Chapter 4Risky Decisions

von NeumannMorgenstern

measures of riskaversion, HARA class

Arrow-Debreugeneral equilibrium,

welfare theorem,representative agent

Radner economiesreal/nominal assets,

market span,risk-neutral prob.,

representative good

von NeumannMorgenstern

measures of riskaversion, HARA class

Finance economySDF, CCAPM,

term structure

Data andthe Puzzles

Empiricalresolutions

Theoreticalresolutions

A very special ingredient: probabilities By defining commodities as being contingent on

the state of the world, we do in principle cover decisions involving risk already.

But risk has a special, additional structure which other situations do not have: probabilities.

We have not explicitly made use of probabilities so far.

The probabilities do affect preferences over contingent commodities, but so far we have not made this connection explicit.

The theory of decisions under risk exploits this particular structure in order to get more concrete predictions about behavior of decision-makers.

The St Petersburg Paradox

A hypothetical gamble

Suppose someone offers you this gamble:

"I have a fair coin here. I'll flip it, and if it's tail I pay you $1 and the gamble is over. If it's head, I'll flip again. If it's tail then, I pay you $2, if not I'll flip again. With every round, I double the amount I will pay to you if it's tail."

Sounds like a good deal. After all, you can't loose. So here's the question:

How much are you willing to pay to take this gamble?

The expected value of the gamble The gamble is risky because the payoff is

random. So, according to intuition, this risk should be taken into account, meaning, I will pay less than the expected payoff of the gamble.

So, if the expected payoff is X, I should be willing to pay at most X, possibly minus some risk premium.

(We will discuss risk premia in detail later.) BUT, the expected payoff of this

gamble is INFINITE!

Infinite expected value

With probability 1/2 you get $1.

With probability 1/4 you get $2.

With probability 1/8 you get $4.

etc.

1

1221

t

payoff

t

yprobabilit

t

1 21

t

0121 2 times

1221 2 times

2321 2 times

The expected payoff is the sum of these payoffs, weighted with their probabilities, so

An infinitely valuable gamble? I should pay

everything I own and more to purchase the right to take this gamble!

Yet, in practice, noone is prepared to pay such a high price.

Why? Even though the

expected payoff is infinite, the distribution of payoffs is not attractive…

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60

probability

$

With 93% probability we get $8 or less, with 99% probability we get $64 or less.

What should we do? How can we decide in a rational fashion about

such gambles (or investments)?

Bernoulli suggests that large gains should be weighted less. He suggests to use the natural logarithm. [Cremer, another great mathematician of the time, suggests the square root.]

1

1)2ln(21

t

payoffofutility

t

yprobabilit

t

gamble of

utility expected)2ln(

Bernoulli would have paid at most eln(2) = $2 to participate in this gamble.

Lotteries and certainty equivalent

Lotteries

A risky payoff is completely described by a list of the possible payoffs and the probabilities associated with it.

We call such a risky payoff a lottery, but we could just as well call it a random variable.

lottery L := [ x1,1 ; … ; xS,S ].

A non-risky payoff is a degenerate lottery [x,1].

Ordinal utility function We assume that agents have preferences

over lotteries.

Let L be the set of all lotteries. A preference relation is defined over elements of L. As before, ~ represents indifference (now between lotteries).

As before, we can represent these preferences with an ordinal utility function V.

Let L,L' є L be two lotteries, then V(L) < V(L') iff L L'.

Defn. risk aversion

The expected payoff of the lottery L = [ x1,1 ; … ; xS,S ] is ssxs =: E{L}.

We say that the agent is risk averse if V(L) < V([E{L},1]) (if L is not degenerate).

In words, the agent strictly prefers the average payoff of the lottery for sure as opposed to the risky lottery.

This means that the agent is willing to forego some payoff on average in exchange for not being exposed to the risk of the lottery.

Certainty equivalent

Consider some lottery L є L. If [x,1] ~ L then we call x the certainty equivalent of L.

Equivalently, V([x,1]) = V(L).

