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Choosing among risky alternatives
using stochastic dominance
When is one risky outcome preferable to another?
Stochastic dominance tries to resolve choice under
weakest possible assumptions.
Generally, SD assumes an expected utility maximizer then
adds further assumptions relative to preference for
wealth and risk aversion.
Basic Assumptions
#1 -
individuals are expected utility maximizers.
#2 -
two alternatives are to be compared and these are
mutually exclusive, i.e., one or the other must be
chosen not a convex combination of both.
#3 -
the stochastic dominance analysis is developed
based on population probability distributions.
1
Choosing among risky alternatives
using stochastic dominance
Background
Expected utility of wealth maximization.
Assume x is the level of wealth while f(x) and g(x)
give the probability of each level of wealth for
alternatives f and g. The difference in the expected
utility between the prospects as follows.


-
-
 u(x) f(x) dx -  u(x) g(x) dx
and this equation can be rewritten as:

 u(x) ( f(x) - g(x) ) dx
-
If f is preferred to g then the sign of the above
equation would be positive. Conversely, if g is
preferred to f, the sign of the above equation be
negative.
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Choosing among risky alternatives
using stochastic dominance
Integration by Parts
Classical calculus technique

 a db  ab
-

-
-  u(x) g(x) dx
-
where a and b are functions of x.
First Degree Stochastic Dominance
Apply the integration by parts formula to
a = u(x)
b = (F(x) - G(x))
X
where
F ( X )   f(x) dx
-
X
G( X )   g(x) dx
-
in turn the differential terms are:
da  u (x)dx
db  (f(x) - g(x) ) dx
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Choosing among risky alternatives
using stochastic dominance
First Degree Stochastic Dominance
Under this substitution the integration of

 u(x) ( f(x) - g(x) ) dx
-
equals

[u(x) ( F(x) - G(x))] -   u' (x) ( F(x) - G(x) ) dx
-
In the left part when F(x) and G(x) are evaluated at
x equals -∞ both equal zero
x equals +∞ both equal one
and plus infinity where they equal one
so the left part equals zero.
Now let us look at the right part which is:

  u' (x) ( F(x) - G(x) ) dx
-
If the overall sign is positive then f dominates g.
We restrict the sign by adding assumptions.
1.) nonsatiation more is preferred to less or u'(x) > 0
for all x.
2) F(x)  G(x) for all x
Under these conditions then implies f dominates g
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Choosing among risky alternatives
using stochastic dominance
First Degree Stochastic Dominance
Given
1) two probability distributions f and g,
2) the assumption that the decision maker
has positive marginal utility of wealth
u’(x)>0 for all x
Then when for all x the cumulative probability under f is
less than or equal to cumulative probability under g with
strict inequality for some x. we say distribution f
dominates distribution g by first degree stochastic
dominance (FSD)
Not revolutionary. Mean of f is greater than that for g and
for every level of probability you make at least as much
money under f as you do under g.
Not very discriminating practically.What one has to
observe is that one crop variety always has to consistently
out perform the other. This may not be the case.
5
Choosing among risky alternatives
using stochastic dominance
First Degree Stochastic Dominance -- Geometry
Figure 1. First Degree
1.2
Cumulative Probability
1
0.8
Distribution f
0.6
Distribution g
0.4
0.2
0
6
8
10
12
14
16
18
20
Wealth
Note here that for all x the area under f(x) is less than the
are under g(x) so f dominates g. Why? Requires for all x
the cdf for f is always to the right of cdf for g or that for
every x the cumulative probability of that level of wealth
or higher is greater under f than g.
Notice in Figure 1 that for a value of x equal to 7 that
there is no meaningful area under the f (x) distribution but
there is under the g(x) distribution so one can make that
much or less under g but will always make more under f.
Note, for any point x where there is an area under both
distributions the area underneath the g distribution is
greater than it is for the f distribution.
6
Choosing among risky alternatives
using stochastic dominance
Second Degree Stochastic Dominance
The FSD stochastic dominance development while
theoretically elegant is not terribly useful so lets go
further
Above FSD derivation says Expected utility of f minus g
can be expressed as

  u' (x) ( F(x) - G(x) ) dx
-
Applying integration by parts
a  u (x)
db  (F(x) - G(x) ) dx
so that:
da  u  (x) dx
b  (F2 (x) - G 2 (x) )
where F2 and G2 are the second integral of the cdfs
X
F2 ( X )   F(x) dx
-
X
G2 ( X )   G(x) dx
-
Under these circumstances if we plug in our integration by
parts formula we get the equation.
- [u' (x) ( F2 (x) - G 2 (x))]

