The Theory of Choice

7/30/2019 The Theory of Choice

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The Theory of Choice:Utility Theory Given Uncertainty

Ing. Ale Kresta, Ph.D.


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Content of the lecture:

Five Axioms of the Choice under Uncertainty

Utility Function

Absolute and Relative Risk Aversion

Valuation

Criteria for Decision-Making


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To develop a theory of rational decision-making in the face ofuncertainty, we need assumptions on individuals behavior(axioms of cardinal utility).

Axiom 1: Comparability (completeness)For the entire set S of uncertain alternatives, an individual can

say for outcomesx, y: (x>y) or (y>x) or (x~y)

Axiom 2: Transitivity (consistency)If (x>y) and (y>z) then (x>z). And if (x~y) and (y~z) then (x~z).


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Axiom 3: Strong independence

We construct a gamble where the individual has a probability of forreceiving outcomexand a probability of (1- ) of receiving outcome z.

We construct the second gamble where the individual has a probabilityof for receiving outcome yand a probability of (1- ) of receivingoutcome z. Then,

if (x~y) then G(x,z:) ~G(y,z:)

Axiom 4: MeasurabilityIfx>y>z then there is a unique probability , such that the individualwill be indifferent between y and a gamble betweenxwith probability and z with probability 1-.

Ifx>y>z, then there exists a unique , such that y~G(x,z:).


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Axiom 5: Ranking

For alternativesx>y>z we can establish a gamble such that anindividual is indifferent between yand a gamble betweenxand z with

certain probability 1, y~

G(x,z:1). Also forx>u>z we can establishgamble such that u~G(x,z:2).

Ifx>y>z andx>u>z, then ify~G(x,z:1) and u~G(x,z:2), it follows that

If1>2 then y>u, if1


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Utility Function

More wealth is preferred to less; marginal utility of wealth is positive MU(W)>0.

We establish a gamble between two prospects, a and b. Probability of receiving

a is , for b it is (1- ); -> G(a,b: ).

Three utility functions with positive marginal utility: (a) risk lover; (b) risk neutral;

(c) risk averter.


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Will we prefer the actuarial value of the gamble (expected/

average outcome) with certainty - or the gamble itself?

Assume the gamble, where you get 30 EUR withprobability 20% and 5 EUR with probability 80%. Theexpected (average) value is thus 10 EUR. Will youchoose the gamble or the value 10 EUR? Or will you bewilling to pay 10 EUR for the gamble?


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Will we prefer the actuarial value of the gamble (expected/

average outcome) with certainty - or the gamble itself?

Assume the gamble, where you get 30 EUR withprobability 20% and 5 EUR with probability 80%. Theexpected (average) value is thus 10 EUR. Will you

choose the gamble or the value 10 EUR? Or will you bewilling to pay 10 EUR for the gamble?

The person preferring the gamble -> risk lover.

The person preferring the actuarial value with

certainty -> risk averter. The person who is indifferent between both -> risk

neutral.


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Risk Averter

W(wealth, income, cash-flow etc.) U(utility)

E(W) = E[G(a,b:)] = 0.8 5 + 0.2 30 = 10 U[E(W)] = U(10) = ln 10 = 2.3

a = 5 EUR U(5) = ln 5 = 1.61b = 30 EUR U(30) = ln 30 = 3.4

E[U(W)] = 0.8 1.61 + 0.2 3.4 = 1.97

Certainty equivalent, CE = 7.17 U(CE) = ln 7.17 = 1.97

Suppose that the utility function U of the person with aversion

to risk is U=ln(W).

Now we can compute the utilities of the gamble and certain

value.


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Risk Averter


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Risk Premium Let us calculate the max amount of wealth a person would be willing to give up in order to

avoid the gamble, called risk premium.

Risk Premium: Difference between expected wealth (given the gamble) and the level of

wealth the individual would accept with certainty if the gamble were removed (= certaintyequivalent wealth).

We had utility function U=ln(W), with current wealth level 10 EUR. Then we have the gamble

G(5,30:80%) -> with prob 80% we will face a decline to 5 EUR, with prob 20% we will increase

wealth by EUR 20.

E[U(G)] = 1.97.

From logarithmic function: 1.97 gives a wealth of EUR 7.17 (CE).

We would accept the gamble, if the CE is 10 EUR (our current wealth) or more.

How much will we pay to avoid the gamble? We will be willing to pay 2.83 EUR (10-

7.17=2.83) = Markowitz risk premium.

If we could buy an insurance against the gamble for less than EUR 2.83, we will buy it.


