1

Lectures in applied economics and finance

Lecture 3: Risk Aversion

Luigi Guiso EIEF2013

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Risk and Portfolio choice

Risk aversion is the single most important determinant of portfolio choice according to standard normative portfolio theoryAll heterogeneity in portfolio allocation between risk and riskless assets should be explained by investors risk attitudes (assuming common priors and beliefs)

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Questions

How can we measure risk aversion?

How does the risk aversion function looks like?Does it decline with wealth?If so, how fast?

What determines the attitude toward risk? Is it just an innate parameter?Can it depend upon observables? And if so which ones

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Outline

Definition of risk aversion (summary)

Risk aversion with background risk

How to elicit individual risk aversion: the different avenues

Determinants of risk aversion

Risk aversion and wealth: the shape of the risk aversion function

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Certainty equivalent and risk premium

W+50 W+150W+100

Lottery expected value U(W+50)

U(W+150)

( )EU W%U(W+100)

WC

Certainty equivalent wealth

Risk premium= Ψ=W+100-WC

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Risk aversion: formally

2 2

And the risk premium is defined as

''( ) ( )'( ) 2 2

x xU W A WU W

σ σΨ = − =

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Background risk

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Risk Aversion Without Background Risk In defining risk attitudes we have implicitly assumed that initial wealth is non random. A risk averse individual is one that, starting from a certain position, refuses a zero mean lotteryWhy do we need to start from a certain endowment?Reason:

If negatively correlated with W a zero mean risk may be attractive to risk averse individuals because provides insuranceIf positively correlated with W, a zero mean risk will be strongly rejected by a risk averse

What if W is random but uncorrelated with the zero mean risk?

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Risk Aversion With Background Risk The introduction of an extra risk that cannot be avoided can make the investor behave in a more risk averse manner This will be so under certain conditions regarding individual preferences Intuition:

If an investor is already facing some risk that he cannot avoid, this makes him less willing to face some risk that he can avoid, that is he may become more risk averse Preferences that have this property are called proper risk averse

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Why does background risk concern us? Investors face several risks that they cannot avoid A prominent one is labor risk

Individuals get firedConditional on working, the wage they receive is not fixed, but can be uncertainLabor risk cannot be insured and thus cannot be avoided

These unavoidable risks can affect portfolio decisions If one is already facing substantial labor market risk he may be less willing to invest in risky assetsSince individuals do face uninsurable risks this may be an important determinant of portfolio choice and reluctance to invest in stocks

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Background Risk FormallyLet v(W) = Eyu(W + %y) (NB expectation taken with respect to %y)

where u( ) = initial utility function%y = background risk; E( %y) = 0, E( %y,W) = 0;Var( %y) =σ y

2

v(W) = indirect utility

We want to compare the risk aversion of u(W) and the risk aversion of the indirect utility v(W) which incorporates the background risk and saywhether :v(W) is more risk averse than u(W)u(W) is more risk averse than v(W)v(W) is as risk averse as u(W)

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Background Risk: DerivationLet Au(W) = −u''(W) / u'(W) = Absolute risk aversion of u(W) Av(W) = −v ''(W) / v '(W) = Absolute risk aversion of v(W)

Taking a second-order Taylor approximation of v(W) = Eyu(W + %y)

around the endowment W (i.e. around %y = 0), we can approximate the indirect utility by v(W) = u(W) + u''(W)σ y

2

Now we can compute the risk aversion of v(W) as

Av(W) = − v ''(W) / v '(W) Au(W)(1+ PuTuσ y

2 )(1+ PuAuσ y

2 )

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Background Risk: Interpretation

The terms and have a specific meaning: ( ) '''( ) / ''( ) Degree of Absolute Prudence of ( )

( ) ''''( ) / '''( ) Degree of Absolute Temperance of ( )

a (more) prudent inve

u u

u

u

P TP W u W u W u W

T W u W u W u W= − == − =

− stor accumulates more precautionary saving in response to income risk

given prudence, a (more) temperant investor, takes less financial risk in response to income risk −

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Background Risk: Condition

2

2

A sufficient condition for background risk to induce a more risk averse behavior is than (1 )

1(1 )

