Transcript Chapter 4

Chapter 4
Risky Decisions
Arrow-Debreu
general equilibrium,
welfare theorem,
representative agent
Radner economies
real/nominal assets,
market span,
risk-neutral prob.,
representative good
von Neumann
Morgenstern
measures of risk
aversion, HARA class
Finance economy
SDF, CCAPM,
term structure
Data and
the Puzzles
Empirical
resolutions
Theoretical
resolutions
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
12 1 times
2
 With probability 1/4 you get $2. 12  times
3
 With probability 1/8 you get $4. 12  times

With probability 1/2 you get $1.

etc.

The expected payoff is the sum of these
payoffs, weighted with their probabilities,

t
so

1
1

t 1
t 1

   2
2


probability
payoff
2
t 1

20
21
22
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…
probability
0.5
0.4
0.3
0.2
0.1
0
0

20
40
60 $
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.]


t 1
t
expected utility
1
t 1

   ln(2 )  ln(2) 
of gamble
2


probabilit y

utility of payoff
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,p1 ; … ; xS,pS ].

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,p1 ; … ; xS,pS ] is spsxs =: 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,p1;x2,p2].

Furthermore, we will also fix the probabilities
(p1,p2), 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,p1;x2,p2]).

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
Any point in
this plane is
a particular
lottery.
x2
Where is the
set of riskfree
lotteries?
45°
x1
If x1=x2,
then the
lottery
contains no
risk.
Indifference curves
Where is the
set of lotteries
with expected
prize E{L}=z?
x2
p
z
45°
z
x1
It's a
straight line,
and the
slope is
given by the
relative
probabilities
of the two
states.
Suppose the
agent is risk
averse. Where
is the set of
lotteries which
are indifferent
to (z,z)?
Indifference curves
???
x2
p
z
45°
z
x1
That's not
right! Note
that there
are risky
lotteries
with smaller
expected
prize and
which are
preferred.
Indifference curves
x2
p
z
So the
indifference
curve must be
tangent to the
iso-expectedprize line.
This is a direct
implication of
risk-aversion
alone.
45°
z
x1
Indifference curves
x2
p
But riskaversion does
not imply
convexity.
This
indifference
curve is also
compatibe
with riskaversion.
z
45°
z
x1
Indifference curves
x2
 V(z,z)
p
z
The tangency
implies that
the gradient
of V at the
point (z,z) is
collinear to p.
Formally,
 V(z,z) = lp,
for some l>0.
45°
z
x1
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,p1; …; xS,pS]) = s ps 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.
p
x
1-p
y
~
1-p
y
p
x

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.
NM axioms:
consequentialism

Consider a lottery whose prices are further
lotteries, L = [L1,p1;L2,p2]. 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.
p1
L1
p11
x1
p12
x2
p21
p2
L2
p22
x3
x4
p1 p11
~
x1
p1 p12
x2
p2 p21
x3
p2 p22
x4
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,p;x,1-p] is still
better than [L2,p;x,1-p]. The common alternative
x should not matter.
p
L1  L2
L1

1-p
L2

1-p
x
p
x
Risk-aversion and concavity

v(x)
Von Neumann and
Morgenstern prove that
with these assumptions,
one can represent a utility
over lotteries V as an
v(x1)
expected utility v.

The shape of the von
Neumann Morgenstern
E{v(x)}
(NM) utility function
contains a lot of
information.

How does the NM utility v(x0)
function of a risk-neutral
agent look like?
x0
Consider a fifty-fifty
lottery with two prizes…
E{x}
x1 x
Risk-aversion and concavity

Risk-aversion means that
the certainty equivalent is
smaller than the expected
prize.

We conclude that a
risk-averse NM
utility function
must be concave.
v(x)
v(x1)
E{v(x)}
v(x0)
x0
v-1(E{v(x)})
E{x}
x1 x
Measures of risk aversion
An insurance problem

Consider an insurance problem:

d amount of damage,

p probability of damage,

m insurance premium for full coverage,

c amount of coverage.
max (1  p)v(w  cm)  pv(w  cm  (1  c)d ).
c
The FOC of this problem is
1 p
v' (w  cm)
d m


.
p v' (w  cm  (1  c)d )
m
An insurance problem

Full coverage (c=1) implies
1 p d m

 m  pd.
p
m
Full coverage is optimal only if the premium is
statistically fair.

