Corollary. If t P G 0 for y` - P - `, then t G 0 for an otherwise arbitrary,
U
twice differentiable risk aÕerse utility function U. Several remarks may be useful at this point.
Ž .
X
Ž . Ž
. 1 If U `
s 0 meaning global strict concavity , then by the second mean value theorem of differential calculus, there must exist a value P
such that Ž
. t
s t P . There is therefore an option style utility function that has exactly the
U
same equivalent margin as the given utility function, strengthening the interpreta- tion of the U
as a set of Autility generatorsB for arbitrary concave utility
P
functions. Ž .
Ž . 2 An alternative expression for the weight w P is,
U
Y
P
Ž .
w P s z
F P ,
Ž . Ž .
Y
E U R
Ž .
Y
Ž .
X
Ž . Ž
. where z
s yEU R
rEU R is the Rubinstein 1973 coefficient of absolute risk aversion. The pattern of weights as P varies will therefore depend on two
factors. The first is the second derivative of the utility function at P relative to the Ž .
average. Consider, for example, the constant relative risk aversion family U R s
Ž .
j
1 q R rj ; j - 0, with j
™0 indicating the logarithmic utility function log Ž
. Ž .
Ž .
jy2
1 q R . For such functions, the weights w P diminish with 1r 1 q P
, so that the log utility function has weights that diminish more slowly relative to other
members of the family, which correspond to greater risk aversion. The second factor is the cumulative probability mass up to P. If U
Y
is diminishing while Ž .
F P is increasing, a trade-off between the two factors is therefore implied, and an Ž .
interior maximum of w P will usually exist. It is generally apparent that in
Ž . certain zones, notably the smallest and highest value of P, t P - 0 over some
small interval will not necessarily conflict with t G 0 for any realistic U.
U
4. The COMD and stochastic dominance
Ž . Ž .
If r dominates R by SSD, then t P G 0, for all P. Hence, if t P - 0 for
some P then r cannot dominate R, so that a necessary test for SSD emerges as a by-product of a COMD analysis. Moreover, one can create a sufficiency test for
SSD. The trick here is to reverse the benchmark, so that R is evaluated against r. If generator equivalent margins are all negative in the framework, then r SSD R.
Ž . w
x With r as benchmark, define g r
s E R N r , the conditional expectation of R, Ž .
given r. The function g r is thus the theoretical regression of R on r, as distinct Ž .
from e R , which is the theoretical regression of r on R. In such terms, the equivalent margin with respect to U of R with respect to r as the benchmark is,
P X
E g r y r U
r
Ž . Ž .
Ž .
r P
t P
s ,
10a
Ž . Ž
.
X r
EU r
Ž .
P
while,
X
E e R
y R U R
Ž . Ž .
Ž .
r P
t P
s ,
10b
Ž . Ž
.
X R
E U R
Ž .
P
Ž . is the equivalent margin of r with R as the benchmark. Operationally, the t
P
R
Ž . is done in terms of the COMD of r
y R ordered by R values, while t P is
r
computed as the COMD of R y r, using the r-values to order the observation
pairs. The principal result is as follows.
Theorem 2. If the COMD with r as benchmark is semi-negatiÕe for eÕery P, then r dominates R by SSD:
SSD
t P
F 0 for all P ´r R.
11
Ž . Ž .
r
Ž .
Proof. As U r is concave,
P
U r
F U R
q r y R U
X
R .
Ž . Ž .
Ž .
Ž .
P P
P
Taking expectations over the joint distribution of r, R gives:
X
w x
E U r
F E U R
q E E r
N R y R U R
Ž . Ž .
Ž .
Ž .
r P
R P
R r
P X
s E U R
q E e R
y R U R
Ž . Ž .
Ž .
Ž .
R P
R P
s E U R
q t P EU
X
R . 12a
Ž . Ž .
Ž . Ž
.
R P
R P
Similarly, U
R F U
r q R y r U
X
r .
Ž . Ž .
Ž .
Ž .
P P
P
Taking expectations with respect to r, R gives: E U
R F E U
r q t
P EU
X
r . 12b
Ž . Ž .
Ž . Ž .
Ž .
R P
r P
r P
Ž .
Ž .
Combining Eqs. 12a and 12b yields the bounds: yt
P EU
X
r F E U
r y E U
R F t
P EU
X
R . 13
Ž . Ž .
Ž . Ž .
Ž . Ž .
Ž .
r P
r P
R P
R P
Ž . Ž .
Ž . Hence, if t P
F 0, all P, it follows that E U r G E U R , and hence from
r r
P R
P
Ž .
SSD
Eq. 5 of Section 3, r R.
I Ž
The operational implication is that one plots both versions alternative bench- .
Ž . marks of the COMD. If there is some P for which t
P - 0, then r cannot be
R
Ž . SSD over R. Or if t P - 0, all P, then r is SSD over R.
r
5. Operational matters
