ECON 337901
FINANCIAL ECONOMICS
Peter Ireland
Boston College
March 17, 2015
These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike
4.0 International (CC BY-NC-SA 4.0) License. http://creativecommons.org/licenses/by-nc-sa/4.0/.
4 Measuring Risk and Risk Aversion
A
B
C
D
E
F
Measuring Risk Aversion
Interpreting the Measures of Risk Aversion
Risk Premium and Certainty Equivalence
Assessing the Level of Risk Aversion
The Concept of Stochastic Dominance
Mean Preserving Spreads
Next: 5 Risk Aversion and Investment Decisions
The Concept of Stochastic Dominance
The concept of stochastic dominance allows us to make
“preference-free” comparisons across risky cash flows, in the
sense that if one asset exhibits stochastic dominance over
another, it will be preferred by any investor no matter what his
or her level of risk aversion.
The Concept of Stochastic Dominance
Consider two assets, with random payoffs Z1 and Z2 :
Payoffs
Probabilities for Z1
Probabilities for Z2
10
100 1000
0.40 0.60 0.00
0.40 0.40 0.20
There may be no state-by-state dominance, if the payoffs
Z1 = 10 and Z2 = 100 can occur together and the payoffs
Z1 = 100 and Z2 = 10 can occur together.
The Concept of Stochastic Dominance
Payoffs
Probabilities for Z1
Probabilities for Z2
10
100 1000
0.40 0.60 0.00
0.40 0.40 0.20
Because E (Z1 ) = 64, σ(Z1 ) = 44, E (Z2 ) = 244, and
σ(Z2 ) = 380, there is no mean-variance dominance either.
The Concept of Stochastic Dominance
Payoffs
Probabilities for Z1
Probabilities for Z2
10
100 1000
0.40 0.60 0.00
0.40 0.40 0.20
But, intuitively, asset 2 “looks” better, because its distribution
takes some of the probability of a payoff of 100 and “moves”
that probability to the even higher payoff of 1000.
Probability theory formalizes this notion, by saying that asset
2 exhibits first-order stochastic dominance over asset 1.
The Concept of Stochastic Dominance
Theorem Consider two assets with random payoffs Z1 and Z2 .
Asset 2 displays first-order stochastic dominance over asset 1
if and only if
E [u(Z2 )] ≥ E [u(Z1 )]
for any nondecreasing Bernoulli utility function u.
Note that this theorem only requires that investors have vN-M
utility and that “more is preferred to less.” They do not even
have to be risk-averse.
The Concept of Stochastic Dominance
First-order stochastic dominance is a weaker condition than
state-by-state dominance, in that state-by-state dominance
implies first-order stochastic dominance but first-order
stochastic dominance does not necessarily imply state-by-state
dominance.
But first-order stochastic dominance remains quite a strong
condition. Since an asset that displays first-order stochastic
dominance over all others will be preferred by any investor
with vN-M utility who prefers higher payoffs to lower payoffs,
the price of such an asset is likely to be bid up until the
dominance goes away.
Mean Preserving Spreads
Graphically, a mean preserving spread takes probability from
the center of a distribution and shifts it to the tails.
Mean Preserving Spreads
Mathematically, one way of producing a mean preserving
spread is to take one random variable X1 and define a second,
X2 , by adding “noise” in the form of a third, zero-mean
random variable Z :
X2 = X1 + Z ,
where E (Z ) = 0.
Mean Preserving Spreads
As an example, suppose that
5 with probability 1/2
X1 =
2 with probability 1/2
+1 with probability 1/2
Z=
−1 with probability 1/2
then

