Risky Attitudes and Risky Comparisons
Juan Carlos Carbajal
(adapted from Essential Microeconomics)
Lecture 08 — ECON 6001
Semester 1, 2015
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Lecture 08
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Road Map
Risky Attitudes
Risk neutrality and risk aversion
Concave functions and the Jensen inequality
Application to insurance decision
Measuring risk aversion
Application to portfolio decision
Risky Comparisons
Comparing lotteries
First-order stochastic dominance
Second-order stochastic dominance
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Risky Attitudes
Attitudes towards Risk
Let X ⊂ R be a set of final monetary outcomes. For simplicity, focus first in the
case of lotteries over finite outcomes, so X = {x1 , . . . , xN }.
Suppose that Anne’s Bernoulli utility function is given by u(x) = x. For any simple
lottery L = (π1 , . . . , πN ), Anne’s vNM expected utility is
X
U (L) =
π n xn .
(1)
n
Note that in the RHS of Eq. (1) we have the expected value associated with L.
Anne is indifferent between simple lotteries with the same expected value. That is,
she is risk neutral.
If x1 = $0, x2 = $1M and x3 = $2M , Anne is indifferent between the lottery that yields
x2 with certainty, and the lottery that assigns equal probability to x1 and x3 .
Holding monetary outcomes fixed, we examine more systematically choices over
lotteries (probability distributions).
We begin by considering changes to the probabilities of outcomes x1 , x2 and x3 .
To fix ideas, let x1 < x2 < x3 .
Since only probabilities of these outcomes change, we ignore all others.
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Risky Attitudes
Risk Neutrality
Suppose that the initial and final lotteries
are L = (π1 , π2 , π3 ) and L0 = L + ∆.
The vector ofP
probability changes (∆1 , ∆2 , ∆3 )
3
must satisfy
1 ∆n = 0.
The change in Anne’s expected utility is
U (L0 ) − U (L) =
X3
1
∆ n xn .
Since ∆2 = −(∆1 + ∆3 ), we write
U (L0 ) − U (L)
= ∆1 (x1 − x2 ) − ∆3 (x2 − x3 ).
Now suppose that L ∼ L0 (as in the figure). Using the fact that U (L0 ) − U (L) = 0,
we find the slope m0 = ∆3 /∆1 of the indifference line through L and L0 using the
previous equation:
∆3
x2 − x1
m0 =
=
.
∆1
x3 − x2
Since the indifference lines are all parallel, the slope of each is m0 .
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Risky Attitudes
Risk Neutrality
Recall that the origin represents the
degenerate lottery that yields outcome x2
with certainty: L[2] = (0, 1, 0).
Suppose that Anne begins with the
degenerate lottery L[2], and consider her
indifference curve through the origin.
The shaded region G0 is the set of
acceptable lotteries for Anne: any such
lottery will (weakly) improve her situation.
Note that any lottery that yields expected value of x2 will be an acceptable lottery
for Anne, as it will leave her indifferent.
For example, if x1 = $0, x2 = $1M and x3 = $2M , then L = ( 12 , 0,
α
In fact, any lottery Lα = ( α
2 , 1 − α, 2 ) is acceptable for Anne.
1
2)
is acceptable.
By contrast, we define a risk-averse individual to be one who will accept fewer
gambles than Anne (who is risk neutral).
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Risky Attitudes
Risk Aversion
Let u(xn ) = un be the Bernoulli utility
function of Bill.
Arguing as earlier, if the lottery L changes
to L0 = L + ∆, then
U (L0 ) − U (L) =
X3
1
∆n un
= ∆1 (u1 − u2 ) − ∆3 (u2 − u3 ).
Along an indifference line the slope is
m =
∆3
u2 − u1
=
.
∆1
u3 − u2
As before, starting from initial lottery L[2], G represents Bill’s set of acceptable
lotteries — the set of points on or above his indifference line through the origin.
If Bill is risk averse, then G ⊂ G0 . Note that this happen if and only if
m =
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u2 − u1
x2 − x1
≥
= m0 .
u3 − u2
x3 − x2
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Risky Attitudes
Characterization of Concave Functions
We can state the definition of a concave function as follows: a function f : R → R is
concave if all x1 < x3 in R, for all x2 ∈ (x1 , x3 ), the point A2 lies above the chord
joining A1 and A3 .
