Download Risk Aversion Lecture

Lecture 4 on Individual Optimization
Risk Aversion
Last lecture we investigated the independence
axiom and the expected utility hypothesis. A
more direct approach is to design experiments
that reveal a person’s attitudes towards risk
and uncertainty, the topic of today’s lecture.
Certainty Equivalence
Instead of pursuing an axiomatic approach to
modeling human behavior under uncertainty,
we could more directly determine their attitude
towards risk.
An important concept is how much a person is
willing to pay to avoid a risk, or equivalently,
how much they are willing to pay to gamble.
(Parenthetically we note that gambling is one
of the biggest recreational activities in the U.S.)
Measuring certainty equivalence
How much a person is willing to pay for a lottery
depends both on the person and also on the
lottery.
Experimental methods have been used to
determine a person’s certainty equivalent to well
specified lotteries.
In this class we shall consider lotteries which
have up to 11 outcomes. They offer the chance to
pay $100 with probability p10, $90 with probability
p9, . . . , and $0 with probability p0.
An experiment
In this experiment you are asked to how much you
are willing to pay for a lottery called L when you
know its probabilities. Call that number b for bid.
We then draw a random number n from a
probability distribution which lies between (that
has support on) 0 and 100.
If n  b, then you pay n in exchange for the
lottery L, and receive the payoff from playing the
lottery.
If b < n, then you neither pay nor receive
anything.
The lotteries
Each of these lotteries can be defined by 10 numbers
(lying the 10 dimensional simplex), where the first
element is p10, the second is p9, and so on.
We only have to write down 10 elements even though
there are 11 outcomes because:
p0 = 1 - p10 - p9 - . . . - p1.
The lotteries you will bid on are listed below:
(0.5, 0, 0, . . . ,0)
(0.0909’, 0.0909’, . . . ., 0.0909’)
(0.1, 0.1, . . . ., 0.1, 0.1)
Verbal explanation
The first lottery pays $100 half the time and $0
half the time.
The second lottery pays $100 one eleventh of the
time, $90 one eleventh of the time, . . ., and $0
eleventh tenth of the time.
The third lottery pays $100 one tenth of the time,
$90 one tenth of the time, . . ., and $10 one
tenth of the time, so always pays out something.
Bidding below
your certainty equivalent
Suppose your certainty equivalent lottery for L is
the value v. How should you choose b?
If you bid b < v you gain v – n whenever n  b,
and zero otherwise.
This bidding strategy guarantees that you never
lose.
However if b < n < v, you miss the gain from
buying the lottery at a price less than your
valuation. If you had bid between n and v instead,
you would have gained v – n.
Bidding above
your certainty equivalent
If you bid b > v, you gain v – n whenever n 
v and you lose n – v if v < n  b.
Otherwise b < n and you neither gain nor lose.
The potential for loss from this strategy can be
mitigated by reducing your bid.
If you bid some number b1 between v and b,
then you still gain v – n whenever n  v and
you don’t lose anything if v  b1 < n  b.
Optimal bidding strategy
The two previous slides have probably
convinced you that the optimal choice is to bid
b = v.
In this case you gain v – n whenever n  v and
zero otherwise.
Therefore experimental subjects who know how
to make an optimal bid will truthfully reveal
their certainty equivalent v for the lottery L.
Expected utility
and certainty equivalence
Suppose the outcomes called x are ordered by your
own preferences from the worst to the best, and let
u(x) denote the utility from the random variable x.
(For example money outcomes are always ordered
that way, since more is preferred to less!)
Assume your preferences obey the expected utility
hypothesis. Then your expected utility from playing
a lottery F is EF[u(x)], where E denotes the
expectations operator.
It follows that your certainty equivalent for F is the
value vF which solves u(vF) = EF[u(x)]
Expected utility
and attitudes towards risk
In this case we can characterize you attitude towards
risk quite simply.
If you are risk neutral, then vF = EF[x] for all F, and
hence we can write u(x) = x.
If you are a risk lover, meaning vF > EF[x] for all F,
then u(x) is convex.
If you are a risk avoider, meaning vF < EF[x] for all F,
then u(x) is concave.
First-order stochastic dominance
Consider two different probability distributions
F(x) and G(x). That is F(x)  G(x) for some real
number x.
We say that F first-order stochastically
dominates G if and only F(x)  G(x) for all x.
This is formally equivalent to saying that if x is
a random variable drawn from G(x), if y is a
random variable that only takes on positive
values, and F(z) is the probability distribution
function for z = x + y, then F first-order
stochastically dominates G.
An example
Notice that the the third lottery first-order
stochastically dominates the second.
First Order Stochastic Dominance
illustrated
First-order stochastic dominance
and expected utility
Now consider those people who obey the
expected utility hypothesis, and order the
outcomes x, a random variable from the
lowest (least preferred) to the highest (most
preferred).
They obtain expected utility EF[u(x)] from
playing lottery F, but the function u(x) differs
from person to person.
Can we rank EF[u(x)] and EG[u(x)] when F
first order stochastically dominates G?
Theorem
If F first-order stochastically dominates G, then
every expected utility maximizer with u(x)
increasing in x, prefers F to G.