-

  u' ' (x) ( F2 (x) - G 2 (x) ) dx
-
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Choosing among risky alternatives
using stochastic dominance
Second Degree Stochastic Dominance
The formula above has two parts. Let us address the right
hand part of it first.
- [u' (x) ( F2 (x) - G 2 (x))]

-

  u' ' (x) ( F2 (x) - G 2 (x) ) dx
-
Right part contains the second derivative of the utility
function times the difference in the integrals of the cdf
with a positive sign in front of it.

  u' ' (x) ( F2 (x) - G 2 (x) ) dx
-
To guarantee that f dominates g the sign of this whole
term must be positive. Second degree stochastic
dominance adds two assumptions
1) the second derivative of the utility function with
respect to x is negative everywhere
2) F2(x) is less than or equal to G2(x) for all x with
strict inequality for some x.
Must also sign the left hand part. First, add the
assumption on nonsatiation U’(X)>0.
This term then multiplies by F2(x) - G2(x) which is zero at
x equals minus infinity and under assumption 2 above is
non-positive at plus infinity.
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Choosing among risky alternatives
using stochastic dominance
Second Degree Stochastic Dominance
Given f and g if an individual has
1) positive marginal utility u'>0;
2) diminishing marginal utility of income
u"<0
and
3) that for all x the value of F2(x) is less
than or equal to that of G2(x) with strict
inequality for some x
Then we can say that f dominates g by a second degree
stochastic dominance.
One aspect of the above assumptions worth mentioning is
that when u" is less than zero and u' is greater than zero,
this implies the Pratt risk aversion coefficient
(-u"(x)/ u'(x)) is positive.
Also, the integral condition allows the cdfs for the
distributions to cross as long as the difference in the areas
before they cross is greater than the difference in their
areas after they cross.
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Choosing among risky alternatives
using stochastic dominance
Second Degree Stochastic Dominance
Figure 2. Second Degree
1.2
Cumulative Probability
1
0.8
Distribution f
0.6
Distribution g
0.4
0.2
0
0
5
10
15
20
25
30
35
Wealth
Figure 2 shows the case where second degree
stochastic dominance would exist. Notice the area
between g and f before x equals 11 exceeds that after x
equals 11.
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Choosing among risky alternatives
using stochastic dominance
Empirical Implementation
One does not usually have full continuous probability
distributions.
Step 1 - take the wealth or x outcomes for all the
probability distributions and array them from
high to low
Step 2 - write the relative frequencies of observations
against each of the x levels for each
probability distribution. Note, some of these
frequencies will usually be zero
Step 3 - divide the frequencies through by the number of
observations under each item
Step 4 - form the cumulative probability distribution by
adding up The algebraic formulae for the
area is:
Fo = Go = 0 , Fi = Fi-1 + fi
Gi = Gi-1 + gi
Step 5 - form the second integral of the probability using
the formulae;
F2,1 = 0 , G2,1 = 0
F2,i = F2,i-1 + Fi * ( xi - xi-1) i > 1
G2,i = G2,i-1 + Gi * ( xi - xi-1) i > 1
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Choosing among risky alternatives
using stochastic dominance