This requires investors to: ) be prudent ( >0), ) be temperant ( >0)an

u u yu u u u

u u y

u u

PTPT P A

P A

a P b T

σσ

+> => >

+

d ) be more temperant than risk averse ( > ) u uc T A

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Background Risk: Remarks

2

Remark 1: not all utility functions satisfy these conditions- Quadratic utility does not satisfy the conditions. Suppose ( ) -

than 0Utility functions that are continuosly differentia

u u

u W aW bWP T== =

− ble and whose derivatives alternate in sign satisfy the property. The CRRA utility has this property and thus satisfies the condition. CARA utility does it weakly.- Example 1: ( ) => =

-

wu u uu W e A P Tθ θ−=− = =

1

Example 2: ( ) => = / < (1 )/ (2 )/1

u u uWu W A W P W T W

γ

γ γ γγ

−

= = + < = +−

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Background Risk: Remarks

2 2

2 2

Remark 2: the condition can be written in a sligthly different form:- Multiply by rigth and left hand side and get

(1 ( / ) ( / ) )1

(1 ( / ) ( / ) )

where , and

u u yv u u u u u

u u y

Wp t y W y

a a p t p ap a y W y

a AW p PW t T

σσ

+= > => >

+

= = = denote relative risk aversion,relative prudence and relative temperance; and / is the income wealth ratio.

Wy W

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Calibrating Background Risk

How large can in practice be the effect of backgroud risk on riskaversion? Important to know because gives us a hint on the effect onportfolio share of stocks- Suppose utility ( ) is CRRA with relatiu W ve risk aversion 3 .Than =1+ 4 and =2+ = 5- let / =0.25 = ratio of labor income to total wealth in the SHIW- / 0.12 (standard deviation of log wage: Guiso et al JPE (2005)

(1 (

u u

y

u uv u

p ty W

y

p t ya a

γγ γ

σ

==

=

+=

2 2 2 2

2 2 2 2

/ ) ( / ) ) (1 4 5 (0.25) (0.12) )3 3 1.007(1 ( / ) ( / ) ) (1 4 3 (0.25) (0.12) )

which is not a big effect. If wage sd=0.2, =3 1.019

y

u u y

v

W yp a y W y

a

σσ

+ × ×= × = ×

+ + × ×

×

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Calibrating Background Risk But for young investors the difference may be substantial - young investors have low accumulated assets and high human capital, so that / is large- they face normally more labor income risk Suppose

y W

- / =2- / 0.20

Than, assuming the other parameters are the same 3 1.327

This is a big effect! The portfolio share in risky assets may be 25 percent lower than if there were no background ris

y

v

y Wy

a

σ =

= ×

k

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Measuring risk aversion

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Inferring risk aversion 1. Following a long tradition in economics one

could rely on revealed preference (RP) theory to measure risk aversion

2. According to revealed preference theory=> preferences are imbedded in behavior

3. Observation of behavior allows one to infer preferences

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How can we use RP to measure risk aversion?

Rely on portfolio theory: individual i share in risky assets is

Hence, observing the portfolio share in risky assets and the variance and mean excess return on the risky assets, one can compute

And obtain an estimate of risk aversion for each individual

2( )E

ii i R

Ra w

ασ

=

ai =RE

α iσ R2

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What we get using RP?Massive heterogeneity in implied risk aversionMedian level=6For 25% AR>17Smallest level 1.6Only 25% have a AR<3.7

Computed for equity premium=0.65; return variance=0.2

020

4060

80Fr

eque

ncy

0 10 20 30 40 50Revealed preference level of risk aversion (among risky asset holders)

Portfolio share in risky assets

Median relative risk aversion=6

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Limits of the RV approach Relies on strong assumption about behavior: all have the same beliefs about returns and variance of returns. Heterogeneity in beliefs is reflected in measured risk aversionIt can only be computed for those who actually invest in risky assets. Many do not and may be they do not because are highly risk averse; but their risk aversion is not computableObviously, one cannot use this approach to test the theory which has strong predictions about the role of risk aversion. Indeed risk aversion can only be inferred under the assumption that the simple theory is true Portfolio choice may first need knowledge of risk aversion in order for the decision to be made: if the decision relies on the advise of a third party – say a bank – the bank cannot rely on observed portfolio choice to obtain information of the investor risk tolerance and provide him with “good” advise

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1. Several strategies to elicit individual attitudes towards risk which rely on answers to specifically designed questions. Major distinction:

