Financial Economics: Making Choices in Risky...

OutlineCriteria for Choice Over Risky Prospects

Preferences and Utility FunctionsExpected Utility Functions

The Expected Utility TheoremThe Allais Paradox

Financial Economics: Making Choices in

Risky Situations

Shuoxun Hellen Zhang

WISE & SOE

XIAMEN UNIVERSITY

March, 2015

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Questions to Answer

How financial risk is defined and measured

How an investor’s attitude toward or tolerance for risk isto be conceptualized and then measured

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Outline

Criteria for Choice Over Risky Prospects

Preferences and Utility Functions

Expected Utility Functions

The Expected Utility Theorem

The Allais Paradox

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State-by-state DominanceMean-variance DominanceSharpe RatioSummary


In the broadest sense, “risk” refers to uncertainty aboutthe future cash flows provided by a financial asset.

A more specific way of modeling risk is to think of thosecash flows as varying across different states of the worldin future periods . . . that is, to describe future cashflows as random variables.

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State-by-state Dominance

Consider three investments

where θ = 1 indicates the bad state and θ = 2 indicates goodstate.Which one do you prefer?

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Investment 3 exhibits state-by-state dominance overinvestments 1 and 2, because it pays as much in all states andstrictly more in at least one state.

Any investor who prefers more to less (nonsatiated inconsumption) would always choose investment 3 above theothers.

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But the choice between investments 1 and 2 is not as clearcut. Investment 2 provides a larger gain in the good state, butexposes the investor to a loss in the bad state.

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Mean-variance Dominance

convert prices and payoffs to percentage returns:

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In probability theory, if a random variable X can take on npossible values, X1; X2; ... ; Xn, with probabilities p1; p2; ...; pn,then the expected value of X is

E (X ) = p1X1 + p2X2 + ... + pnXn

the variance of X is

σ2(X ) = p1[X1−E (X )]2+p2[X2−E (X )]2+...+pn[Xn−E (X )]2

and the standard deviation of X is σ(X ) = [σ2(X )](1/2).

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E (r1) = 0.5 ∗ 20 + 0.5 ∗ 5 = 12.5

σ1 = [0.5 ∗ (20− 12.5)2 + 0.5 ∗ (5− 12.5)2](1/2) = 7.5

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bad state good state E (r) σInvestment 1 5% 20% 12.5% 7.5%Investment 2 -50% 60% 5% 55%Investment 3 5% 60% 32.5% 27.5%

Investment 1 exhibits mean-variance dominance overinvestment 2, since it offers a higher expected return withlower variance.

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bad state good state E (r) σInvestment 1 5% 20% 12.5% 7.5%Investment 2 -50% 60% 5% 55%Investment 3 5% 60% 32.5% 27.5%

But notice that by the mean-variance criterion, investment 3dominates investment 2 but not investment 1, even though ona state-by-state basis, investment 3 is clearly to be preferred.Mean-variance dominance neither implies nor is implied bystate-by-state dominance.

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Mean-variance Dominance can be expressed in the form of acriterion for selecting investments of equal magnitude

For investments of the same Er , choose the one withlower σ

For investments of the same σ, choose the one withgreatest Er

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Sharpe Ratio

Consider two more assets:

Which one do you prefer?Neither exhibits state-by-state dominance, nor themean-variance dominance.

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Sharpe Ratio

Consider two more assets:

William Sharpe (US, b.1934, Nobel Prize 1990) suggested thatin these circumstances, it can help to compare the two assets’Sharpe ratios, defined as E (r)/σ(r).Comparing Sharpe ratios, investment 4 is preferred toinvestment 5.

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Sharpe Ratio

But using the Sharpe ratio to choose between assetsmeans assuming that investors “weight” the mean andstandard deviation equally, in the sense that a doubling ofσ(r) is adequately compensated by a doubling of E (r).Investors who are more or less averse to risk will disagree.

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State-by-state dominance is the most robust criterion, butoften cannot be applied.

Mean-variance dominance is more widely-applicable, butcan sometimes be misleading and cannot always beapplied.

The Sharpe ratio can always be applied, but requires avery specific assumption about consumer attitudestowards risk.

We need a more careful and comprehensive approach tocomparing random cash flows.

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PreferencesPreferences and Utility Functions


Of course, economists face a more general problem of thiskind.