Equivalently, the slope of the chord joining A1 and A2 exceeds the slope of the
chord joining A2 and A3 .
Lemma f : R → R is concave if and
only if for all x1 < x3 in R, for all
x2 ∈ (x1 , x3 ),
f (x2 ) − f (x1 )
f (x3 ) − f (x2 )
≥
x2 − x1
x3 − x2
This is precisely the condition we found for the slopes of Bill’s indifference lines.
We formalize this intuition for general gambles.
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Risky Attitudes
Risk Aversion and Risk Neutrality
Let X ⊆ R be a convex set — sometimes we work with X = R+ , sometimes with a
compact interval X = [x, x]. X is the set of final (monetary) outcomes.
Let u : X → R be a Bernoulli utility function over monetary outcomes. We assume
u(·) is continuous and increasing.
Recall that a lottery now is a probability distribution FR over X. The lottery F
generates an expected payment or expected value of x dF (x).
Definition A decision maker exhibits risk
R aversion if for any lottery F , the
degenerate lottery that yields the amount
as the lottery F itself.
x dF (x) with certainty is at least as good
If the decision maker is always indifferent between these lotteries, then we say she
exhibits risk neutrality.
If indifference holds only when F is a degenerate lottery, then we say that the
decision maker exhibits strict risk aversion.
In our example, Bill prefers the degenerate lottery that yields outcome x2 = $1M for sure
to the lottery L = ( 12 , 0, 21 ) whose expected value is also $1M .
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Risky Attitudes
Risk Aversion and Risk Neutrality
From this definition, Bill is risk averse if and only if for every lottery F ,
Z
Z
x dF (x) ≥
u(x) dF (x).
u
Eq. (2) states that the degenerate lottery that yields amount
preferred to the lottery F .
R
(2)
x dF (x) for sure is
On the other hand, the certainty equivalent of F , denoted by c(F, u), gives precisely
the amount of money that makes Bill indifferent:
Z
u c(F, u) =
u(x) dF (x).
Proposition The following properties are equivalent:
(1) The decision maker is risk averse.
(2) The Bernoulli utility Rfunction u(·) is concave.
(3) For all F , c(F, u) ≤ x dF (x).
Proof (1) ⇐⇒ (2) follows from the definition of risk aversion, and the fact that
Eq. (2), known as the Jensen Inequality, characterizes concave functions.
(2) ⇐⇒ (3) follows from the Jensen inequality and the fact that u(·) is increasing.
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Risky Attitudes
Concave and Convex Functions
Let Y ⊆ R be a convex set, and recall that a convex combination of elements
y1 , . . . , yN in Y is a vector of the form
X
X
α n yn ,
where αn ≥ 0 for all n and
αn = 1.
n
n
Proposition (Jensen Inequality) The function ψ : Y → R P
is concave if and
only if for all y1 , . . . , yN in Y , all α1 , . . . , αn with αn ≥ 0 and
X
X
ψ
α n yn ≥
α n ψ yn .
n
n
αn = 1, one has
n
(3)
The function ψ is convex if and only if the above inequality is reversed.
Note that this is the generalization of Eq. (2).
Proof (sketch)
Clearly, if Eq. (3) is satisfied, then ψ is concave.
The proof in the other direction follows an inductive argument. We know that
Eq. (3) holds for any convex combination of two elements y1 and y2 in Y . Assume it
holds for N − 1 elements, and show that it holds for N elements.
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Risky Attitudes
Application: insuring a risk-neutral decision maker
Anne has initial wealth w > 0, but she faces a loss of D with probability π ∈ (0, 1).
Anne can buy an insurance policy on the following terms: a unit of insurance pays
$1 dollar if the loss occurs and costs p > 0. Let α ≥ 0 be the units of insurance that
Anne buys — determine the optimal α∗ .
Outcome 1 (no loss): x1 = w − pα, with probability (1 − π).
Outcome 2 (loss): x2 = w − pα − D + α, with probability π.
Anne is risk neutral, so her expected utility from buying α units of insurance is
(1 − π)(w − pα) + π(w − pα − D + α) = w − πD + α(π − p).
