4th session ppt

8/10/2019 4th Session PPT

1/22

Risks and uncertainity

Insurable risks

Non-insurable risks

Investment decisions under risk:

Strategies and state of nature. Outcome and pay-off matrix.

Risk-Return evaluation.

Risk preference.

risk seekerrisk averter

risk indifferent


2/22

Adjustments for risk and criteria for decisions:

1. Finite-horizon method

2. Risk discounting model.

3. The shackle approach.

4. The probability theory approach.

5. Sensitivity analysis.

6. Simulation.7. Hedging.

8. Decision tree.


3/22

Investment decisions under

uncertainity

The maximin criteria.

The minimax regret criteria.

The hurwicz alpha index. The maximax criteria.

The laplace criterion.


4/22

Treatments of Risks and Uncertainty in

Projects

The availability ofpartialor imperfect

information about a problem leads to two

new category of decision-making techniques

Decisions under risk (In terms of a probability function)

Decisions under Uncertainty (No probability function is

secure)


5/22

Decisions under risk

Decisions under risk are usually based on one

of the following criteria

Expected Value

Combined Expected value and variance

Known Aspiration level

Most likely occurrence of a future state


6/22

Expected Value Criterion

Expressed in terms of either actual money or its utility

Decision Makers attitude towards the worth or utility of

money is important

The final decision should ultimately be made by consideringall pertinent factors that affect the decision makersattitude

towards the utility of money

The drawback of this is that use of expected value criterion

may be misleading fro the decisions that are applied only a

few number of times i.e small sample sizes


7/22

Example 1

A preventive maintenance policy requires making decisions about when a

machine (or a piece of equipment) should be serviced on a regular basis in

order to minimize the cost of sudden breakdown

The decision situation is summarized as follows. A machine in a group of nmachines is serviced when it breaks down. At the end of T periods,

preventive maintenance is performed by servicing all n machines. The

decision problem is to determine the optimum T that minimizes the total

cost per period of servicing broken machines and applying preventive

maintenance


8/22

Let ptbe the probability that machine would break down in period t

Let ntbe the random variable representing the number of broken machines in

the same period.

C1is the cost of repairing a broken machine C2 the preventive maintenance of the machine

The expected cost per period can be written as

Where E{nt}is the expected number of broken machines in period t.

nt is a binominal random variable with parameter (n,pt), E{nt}=npt

The necessary condition for T*to minimize EC(T) are

EC(T*-1)>= EC(T*) and EC(T*+1)>= EC(T*)

T

ncnEcTEC

T

tt

1

121

)(


9/22

To illustrate the above formulation, suppose

c1=Rs.100, c2=Rs.10 and n=50

The values of pt and EC(T) are tabulated below

T*

T pt Cumulative pt EC(T)

1 0.05 0 500

2 0.07 0.05 375

3 0.10 0.12 366.7

4 0.13 0.22 400

5 0.18 0.35 450


10/22

Expected Value-Variance Criterion

We indicated that the expected value criterion is suitable formaking long-run decisions

To make it work for the short-run decision problems Expected

Value-Variancecriterion is used

A possible criterion reflecting this objective is Max E[Z]-k*var[z]

Where z is a random variable for profit and k is a constant

referred to as risk aversion factor

Risk aversion factor k is an indicator of the decision makersattitude towards excessive deviation from the expected

values.


11/22

Applying this criteria to example 1 we get

Ct is the variance of EC(T) This criteria has resulted in a more conservative decision that applies

preventive maintenance every period compared with every third period

previously

T pT pT2 Cum. pT Cum. pT2 EC(T)+varcT

1 0.05 0.0025 0 0 500

2 0.07 0.0049 0.05 0.0025 6312

3 0.10 0.0100 0.12 0.0074 6622

4 0.13 0.0169 0.22 0.0174 6731

5 0.18 0.0324 0.35 0.0343 6764


12/22

Aspiration Level Criterion

This method does not yield an optimal decision in the sense

of maximizing profit or minimizing cost

It is a means of determining acceptable courses of action

Most Likely Future Criterion

Converting the probabilistic situation into deterministic

situation by replacing the random variable with the singlevalue that has the highest probability of occurrence


13/22

Decisions under uncertainty

They assume that there is no probability distributionsavailable to the random variable.

The methods under this are

The Laplace Criterion The Minimax criterion

The Savage criterion

The Hurwicz criterion


14/22

Laplace Criterion

This Criterion is based on what is known as theprinciple of

insufficiency

ai is the selection yielding the largest expected gain

Selection of the action ai

*corresponding

where 1/n is the probability that

ia

n

j

jiavn

1

,1

max


15/22

Example 2

A recreational facility must decide on the level of supply it must stock to meet the needsof its customers during one of the holiday. The exact number of customers is not known,but it is expected to be of four categories:200,250,300 or 350 customers. Four levels ofsupplies are thus suggested with level i being ideal (from the view point of the costs) if thenumber of customer falls in category i. Deviation from these levels results in additionalcosts either because extra supplies are stocked needlessly or because demand cannot besatisfied. The table below provides the costs in thousands of dollars

a1, a2, a3 and a4 are the supplies level

Customer Category

3 4

a1 5 10 18 25

a2 8 7 8 23

a3 21 18 12 21

a4 30 22 19 15


16/22

Solution by Laplace Criterion

E{a1} = (1/4)(5+10+18+25) = 14.5

E{a2} = (1/4)(8+7+8+23) = 11.5

E{a3} = (1/4)(21+18+12+21) = 18.0

E{a4} = (1/4)(30+22+19+15) = 21.5

Thus the best level of inventory according to

Laplace criterion is specified by a2.


17/22

Minimax (Maxmini) Criterion

This is the most conservative criterion since it is based on

making the best out of the worst possible conditions

If the outcome v(ai,j)represents loss for the decision maker,

then, for, aithe worst loss regardless of what jmay be is max

j [v(ai,j)]

The minimax criterion then selects the action ai associated

with min ai max j [v(ai,j)]

Similarly if v(ai,j)] represents gain, the criterion selects the

action aiassociated with max ai min j [v(ai,j)]

This is called themaxmini criterion


18/22

Applying this criterion to the Example 2

Thus the best level of inventory according to this criterion isspecified by a3

Customer Category

MaxSupply 3 4

a1 5 10 18 25 25

a2 8 7 8 23 23

a3 21 18 12 21 21

a4 30 22 19 15 30

Minimax value


19/22

Savage Minimax Regret criterion

This is an extremely conservative method

The Savage Criterion introduces what is called as regret matrix

which is defined as

r(ai,j)={jijk

aavav

k

,,max

jkaji avav k ,min,

if v is profit

if vis loss


20/22

Applying this criteria to Example 2

The regret matrix is shown below

Thus the best level of inventory according to this criterion is specified by a2

Customer Category

MaxSupply 1 2 3 4

a1 0 3 10 10 10

a2 3 0 0 8 8

a3 16 11 4 6 16

a4 25 15 11 0 25

minimax


21/22

Hurwicz Criterion

This Criterion represents a range of attitudes from the most optimistic to the most

pessimistic

The Hurwicz criterion strikes a balance between extreme pessimism and extreme

optimism by weighing the above two conditions by the respective weights and

1- , where 0


22/22

Applying this criterion to Example 2 Set =0.5

Resolving with =0.75 for selecting between a1 and

a2

min max min+(1-)max

5 25 15

7 23 15

12 21 16.5

15 30 22.5

minimum

4th session ppt

Documents