Even if we accept that more (of everything) is preferredto less, how do consumers compare different “bundles” ofgoods that may contain more of one good but less ofanother?

Microeconomists have identified a set of conditions thatallow a consumer’s preferences to be described by autility function.

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Preferences

Let a, b, and c represent three bundles of goods.

These may be arbitrarily long lists, or vectors (a ∈ RN),indicating how much of each of an arbitrarily largenumber of goods is included in the bundle.

A preference relation � can be used to represent theconsumer’s preferences over different consumptionbundles.

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Preferences

a � b, indicates that the consumer strictly prefers a to b

a ∼ b indicates that the consumer is indifferent between aand b

a � b indicates that the consumer either strictly prefers ato b or is indifferent between a and b

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Assumptions on Preferences

A.1. The preference relation is assumed to be complete:For any two bundles a and b, either a � b, b � a, orboth, and if both hold, a ∼ b.

The consumer has to decide whether he or she prefers onebundle to another or is indifferent between the two.Ambiguous tastes are not allowed.

A.2. The preference relation is assumed to be transitive:For any three bundles a, b and c , if a � b, b � c , thena � c .

The consumer’s tastes must be consistent in this sense.Together, (A.1.) and (A.2.) require the consumer to be fullyinformed and rational.

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A.3. The preference relation is assumed to be continuous:if an and bn are two sequences of bundles such thatan → a, bn → b and an � bn for all n, then a � b.

Very small changes in consumption bundles cannot lead tolarge changes in preferences over those bundles.

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Remark

An two-good example that violates (A.3.) is the case oflexicographic preferences:

a = (a1, a2) � b = (b1, b2) if a1 > b1,

or a1 = b1 and a2 > b2.

It is not possible to represent these preferences with a utilityfunction, since the preferences are fundamentallytwo-dimensional and the value of the utility function has to beone-dimensional.

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Theorem

If preferences are complete, transitive, and continuous, thenthey can be represented by a continuous, time-invariant,real-valued utility function. That is, if (A.1.)-(A.3.) hold,there is a continuous function u : Rn 7→ R such that for anytwo consumption bundles a and b,

a � b if and only if u(a) ≥ u(b)

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Note that if preferences are represented by the utility functionu

a � b if and only if u(a) ≥ u(b)

then they are also represented by the utility function v, where

v(·) = F (u(·))

where F : R 7→ R is any increasing function. The concept of

utility as it is used in standard microeconomic theory isordinal, as opposed to cardinal.

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Cardinal V.S. Ordinal

An ordinal utility function describing a consumer’s preferencesover two goods can be written as u(x , y), the samepreferences could be expressed as another utility function thatis an increasing transformation of u: g(x , y) = f (u(x , y)).Utility functions g and u give rise to identical indifferencecurve mappings.

A cardinal utility function that preserves preference orderingsuniquely up to positive affine transformations. Two utilityindices are related by an affine transformation, i.e. u and vsatisfies a relationship of the form v(x) = au(x) + b, where aand b are constants.

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Under certainty, the “goods” are described byconsumption baskets with known characteristics.

Under uncertainty, the “goods” are random(state-contingent) payoffs.

The problem of describing preferences over thesestate-contingent payoffs, and then summarizing thesepreferences with a utility function, is similar in overallspirit but somewhat different in its details to the problemof describing preferences and utility functions undercertainty.

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Consider shares of stock in two companies:

bad state good stateIBM 100 150RDS 100 150

where the good state occurs with probability π and the badstate occurs with probability 1− π.

We assume that if the two assets provide exactly the samestate-contingent payoffs, then investors will be indifferentbetween them.

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We also assume that investors will prefer any asset thatexhibits state-by-state dominance over another. Thus, if u(x)measures utility from the payoff x in any particular state, wewill assume that u is increasing.

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Here, there is no state-by-state dominance, but it seemsreasonable to assume that a higher probability π will makeinvestors tend to prefer IBM, while a higher probability 1− πwill make investors tend to prefer RDS.

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A criterion that reflects both of these properties was suggestedby Blaise Pascal (France, 1623-1662): base decisions on theexpected payoff,

E (x) = πxG + (1− π)xB ;

where xG and xB , with xG > xB , are the payoffs in the goodand bad states.