If the insurance’s price is actuarially fair, we have p = π and thus any α ≥ 0 will be
optimal for Anne.
If on the other hand p > π, then she does not buy insurance at all (α∗ = 0).
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Risky Attitudes
Application: insuring a risk-averse decision maker
Bill faces the same situation as Anne, except that he is strictly risk averse. Bill’s
expected utility maximization problem can be written as
n
o
max (1 − π) u(w − pα) + π u(w − pα − D + α) .
α≥0
The FOC of this problem is
− p(1−π)u0 (w −pα∗ ) + (1−p)πu0 (w −D +(1−p)α∗ ) ≤ 0,
w/ equality if α∗ > 0.
If the insurance’s price is actuarially fair, we have p = π and thus the FOC become
u0 (w − D + (1 − π)α∗ ) ≤ u0 (w − πα∗ ),
w/ equality if α∗ > 0.
Note that here α∗ = 0 yields to u0 (w − D) ≤ u0 (w), but since u(·) is strictly
concave, this is impossible.
Thus, under an actuarially fair insurance price, α∗ > 0 and from the FOC we obtain
w − D + (1 − π)α∗ = w − πα∗
=⇒
α∗ = D.
One can show that if p > π, then Bill will not be fully insured: 0 ≤ α∗ < D.
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Risky Attitudes
Greater Aversion to Risk
Bill and Denise have Bernoulli utility functions u(·) and v(·) = ψ(u(·)), respectively.
If ψ(·) is increasing and strictly concave, then Denise’s Bernoulli utility function is a
strictly concave transformation of the Bill’s:
v(x) = ψ(u(x)),
for all x.
We argue that Denise must have a smaller set of acceptable gambles — Denise is
more risk averse than Bill.
Appealing to our previous Lemma, ψ(·)
is strictly concave if and only if, for all
u1 < u2 < u3 ,
ψ(u2 ) − ψ(u1 )
u2 − u1
>
.
ψ(u3 ) − ψ(u2 )
u3 − u2
That is,
mD =
v2 − v1
u2 − u1
>
= mB .
v3 − v2
u3 − u2
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Risky Attitudes
Measuring Risk Aversion
Because Denise’s Bernoulli utility function is a strictly concave transformation of
u(·), she is everywhere more risk averse than Bill. Taking derivatives,
v 0 (x) = ψ 0 (u(x)) u0 (x).
Taking logs,
ln v 0 (x) = ln ψ 0 (u(x)) + ln u0 (x).
Differentiating again, multiplying by −1, and rearranging obtains
−
v 00 (x)
ψ 00 (u(x)) 0
u00 (x)
= −
u (x) − 0
v 0 (x)
ψ 0 (u)
u (x)
Since ψ is increasing and strictly concave, the first term on the right-hand side is
positive. Thus, Denise is unambiguously more risk averse if and only if
−
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v 00 (x)
u00 (x)
> − 0
v 0 (x)
u (x)
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for all x.
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Risky Attitudes
Absolute Risk Aversion
Definition Given a twice-differentiable, increasing Bernoulli utility function u(·) for
money, the Arrow-Pratt coefficient of absolute risk aversion at x is defined by
rA (x, u) = −
u00 (x)
.
u0 (x)
We found above that more risk aversion everywhere is associated with either (i)
concave transformations, or (ii) higher absolute risk aversion coefficient.
Proposition Given two Bernoulli utility functions u(·) and v(·), say that v(·)
exhibits more risk aversion everywhere if and only if any of the next conditions hold:
(1) rA (x, v) ≥ rA (x, u) for all x.
(2) There is an increasing concave function ψ(·) such that v(x) = ψ(u(x)) for all x.
(3) c(F, v) ≤ c(F, u) for all lotteries F .
Proof The equivalence between (1) and (2) has been shown already. To show that
(2) and (3) are equivalent, let F be given. Then,
Z
Z
v c(F, v) =
v(x) dF (x) =
ψ(u(x)) dF (x)
Z
≤ ψ
u(x) dF (x) = ψ u c(F, u)
= v c(F, u) .
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Risky Attitudes
CARA and DARA
Example Consider the Bernoulli utility function u(x) = − a e−cx + b, where a, c > 0
and b ∈ R are given. This class of functions exhibits constant absolute risk aversion:
rA (x, u) = c,
for all x.
