Frank Cowell : Risk

RISKMICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential: Consumption and UncertaintyProbability Distributions

Prerequisites

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Frank Cowell : Risk

Risk and uncertainty In dealing with uncertainty a lot can be done without

introducing probabilityNow we introduce a specific probability model

• This could be some kind of exogenous mechanism• Could just involve individual’s perceptions

Facilitates discussion of risk Introduces new way of modelling preferences

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Frank Cowell : Risk 3

ProbabilityWhat type of probability model?A number of reasonable versions:

• public observable• public announced• private objective • private subjective

Need a way of appropriately representing probabilities in economic models

See the presentation Probability Distributions

July 2015

Lottery

government policy?

coin flip

emerges from structure of preferences

Frank Cowell : Risk

Overview

Risk comparisons

Special Cases

Lotteries

Risk

Shape of the u-function and attitude to risk

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Frank Cowell : Risk

Risk aversion and the function uWith a probability model it makes sense to discuss risk

attitudes in terms of gamblesCan do this in terms of properties of “felicity” or

“cardinal utility” function u• Scale and origin of u are irrelevant• But the curvature of u is important

We can capture this in more than one wayWe will investigate the standard approaches……and then introduce two useful definitions

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Frank Cowell : Risk

Risk aversion and choice Imagine a simple gambleTwo payoffs with known probabilities:

• xRED with probability pRED

• xBLUE with probability pBLUE

• Expected value Ex = pREDxRED + p BLUE x BLUE

A “fair gamble”: stake money is exactly Ex Would the person accept all fair gambles?Compare Eu(x) with u(Ex)

depends on shape of u

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Frank Cowell : Risk

Attitudes to risk

u u(x)

xBLUExxREDE x

Risk-loving

uu(x)

xBLUExxREDEx

Risk-neutral

u(x)

xBLUExxREDE x

u

Risk-averse

Shape of u associated with risk attitude Neutrality: will just accept a fair gamble Aversion: will reject some better-than-fair gambles Loving: will accept some unfair gambles

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Frank Cowell : Risk

Risk premium and risk aversion

xBLUE

xREDO

pRED – ____pBLUE

Certainty equivalent income

A given income prospect

Slope gives probability ratio

Exx

Mean incomeThe risk premium

· P0

· P

Risk premium: Amount that you would sacrifice to eliminate the risk Useful additional way of characterising risk attitude

–

example

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Frank Cowell : Risk

An example… Two-state model Subjective probabilities (0.25, 0.75) Single-commodity payoff in each case

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Frank Cowell : Risk

Risk premium: an example

u

u(x)

xBLUExxRED

u(xBLUE)

u(xRED)

Ex

u(Ex)

x

Eu(x)amount you would sacrifice to eliminate the risk

u(Ex)

E xx

Expected payoff & U of expected payoff Expected utility and certainty-equivalent The risk premium again

Utility values of two payoffs

Eu(x)

·

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Frank Cowell : Risk

Change the u-functionu

xBLUExxRED

u(xBLUE)

u(xRED)

Exx

The utility function and risk premium as beforeNow let the utility function become “flatter”…

u(xBLUE)

x

Making the u-function less curved reduces the risk premium……and vice versaMore of this later

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Frank Cowell : Risk

An index of risk aversion?Risk aversion associated with shape of u

• second derivative• or “curvature”

But could we summarise it in a simple index or measure?

Then we could easily characterise one person as more/less risk averse than another

There is more than one way of doing this

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Frank Cowell : Risk

Absolute risk aversionDefinition of absolute risk aversion for scalar payoffs

uxx(x)a(x) := ux(x)

For risk-averse individuals a is positiveFor risk-neutral individuals a is zero Definition ensures that a is independent of the scale

and the origin of u• Multiply u by a positive constant…• …add any other constant…• a remains unchanged

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Frank Cowell : Risk

Relative risk aversionDefinition of relative risk aversion for scalar payoffs:

uxx(x) r(x) := x ux(x)

Some basic properties of r are similar to those of a:• positive for risk-averse individuals• zero for risk-neutrality • independent of the scale and the origin of u

Obvious relation with absolute risk aversion:• r(x) = x a(x)

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Frank Cowell : Risk

Concavity and risk aversion

u

u(x)

payoff

utili

ty

x

û(x)

Draw the function u againChange preferences: φ is a concave function of u Risk aversion increases

More concave u implies higher risk aversion

now to the interpretations

lower risk aversion

higher risk aversion

û = φ(u)

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Frank Cowell : Risk

Interpreting a and rThink of a as a measure of the concavity of uRisk premium is approximately ½ a(x) var(x)Likewise think of r as the elasticity of marginal u In both interpretations an increase in the “curvature” of

u increases measured risk aversion• Suppose risk preferences change…• u is replaced by û , where û = φ(u) and φ is strictly concave• Then both a(x) and r(x) increase for all x

An increase in a or r also associated with increased curvature of IC…

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Frank Cowell : Risk

Another look at indifference curvesu and p determine the shape of IC Alf and Bill differ in risk aversion

xBLUE

xREDO

xBLUE

xREDO

Alf, Charlie differ in subj probability

Bill

Alf

Alf

Charlie

Same us but different psSame ps

but different us

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Frank Cowell : Risk

Overview

Risk comparisons

Special Cases

Lotteries

Risk

CARA and CRRA

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Frank Cowell : Risk

Special utility functions?