We denote x with CE(L).

CE(L) is the risk-free payoff that is equivalent to the risky payoff contained in lottery L.

Risk premium

An alternative way to define risk aversion is to require CE(L) < E{L} (if L is not degenerate).

We call the difference,

E{L} – CE(L) =: RP(L)

the risk premium.

It is the maximum premium the agent is willing to pay for an insurance that shields him from the risk contained in L.

The utility function V In order to be able to draw indifference curves we

will restrict attention to lotteries with only two possible outcomes, [x1,1;x2,2].

Furthermore, we will also fix the probabilities (1,2), so that a lottery is fully described simply by the two payoffs (x1,x2). So a lottery is just a point in the plane.

From the ordinal utility function V we define a new function V that takes only the payoffs as an argument, V(x1,x2) = V([x1,1;x2,2]).

V is very much like a utility function over two goods that we have used in chapter 2. This makes it amenable to graphical analysis.

Indifference curves

x1

x2

Any point in this plane is a particular lottery.

Where is the set of risk-free lotteries?

If x1=x2, then the lottery contains no risk.

45°

Indifference curves

x1

x2

Where is the set of lotteries with expected prize E{L}=z?

It's a straight line, and the slope is given by the relative probabilities of the two states.

45°

z

z

Indifference curves

x1

x2

Suppose the agent is risk averse. Where is the set of lotteries which are indifferent to (z,z)?

That's not right! Note that there are risky lotteries with smaller expected prize and which are preferred.

45°

z

z

??????

Indifference curves

x1

x2 So the indifference curve must be tangent to the iso-expected-prize line.

This is a direct implication of risk-aversion alone.45°

z

z

Indifference curves

x1

x2

45°

z

z

But risk-aversion does not imply convexity.

This indifference curve is also compatibe with risk-aversion.

Indifference curves

x1

x2

45°

z

z

V(z,z)The tangency implies that the gradient of V at the point (z,z) is collinear to .

Formally, V(z,z) = , for some >0.

Expected utility representation

What we are after: an expected utility representation

The ordinal utility function V is a cumbersome object because its domain is a large set, L, the set of all lotteries.

A utility function on prizes (and the expected value of such a utility function, given a lottery), would be much easier to work with.

So we look for a representation of the form

V([x1,1; …; xS,S]) = s s v(xs).

NM axioms: state independence Von Neumann and Morgenstern have presented a

model that allows the use of an expected utility under some conditions.

The first assumption is state independence.

It means that the names of the states have no particular meaning and are interchangeable.

Note that this rules out the case where you have different preferences over wealth if it rains or if the weather's bright.

x

y

1-

y

x

1-

~

NM axioms: consequentialism Consider a lottery whose prices are further

lotteries, L = [L1,1;L2,2]. This is a compound lottery.

The axiom requires that agents are only interested in the distribution of the resulting prize, but not in the process of gambling itself.

L1

1

2

x111

L2

12

21

22

x2

x3

x4

1 11 x1

x2

x3

x4

1 12

2 21

2 22

~

NM axioms: irrelevance of common alternatives This axiom says that the ranking of two lotteries

should depenend only on those outcomes where they differ.

If L1 is better than L2, and we compound each of these lotteries with some third common outcome x, then it should be true that [L1,;x,1-] is still better than [L2,;x,1-]. The common alternative x should not matter.

L1

x

1-

L2

x

1-

L1 L2

Von Neumann and Morgenstern prove that with these assumptions, one can represent a utility over lotteries V as an expected utility v.

The shape of the von Neumann Morgenstern (NM) utility function contains a lot of information.

E{v(x)}

Risk-aversion and concavity

x

v(x)Consider a fifty-fifty lottery with two prizes…

x0 x1

How does the NM utility function of a risk-neutral agent look like?

v(x0)

v(x1)

E{x}

Risk-aversion means that the certainty equivalent is smaller than the expected prize.

