising between any performance integrating risk and uncertainty.
  1. Therefore, selection among different ways of accomplishment that vary in both anticipated value and risk, needs to be scrutinised the individuals preference towards risk.

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