 u  z  f  z dz

 
x, y
 x, y dxdy

    u x  y  f x
   

x y dx  f
y


  u x  y  f
  


    u x  f x
   


x y dx  f x  y dy
y

 

  u  x  f  x, y dxdy
xy
  


 u  x  f  x dx
x

 y dy
Second-order stochastic dominance
Consider two different probability distributions F(x)
and G(y) with the same mean. That is EF[x]= EG[x].
We say that F second-order stochastically dominates
G if and only for all t:
t
t
0
0
 F xdx   Gxdx
This is saying that if x is a random variable drawn
from G(x), if y is a random variable with mean 0, and
that probability distribution function for z = x + y is
F(z) , then F second-order stochastically dominates G.
Example
X
Second-order stochastic dominance
and expected utility
Assume a person follows the expected utility
hypothesis, and thus obtains expected utility
EF[u(x)] from playing a lottery F.
We now assume u(x) is concave increasing in x,
implying that the person is risk averse, but we
impose no other restrictions on u(x).
Suppose EF[x]= EG[x]. Then a risk averse person
prefers F to G, that is vG  vF if and only if F
second-order stochastically dominates G.
Another theorem
Suppose F and G have the same mean. Then a risk
averse person prefers F to G if and only if F
second-order stochastically dominates G.

 uz g z dz
z

 

  u x  y  f x, y dxdy
x, y
  


    u  x  y  f y x  y x dy  f x  x dx
   




  u    x  y  f y x  y x dy  f x  x dx
 




  u   xf y x  y x dy  f x  x dx
 



  u  x  f x  x dx

Investing in a risky asset
We now modify the lottery experiment. Instead of bidding
on a lottery, you are asked to choose how many units of
each independently distributed lottery you would like to
buy
A unit of each lottery costs $50, and you can spend up to
$50,000. Your total payoff are your lottery winnings plus
unspent cash.
The payoff from the lotteries you are listed below:
- Firm A : [$10 with probability 0.5 and $100 with
probability 0.5]
- Firm B : [$10 with probability 0.1, $20 with probability
0.10, . . ., $100 with probability 0.1]
- Firm C : [$20 with probability 0.2, $30 with 0.1, $40 with
probability 0.1,. . ., $100 with probability 0.1]
Optimal Insurance
A risk averse person fully insures himself
against a calamity if actuarially fair
insurance is available.
This follows from the fact that for risk
averters the certainty equivalent of a
gamble is greater than its expected value.
By the same argument we have used on
portfolio choices, a risk averter will not fully
insure himself if the rates are not actuarially
fair. He will retain ownership over a portion
of the lottery.