E{v(x)}

Risk-aversion and concavity

x

v(x)

x0 x1

We conclude that a risk-averse NM utility function must be concave.

v(x0)

v(x1)

E{x}

v-1(E{v(x)})

Measures of risk aversion

An insurance problem Consider an insurance problem:

d amount of damage, probability of damage, insurance premium for full coverage, c amount of coverage.

).)1(()()1(max dccwvcwvc

The FOC of this problem is

.))1(('

)('1

d

dccwvcwv

An insurance problem Full coverage (c=1) implies

Full coverage is optimal only if the premium is statistically fair.

.1

dd

Suppose the premium is not fair. Let = (1+m)d, and m>0 be the insurance company's markup.

Then,

d1

))1((')('FOC the by dccwvcwv

.1)1( cdccwcw

An insurance problem If the insurance premium is not fair, it is optimal not

to fully insure. In fact, if the premium is large enough (m0), no

coverage is optimal. The FOC, with substituted by (1+m)d, is

We extract m0 by setting c=0,

.)1(

)1())1()1(('

))1(('1dm

dmddcdmcwv

dmcwv

,)1(

)1()('

)('1

0

0

dmdmd

dwvwv

.)(')(')1())(')(')(1(

0 dwvwvwvdwv

m

and then solve for m0

An insurance problem

If =(1+m0)d, the agent is just indiff between insuring and carrying the whole risk.

Thus, w-(1+m0)d is the certainty equivalent.

It is clear that m0 vanishes as the risk becomes smaller, d 0.

But the relative speed of convergence is not so clear: how fast does m0 vanish compared to d?

.)(')(')1())(')(')(1(

0 dwvwvwvdwv

m

?lim 0

0

dm

d

Absolute risk aversion

dm

d

0

0lim .

)(')(')1(/))(')('(

lim1

0 dwvwvddwvwv

d

For symmetric risks (=1/2) we thus get

Note that this part is just the second derivative of v…

…and this part converges to the first derivative.

)(:)(')(''

lim 0

0wA

wvwv

dm

d

This is the celebrated coefficient of absolute risk aversion, discovered by Pratt and by Arrow.

We see here that it is a measure for the size of the risk premium for an infinitesimal risk.

CARA, IARA, DARA Consider some utility function and its associated

absolute risk aversion as a function of wealth, A(w).

We say that absol risk aversion is decreasing (DARA) if A'(w)<0; it is constant (CARA) if A'(w)=0; it is increasing (IARA) if A'(w)>0.

Consider then two people, both with the same utility function, but one is very poor, the other very rich.

Suppose also that both agents are subject to the same risk of losing, say, $10.

How of the two is more willing to insure against this risk?

CARA, IARA, DARA It seems reasonable to assume that the rich

person is not willing to insure, because he will not notice the difference of $10.

The very poor person, on the other hand, may be very eager to insure, because $10 amounts to a substantial part of his wealth.

If this is the case, then utility is DARA.

IARA would imply that the rich person buys insurance against this risk, but the poor does not, which seems not plausible.

Relative risk aversion We have considered additive risks so far: lose $10

with some probability, or experience a damage d with probability .

But some risks are multiplicative (risking 10% of your wealth). This is particularly true in financial markets: if you invest 20% of your wealth in shares, then you are exposed to a multiplicative risk.

Relative risk aversion rather than absolute risk aversion is the correct measure to estimate the willingness to insure against infinitesimal multiplicative risks.

Relative risk aversion

Consider the rich and poor people of before again, but now suppose they are both subject to a 50% risk of losing 10% of their wealth.

Which of the two is willing to pay a larger share of his wealth to insure against this risk?

This seems unclear a priori. If both have the same willingness to pay (as a percentage of their wealth), the utility function exhibits constant relative risk aversion (CRRA).

The other possibilities are IRRA and DRRA.

R(w) := w A(w).

Prudence The measures of risk aversion inform us about the

willingness to insure …

… but they do not inform us about the comparative statics.

How does the behavior of an agent change when we marginally increase his exposure to risk?

An old hypothesis (going back at least to J.M.Keynes) is that people should save more now when they face greater uncertainty in the future.