The expected payoff rises when either the payoff in either staterises or the probability of the good state goes up.

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Nicolaus Bernoulli (Switzerland, 1687-1759) pointed to aproblem with basing investment decisions exclusively onexpected payoffs: it ignores risk. To see this, specialize theprevious example by setting π = 1− π = 0.5 but add, as well,a third asset:

bad state good stateIBM 100 150RDS 90 160T-bill 125 125

The expected payoff of all three assets are E (x) = 125,but theT-bill is less risky than both stocks, and IBM stock is less riskythan RDS stock.

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Sony is dominated by both IBM and RDS. But the choice between thelatter two can now be described in terms of an improvement of $10 overthe Sony payoff, either in state 1 or in state 2. Which is better?The relevant feature is that IBM adds $10 when the payoff is low ($90),while RDS adds the same amount when the payoff is high ($150). Mostpeople would think IBM more desirable, and with equal stateprobabilities, would prefer IBM.

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Gabriel Cramer (Switzerland, 1704-1752) and Daniel Bernoulli(Switzerland, 1700-1782) suggested that more reliable comparisons couldbe made by assuming that the utility function u over payoffs in any givenstate is concave as well as increasing.This implies that investors prefer more to less, but have diminishingmarginal utility as payoffs increase.

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About two centuries later, John von Neumann (Hungary,1903-1957) and Oskar Morgenstern (Germany, 1902-1977)worked out the conditions under which investors’ preferencesover risky payoffs could be described by an expected utilityfunction such as

U(x) = E [u(x)] = πu(xG ) + (1− π)u(xB);

where the Bernoulli utility function u is increasing and concaveand the von Neumann-Morgenstern utility function U is linearin the probabilities.

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LotteryAssumptionsThe Expected Utility Theorem

Simple lottery

The simple lottery (x ; y ; π) offers payoff x with probability πand payoff y with probability 1− π.

In this definition, x and y can be monetary payoffs, as in thestock and bond examples from before.

Alternatively, they can be additional lotteries!

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Compound lottery

The compound lottery (x ; (y , z , τ); π) offers payoff x withprobability π and lottery (y , z , τ) with probability 1− π.

Notice that a simple lottery with more than two outcomes canalways be reinterpreted as a compound lottery where eachindividual lottery has only two outcomes.

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Compound lottery

Notice that a simple lottery with more than two outcomes canalways be reinterpreted as a compound lottery where eachindividual lottery has only two outcomes.

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Compound lottery

So restricting ourselves to lotteries with only two outcomesdoes not entail any loss of generality in terms of the numberof future states that are possible.

But to begin describing preferences over lotteries, we need tomake additional assumptions.

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Axioms

C.1.a. A lottery that pays off x with probability one is thesame as getting x for sure: (x , y , 1) = x .

C.1.b. Investors care about payoffs and probabilities, butnot the specific ordering of the states:(x , y , π) = (y , x , 1− π)

C.1.c. In evaluating compound lotteries, investors careonly about the probabilities of each final payoff:(x ; z ; π) = (x , y , π + (1− π)τ) if z = (x , y , τ)

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Axioms

C.2. There exists a preference relation � defined onlotteries that is complete and transitive.

Again, this amounts to requiring that investors are fullyinformed and rational.

C.3. The preference relation � defined on lotteries iscontinuous.

Hence, very small changes in lotteries cannot lead to very largechanges in preferences over those lotteries.

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Axioms

By the previous theorem, we already know that (C .2) and(C .3) are sufficient to guarantee the existence of a utilityfunction over lotteries and, by (C .1a), payoffs received withcertainty as well.

What remains is to identify the extra assumptions thatguarantee that this utility function is linear in the probabilities,that is, of the von Neumann-Morgenstern (vN-M) form.

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Axioms

C.4. Independence axiom: For any two lotteries(x ; y ; π) and (x ; z ; π) , y � z if and only if(x ; y ; π) � (x ; z ; π).

This assumption is controversial and unlike any made intraditional microeconomic theory: you would not necessarilywant to assume that a consumer’s preferences oversub-bundles of any two goods are independent of how much ofa third good gets included in the overall bundle. But it isneeded for the utility function to take the vN-M form.

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Axioms

C.5. There is a best lottery b and a worst lottery w .