2. Qualitative indicators: allow to sort individuals between more and less risk tolerant

3. Quantitative: allow to obtain a measure of Arrow-Pratt risk aversion on the basis of some assumption about the utility function

Strategies to elicit risk attitudes

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Qualitative Measures of Risk Attitudes

Often used in psychologyAlso relied upon in the financial industry to elicit an investor risk tolerance and provide financial adviseZuckerman SSS (sensation-seeking) Scale

Commonly used in psychologyRisk seeking seen as a personality trait

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Zuckerman SSS scale

Zuckerman distinguished four subscales:

1. Disinhibition (Dis): measures disinhibited behavior

2. Thrill and Adventure Seeking (TAS): measures the desire for risky sports or activities (e.g. mountain climbing, parachuting)

3. Experience Seeking (ES): measures the desire to expand one’s experience through the mind and senses and an un-conforming life style

4. Boredom Susceptibility (BS): measures the aversion to sameness, lack of change, or unpredictability in activities or friends and a general restlessness when there is no novelty or change.

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Zuckerman questions

Four types of questions Respondents choose either A or BAnswers differ by risk attitudeA scale (score) is obtained by combining the answers: the higher the score the more risk tolerant

A. I would like to try parachute jumping.B. I would never want to try jumping out of a plane, with or without a parachute.

A. The worst social sin is to be rude.B. The worst social sin is to be a bore.

A. Even if I had the money, I would not care to associate with flighty rich people who frequently appear in the newspapers and tabloids.B. I could conceive of myself seeking pleasures around the world with the sort of people who are frequently covered in newspapers and celebrity magazines.

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Zuckerman Scale

Men (N = 133) (Std. Dev.)

Women (N = 99)

(Std. Dev.)

Means Test t-stat.

(p-value) Total

21.90 (6.05)

20.12 (5.87)

2.24 (0.03)

Disinhibition

5.23 (2.87)

4.90 (2.97)

0.874 (0.40)

Boredom Susceptibility

3.68 (1.97)

3.06 (1.89)

2.43 (0.02)

Thrill and Adventure Seeking 7.62 (2.23)

6.61 (2.49)

3.27 (0.001)

Experience Seeking

5.36 (2.26)

5.54 (1.95)

-0.55 (0.52)

Higher score value more risk tolerant

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Qualitative Measures of Risk: view of risk - 1

Ask individuals their view about risk (Unicredit survey). “How would you classify risk among the following two

alternatives ? 1. Risk is an uncertain event from which one can extract a

profit2. Risk is an uncertain event from which one should seek

protection.

Most respondents (71%) answer (2), considering risk a threat rather than an opportunity

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Qualitative Measures of Risk: view of risk - 2 Alternatively ask individuals

how they view themselves (Socio Economic Panel Dohmen et. al., 2006)

“Are you generally a person who is fully prepared to take risks or do you try to avoid taking risk? Please tick a box on the scale, where the value 0 means “unwilling to take risk” and the value 10 means “fully prepared to take risk”

The vast majority dislikes riskUnwilling Fully

prep ared

0 1 2 3 4 5 6 7 8 9 10

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Qualitative Measures of Risk: choice based self- classification

Ask individuals how much risk they would undertake: SCF, SHIW, Unicredit survey "Which of the following statements comes closest to the amount of financial risk that you are willing to take when you make your financial investment?

1. a very high return, with a very high risk of loosing the money

2. high return and high risk 3. moderate return and moderate risk4. low return and no risk

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Sample Distribution of Qualitative Risk aversion

“ In managing your financial investment, you think you are a person that is interested in investments that offer the possibility of: (1) a high return, with a high risk of loosing the capital; (2) a good return, and a reasonable return; (3) a moderate return, but at the same time a good degree of safety; (4) a low return, without any risk of loosing the capital.”0

.1.2

.3.4

.5

Willing to take..

high return and risk medium return and riskmoderate return and risk low return and low risk

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Qualitative measures: featuresMost individuals view risk as a danger; some think it is an opportunityMost individuals are classified as risk averseThese questions are simple and people seem to understand them

When asked in large surveys there are very few non-responses

But: they do not distinguish between aversion to riskand risk perceptions: some may be more averse because they perceive more risk (attach higher probability to adverse events). That is, probabilities are not held constant across respondents