Note that the attitude towards risk under a CARA utility function is independent of
wealth — the Arrow-Pratt coefficient of absolute risk aversion is constant for all x.
Yet it is a commonly assumed that wealthier individuals are more willing to take
risks, not just because they may have a different Bernoulli utility function, but
because they can afford it.
A Bernoulli utility function is said to exhibit decreasing absolute risk aversion if
rA (x, u) is a decreasing function of x.
Example
Consider the Bernoulli utility function u(x) = a ln x, for a > 0. Then
rA (x, u) =
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1
,
x
for all x > 0.
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Risky Attitudes
Application: portfolio choice
The following is a simple model of portfolio investment choice.
Bill must decide how much to invest in a riskless asset that has a gross yield of
1 + s, and how much in a risky asset with a gross yield 1 + r.
To model the risky asset, suppose that there are N possibles states of nature, each
occurring with probability πn . The gross yield of the risky asset is 1 + rn in state n.
Bill invests α dollars in the risky asset and w − α in the riskless asset.
His final consumption (in monetary terms) in state n is then
xn = (w − α)(1 + s) + α(1 + rn ) = w(1 + s) + α θn ,
where θn = rn − s is the excess yield of the risky asset in state n = 1, . . . , N .
Bill’s expected utility is then written as
XN
n
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πn u(xn ) =
XN
n
πn u w(1 + s) + α θn .
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Risky Attitudes
Application: portfolio choice
Bill’s marginal gain from increasing α is
XN
n
πn u0 w(1 + s) + α θn θn .
(4)
Since he is strictly risk-averse, u(·) is strictly concave, so Bill’s objective function is
a strictly concave function of α. It follows that there is a single turning point.
Moreover, at α = 0 one has xn = w(1 + s) for all n, thus from Eq. (4),
XN
u0 w(1 + s)
π n θn > 0
n
⇐⇒
XN
n
πn θn > 0.
Thus unless Bill is infinitelyPrisk averse, he will purchase some of the risky asset as
long as its expected yield n rn is strictly higher than the yield of the safe asset.
To understand this, note from Eq. (4) that Bill weights marginal claims in state n via
his marginal Bernoulli utility in that state and then takes expectations. With no risk,
each of the marginal utilities is the same. Thus, the decision of whether to invest at
all is the same as the decision of a risk-neutral individual.
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Risky Attitudes
More risk-aversion leads to less risk taking
Of course, for any positive investment in the risky asset, the marginal utilities
change (and these changes depend on the degree of risk aversion).
Denise faces an identical portfolio choice problem except that she is more risk averse
than Bill: v(x) = ψ(u(x)), where ψ(·) is an increasing strictly concave function.
Let α∗ > 0 be Bill’s optimal investment in the risky asset, and for each state
n = 1, . . . , N , define x∗n = w(1 + s) + θn α∗ . The FOC for Bill is expressed as
XN
n
πn u0 (x∗n )θn = 0.
(5)
We claim that if Denise chooses the same portfolio, her expected marginal utility is
negative at α∗ : she over-invested in the risky asset.
Perhaps by renaming them, order the states so that θ1 > θ2 > . . . > θN .
Let m be the largest state for which the excess yield is positive. This translates to
x∗n > x∗m for all n = 1, . . . , m − 1, and x∗n < x∗m for all n = m + 1, . . . , N .
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Risky Attitudes
More risk-aversion leads to less risk taking
Denise’ marginal expected utility from α evaluated at Bill’s optimal investment α∗ is:
N
X
πn v 0 (x∗n )θn =
n=1
N
X
πn ψ 0 u(x∗n ) u0 (x∗n )θn
n=1
=
m
X
N
X
πn ψ 0 u(x∗n ) u0 (x∗n )θn −
πn ψ 0 u(x∗n ) u0 (x∗n )(−θn ).
1
m+1
Each term in both summations is non-negative. Moreover, the strict concavity of
ψ(·) implies that
ψ 0 u(x∗n ) < ψ 0 u(x∗m ) ,
for all n = 1, . . . , m − 1, and
0
∗
0
∗
ψ u(xn ) > ψ u(xm ) ,
for all n = m + 1, . . . , N .
