chapter three

Chapter Three:

The Theory of Choice : Utility Theory Given Uncertainty

Kawser Ahmed ShibluLecturer

Department of FinanceJagannath University.

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Economics is the study of how people & societies choose to allocate scarce resources and distribute wealth among one another and over time. So we have to know the objects of choice & the method of choice.

In this chapter, we focus on the theory of how people make choices when faced with uncertainty i.e. the method of choice. In later, we will be familiar with the objects of choice faced by an investor.

Generally, in an economic society, prices provide a system of signals for optimal allocation. But there are still other means or factor of allocation. For example, the consumption behavior of the investors.

The theory of choice starts with the microeconomic price theory which is the choice of various bundles of commodities at an instant in time.

Next come the utility theory of choice which is one-period consumption/investment decision i.e. consume now or save (invest).

Finally comes the theory of investor choice which is the choice between timeless risky alternatives.

Introduction

The Theory of Choice: Utility Theory Given Uncertainty

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The theory of investor choice begins with five axioms or assumptions about the behavior of individuals when faced with the task of ranking risky alternatives. These assumptions are known as axioms of cardinal utility.

These axioms provide the minimum set of conditions for consistent and rational behavior.

Introduction


Oranges

Apples C1

C0

Return

Risk-(1+r)

Price Theory Utility Theory of Choice

Theory of Investor Choice

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

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).

Axiom 3: Strong independence We construct a gamble where the individual has a probability of for receiving α

outcome x and a probability of (1- ) of receiving α mutually exclusive outcome z i.e. we can write the gamble as G(x,z: ). Again we can construct the second gamble αwhere the individual has a probability of for receiving outcome y and a αprobability of (1- ) of receiving the same α mutually exclusive outcome z i.e. we can write the gamble as G(y,z: ). Then, if (x~y) then G(x,z: ) ~G(y,z: ).α α α

But this axiom is the hardest to accept. Lets see the example. Let, Outcome x: winning left shoe, Outcome y: winning right shoe & Outcome z:

winning right shoe. 1st gamble: G(x,z:0.50) i.e. a left shoe or a right shoe. Again 2nd gamble: G(y,z:0.50) i.e. a right shoe or a right shoe.

We may indifferent between a left shoe (outcome x) & a right shoe (outcome y) separately i.e. x~y. But this axiom states that we also indifferent between above gambles. Is it correct???? No. So mutually exclusiveness of z is critical to the axiom.

Five Axioms of Choice under Uncertainty


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Axiom 4: Measurability If x>y>z then there is a unique probability , such that the individual will be α

indifferent between y and a gamble between x with probability and z with αprobability (1- ) i.e. If x>y>z, then there exists a unique , such that y~G(x,z: ).α α α

Let, Outcome x: $ 16, Outcome y: $ 10 & Outcome z: $ 6. so x>y>z. So for which unique probability of x, G(x,z: ) will give $ 10 which is exactly equal to α

y. The answer is 0.40. Check it. Let,15~G(20,12: ), now =???α α

Axiom 5: Ranking For alternatives x>y>z we can establish a gamble such that an individual is

indifferent between y and a gamble between x and z with certain probability α1, y~G(x,z:α1). Also for x>u>z we can establish gamble such that u~G(x,z:α2).

If x>y>z and x>u>z, then if y~G(x,z:α1) and u~G(x,z:α2), it follows that If α1>α2 then y>u, if α1<α2 then y<u and if α1=α2 then y~u.

Let, Outcome x: $ 16, Outcome z: $ 6; 16>Outcome y or u>6 i.e. x>y>z & x>u>z. If α1= 0.60 & α2= 0.40 then y>u. Again if α1= 0.40 & α2= 0.60 then u>y.

These axioms lead to the following assumptions on human behavior:I. Individuals always make completely rational decisions. For example, “I prefer

iphone to Samsung and Samsung to Nokia but Nokia to iphone”- is not rational.II. People are able to make these rational decisions among thousands of alternatives.

Five Axioms of Choice under Uncertainty


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Now the question arise: “How do individuals rank various combinations of risky alternatives?” or “How can we establish a utility function that allows the assignment of a unit

measure (a number) to various alternatives so that we can rank them?”

The answer leads us two properties of utility functions-I. Utility function will be order preserving i.e. if utility of x is greater than that of y,

U(x)>U(y), then it can be said outcome x is really preferred to outcome y, x>y.II. Expected utility can be used to rank combinations of risky alternatives i.e.