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Quantitative Measures of Risk Attitudes

1. To get a quantitative measure of the degree of risk aversion one needs information on an individual willingness to pay for some given lottery that is independent of the individual endowment

2. One can thus set up a hypothetical lottery/risk prospect and ask an individual his willingness to pay.

3. Using that and some utility function one can infer his risk aversion, assuming individuals are expected utility maximizers

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IllustrationExamplea firm is selling a security that over a year yields

50000 EUR with probability 1/2

0 EUR with probability 1/2 you can buy the security. What is the maximum price you would be willing to

R⎧

= ⎨⎩

%

pay? Let us say, your price (willingness to pay) is . Then it must be the case that

( ) ( - ) The willingness to pay reveals the risk tolerance of the individualAssume utility is exponential

P

u w Eu w R P= + %

( ) ( 50000 ) ( )

(50000 )

with risk aversion . Then:1 1 2 2

=> 2= Hence given the degree of absolute risk aversion, is uniquely defined. Probabilities are held co

w w R P w P w P

P P

e Ee e e

e eP

θ θ θ θ

θ θ

θ

θ

− − + − − + − − −

− −

− = − = − −

+

%

nstant, so approach identifies risk aversion

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Implementing this strategy

•

Guiso & Paiella (2008) follow this strategy to obtain a measure of absolute risk aversion from the 1995 Bank of Italy Survey of Household Income and Wealth (SHIW)•

In the 1995 wave each survey participant is offered a hypothetical security and is asked to report the maximum price that he would be willing to pay for it.

“We would like to ask you a hypothetical question that we would like you to answer as if the situation was a real one. You are offered the opportunity of acquiring a security permitting you, with the same probability, either to gain Euros 5,000 or to lose all the capital invested. What is the most that you are prepared to pay for this security?”

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Implications for risk aversion

NB: the security is such that you win 5,000 but may loose P

Hence: Expected value of security: 1/2(5000- Pi ) Pi = 5,000 Eur →

risk neutrality

Pi < 5,000 Eur →

risk aversion Pi > 5,000 Eur →

risk loving

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Measuring risk aversion

They then assume exponential utility and obtain an estimate of the absolute risk aversion of each respondent

Remarks:1. The calculation of the measure of absolute risk aversion

does not depend much on the specific assumption about the utility function =>same findings if CRRA is assumed

2. However, a quadratic approximation to ui (wi )=0.5 ui (wi +5000)+ 0.5 ui (wi -Pi )=E ui (wi +Ri ) would bias risk aversion at low levels of P

3. Since probabilities of the various events are held constant, the methodology identifies risk aversion, not differences in risk perceptions

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Problems This approach too has its own problems

1. The question is difficult to understand => high non responses (over 50% in the SHIW)

2. People tend to understate willingness to pay and to overstate willingness to accept. Large literature showing this. Hence risk aversion may be highly overstated. If one is interested in the cross sectional pattern may not be a problem. If one is interested in the size of A, it is a problem

3. It is not incentive compatible: why should individuals reveal their true willingness to pay? One can set up mechanisms to induce individuals to tell the truth

4. Answers may depend on the size of the lottery (Holt&Laury, 2002)

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Consider two lotteries, A and BBoth have two outcomes, high and low payoffB is riskier than A: the difference between the high and low payoff is larger for B than for ADepending on the probabilities of the low and high payoffs A can be preferred to B and vice versa by a risk averse individual; If probability of high payoff high, B more attractive than AA high risk averse requires a high probability of the high payoff to prefer B to AStrategy: hold the payoffs constant and vary the probability of the high payoff and see what individual choose

An alternative avenue: Holt/Laury (2002) Instrument

Payoff1.6 2.00.1 3.85

Lottery A Payoffs

Lottery B Payoffs

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Example of decision screen (risk neutral choice decisions 4 and 5)

Risk Averse

Male Female

0 2.11 0.00 1 3.52 3.92 2 4.93 3.92 3 11.97 14.71 4 22.54 21.57 5 19.72 35.29 6 23.24 14.71 7 8.45 3.92 8 0.70 0.00 9 0.70 0.00 10 2.11 1.96 Risk Seeking

Summary of first risky choice (Percentages)

An alternative avenue: Holt/Laury (2002) Instrument

Probability of low payoff

Option APayoffs

(Euros)

