Pricing Under Risk and Uncertainty

7/27/2019 Pricing Under Risk and Uncertainty

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PRICING UNDERRISK AND

UNCERTAINTY


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INTRODUCTIONReal world is full of uncertainties and risk.

People face uncertainty as regards to future income.e.g. Taking loan to buy or build a house, factory or acar and plan to pay for them out of their future

incomes.


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People have often to choose

how much risk to bear.e.g. Investment of money insavings a/c. or banksdeposits or investment in

shares of some company orin mutual funds.

To make a choice among

these alternatives, we needto measure the risk so as tocompare the riskiness ofalternative choices.


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THE CONCEPT OF RISK ANDUNCERTAINTYRISK :

The risk refers to asituation when theoutcome of a decision is

uncertain but when theprobability of eachpossible outcome isknown or can be

estimated.


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UNCERTAINTY :

The uncertainty refers to the situationwhen there is more than one possibleoutcome of a decision but where theprobability of occurrence of each

particular outcome is not known or evencannot be estimated.


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In Decision Making Involving Risk and Uncertaintythe three terms are quite often used.

(i) strategy ,(ii) state of nature and(iii) outcome.


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(i) STRATEGY:A strategy refers to one

of several alternative courses of

actions or plans that can be

implemented to achieve the desiredgoal.


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(ii) STATE OF NATURE :

It refers to the conditions that prevail in

future and which have a significant

effect on the success or failure of the

strategy.


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(iii) OUTCOME :

It refers to the results which are usuallyin the form of profit that come about asa result of implementation of a strategy.

Risk refers to the amount of variabilityin the outcome as a result of the

adoption of a particular strategy.


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MEASURING RISK : PROBABILITYOF AN OUTCOMETo measure the degree of risk, we need to know the

probability of each possible outcome of a decision. Theprobability means the likelihood of occurring of an event. Thus,

if probability of an outcome occurring is or .25 this means that

there is 1 chance in 4 or 25% chance for the outcome to occur.


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There are two concepts of probability dependingon how it is measured

(i) Frequency concept of probability :It is based on past information. If a situation isrepeated over a large number of times , say M,

and if outcome , say X , occurs m times , then

P(X) = m/M

(ii) Concept of subjective probability :

It is based on personal judgement , experience orknowledge.


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MEASURING RISK WITHPROBABILITYDISTRIBUTIONProbability distribution describes theoccurrence of all possible outcome of an eventand probability of occurrence of each outcome.

It is worth noting that the sum of probabilitiesof all possible outcomes must equal unity

because probabilities of all outcomes together

must equal certainty.


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Table 35.1 :

We give all possible cash flows that will occurfrom an investment project A in the next yearsand the probabilities of their occurrence.


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Table 35.2 :

We give the cash flows that will occur in the nextyear and their associated probabilities from aninvestment project B.


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Figure 35.1 Figure 35.2


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The concept of probability distribution is required forevaluating and comparing investment projects whenmanagers have to take decision under conditions of risk.

From the probability distribution of outcomes , we cancalculate two values which are essential for decisionmaking under conditions of risk.


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They are :

(1)expected value of all possibleoutcomes (cash flows) and

(2) a value that measure the degree ofrisk involved (variability of an

outcome) .


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(1) EXPECTED VALUE :

If there are two possible outcomes with payoffs ofX1 and X2 and the probability of each possibleoutcome denoted by P1 and P2 , then theexpected value of investment is given by

E(X) = P1X1 + P2X2

Similarly , if there are n possible outcomes , then the

expected value is

E(X) = P1X1 + P2X2 + P3X3 + _ _ _ _PnXn


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In addition to the expected value , the probabilitydistribution of outcomes also helps us in measuring riskinvolved in a project . The variability of outcomesmeasures the degree of risk involved in any choice of aproject or strategy from the various lternative projects or

strategies.