6



4
X2 = X1 + Z =
3



1
with
with
with
with
probability
probability
probability
probability
1/4
1/4
1/4
1/4
E (X1 ) = E (X2 ) = 3.5, but if these are random payoffs, asset 2
seems riskier.
Mean Preserving Spreads
The following theorem relates the concept of a mean
preserving spread to a weaker concept of second-order
stochastic dominance.
Theorem Let X1 and X2 , with E (X1 ) = E (X2 ), be random
payoffs on two assets. Then the following two statements are
equivalent: (i) X2 = X1 + Z for some random variable Z with
E (Z ) = 0 and (ii) asset 1 displays second-order stochastic
dominance over asset 2.
Mean Preserving Spreads
Theorem Consider two assets with random payoffs Z1 and Z2 .
Asset 2 displays second-order stochastic dominance over asset
1 if and only if
E [u(Z2 )] ≥ E [u(Z1 )]
for any nondecreasing and concave Bernoulli utility function u.
This theorem imply that any risk-averse investor with vN-M
preferences will avoid “pure gambles,” in the form of assets
with payoffs that simply add more randomness to the payoff of
another asset.
Mean Preserving Spreads
Second-order stochastic dominance is a weaker condition than
first-order stochastic dominance, in that first-order stochastic
dominance implies second-order stochastic dominance but
second-order stochastic dominance does not necessarily imply
first-order stochastic dominance.
But second-order stochastic dominance remains a strong
condition. Since an asset that displays second-order stochastic
dominance over all others will be preferred by any risk-averse
investor with vN-M utility, the price of such an asset is likely
to be bid up until the dominance goes away.
5 Risk Aversion and Investment Decisions
A Risk Aversion and Portfolio Allocation
B Portfolios, Risk Aversion, and Wealth
Risk Aversion and Portfolio Allocation
Let’s now put our framework of decision-making under
uncertainty to use.
Consider a risk-averse investor with vN-M expected utility who
divides his or her initial wealth Y0 into an amount a allocated
to a risky asset – say, the stock market – and an amount
Y0 − a allocated to a safe asset – say, a bank account or a
government bond.
Risk Aversion and Portfolio Allocation
Y0 = initial wealth
a = amount allocated to stocks
˜r = random return on stocks
rf = risk-free return
Y˜1 = terminal wealth
Y˜1 = (1 + rf )(Y0 − a) + a(1 + ˜r )
= Y0 (1 + rf ) + a(˜r − rf )
Risk Aversion and Portfolio Allocation
The investor chooses a to maximize expected utility:
max E [u(Y˜1 )] = max E {u[Y0 (1 + rf ) + a(˜r − rf )]}
a
a
If the investor is risk-averse, u is concave and the first-order
condition for this unconstrained optimization problem, found
by differentiating the objective function by the choice variable
and equating to zero, is both a necessary and sufficient
condition for the value a∗ of a that solves the problem.
Risk Aversion and Portfolio Allocation
The investor’s problem is
max E {u[Y0 (1 + rf ) + a(˜r − rf )]}
a
The first-order condition is
E {u 0 [Y0 (1 + rf ) + a∗ (˜r − rf )](˜r − rf )} = 0.
Note: we are allowing the investor to sell stocks short (a∗ < 0)
or to buy stocks on margin (a∗ > Y0 ) if he or she desires.
Risk Aversion and Portfolio Allocation
Theorem If the Bernoulli utility function u is increasing and
concave, then
a∗ > 0 if and only if E (˜r ) > rf
a∗ = 0 if and only if E (˜r ) = rf
a∗ < 0 if and only if E (˜r ) < rf
Thus, a risk-averse investor will always allocate at least some
funds to the stock market if the expected return on stocks
exceeds the risk-free rate.
Risk Aversion and Portfolio Allocation
As an example, suppose u(Y ) = ln(Y ), as suggested by Daniel
Bernoulli. Recall that for this utility function, u 0 (Y ) = 1/Y .
Then assume that stock returns can either be good or bad:
rG with probability π
˜r =
rB with probability 1 − π
where rG > rf > rB defines the “good” and “bad” states and
πrG + (1 − π)rB > rf ,
so that E (˜r ) > rf and the investor will choose a∗ > 0.
Risk Aversion and Portfolio Allocation
The first-order condition
E {u 0 [Y0 (1 + rf ) + a∗ (˜r − rf )](˜r − rf )} = 0
specializes to
π(rG − rf )
(1 − π)(rB − rf )
+
= 0.
∗
Y0 (1 + rf ) + a (rG − rf ) Y0 (1 + rf ) + a∗ (rB − rf )
Risk Aversion and Portfolio Allocation
(1 − π)(rB − rf )
π(rG − rf )
+
=0
∗
Y0 (1 + rf ) + a (rG − rf ) Y0 (1 + rf ) + a∗ (rB − rf )
π(rG − rf )[Y0 (1 + rf ) + a∗ (rB − rf )]
= −(1 − π)(rB − rf )[Y0 (1 + rf ) + a∗ (rG − rf )]
a∗ (rG − rf )(rB − rf )
= −Y0 (1 + rf )[π(rG − rf ) + (1 − π)(rB − rf )]
Risk Aversion and Portfolio Allocation
a∗ (rG − rf )(rB − rf )
= −Y0 (1 + rf )[π(rG − rf ) + (1 − π)(rB − rf )]
implies
a∗
(1 + rf )[π(rG − rf ) + (1 − π)(rB − rf )]
=−
,
Y0
(rG − rf )(rB − rf )
which is positive, since rG > rf > rB and
E (˜r ) − rf = π(rG − rf ) + (1 − π)(rB − rf ) > 0.
Risk Aversion and Portfolio Allocation
a∗
(1 + rf )[π(rG − rf ) + (1 − π)(rB − rf )]
=−
,
Y0
(rG − rf )(rB − rf )
In this case, a∗ :
Rises proportionally with Y0 .
Increases as E (˜r ) − rf rises.
Falls as rG and rB move father away from rf ,
holding E (˜r ) constant; that is, in response to a
mean preserving spread.