Option B (Euros)

Expected payoff difference (E(A) -E(B))

Choice

9/10 [1.6, 2.0] [0.10, 3.85] 1.178/10 [1.6, 2.0] [0.10, 3.85] 0.837/10 [1.6, 2.0] [0.10, 3.85] 0.506/10 [1.6, 2.0] [0.10, 3.85] 0.16 (Risk

neutral choice)

5/10 [1.6, 2.0] [0.10, 3.85] -0.184/10 [1.6, 2.0] [0.10, 3.85] -0.513/10 [1.6, 2.0] [0.10, 3.85] -0.852/10 [1.6, 2.0] [0.10, 3.85] -1.181/10 [1.6, 2.0] [0.10, 3.85] -1.520/10 [1.6, 2.0] [0.10, 3.85] -1.85

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Adds precision to the elicitation strategyHL show that it is robust to whether the lottery involves money (incentive compatible) or is hypotheticalChoice is sensitive to the size of the payoffs: as the payoff increases, individuals become more risk averseBut there are limitations: It is ok to ask in a face-to-face experiment, perhaps a bit involved in a general survey

Holt/Laury Instrument: Comments

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Several ways to elicit risk aversion: can we compare them? Dohmen et. al. do this. Use the SEP for Germany

Very large sample of German households (20,000 individuals)Elicit risk aversion in several ways:General qualitative question on willingness to take risks in generalFive questions on willingness to take risks in specific hypothetical domains A general lottery question asking willingness to payAn experimental question on a sub-sample of households involving money relevant choices

Which strategy is better?

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There is considerable heterogeneity in risk attitudes however measuredThe various measures are relatively highly correlated, even when referred to different domains. This is consistent with the idea that the risk attitude is a single individual trait, captured for instance by the Arrow-Pratt measureDifferences across individuals are partly explained by observables These observables have similar effects on the various measuresMeasures that involve money and those based on hypothetical questions give similar answersThese measures have predictive power on behavior under risk (portfolio choice, migration, smoking etc.)The best predictor is the general qualitative question (which iseasy to ask)

What do they find?

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How does the risk aversion function looks like?

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How does the risk aversion function looks like?

•In many instances it is crucial to know how risk aversion varies with the individual endowment. Should the rich old a larger share in risky assets? The answer depends on how risk aversion varies with the endowment

•Common sense, introspection, casual observation and a priori reasoning have all been used to draw conclusions on the nature of the relation

•Shared opinion: Absolute Risk Aversion should decline with wealth.

•But does it actually does and how fast?

•If one accepts that preferences are CRRA, then ARA is decreasing and convex and the speed at which it declines is unitary in wealth: a doubling of wealth is associated with a halving of absolute risk aversion (=> absolute risk tolerance is increasing and linear in wealth)

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Once a measure of absolute risk aversion is available one can study its shape. Guiso and Paiella (2008) undertake this task

To estimate the relationship between ARA and consumer endowment, they focus on risk averse individuals and use the following specification

This a generalization of ARA under CRRA preferencesβ =1→ CRRA utility, with Relative risk aversion coefficient equal to aeγH+ν

A(W) decreases in W for β >0. If β > 0, is convex in WH is a vector of characteristics that affect risk aversion, v a random error

( )HaeA W

W

γ υ

β

+

=

Studying the risk aversion function

48

Findings

Absolute risk aversion declines with the endowment=> reject CARADeclines less than implied by CRRA=> reject CRRARelative risk aversion increases with wealth=> wealthier more sensitive to relative risks Some demographic matter

Variable (1) (2)

Log( W ) -0.647*** -0.637***

Age - 0.002Gender (woman) - 0.174*

(0.098)Junior high school diploma

- -0.176*

Siblings - 0.001

Fatherself-employed

- 0.046

Father publicsector employee

- 0.024

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The estimated risk tolerance endowment relation

45

67

8

0 100000 200000 300000 400000non-durable expenditure

beta IV beta OLS

Picture shows the estimated relation between risk tolerance and the endowment in Guiso&Paiella

CRRA=> linear relation

RISK TOLERANCE AND CONSUMER ENDOWMENT

( ) H

WT wae

β

γ υ+=

50

Holt&Laury

Holt and Laury argue that their results are consistent with a utility function like the following

Show the underlying absolute risk aversion function

Is it decreasing with W? What does it imply for the relation between relative risk aversion and the endowment?