U[G(x,y: )]= *U(x) + (1- )*U(y)α α α

Property-1: Utility function are order preserving. Let, S= set of risky outcomes, a= highest possible outcome, b= lowest possible

outcome, x & y= intermediate outcomes where a>x>b & a>y>b According to axioms 4 (measurability), for unique probability of x & y, we can

construct the following two gamble:x~G(a,b:αx) & y~G(a,b:αy)

Again according to axioms 5 (ranking), we can say, If αx>αy then x>y, if αx<αy then x<y and if αx=αy then x~y.

In this way, we can develop an order-preserving utility function.

Developing Utility Function


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Property-2: Expected utility can be used to rank risky alternatives i.e. U[G(x,y: )]= *U(x) + (1- )*U(y)α α α Lets, represent the previous gambles in decision tree:



a

b

x ~

αx

1-αx

a

b

y~

αy

1-αy

x~G(a,b:αx) y~G(a,b:αy)

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Now, consider a new choice of z and construct a gamble with respect to x & y i.e. z~G(x,y:βz). Now represent this gambles in decision tree:

Now if we construct the decision tree of the last gamble with respect to a & b, then the gamble will be: z~G[a, b: βz* αx+ (1- βz)* αy



z ~

βz

1-βz

a

b

x ~

αx

1-αx

a

b

y ~

αy

1-αy

z~G(x,y:βz)

a

b

z ~

β z* α x

+ (1- β z)* α y

βz *(1- α

x )+ (1- βz )* (1-α

y )

z~G(a,b:βz* αx+ (1- βz)* αy)

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According to axioms 4 & 5, utilities of x & y can be represented by their probabilities i.e. Ux= αx and Uy= αy. So the above gamble can be written as :

z~G[a, b: βz* Ux+ (1- βz)* Uy] The unique probability of outcome z can be used as a measure of its utility

relative to a & b. We can write:

Uz= βz* Ux+ (1- βz)* Uy

That is the utility of z is the probability of x times its utility plus the probability of y times its utility. This is expected utility that represents a linear combination of the utilities of outcomes.

In general, E[U(w)]= ∑ρ*U(W)



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Given the axioms of rational investor behavior & knowing that all investors always prefer more wealth to less, we can say that investors always want to maximize their expected utility of wealth i.e. maximize E[U(w)].

So all investors calculate the expected utility of wealth for all possible alternative and choose the outcome that maximize their expected utility of wealth.

Constructing a utility function How we can construct the utility function? Let, loss of $1000 leads you

to utility of -10 utiles and you faced a gamble of sure 0 and a gamble with probability 0.60 of winning $1000 and probability (1-0.60) of losing $1000. what is the utility of winning $1000?

By repeating the problem for different payoffs, it is possible to develop a utility function. But interpersonal comparison of utility function is impossible. Because giving $1000 to two persons, we cant say who is the happier.



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After developing utility function, now we give focus on determining the risk tolerance level of an investor i.e. whether an investor is a risk lover or risk neutral or risk averter. This will be done with the help of risk premium.

Here, there are two assumptions:I. More wealth is preferred to less &II. Marginal utility of wealth is positive MU(W)>0.

Assume, we establish a gamble between two prospects a and b. Probability of receiving a is , for b it is (1- ) & the gamble is α α G(a,b:

).α Now the question arise-

“Will we prefer the actuarial value of the gamble (expected or average outcome) with certainty or the gamble itself?”

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.

Establishing a Definition of Risk Aversion


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Assume the gamble, where you get 30 EUR with probability 20% and 5 EUR with probability 80%. The expected (average) value is thus 10 EUR. Will you choose the gamble or the value 10 EUR Under the following assumptions-I. Suppose that the utility function is U(W)=ln(W). Comment the answer.II. Suppose that the utility function is U=0.04*W2. Comment the answer.III. Suppose that the utility function is U=0.5*W. Comment the answer.

Establishing a Definition of Risk Aversion:Class Practice


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Lets see the answer under assumption-1.

As you receive more utility from the actuarial value of the gamble obtained with certainty than from taking the gamble itself, you are a risk averter.

Generally speaking, if utility of expected wealth is greater than the expected utility of wealth, then the investor is risk averter.