Empirical Implementation
First Degree Stochastic Dominance Example
suppose we wish to compare
(1,2,2,2,4,4,4,4,5,5,5,5,6,6,7,7,7,9,9,10)
with
(1,1,2,2,2,2,2,3,4,4,4,4,4,5,5,5,5,5,5,5)
x
Freq
f
Freq g
Pdf f
(fi)
Pdf g
(gi)
CDFf
(Fi)
CDF g
(Gi)
1
2
3
4
5
6
7
8
9
10
1
3
2
5
1
5
7
0.05
0.15
0.00
0.20
0.20
0.10
0.15
0.00
0.10
0.05
0.10
0.25
0.50
0.25
0.35
0.0
0.0
0.0
0.0
0.0
.05
.20
.20
.40
.60
.70
.85
.85
.95
1.00
.10
.35
.40
.65
1.00
1.00
1.00
1.00
1.00
1.00
Mean
Std Err
5.2
2.42
4
4
2
3
2
1
3.5
1.43
12
Intcdf f
(F2i)
0.05
0.25
0.45
0.85
1.45
2.15
3.00
3.85
4.80
5.80
Intcdf g
(G2i)
0.1
0.45
0.85
1.5
2.15
3.15
4.15
5.15
6.15
7.15
Table 2. Second Degree Stochastic Dominance Example
suppose we wish to compare
(1,2,2,2,4,4,4,4,5,5,5,5,6,6,7,7,7,9,9,10)
with
(1,1,2,2,2,2,2,3,4,4,6,6,6,6,6,6,6,7,7,7)
x
Freq f
Freq g
Pdf f
(fi)
Pdf g
(gi)
CDF f
(Fi)
CDF g
(Gi)
Intcdf f
(F2i)
Intcdf g
(G2i)
1
1
2
0.05
0.10
0.05
0.10
0.05
0.10
2
3
5
0.15
0.25
0.20
0.35
0.25
0.45
1
0.00
0.50
0.20
0.40
0.45
0.85
2
0.20
0.10
0.40
0.50
0.85
1.35
0.20
0.10
0.60
0.50
1.45
1.85
3
4
4
5
4
6
2
7
0.10
0.35
0.70
0.85
2.15
2.70
7
3
3
0.15
0.15
0.85
1.00
3.00
3.70
0.00
0.00
0.85
1.00
3.85
4.70
8
9
2
0.10
0.00
0.85
1.00
4.80
5.70
10
1
0.05
0.00
1.00
1.00
5.80
6.70
Mean
Std Err
5.2
2.42
4.3
2.29
13
Table 3. No Stochastic Dominance
suppose we wish to compare
(1.95,2,2,2,4,4,4,4,5,5,5,5,6,6,7,7,7,9,9,10)
with
(2,2,2,2,2,2,2,3,4,4,5,5,5,5,5,5,5,7,7,7)
x Freq f
1.95
2
3
4
5
6
7
8
9
10
Mean
Std Err
1
3
4
4
2
3
2
1
5.25
2.35
Freq
g
Pdf f
(fi)
0
7
1
2
7
5%
15%
0%
20%
20%
10%
15%
0%
10%
5%
3
Pdf g CDF f
(gi)
(Fi)
0%
35%
5%
10%
35%
0%
15%
0%
0%
0%
4.05
1.86
14
5%
20%
20%
40%
60%
70%
85%
85%
95%
100%
CDF Intcdf Intcdf g
g f
(G2i)
(Gi) (F2i)
0%
0.10
0.00
35%
0.11
0.02
40%
0.31
0.42
50%
0.71
0.92
85%
1.31
1.42
85%
2.01
2.27
100%
2.86
3.27
100%
3.71
4.27
100%
4.66
5.27
100%
5.66
6.27
Choosing among risky alternatives
using stochastic dominance
Moment Based Stochastic Dominance Analysis
if one assumes normality then the SSD rule can be
transformed to
uf  ug
σf  σg
with at least one strict inequality where uf, ug, σf and σg are
the mean and variance parameters of the f and g data that is
assumed to be normally distributed.
under log normal distributions we get the rule
uf 
 2f
2
 ug 
 g2
2
under Gamma distributions we get the rule
f
 max( 1,  g /  f )
g
15
Choosing among risky alternatives
using stochastic dominance
Problems With Stochastic Dominance
Non-Discrimination - Low Crossings
If the distribution shows a vast improvement under all the
observations but the lowest one as in Table 3, then
stochastic dominance will not hold in any form. The real
question is how risk adverse will individuals be?
Portfolio Effects
A second assumption of stochastic dominance is the
assumption that the alternatives are mutually exclusive.
When one does stochastic dominance one ignores the
possibility that the alternatives could be diversified.
Sample Size
A third problem with stochastic dominance is sampling
distributions. Namely, when one goes out and finds data,
one does not find population data and one usually finds a
sample.
16
Crossings and Dominance Failures
Generalized Stochastic Dominance
One Extension of stochastic dominance that has been
utilized is generalized stochastic dominance (GSD). One
again starts from the variant of the expected utility function:

  u' (x) ( F(x) - G(x) ) dx
-
Meyer investigated the magnitude of this expression under
the conditions that the Pratt risk aversion coefficient falls
into an interval:
r1 (x) 
u" ( x )
 r2 (x)
u' ( x )
Meyer poses an optimal control format for this examination
where the variable is u(x)

Max   u' (x) ( F(x) - G(x) ) dx
-
u" (x)
) u' (x)
u' (x)
u" ( x )
r1 (x) 
 r2 (x)
u' ( x )
s.t. ( u' (x))'  (
17
Choosing among risky alternatives
using stochastic dominance
Crossings and Dominance Failures
Generalized Stochastic Dominance

Max   u' (x) ( F(x) - G(x) ) dx
-
u" (x)
) u' (x)
u' (x)
u" ( x )
r1 (x) 
 r2 (x)
u' ( x )
s.t. ( u' (x))'  (
When this problem is solved it looks for the choice of utility
function which has r(x) constrained in the interval. The
objective function is the expected utility difference, which
if positive means f dominates g. When we maximize we
find the greatest expected utility difference over all possible
utility choices such that r(x) is in that interval. If the
greatest utility difference is negative therefore f must
dominate g.
Meyer recognized that this is a simple optimal control
problem since it is linear in the control variables. The
problem has what it is called a Bang-Bang solution.

 r1 (x * ) if

r(x * )  
 r2 (x * ) if




u'
(x)
(
F(x)
G(x)
)
dx

0
*

x


* u' (x) ( F(x) - G(x) ) dx  0 
x


18
Crossings and Dominance Failures
Generalized Stochastic Dominance
Meyer originally wrote a computer program to do this but
implements it with u(xi) = - e-rxi
This does not imply that the risk aversion parameter is
constant but rather that it could be increasing, decreasing or
of any other form as long as it remains in between the two
bounds.
GSD generalizes the other stochastic dominance forms
when r1 = 0 and r2 =  we get second degree while r1 = -
and r2 =  is the same as first degree.
This was been a fairly heavily used technique in the 1990’s.
The biggest problem in using that technique was always
finding the r1, r2 values. I wrote a code that starts from r1
and finds biggest r2 or vice versa. MEYEROOT on web
page agrinet.tamu.edu/mccarl
19
Crossings and Dominance Failures
BRAC
Yet another approach has been used to deal with
crossings. Hammond showed that given two alternatives
which cross once that under constant absolute risk aversion
there is a break-even risk aversion coefficient (BRAC) that
differentiates between those two alternatives.
Hammond also noted the expected utility problem
given a constant absolute RAC (r) is

-e
- rx
f(x) dx
-
is a form of the mathematical statistics moment generating
function
20
Choosing among risky alternatives
using stochastic dominance
Crossings and Dominance Failures BRAC
The moment generating function under normality given the
risk aversion parameter r for distribution f is as follows:
m( r )  e

 f r2
-  ru f 

2





If we go to solve this for the break-even risk aversion
parameter, first thing we would do is set the expected
utilities equal:
e

 2f r 2
-  r u f 
2






e

 2g r 2
-  r u g 
2






Or
ru f 
 2 r2
f
2
 ru g 
 2r2
g
2
this can be manipulated to
  2f  g2 
  r (u f  u g )  0
r 