(1 )

1 Weβα

α

−−−

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Determinants of attitudes toward risk

52

What determines risk aversion

What determines the attitude toward risk? Is it just an innate parameter?Can it depend upon observables? And if so which ones

If we interpret risk aversion as the willingness an individual has to bear risk, it is clear that it is not necessarily only a genetic feature that characterizes one individuals preferences. The environment may matter

Background riskLiquidity constraints: individuals that have a more difficult access to the credit market are less willing to bear risk

Balanced view: depends on both

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A role for characteristics?

Which characteristics matter?

One needs to be careful: preferences affect choice! Hence any variable that can be chosen can be correlated with risk attitudes, but causality may go from risk aversion to the characteristicPick up exogenous characteristics

AgeSex Features of the parents (e.g. their education)Careful when measuring Background risk or Liquidity constraints

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A summary of evidence

Women are more risk averse than men: genetic or environment? Difficult to sort out, may be bothAge seems to matter: the younger are less risk averse: hard to attribute to genetic traits: should be invariant Height seems to matter: taller individuals are less risk averse. Open to interpretationPlace of birth matters, but hard to find specific characteristics of the placeBackground risk and Liquidity constraints result in lower risk tolerance Risk aversion transmits across generations: sons of risk averse individuals are more risk averse! Education or genetics (Thomas Dohmen et. al.)

55

Evidence from Guiso&Paiella: (log) risk tolerance on the left hand side

Variable IV IV IV(1) (2) (3)

Log( W ) 0.764** 0.798** 0.750*Variance of shocks to per-

capita GDP-20.198** - -20.070***

Dummy for liquidity constraints

- -0.356*** -0.356***

Age/10 0.017 0.001 0.001Siblings 0.007 0.018 0.024Father self-employed 0.052 0.041 0.047Father public sector employee -0.047 -0.068 -0.064

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Time varying risk aversion

Do preferences for risk evolve over time? Or rather is risk preference a fixed attribute? If they change, why do they do?Important in assets pricing to explain why they change over the business cycle. If background risk is time varying then also risk aversion may be time varying. But that may not be enough:

Stockholding of the wealthy also change and they are unlikely to face much human capital risk

57

Time varying risk aversion

Approach taken: allow preferences to generate relative risk aversion that declines with wealth=> habits (see lecture 5) That shifts the questions: are habit preferences reasonable? An alternative is first to establish the fact and then try to ask whyFinancial crisis is a good opportunityGuiso, Sapienza nd Zingales (2010) ask risk aversion questions in a survey of Italian investors before and after the crisisQuestion asked to the same pool of people

58

Time varying risk aversion: qualitative indicator

.018

.2718

.5511

.1592

0.2

.4.6

prop

ortio

n

0 1 2 3 4risk aversion

.0135

.1231

.4369.4264

0.1

.2.3

.4pr

opor

tion

0 1 2 3 4risk aversion

2007 2009

No risk

Norisk

High RET & High RIS

i

Medium RET& RIS

Moderate RET &RIS

59

Time varying risk aversion: quantitative measure

4257.06

2816.07

01,

000

2,00

03,

000

4,00

0R

isk

aver

sion

indi

cato

r

mean of ra06 mean of ra09

4000

1500

01,

000

2,00

03,

000

4,00

0R

isk

aver

sion

indi

cato

r

p 50 of ra06 p 50 of ra09

mean median

2007 200920092007

Suppose you are in a room and to get out you can chhose betwen 2 doors. If you choose the rigth door you get 10.000 euro, if you choose the wrong one, you get nothing. You do not know which door is which. You can take a third door and a get a sure sum. If you are offered EUR 100 to exit the third door would you exit from it or rather chhose among the other two? If “Yes”, stop; if “NO” continue with another larger sure sum .

60

Summing up

Not surprising, eliciting risk aversion is hard Yet, attempts made are encouraging Results thus far suggest that available measures are good at sorting individuals on the basis of their willingness to bear risk; less so at measuring “the”degree of risk aversion A parallel line of research attempts at eliciting probability distribution of future events, such as the return on stocks Variation in beliefs about risk and return, may together with differences in risk tolerance, be important determinant of portfolio choices