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Decision Making Under Risk

  Decision Making Under Risk

Illustration 111

An investor has to select among two investment projects E and F each needs an initial expenditure of $5 millions. The present value of feasible cash flows from them and their associate probabilities are provided as under:

Investment Project E
Value in Million $

Investment Project F
Value in Million $

Cash Flow

30

70

40

50

60

Probability

0.55

0.45

0.4

0.3

0.3

(1) Compute the anticipated value of every investment project
(2) Compute the level of risk of each investment project
(3) In which project the industry must undertake investment

Solution

  1. Anticipated Value of Project E

  E (X) of Project E       =          P1X1 + P2X2

                                    =          0.55 * 30 + 0.45 * 70

                                    =          16.5     +          31.5

                                    =          48

Anticipated Value of Project F

E (X) of Project F       =          P1X1 + P2X2 + P3X3

                                    =          0.4 * 40 + 0.3 * 50 + 0.3 * 60

                                    =          16        +          15        +          18

                                    =          49
 
The anticipated value of Projects E and F are almost equal 48, 49. As each project costs $5 millions, the anticipated value of profits of each will be $48 millions.

  1. Risk can be measured by computing the standard deviation of every probability allocation.

                            n
 σ of E  =          √ Σ       (Xi – E (Xi)) ^ 2 Pi
                          t=1

            =          √ (X1 – E (X1)) ^2 + (X2 – E (X2)) ^2

            =     √   (30 – 48) ^2     +          (70 – 48) ^2

            =     √   (-12) ^2           +          (22) ^2

            =     √   144                  +          484

            =     √   628      =          25.06

σ of F              =     √   (40 – 49) ^2 + (50 – 49) ^2 + (60 – 49) ^2

                        =     √   (-9) ^2 +          (-1) ^2 +          (11) ^2

                        =     √   81        +          1          +          121

                        =     √   203      =          14.25

It is unambiguous from the values of Standard deviation that investment in Project E is more risky.

  1. As anticipated value of cash flows of each investment is the same but investment project E is more risky, the industry will undertake investment in project F.

Illustration 112

A manager of an industry is countenanced with the difficulty in which of the commodities G and H should he invest. The market analyses the following net present values of all future profits under three feasible states of the economy.

State of the Economy

Investment in Commodity G
Probability                    Profit

Investment in Commodity H
Probability                    Profit

Inflation

0.3

20

0.4

40

Moderate

0.4

60

0.2

30

Depression

0.3

0

0.4

20

The manager’s utility function for money is U = 200M – M ^2 where M symbolises for money profits.

  1. Ascertain whether manager is risk hesitant, risk neutral , seeker
  1. If manager’s objective is to maximise money profits irrespective of risk, in which commodity he must invest?
  1. What is the relative level of risk involved in each investment?
  1. If the manager’s aim were to optimise utility in which commodity he must invest?

Solution

Taking the first derivative of (1) which will indicate the marginal utility of money, we have

                                    dU       =          200 – 2M
                                    dM

Taking the second derivative

                                    D^2 U =          -2
                                      dM

Taking negative value of the second derivative implies that marginal utility of money for the manager diminishes. Hence he is a risk hesitant.

(2) To determine in which product will capitulate him more profits we ascertain the anticipated values of money profits for investment in the two commodities.

Investment in Commodity G

State of Economy

Pi

πi

Pi πi

      _
π - π

_
(π – π) ^2

_
     (π – π) ^2 * Pi

Inflation

0.3

20

6

-10

100

30

Moderate

0.4

60

24

30

900

360

Depression

0.3

0

0

-30

900

270

     

_
π = 30

   

660

Anticipate Profits from G       =          30

                                                               n                           _
Standard Deviation of G        =          √ Σ       =          (πi – π) ^2 * Pi
                                                              t=1

                                                                        =          √ 660   =          25.70

Relative risk is determined by co-efficient of variation         =          σ         
                                                                                                            _
                                                                                                            π

                                                                                                =          25.70   = 0.85
                                                                                                               30

Investment in Commodity H

State of Economy

Pi

πi

Pi πi

      _
π - π

 _
(π – π) ^2

_
      (π – π) ^2 * Pi

Inflation

0.4

40

16

10

100

40

Moderate

0.2

30

6

0

0

0

Depression

0.4

20

8

-10

100

40

     

 _
 π = 30

   

80

Anticipate Profits from H       =          30

                                                               n                           _
Standard Deviation of H        =          √ Σ       =          (πi – π) ^2 * Pi
                                                              t=1

                                                                        =          √ 80     =          0.94

Relative risk is determined by co-efficient of variation         =          σ        
                                                                                                            _
                                                                                                            π

                                                                                                =          0.94     = 0.031
                                                                                                              30

It adopts from above that if manager’s objective is to optimise profits he will select to invest in commodity H as anticipated value of profits for it is 30, however it is the same 30 in the case of Commodity G too.

While if we consider the co-efficient variance, then too, commodity H is effective since it is comparatively lesser than commodity G.

(3) Relative risk is gauged by coefficient of variation V which in the current case will show standard deviation per dollar of profits. Coefficient of variation of investment in commodity

                                    G         =          σ          =          25.70   =          0.85
                                                            π                         30

Coefficient of variation of investment in commodity H

                                    H         =          σ          =          0.94     =          0.031
                                                            π                        30

Therefore, investment in commodity G integrates more risk, though it capitulate higher anticipated money profits.

(4) To determine anticipated utility we first compute utilities of money profits and then increment them by their related probabilities. For computing utilities procured from money profits, we use the provided utility function U = 200M – M^2.

Computing Anticipated Utility from Investment in G

πi

Ui

Pi

PiUi

20

100*20 – 20^2 = 1600

0.3

480

60

100*60 – 60^2 = 2400

0.4

960

0

0

0.3

0

 

Σ PiUi = 1440

Computing Anticipated Utility from Investment in H

πi

Ui

Pi

PiUi

40

100*40 – 40^2 = 2400

0.4

960

30

100*30 – 30^2 = 2100

0.2

420

20

100*20 – 20^2 = 1600

0.4

640

 

Σ PiUi = 2020

Therefore, if manager’s aim is to optimise utility, he will invest in commodity H.

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