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Risk Lover

W(wealth, income, cash-flow etc.) U(utility)

E(W) = E[G(a,b:)] = 0.8 5 + 0.2 30 = 10 U[E(W)] = U(10) = 0.04 10 10 = 4

a = 5 EUR U(5) = 0.04 5 5 = 1

b = 30 EUR U(30) = 0.04 30 30 = 36

E[U(W)] = 0.8 1 + 0.2 36 = 8

Certainty equivalent, CE = 14.14 U(CE) = 0.04 14.14 14.14 = 8

Suppose that the utility function U of the person looking for the

risk is U=0.04W2.


value.


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Risk Neutral

W (wealth, income, cash-flow etc.) U (utility)

E(W) = E[G(a,b:)] = 0.8 5 + 0.2 30 = 10 U[E(W)] = U(10) = 0.5 10 = 5

a = 5 EUR U(5) = 0.5 5 = 2.5

b = 30 EUR U(30) = 0.5 30 = 15

E[U(W)] = 0.8 2.5 + 0.2 15 = 5

Certainty equivalent, CE = 10 U(CE) = 0.5 10 = 5

Suppose that the utility function U of the person neutral to the

risk is U=0.5W.


value.


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Utility Function We have to compare the actuarial value (average, expected) of the gamble

obtained with certainty and the gamble itself:

if U[E(W)]>E[U(W)] then we have risk aversion individual

(concave utility function),

if U[E(W)]=E[U(W)] then we have risk neutral individual

(linear utility function),

if U[E(W)]


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The higher the curvature of utility function U, the higherthe risk aversion (and also the risk premium).

Risk aversion can be measured by the ArrowPratt absolute

risk-aversion, also known as the coefficient ofabsolute riskaversion (ARA), defined as

=

.

In simple terms, what we are measuring above is the actual

dollar amountan individual will choose to hold in riskyassets, given a certain wealth level W. For this reason, themeasure described above is referred to as a measure ofabsolute risk-aversion.


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If we want to measure thepercentage of wealthheld in risky assets, for a given wealth level W, we

simply multiply the Arrow-pratt measure ofabsolute risk-aversion by the wealth W, to get ameasure ofrelative risk-aversion.

The Arrow-Pratt measure of relative risk-aversionor coefficient ofrelative risk aversion(RRA) isdefined as

=

.


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Absolute and Relative Risk AversionType of Risk-Aversion Description

Increasing absolute risk-aversionAs wealth increases, hold fewer dollars in risky

assets

Constant absolute risk-aversion As wealth increases, hold the same dollaramount in risky assets

Decreasing absolute risk-aversionAs wealth increases, hold more dollars in risky

assets

Type of Risk-Aversion Description

Increasing relative risk-aversion As wealth increases, hold a smaller percentageof wealth in risky assets

Constant relative risk-aversionAs wealth increases, hold the same percentage

of wealth in risky assets

Decreasing relative risk-aversionAs wealth increases, hold a larger percentage

of wealth in risky assets


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Absolute and Relative Risk

Aversion an Example Suppose the logarithmic utility function U=ln(W).

Thus marginal utility is =1

.

The change in marginal utility with respect to the change in wealth is then

= 1

.

=

=

=1

.

=

= W

=1.

We can see that:

MU of wealth is positive and decreases with increasing wealth,

the measure of ARA decreases with increasing wealth,

RRA is constant.


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Valuation In valuation we transform the future random value Vt+1 to present

value.We can utilize one from the following methods: Risk Adjusted Costof Capital (RACC) and Certainty Equivalent Method (CEM).

According to RACC method we compute the acturial (expected,average) future value and discount it by riks adjusted cost of capital,

=

1+ .

According to CEM method the future risky values are transformedto certainty equivalent (CE) a this is disconted by riskfree return.

=

1+.


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Valuation - Example We have a project which will cost in the first year 150 EUR.

The cash-flow in the second year is random variable with four

possible scenarios with corresponding probabilities, see thetable below. Risk-free rate is 5% and risk adjusted cost of

capital is 9.2%. The utility function is U=ln(CF). Compute the

value of the project both by RACC and CEM methodology.

scenario probability cash flow

1 10% 100

2 25% 150

3 40% 200

4 25% 250


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Valuation Example RACC We compute the mean (weighted average) of cash-flow in the

second year:

E(CF1

) = 0.1 100 + 0.25 150 + 0.4 200 + 0.25 250 = 190 EUR

=

1+ =

150

1+0.092

190

1+0.092 = 24 EUR


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Valuation Example CEM First of all, we have to compute the certainty equivalent.

= 1 ,

where = 0.1 4.6 + 0.25 5 + 0.4 5.3 + 0.25 5.5 =

5.21.

= = 5.21 = 183 EUR

= 1+

=150

1+0.05 183

1+0.05 = 24 EUR

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The Theory of Choice