The utility curve of this risk averter investor will be like as follow:



Expected Case In case of Gamble E(W) =0.8*5 + 0.2*30 = 10 E[U(W)] = 0.8* U(5) + 0.2 *U(30) U[E(W)] = U(10) = ln 10 = 2.3 E[U(W)] = 0.8* 1.61 + 0.2 *3.4 = 1.97

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As you receive less utility from the actuarial value with certainty than from taking the gamble itself, you are a risk lover.

Generally speaking, if utility of expected wealth is less than the expected utility of wealth, then the investor is risk lover.


As you receive equal utility from the actuarial value with certainty to from taking the gamble itself, you are a risk neutral.

Generally speaking, if utility of expected wealth is equal to the expected utility of wealth, then the investor is risk neutral.



Expected Case In case of GambleE(W) =0.8*5 + 0.2*30 = 10 E[U(W)] = 0.8* U(5) + 0.2 *U(30) U[E(W)] = U(10) = 0.04*10*10 = 4 E[U(W)] = 0.8*(0.04*5*5)+ 0.2 *(0.04*30*30)

E[U(W)] = 8

Expected Case In case of GambleE(W) =0.8*5 + 0.2*30 = 10 E[U(W)] = 0.8* U(5) + 0.2 *U(30) U[E(W)] = U(10) = 0.50*10= 5 E[U(W)] = 0.8*0.50*5+ 0.2 *0.50*30

E[U(W)] = 5

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We have to compare the actuarial value (average, expected) of the gamble obtained with certainty and the gamble itself:I. if U[E(W)]>E[U(W)] then we have risk aversion individual (concave utility function), II. if U[E(W)]=E[U(W)] then we have risk neutral individual (linear utility function), III. if U[E(W)]<E[U(W)] then we have risk seeking individual (convex utility function).

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



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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 i.e. Difference between expected wealth given the gamble and the level of wealth the

individual would accept with certainty if the gamble were removed i.e. certainty equivalent wealth. symbolically U[E(W)]-E[U(W)]

Let, we had utility function U(W)=ln(W) with current wealth level 10 EUR. Then we have the gamble G(5,30:80%) i.e. with probability 80% of a decline to 5 EUR & probability 20% of increase wealth by EUR 20.

E[U(G)] = 1.97. From logarithmic function: 1.97 gives a wealth of EUR 7.17 (Certainty Equivalent). This is the value of the gamble.

If we accept the gamble & then the expected wealth is EUR 10 calculated as follows : Current wealth adjusted for gain or loss Or Outcomes weighted by their probability

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.

The cost of gamble = current wealth minus Certainty Equivalent.



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Assume you are facing a gamble of 10% chance of winning $10 and 90% chance of winning $100 with current wealth of $10. Compute the following:a) What is expected wealth?b) What is Certainty equivalent wealth?c) Calculate the risk premium?d) Calculate the cost of gamble?

Establishing a Definition of Risk Aversion:Class Practice


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You have a logarithmic utility function U(W)=ln(W) and your current level of wealth is $5000.a) Suppose you are exposed to a situation that results in a 50/50 chance of

winning or losing $1000. if you can buy insurance that completely removes the risk for a fee of $125, will you buy it or take the gamble?

b) Suppose you accept the gamble outlined in (a) and lose so that your wealth is reduced to $4000. if you faced with the same gamble and have the same offer of insurance as before, will you buy the insurance the second time around?

Understanding ability testing


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For a risk averse investors, I. Risk premium is always positive but the cost of gamble can be

positive, negative or zero. II. Their utility function are concave and increasing i.e.

a) they always prefer more to less (marginal utility of wealth is positive i.e. MU(W)>0). symbolically U’(w)>0

b) Their marginal utility of wealth decreases as they have more and more wealth i.e. marginal utility of wealth is increasing at a decreasing rate or (dMU(W)/dW<0). symbolically U’’(w)<0



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Pratt-Arrow measure of risk premium:= 0.50*Π σ2*[-U’’(W)/ U’(W)]

Here, since (0.50*σ2 ) is always positive so the sign of risk premium is always determined by [-U’’(W)/ U’(W)].

So risk aversion can be measured by the Pratt-Arrow absolute risk-aversion, also known as the coefficient of absolute risk aversion (ARA), defined as-

ARA=[-U’’(W)/ U’(W)]

The actual dollar amount an individual will choose to hold in risky assets, given a certain wealth level W is measuring above. For this reason, the measure described above is referred to as a measure of absolute risk-aversion.