The idea is called precautionary saving and has intuitive appeal.

Prudence But it does not directly follow from risk aversion

alone. In fact, it involves the third derivative of the utility

function. Kimball (1990) defines absolute prudence as

P(w) := –v'''(w)/v''(w). He shows that agents have a precautionary

saving motive if any only if they are prudent. This finding will be important later on when we do

comparative statics of interest rates. Prudence seems uncontroversial, because it is

weaker than DARA.

Reasonable specifications and the HARA class

Uncontroversial properties

Monotonicity seems uncontroversial. After all, most people prefer more wealth to less.

Risk aversion is only slightly more controversial. Of course, there may be some risk seekers. But most people shun risks.

DARA also seems reasonable.

And since DARA implies prudence, we should also expect people to be prudent.

Animal experiments

Kagel et al. (1995) have performed interesting experiments with animals.

Payoff is food.

Very good control over "wealth" (nutritional level) of animals.

They find evidence for risk aversion and DARA.

In fact, they find some weak evidence for a utility function of the form (x-s)1-/(1-), where x is consumption and s>0 can be interpreted as subsistence level (see chapter 6.3 of their book).

This utility function exhibits DRRA as well.

Empirical estimates Friend and Blume (1975): study U.S. household

survey data in an attempt to recover the underlying preferences. Evidence for DARA and almost CRRA, with R 2.

Tenorio and Battalio (2003): TV game show in which large amounts of money are at stake. Estimate rel risk aversion between 0.6 and 1.5.

Abdulkadri and Langenmeier (2000): farm household consumption data. They find significantly greater risk aversion.

Van Praag and Booji (2003): large survey done by a dutch newspaper. They find that rel risk aversion is close to log-normally distributed, with a mean of 3.78.

Introspection

In order to get a feeling for what different levels of risk aversion actually mean, it may be helpful to find out what your own personal coefficient of risk aversion is.

You can do that by working through Box 4.6 of the book, or by using the electronic equivalent available from the website.

LAUNCH

Evolutionary stability A completely different take on the problem is provided by

evolutionary finance.

Natural selection, so the argument, favors agents with relative risk aversion close to unity.

The reason is that such agents maximize the growth rate of their wealth, and thus eventually dominate the market.

Thus, either you maximize ln(y), or you are eventually marginalized.

With this line of argument, experiments, empirics or introspection are not needed, because in the long run the market is necessarily dominated by lof utility functions.

Literature: Hakansson (1971), Blume and Easly (1992), Sinn and Weichenrieder (1993), Sinn (2002).

The HARA class

Usual utility functions

name formula A R P a b

affine 0+1y 0 0 undef undef undef

quadratic 0y – 1y2 incr incr 0 0/(21) –1

exponential –e–y/ incr 1/ 0

power y1–/(1–) decr decr 0 1/

Bernoulli ln y decr 1 decr 0 1

One finds only a handful of specifications in the literature.

A, R, and P denote absolute risk aversion, relative risk aversion, and prudence. a and b will be explained later.

The HARA class

All the functions of the previous tables belong to the hyperbolic absolute risk aversion, or HARA, class. (An alternative name is linear risk tolerance, or LRT.)

Let absolute risk tolerance be defined as the reciprocal of absolute risk aversion, T := 1/A.

u is HARA if T is an affine function,

T(y) = a + by. The slope b is is sometimes also called

cautiousness.

The HARA class Robert Merton has shown that a utility function v is

HARA if and only if it is an affine trasformation of this,

otherwise.

,0 if

,1 if

,)()1(

,

),ln(

:)(/)1(1

/

b

b

byab

ae

ay

yvbb

ay

The sign of b is closely related to absolute risk averion as a function of wealth. We have,

DARA b>0; CARA b=0; IARA b<0. Also, v is CRRA if and only if a=0.

Mean-Variance

Mean-variance is simpler

Early researchers in finance, such as Markowitz and Sharpe, used just the mean and the variance of the return rate of an asset to describe it.