 2
2 

2
which yields two roots
r 0
r
2 (u f  u g )
( 2f   g2 )
Notice then for any two normally distributed prospects we
can find a break-even risk aversion parameter using this
formula
21
Choosing among risky alternatives
using stochastic dominance
Crossings and Dominance Failures
BRAC
I wrote a program to implement Hammond’s approach with
an empirical discrete distribution of unknown form.
It is called RISKROOT and is available on the web page
agrinet.tamu.edu/mccarl.
RISKROOT takes data for two alternatives and searches
for the break-even risk aversion parameters between those
two alternatives by solving the following equation for all
applicable values of r.
- e
-rx i
(f i (x) - g i (x) )  0
i
22
Choosing among risky alternatives
using stochastic dominance
Crossings and Dominance Failures
BRAC Example
Use data from above 3 examples
OUTPUT FROM RISKROOT - CONSTANT RISK AVERSION ROOT FINDER
Example 1
DISTRIBUTION 1 NAME IS CASE 1
DISTRIBUTION 2 NAME IS CASE 2
THE DISTRIBUTIONS DO NOT CROSS -- 1 IS DOMINANT
Example 2
THE DISTRIBUTION CDFS CROSS 2 TIMES
1 HAS BEEN FOUND DOMINANT BETWEEN
0 2.2568226094
1 HAS BEEN FOUND DOMINANT BETWEEN
0 -2.2568226094
Example 3
SUMMARY STATISTICS ON THE DATA
DISTRIBUTION MEAN
STDDEV MIN MAX
CASE 1
5.25
2.35
1.95 10.00
CASE 2
4.40
2.01
2.00 7.00
RAC IS LIMITED TO BE BETWEEN +/-.238779E+01 BASED ON MCCARL
AND BESSLER
THE DISTRIBUTION CDFS CROSS 3 TIMES
1 HAS BEEN FOUND DOMINANT BETWEEN
0
2.3877865878
TROUBLE -- FOUND 1 DOMINANT AT HIGHEST RAC
-- SHOULD FIND RAC LARGE ENOUGH THAT 2 DOMINATED
1 HAS BEEN FOUND DOMINANT BETWEEN
0 -2.3877865878
23
Choosing among risky alternatives
using stochastic dominance
BRAC Properties
1. The number of roots you find is determined by the
number of crossings the cdf’s exhibit. If no crossings then
first degree stochastic dominance
2., Multiple roots possible with multiple crossings
Intuitively, consider distribution with lower min and higher
max and another with higher mean. At high risk aversion,
distribution with higher min preferred, while at high neg
risk higher max preferred. At moderate risk aversion
distribution with higher mean preferred.
3. Maximum BRAC dependent upon a formula derived
from McCarl and Bessler. 2/σ
4. BRAC always found at r=0 amd r= infinity
 - e-rx i (f i (x) - gi (x) )  0
i
5. RISKROOT BRAC’s determine results of Meyer GSD.
One cannot span a BRAC with GSD. RISKROOT BRAC
gives much stronger results telling exactly where the
preference shifts rather than having to guess
24
Choosing among risky alternatives
using stochastic dominance
Sampling
Pope and Ziemer investigated sampling error. Not a lot
can be said beyond the following
1) when distribution means and variances get close
together that the probability of improper
dominance conclusions can become quite high.
2) using the moment based stochastic dominance
rules is inferior to using the empirical distribution
based stochastic dominance rules.
3) the smaller the sample size the more likely one is
to have errors.
25
Choosing among risky alternatives
using stochastic dominance
Portfolios
A problem in stochastic dominance involves looking at
two stochastic prospects which are not mutually exclusive
but which may be correlated
So what happens with the correlation.
McCarl, Knight et al. looked at portfolio problem
Question now -- The conditions under which when one
finds that f dominates g that the prospect will also dominate
all combinations of the f and g.
Procedure for investigation that is used in that paper is
based on two moment based stochastic dominance rules.
26
Choosing among risky alternatives
using stochastic dominance
Portfolios
Normally distributed f dominates g whenever the following
uf  ug
σf  σg
two conditions are discovered.
Suppose f dominates g via the above when will f dominate
prospect h which is a convex combination of f and g.
Convex combination is written according to the following
formula:
h = λf + (1 - λ)g
where λ varies between 0 and 1.
From mathematical statistics that when we form prospect h
then its mean and variance are given by
u h   u f  (1 -  ) u g
 2  2  2  (1 -  )  g2  2   (1 -  )  f  g