Suppose the premium is not fair. Let m =
(1+m)pd, and m>0 be the insurance company's
markup.
1 p d m

Then,
p
m
by the FOC

 v' (w  cm)  v' (w  cm  (1  c)d)
 w  cm  w  cm  (1  c)d  c  1.
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 m substituted by (1+m)pd, is
1 p
v' (w  c(1  m)pd )
d  (1  m)pd


.
p v' (w  c(1  m)pd  (1  c)d )
(1  m)pd

We extract m0 by setting c=0,
1 p
v' (w)
d  (1  m0 )pd


,
p v' (w  d)
(1  m0 )pd
and then
solve for m0
(1  p)(v' (w  d )  v' (w))
 m0 
.
(1  p)v' (w)  pv' (w  d )
An insurance problem
(1  p)(v' (w  d )  v' (w))
m0 
.
(1  p)v' (w)  pv' (w  d )

If m=(1+m0)pd, the agent is just indiff between
insuring and carrying the whole risk.

Thus, w-(1+m0)pd is the certainty equivalent.

It is clear that m0 vanishes as the risk becomes
smaller, pd  0.

But the relative speed of convergence is not so
clear: how fast does m0 vanish compared to pd?
m0
lim
?
d 0 pd
Absolute risk aversion
1 p
(v' (w)  v' (w  d )) / d
m0
 lim
.
lim

d

0
d  0 pd
p
(1  p)v' (w)  pv' (w  d )
…and this part
converges to the
first derivative.
Note that this part
is just the second
derivative of v…
For symmetric risks (p=1/2) we thus get
m0
v' ' (w)
lim

: A(w)
d 0 pd
v' (w)
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 p.

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
R(w) := w A(w).

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.
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-g/(1-g), 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
One finds only a handful of specifications in
the literature.
name
formula
A
R
P
a
b
affine
g0+g1y
0
0
undef
undef
undef
quadratic
g0 y – g1y2
incr
incr
0
g0/(2g1)
–1
exponential
–e–gy/g
g
incr
g
1/g
0
power
y1–g /(1–g)
decr
g
decr
0
1/g
Bernoulli
ln y
decr
1
decr
0
1
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,
ln(y  a),

y / a
v(y ) :  ae
,
(b  1)1(a  by )(b 1) / b ,


if b  1,
if b  0,
otherwise.
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
E {v(y)}  aE{y}  bE {y 2}
 aE{y}  b(E {y}2  var(y)).
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
mi = E{yi} and si2 = E{(yi–mi)2} denote the mean
and the variance of yi.
Mean-variance:
linear distribution classes

Define then the random variable x = (yi–mi)/si.
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.

We want to evaluate the expected utility of yi ,

 v(z)dFi (z).
Mean-variance:
linear distribution classes
But yi = mi + si x, thus


 v(z)dFi (z)   v(mi  si z)dF (z)
: u(m i , si ).
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
( x  x0 )1
( x  x0 )2
f ( x )  f ( x0 )  f ' ( x0 )
 f ' ' ( x0 )

1!
2!
( x  x0 )3
f ' ' ' ( x0 )

3!

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,
E {x 2 }
E {v(w  x )}  v(w)  v' (w)E {x}  v' ' (w)
2
var( x )
 v(w)  v' ' (w)
.
2
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.
v(c)  v(w)  v' (w)(c  w).
Mean-variance: small risks

Since E{v(w+x)} = v(c), this simplifies to
var( x )
w  c  A(w)
.
2

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.
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
var(g)
1  k  R(w)
,
2
where k is the certainty equivalent growth rate,
v(kw) = E{v(gw)}.

The coefficient of relative risk aversion is
relevant for multiplicative risk, absolute risk
aversion for additive risk.