Sometimes convenient to use special assumptions about risk• Constant ARA• Constant RRA

By definition r(x) = x a(x) Differentiate w.r.t. x:

dr(x) da(x) = a(x) + x dx dx

So one could have, for example:• constant ARA and increasing RRA• constant RRA and decreasing ARA• or, of course, decreasing ARA and increasing RRA

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Frank Cowell : Risk

Special case 1: CARAWe take a special case of risk preferencesAssume that a(x) = a for all xFelicity function must take the form

1 u(x) := eax

aConstant Absolute Risk AversionThis induces a distinctive pattern of indifference

curves…

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Frank Cowell : Risk

Constant Absolute RACase where a = ½Slope of IC is same along 45° ray (standard vNM)For CARA slope of IC is same along any 45° line xBLUE

xREDO

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Frank Cowell : Risk

CARA: changing a

Case where a = ½ (as before)

Change ARA to a = 2

xBLUE

xREDO

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Frank Cowell : Risk

Special case 2: CRRA Another important special case of risk preferences Assume that r(x) = r for all r Felicity function must take the form

1 u(x) := x1 r

1 r Constant Relative Risk Aversion Again induces a distinctive pattern of indifference curves…

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Frank Cowell : Risk

Constant Relative RA

Case where r = 2Slope of IC is same along 45° ray (standard vNM)For CRRA slope of IC is same along any rayICs are homothetic

xBLUE

xREDO

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Frank Cowell : Risk

CRRA: changing r

xBLUE

xREDO

Case where r = 2 (as before)

Change RRA to r = ½

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Frank Cowell : Risk

CRRA: changing p

xBLUE

xREDO

Case where r = 2 (as before)

Increase probability of state RED

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Frank Cowell : Risk

Overview

Risk comparisons

Special Cases

Lotteries

Risk

Probability distributions as objects of choice

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Frank Cowell : Risk

Lotteries Consider lottery as a particular type of uncertain prospect Take an explicit probability model Assume a finite number of states-of-the-world Associated with each state w are:

• A known payoff xw ,• A known probability pw ≥ 0

The lottery is the probability distribution over the “prizes” xw, w=1,2,…,• The probability distribution is just the vector p:= (p1,,p2 ,…,,p) • Of course, p1+ p2 +…+p = 1

What of preferences?

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Frank Cowell : Risk

The probability diagram: #W=2

pBLUE

pRED(1,0)

(0,1) Cases where 0 < p < 1

Probability of state BLUE

Cases of perfect certainty

Probability of state RED

pRED +pBLUE = 1

The case (0.75, 0.25)

• (0, 0.25)

(0.75, 0)

Only points on the purple line make senseThis is an 1-dimensional example of a simplex

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Frank Cowell : Risk

The probability diagram: #W=3

0

pBLUE

pRED

pGREEN

Third axis corresponds to probability of state GREEN

(1,0,0)

(0,0,1)

(0,1,0)

There are now three cases of perfect certaintyCases where 0 < p < 1

pRED + pGREEN + pBLUE = 1

• (0, 0, 0.25)

(0.5, 0, 0)

(0, 0.25 , 0)

The case (0.5, 0.25, 0.25)

Only points on the purple triangle make senseThis is a 2-dimensional example of a simplex

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Frank Cowell : Risk

Probability diagram #W=3 (contd.)

(1,0,0)

(0,0,1)

(0,1,0) .

• (0.5, 0.25, 0.25)

All the essential information is in the simplex

Display as a plane diagramThe equi-probable case

The case (0.5, 0.25, 0.25)

• (1/3,1/3,1/3)

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Frank Cowell : Risk

Preferences over lotteries Take the probability distributions as objects of choice Imagine a set of lotteries p°, p', p",… Each lottery p has same payoff structure

• State-of-the-world w has payoff xw

• … and probability pw° or pw' or pw" … depending on which lottery

We need an alternative axiomatisation for choice amongst lotteries p

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Frank Cowell : Risk

Axioms on preferencesTransitivity over lotteries

• If p°p' and p' p" …• …then p°p"

Independence of lotteries• If p° p' and l(0,1)…• …then lp° + [1l]p" lp' + [1l] p"

Continuity over lotteries• If p°p'p" then there are numbers l and m such that• lp° + [1l]p" p'• p' mp° + [1m]p"

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Frank Cowell : Risk

Basic result Take the axioms transitivity, independence, continuity Imply that preferences must be representable in the form of a

von Neumann-Morgenstern utility function:

å pw u(xw) w W

or equivalently:

å pw uw w W

where uw := u(xw)

So we can also see the EU model as a weighted sum of ps

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Frank Cowell : Risk

p-indifference curves

Indifference curves over probabilities

Effect of an increase in the size of uBLUE

(1,0,0)

(0,0,1)

(0,1,0) .

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Frank Cowell : Risk

What next? Simple trading model under uncertainty Consumer choice under uncertainty Models of asset holding Models of insurance This is in the presentation Risk Taking

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