h
f
27
Choosing among risky alternatives
using stochastic dominance
Portfolios
Now since f dominates g that the following two equations
uf  ug
or
σf  σg
are satisfied.
Now does mean dominance occur
u h   u f  (1 -  ) u g
Note since ug < uf
then
u h   u f  (1 -  ) u g   u f  (1 -  ) u f  u f
or
u h  uf
So dominance can occur since uh is less than or equal to uf.
28
Choosing among risky alternatives
using stochastic dominance
Portfolios
Now how about variance
 2  2  2  (1 -  )  g2  2   (1 -  )  f  g
h
f
Suppose we make a substitution namely since the variance
of f is smaller than the variance of g, we can write.
K σf = σg
K1
which renders our equation into the form
 2  2  2  (1 -  ) K 2 f2  2   (1 -  ) K f2
h
f
  f (2  (1 -  ) K 2  2   (1 -  ) K )
2
Now suppose we look for break even setting σf2 = σh2 or
 f2   2 (2  (1 -  ) K 2  2   (1 -  ) K )
f
or
(2  (1 -  ) K 2  2   (1 -  ) K )  1
If we collect terms into a quadratic format and use the
classical quadratic formula, we find two roots
 1
(K 2  1)
  2
(K  2  K  1)
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Choosing among risky alternatives
using stochastic dominance
Portfolios
To preclude convex combinations we wish only to find
λ1
This implies
(K 2 1)
1
(K 2  2  K  1)
or
(K 2 1)  (K 2  2  K  1)
or
2 K 2
which can be simplified to
ρ  1/K = σf/σg
where ρ is the correlation coefficient.
Thus, we have the restriction that the correlation coefficient
must be greater than or equal to the ratio of the variances.
We now have a condition under which we are certain that if
f dominates g via a second degree stochastic dominance
then f will dominate all potential convex combinations of f
and g.
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Choosing among risky alternatives
using stochastic dominance
Portfolios
ρ  1/K = σf/σg
This equation has several implications.
1. If items are perfectly correlated then we are safe
2. If ρ is zero or negative then there is no way that one can
ever guarantee that all the convex combinations are
dominated.
McCarl, Knight et al. do rather extensive evaluation on this
rule in mind that it works in a very high proportion if the
cases for normal and non-normal cases.
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Choosing among risky alternatives
using stochastic dominance
Portfolios
We developed a second criteria for dominance. This starts
from a rule that is sensitive to the differences in mean and
just expresses the certainty equivalents
uf 
r 2
r
 f  u g   2g
2
2
under this rule following the same approach as talked
through above, they find the following condition f will
dominate all combinations of f and g whenever


f
g

u
f
 ug
r f g

This can be transformed using the rule that r is twice the
value σ ( r  2Z ) as explained in McCarl and Bessler to
become


f
g

u
 ug
2Zg
f

what this rule shows is that the maximum acceptable
correlation coefficient becomes smaller as the means
become more disparate.
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Choosing among risky alternatives
using stochastic dominance
Portfolios
What these rules can be used for is to examine when
one has two potentially diversified alternatives whether can
successfully do dominance analysis between the two
without considering diversifications.
Namely, if the rules are satisfied one is safe. But if the rule
is not satisfied then one needs to potentially consider
diversifications.
One can also use the formula for λ as expressed above
giving a particular ratio of the standard errors and a
correlation coefficient to find the largest possible
diversification that should be considered. For example, if
one plugs in the ratio of Z = or 2 where σg is twice as big as
σf, with a correlation of .5 then one can use the formula to
find that the diversification that should be considered is
something between 100% of f and 43% of f and one then
can lay a grid out where one might consider 100, 90, 80, 70,
60, 50 and 43% of f and corresponding values of g and then
do stochastic dominance over all those alternatives.
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