(2) RISK AND PROBABILITY DISTRIBUTION :


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The greater the variability or

dispersion of various outcomes fromthe expected value of payoffs meansthe greater risk involved. Thevariability of outcomes may bemeasured by the average deviationof actual values of payoffs of variousoutcomes from the expected value of

payoff with probability of each beingused as weights.


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Let X1and X2 are the payoffs of two outcomesand the probability of each is P1 and P2 , thenthe average deviation (V) as a measure of risk isgiven by


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Most widely used measure of dispersionor variability is the standard deviation.


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Since expected value E(X) is alsowritten as mean , the standarddeviation is also written as


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STANDARD DEVIATION ANDPROBABILITY DISTRIBUTION

Probability distribution of outcomes is assumed to be one ofstandard normal distribution which is symmetric around theexpected value and also there is 50% possibility that outcome willbe above the expected value and there is 50% possibility that

outcome will be less than 50. The probability of a particularoutcome occurring depends on how many standard deviation itis away from expected value.

In the probability distribution given in Table 35.3 , the expected

value (mean) from all five outcomes is Rs. 50 lakhs and standarddeviation is


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Table 35.3 Table 35.4


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CONTINUOUS PROBABILITYDISTRIBUTION AND NORMAL CURVEFig 35.3


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MEASURINGPROBABILITY OFOUTCOME LYINGWITHIN PARTICULAR

RANGE


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In Table 35.1 we are interested to know what will be the

probability of cash flow of Rs. 62 lakhs from theproject A will be within range of Rs. 50 lakhs and Rs.60 lakhs. Assuming that cash flows are normallydistributed with expected value equal to Rs. 50 lakhsand standard deviation of 10.59. We first find the value

of Z as under


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THE COEFFICIENT OF VARIATIONThe Relative measure of risk : When theexpected values of two projects are equal or

very close to each other , the S.D. is anappropriate measure of riskiness of projects.However, when the expected values of theinvestment projects are quite different , then

we make use of coefficient of variation whichis a relative measure of risk.


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The coefficient of variation measures risk

relative to the expected values of theprojects i.e. cov measures risk per rupeeof the expected value. Coefficient of

variation is obtained from dividing the

s.d. of probability distribution by theexpected value(mean).

Coefficient of variation :


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Table 35.5


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DECISION MAKING UNDER RISKFor making a rational decision , the following 3 things should be

determined :

(1)The expected values of payoffs associated with

various outcomes be calculated.

(2) The degree of risk of various strategies bemeasured by estimating the standard deviation

of the average deviation of payoffs of variousoutcomes from the expected value.

(3) Information about the decision makerregarding his preference towards risk be

obtained.


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e.g. Suppose an individual

is considering two types ofinvestment , say A and B .Each type of investmentrequires an initial cost of

Rs. 1 lakh and have a life of5 years. The monetaryreturn on these two types ofinvestment depends on therate of inflation in the next5 years.


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Table 35.6


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ATTITUDE TOWARDS RISKASSUMPTIONS :

To explain the preference towards risk we will

consider a single composite commodity , namely ,

money income. An individuals money income

represents the market basket of goods that he can

buy. It is assumed that the individual knows the

probabilities of making or gaining money income indifferent situations. But the outcomes or payoffs

are measured in terms of utility rather than rupees.


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RISK AVERTER : Figure 35.4

For a riskaverse

individualmarginalutility of

moneydiminishes ashe has more

money.


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RISK SEEKER : For a risk seeker

individual marginal utility of

money increases as money withhim increases.

Fig 35.5

RISK NEUTRAL : For a risk neutral

individual marginal utility of money

remains constant as he has moremoney.

Fig 35.6


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MANAGEMENT OF RISK : REDUCINGRISKWe have seen now above that though some individuals are

risk-seekers , most of the individuals are risk-averse and try to

reduce risk or uncertainty they face. There are three methods

by which individuals or consumers can reduce risk. They are :

(1)DIVERSIFICATION

(2) INSURANCE

(3) GATHERING

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Pricing Under Risk and Uncertainty