Characterizing the prospects of a gamble with its mean and variance is often easier than using an NM utility function, so it is popular.

But is it compatible with NM theory? The answer is yes … approximately …

under some conditions.

Mean-variance: quadratic utility

Suppose utility is quadratic, v(y) = ay–by2.

Expected utility is then

)).var(}{(}{

}{}{)}({2

2

yyEbyaE

ybEyaEyvE

Thus, expected utility is a function of the mean, E{y}, and the variance, var(y), only.

Mean-variance: joint normals Suppose all lotteries in the domain have normally

distributed prized. (They need not be independent of each other).

This requires an infinite state space.

It is a fact of mathematics that any combination of such lotteries will also be normally distributed.

The normal distribution is completely described by its first two moments.

Therefore, the distribution of any combination of lotteies is also completely described by just the mean and the variance.

As a result, expected utility can be expressed as a function of just these two numbes as well.

Mean-variance: linear distribution classes Generalization of joint nomals.

Consider a class of distributions F1, …, Fn with the following property:

for all i there exists (m,s) such that Fi(x) = F1(a+bx) for all x.

This is called a linear distribution class.

It means that any Fi can be transformed into an Fj by an appropriate shift (a) and stretch (b).

Let yi be a random variable drawn from Fi. Let i = E{yi} and i

2 = E{(yi–i)2} denote the mean and the variance of yi.

Mean-variance: linear distribution classes Define then the random variable x = (yi–i)/i.

We denote the distribution of x with F.

Note that the mean of x is 0 and the variance is 1, and F is part of the same linear distribution class.

Moreover, the distribution of x is independent of which i we start with.

.)()(

zdFzv i

We want to evaluate the expected utility of yi ,

Mean-variance: linear distribution classes

).,(:

)()()()(

ii

iii

u

zdFzvzdFzv

But yi = i + i x, thus

The expected utility of all random variables drawn from the same linear distribution class can be expressed as functions of the mean and the standard deviation only.

Mean-variance: small risks The most relevant justification for mean-variance

is probably the case of small risks. If we consider only small risks, we may use a

second order Taylor approximation of the NM utility function.

A second order Taylor approximation of a concave function is a quadratic function with a negative coefficient on the quadratic term.

In other words, any risk-averse NM utility function can locally be approximated with a quadratic function.

But the expectation of a quadratic utility function can be evaluated with the mean and variance. Thus, to evaluate small risks, mean and variance are enough.

Mean-variance: small risks Let f : R R be a smooth function. The

Taylor approximation is

!3)(

)('''

!2)(

)(''!1

)()(')()(

30

0

20

0

10

00

xxxf

xxxf

xxxfxfxf

So f(x) can approximately be evaluated by looking at the value of f at another point x0, and making a correction involving the first n derivatives.

We will use this idea to evaluate E{u(y)}.

Mean-variance: small risks Consider first an additive risk, i.e. y = w+x where

x is a zero mean random variable. For small variance of x, E{v(y)} is close to v(w). Consider the second order Taylor approximation,

.2

)var()('')(

2}{

)(''}{)(')()}({2

xwvwv

xEwvxEwvwvxwvE

Let c be the certainty equivalent, v(c)=E{v(w+x)}.

For small variance of x, c is close to w, but let us look at the first order Taylor approximation.

).)((')()( wcwvwvcv

Mean-variance: small risks Since E{v(w+x)} = v(c), this simplifies to

w – c is the risk premium.

We see here that the risk premium is approximately a linear function of the variance of the additive risk, with the slope of the effect equal to half the coefficient of absolute risk.

.2

)var()(

xwAcw

Mean-variance: small risks The same exercise can be done with a

multiplicative risk. Let y = gw, where g is a positive random variable

with unit mean. Doing the same steps as before leads to

where is the certainty equivalent growth rate, v(w) = E{v(gw)}.

The coefficient of relative risk aversion is relevant for multiplicative risk, absolute risk aversion for additive risk.

,2

)var()(1

gwR