This assumption will automatically hold if there are only afinite number of possible payoffs and if the independenceaxiom holds.

C.6. (implied by (C .3)) Let x , y , and z satisfy x � y � z .Then there exists a probability π such that (x ; z ; π) ∼ y .

C.7. (implied by (C .4)) Let x � y . Then(x ; y ; π1) � (x ; y ; π2) if and only if π1 > π2.

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The Expected Utility Theorem

Theorem

Expected Utility Theorem

Consider a preference ordering, defined on the space oflotteries, that satisfies axioms (C .1)− (C .7), then there existsa utility function U defined over lotteries, with Bernoulli utilityfunction, such that

U((x , y , π)) = πu(x) + (1− π)u(y)

Note that we can prove the theorem simply by “constructing”the utility functions U and u with the desired properties.

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Proof

Begin by settingU(b) = 1; U(w) = 0

For any lottery z besides the best and worst, (C .6) impliesthat there exists a probability πz such that (b,w , πz) ∼ z and(C .7) implies that this probability is unique. For this lottery,set

U(z) = πz

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Proof

Condition (C .7) also implies that with U so constructed,z � z ′ implies

U(z) = πz > πz ′ = U(z ′)

and z ∼ z ′ implies

U(z) = πz = πz ′ = U(z ′)

so that U is a utility function that represents the underlyingpreference relation �.

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Proof

Now let x and y denote two payoffs.

By (C .1a), each of these payoffs is equivalent to a lottery inwhich x or y is received with probability one.

With this in mind, let

u(x) = U(x) = πx

u(y) = U(y) = πy

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Proof

Finally, let π denote a probability and consider the lotteryz = (x , y , π).

Condition (C .1c) implies,

(x , y , π) ∼ ((b,w , πx), (b,w , πy ), π) ∼ (b,w , ππx+(1−π)πy )

this last expression is equivalent to

U(z) = U((x , y , π)) = ππx+(1−π)πy = πu(x)+(1−π)u(y)

confirming that U has the vN-M form.

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Remark

Note that the key property of the vN-M utility functionU(z) = U((x , y , π)) = πu(x) + (1− π)u(y). its linearityin the probabilities π and 1− π, is not preserved by alltransformations of the form

V (z) = F (U(z))

where F is an increasing function.

In this sense, vN-M utility functions are cardinal, notordinal.

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Remark

On the other hand, given a vN-M utility functionU(z) = U((x , y , π)) = πu(x) + (1− π)u(y). consider anaffine transformation

V (z) = αU(z) + β

and define

v(x) = αu(x) + β

v(y) = αu(y) + β

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Remark

U(z) = U((x , y , π)) = πu(x) + (1− π)u(y)

V (z) = αU(z) + β

v(x) = αu(x) + β; v(y) = αu(y) + β

V ((x , y , π)) = αU((x , y , π)) + β

= α[πu(x) + (1− π)u(y)] + β

= π[αu(x) + β] + (1− π)[αu(y) + β]

= πv(x) + (1− π)v(y)

In this sense, the vN-M utility function that represents anygiven preference relation is not unique.

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The Allais Paradox

As mentioned previously, the independence axiom has beenand continues to be a subject of controversy and debate.

Maurice Allais (France, 1911-2010, Nobel Prize 1988)constructed a famous example that illustrates why theindependence axiom might not hold in his paper “LeComportement de L’Homme Rationnel Devant Le Risque:Critique Des Postulats et Axiomes De L’Ecole Americaine,”Econometrica Vol.21 (October 1953): pp.503-546.

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The Allais Paradox

Consider two lotteries:

L1 = { $10000 with probability 0.1$0 with probability 0.9


Which would you prefer?People tend to say L2 � L1.

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The Allais Paradox

But consider another two lotteries:

L3 = { $10000 with probability 1$0 with probability 0


Which would you prefer?The same people who say L2 � L1

often say L3 � L4

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The Allais Paradox



L3 = { $10000 with probability 1$0 with probability 0


But L1 = (L3; 0; 0.1) and L2 = (L4; 0; 0.1) so the independenceaxiom requires L3 � L4 ⇔ L1 � L2

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The Allais Paradox

The Allais paradox suggests that feelings about probabilitiesmay not always be “linear,” but linearity in the probabilities isprecisely what defines vN-M utility functions.

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