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
MORE INFORMATION
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