To measure the percentage of wealth held in risky assets, for a given wealth level W, we simply multiply ARA by the wealth W, to get the Arrow-Pratt measure of relative risk-aversion or coefficient of relative risk aversion (RRA) i.e.

RRA=W*[-U’’(W)/ U’(W)]

Pratt-Arrow Risk Aversion Calculation


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For following quadratic utility function: U(W)=aW-bW2. ARA & RRA are as follows-

ARA= [2b/(a-2bw)] & RRA= [2b/((a/W)-2b)]

The quadratic utility function exhibits increasing ARA and increasing RRA. But these don’t make sense!!! Why???

This problem of quadratic utility function is solved by Friend & Blume by using power utility function U(W)=-W-1

Thus marginal utility is U’(W)=W-2 and the change in marginal utility with respect to the change in wealth is then U’’(W)=-2W-3 . ARA & RRA are as follows-

ARA= (2/W) & RRA= 2

This result is consistent as this indicates- marginal utility of wealth is positive, it decrease with increasing wealth, the measure of ARA decreases with increasing wealth and RRA is constant.

Pratt-Arrow Risk Aversion Calculation


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An individual with logarithmic utility function U(W) = ln (W) and a level of wealth of $20000 is exposed to two different risks:1. A 50/50 chance of winning or losing $10 and2. An 80% chance of losing $1000 and a 20% chance of losing $10000.

What is the risk premium under both Pratt-Arrow and Markowitz measure of risk premium for both of the above situations? Given that, U’(W)= W-1

Key Points: When risk is small and actuarially neutral then Pratt-Arrow

measure of risk premium is closely approximate to Markowitz measure of risk premium.

But Markowitz measure of risk premium is superior for large or asymmetric risks.

Pointing the problem with Pratt-Arrow


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An asset is said to be stochastically dominant over another if an individual receives greater wealth from it in every state of nature. This definition is known as First-order stochastic dominance.

Mathematically, asset x, with cumulative probability distribution Fx(W), will be stochastically dominant over asset y, with cumulative probability distribution Gy(W), if

Fx(W) ≤ Gy(W) for all W

Fx(W) < Gy(W) for some W

Stochastic Dominance: First-order


In other words, cumulative probability distribution for asset y always lies to the left of the cumulative probability distribution for x.The figure shows - cumulative probability distribution for asset y always lies to the left of the cumulative probability distribution for x.

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An asset x to be stochastically dominant at second order asset y for all risk averse investors, the accumulated area under the probability distribution of y must be greater than the accumulated area for x i.e.

ƒ-∞w[Gy(W)-Fx(W) ] dW≥0 for all W

Gy(W) ≠Fx(W) for some W

Stochastic Dominance: Second-order


Second order stochastic dominance requires the difference in the areas under the cumulative density function be positive i.e. the sum of the differences between two cumulative density functions is always greater than or equal to zero.

So, here cumulative density function can cross.

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Mean variance criteria give different decision which is fully depends on the investor’s level of risk tolerance. Investor -1 choose combination of A & B, Investor -2 choose B & Investor -3 choose A.

Note that, EPS of firm B is better in all state of nature.

A paradox:Usefulness of Stochastic Dominance


State of nature Worst Bad Average Good BestProb. 0.2 0.2 0.2 0.2 0.2

EPS (Firm A) 3 4 5 6 7EPS (Firm B) 3 5 7 9 11

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Look the EPS distribution in the previous table, there firm B give higher EPS in every state of nature, so firm B is dominant at first order.

If GA(W)-FB(W) ≥0 or FB(W)- GA(W) ≤0, then firm B is dominant at second order. The above table represents the firm B’s dominance at second order.

So we can say that mean-variance criteria cant not be give the exact decision whereas stochastic dominance can.

A paradox:Usefulness of Stochastic Dominance


EPS P(B) P(A) F(B) G(A) G(A) - F(B) ∑G(A) - F(B)3 0.20 0.20 0.20 0.20 0.00 0.004 0.00 0.20 0.20 0.40 0.20 0.205 0.20 0.20 0.40 0.60 0.20 0.406 0.00 0.20 0.40 0.80 0.40 0.807 0.20 0.20 0.60 1.00 0.40 1.208 0.00 0.00 0.60 1.00 0.40 1.609 0.20 0.00 0.80 1.00 0.20 1.80

10 0.00 0.00 0.80 1.00 0.20 2.0011 0.20 0.00 1.00 1.00 0.00 2.00

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Best of Luck!!!


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