Hence, we obtain
N
X
πn v 0 (x∗n )θn <
n=1
m
X
N
X
πn ψ 0 u(x∗m ) u0 (x∗n )θn −
πn ψ 0 u(x∗m ) u0 (x∗n )(−θn )
1
m+1
N
X
= ψ 0 u(x∗m )
πn u0 (x∗n )θn = 0.
n=1
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Risky Comparisons
Stochastic Dominance
The prior comparisons were in terms of attitudes towards risk among different
individuals.
Now we want to consider comparisons of lotteries or payoff distributions, restricting
attention to distributions F over X = R+ such that F (0) = 0 and F (x) = 1 for
some x — although the support [0, x] may differ for each distribution.
There are two natural ways to compare distributions over monetary outcomes.
(1) According to the level of returns: we want to see when we can say that F yields
unambiguously higher returns than G.
(2) According to the dispersion of the returns: we want to see when we can say that
F is unambiguously less risky than G.
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Risky Comparisons
First Order Stochastic Dominance
It seems natural to say that F yields unambiguously higher returns than G if every
expected utility maximizer would choose F over G.
Definition The distribution F first-order stochastically dominates G if for every
non-decreasing function u : R+ → R, one has
Z
Z
u(x) dF (x) ≥
u(x) dG(x).
From the definition of FOS dominance, the
expected return from lottery F is higher
than the expected return from G.
A related notion of yielding unambiguously
higher returns is that for any targeted
monetary level x, the probability of getting
at least x is higher under F than under G.
Proposition The distribution F first-order stochastically dominates G if and only
if F (x) ≤ G(x), for all x ≥ 0.
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Risky Comparisons
First Order Stochastic Dominance
Proof Suppose first that F first-order stochastically dominates G. Fix any x̂ and
consider the non-decreasing function u(·) defined by u(x) = 0 for all x < x̂ and
u(x) = 1 for all x > x̂. For such function, one has that
Z ∞
Z
u(x) dF (x) =
dF (x) = 1 − F (x̂),
x̂
and similarly for G. Since F first-order stochastically dominates G, it follows that
1 − F (x̂) ≥ 1 − G(x̂), or F (x̂) ≤ G(x̂), as desired.
For
we show that F (x) ≤ G(x) for all x implies that
R the other direction,
R
u(x) dF (x) ≥ u(x) dG(x) for all non-decreasing differentiable functions u(·).
Z
Z
Z
h
i
u(x) dF (x) − u(x) dG(x) =
u(x) d F (x) − G(x)
Z h
h
i
i∞
= u(x) F (x) − G(x)
−
F (x) − G(x) u0 (x) dx
0
Z h
i
0
=
G(x) − F (x) u (x) dx ≥ 0.
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Risky Comparisons
Second Order Stochastic Dominance
Assume that F and G are two lotteries with the same expected return (same mean).
It seems natural to say that F yields unambiguously higher risks than G if every
risk-averse expected utility maximizer would choose F over G.
Definition The distribution F second-order stochastically dominates G if for
every non-decreasing concave function u : R+ → R, one has
Z
Z
u(x) dF (x) ≥
u(x) dG(x).
Example Consider the following compound lottery. In the first stage x is chosen
according to F . In the second stage, for each realized x we randomize between x + z
and x − z with equal probability.
The resulting reduced lottery, which we denote by G, is called a mean-preserving
spread of F . Notice that by construction F and G have the same mean, but G is
seems to be riskier than F .
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Risky Comparisons
Second Order Stochastic Dominance
Indeed, a risk-averse agent has
Z h
Z
i
1
u(x) dG(x) =
u(x + z) + 12 u(x − z) dF (x)
2
Z
Z
u(x) dF (x).
≤
u 12 (x + z) + 21 (x − z) dF (x) =
It turns out that we can generalize this construction. That is, given a lottery F , we
can randomize each possible outcome x further so that the payoff is y = x + zx ,
where zx has a distribution Hx with zero mean.
The reduced lottery G associated with this compound lottery is still a mean
preserving spread:
Z
Z Z
Z
y dG(y) =
(x + zx ) dHx (z) dF (x) =
x dF (x).
Proposition Let F and G be two distributions with the same mean. Then F
second-order stochastically dominates G if and only if G is a mean-preserving
spread of F .
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