Decreasing Absolute Risk Aversion, Prudence and Increased Downside Risk Aversion*
August 2011
Liqun Liu
Private Enterprise Research Center
Texas A&M University
College Station, TX 77843
[email protected]
Jack Meyer
Department of Economics
Michigan State University
East Lansing, MI 48824
[email protected]
Abstract: Downside risk increases have previously been characterized as changes preferred by
all decision makers u(x) with u'''(x) > 0. For risk averse decision makers, u'''(x) > 0 also defines
prudence. This paper finds that downside risk increases can also be characterized as changes
preferred by all decision makers displaying decreasing absolute risk aversion (DARA) whenever
all random variables have equal means. Building on these findings, the paper proposes using
“more decreasingly absolute risk averse” or “more prudent” as alternative definitions of
increased downside risk aversion. These alternative definitions generate a transitive ordering,
while the existing definition based on a transformation function with a positive third derivative
does not. Other properties of the new definitions of increased downside risk aversion are also
presented.
Key Words: DARA; Downside risk; Downside risk aversion; Decreasing absolute risk aversion,
Prudence
JEL Classification Codes: D81
1
1. Introduction
More than thirty years ago, Menezes, Geiss and Tressler (MGT) (1980) present a
definition of an increase in downside risk. This definition specifies in a formal way the
unambiguous shifting of risk from the right to the left in the support of a random variable
keeping the total risk constant. MGT indicate that preferences for or aversion to such risk shifts
should depend on whether risk aversion is increasing or decreasing.1 The main theorem they
present, however, shows that one cumulative distribution function (CDF) is an increase in
downside risk from another if and only if the latter is preferred to the former by all decision
makers whose utility function has a positive third derivative. Of course, it is well known that a
positive third derivative for the utility function is a necessary, but not sufficient condition for
decreasing absolute risk aversion (DARA). Thus, the connection that MGT establish is between
increases in downside risk and the slope of u''(x) rather than the slope of risk aversion.
Quite naturally, the ensuing literature extending the work of MGT has also focused on
the sign of the third derivative of utility. This is true when defining what it means for one
decision maker to be more downside risk averse than another, when attempting to measure the
strength or degree of downside risk aversion, and when defining what an expected utility
preserving increase in downside risk is. Kimball (1990), Keenan and Snow (2002), Modica and
Scarsini (2005), Jindapon and Neilson (2007), Crainich and Eeckhoudt (2008) and Keenan and
Snow (2009) are among those who carry out these extensions.
There are a number of unresolved issues in this literature, including how to define an
increase in downside risk aversion as well as how to measure its intensity. Various researchers
have proposed different things, including at least one definition of when one decision maker is
2
more downside risk averse than another, and at least three different measures of the intensity of
downside risk aversion. At this point, however, there is no agreed upon measure of the intensity
of downside risk aversion, and the proposed definition of increased downside risk aversion has
serious defects. The analysis here explains in part why this is the case, and suggests possible
alternatives.
The beginning point of this analysis is the observation that the existing definition of an
increase in downside risk aversion presented by Keenan and Snow (2002, 2009) has a serious
defect. Keenan and Snow define v(x) to be more downside risk averse than u(x) whenever
v(x) = (u(x)) and '''(u)  0. While this definition appears to be a very natural one, and is well
defended in Keenan and Snow’s analysis, the partial order over utility functions characterized by
this definition is not transitive. That is, w(x) more downside risk averse than v(x) and v(x) more
downside risk averse than u(x) does not imply w(x) is more downside risk averse than u(x) by
this definition of increasing downside risk aversion. 2 This lack of transitivity makes it
impossible to find a measure of the strength or intensity of downside risk aversion to associate
with this definition of increasing downside risk aversion.
In order to add to rather than just be critical of the discussion surrounding increased
downside risk aversion, this definition must be replaced by one which is transitive. The first step
taken here to do this is to establish a connection between increases in downside risk and the
slope of absolute risk aversion, the link that MGT fail to identify. Once this link is established, it
is natural to use the slope of absolute risk aversion as a possible way to define a partial order
indicating when one decision maker is more downside risk averse than another, and also to
develop a measure of downside risk aversion.
3
The connection between DARA and increases in downside risk that is demonstrated here
is that CDF F(x) is preferred or indifferent to an equal-mean CDF G(x) for all utility functions
displaying DARA if and only if G(x) is an increase in downside risk from F(x). This finding is
demonstrated using a general theorem that holds for any mean preserving change in a random
variable. The general theorem indicates that a mean preserving change increases (decreases)
expected utility for all decision makers with u'''(x) > 0 if and only if the change also increases
(decreases) expected utility for all decision makers displaying DARA. This generalizes a finding
by Fishburn and Vickson (1978) concerning stochastic dominance. Fishburn and Vickson show
that third degree stochastic dominance (TSD) and DARA stochastic dominance are equivalent
concepts when the means of the random alternatives are equal to one another.
Once DARA is associated with aversion to downside risk increases, the slope of absolute
risk aversion is then used to provide an alternate definition of increased downside risk aversion.
This definition generates a transitive, asymmetric partial order over utility functions.
Corresponding to this partial order is both a measure of the intensity of downside risk aversion,
and also a definition of an expected utility preserving downside increase in risk. Greater
downside risk aversion according to the slope of the absolute risk aversion measure is
demonstrated to be a necessary and sufficient condition for aversion to such an expected utility
preserving increase in downside risk.
Following the use of the slope of absolute risk aversion as a basis for defining a partial
order and measure for downside risk aversion, other definitions and results are given using the
well known concept of prudence instead. Similar steps are taken when doing this. First, the
literature has already established the connection between prudence and aversion to downside risk
4
increases and “more prudent” has previously been suggested as a possible way to define a partial
order and measure for increased downside risk aversion. This prudence related definition yields
a transitive, asymmetric partial order over utility functions. What is added in this analysis is yet
a third definition for an expected utility preserving downside risk increase. For this definition,
being more prudent is a necessary and sufficient condition for aversion to an expected utility
preserving increase in downside risk.
The paper is organized as follows. First the results concerning increases in downside risk
are briefly reviewed and the notation and assumptions used in the paper are established. The
definition given by MGT and the theorems associating aversion to downside risk increases with
the third derivative of utility are presented. Next, the Keenan and Snow definition of increasing
downside risk aversion is reviewed and its failure to lead to a transitive or asymmetric partial
order over risk preferences is discussed. Following this, a theorem is presented that shows that
aversion to an increase in downside risk is also characterized by DARA. DARA is then used to
define one decision maker being more downside risk averse than another. Next, an alternate way
to extend MGT’s definition of a downside increase in risk to an expected utility preserving
downside increase in risk is given. A theorem is provided connecting the new definition of
increasing downside risk aversion with this definition of an expected utility preserving increase
in downside risk. Following this, it is noted that prudence also characterizes downside risk
aversion and more prudent is used to define increasing downside risk aversion. An expected
utility preserving increase in risk is defined so that an increase in downside risk is avoided by all
decision makers more prudent than the reference person. The paper concludes with discussion of
5
the findings in the paper, and the comparison of using the slope of absolute risk aversion with
size of prudence when defining an increase in downside risk aversion.
2. Downside Risk Increases, Downside Risk Aversion and More Downside Risk Aversion
Menezes, Geiss and Tressler (1980) define an increase in downside risk using
combinations of Rothschild and Stiglitz (1970) mean preserving spreads and contractions. The
definition they present requires that these increases and decreases in risk occur in such a way that
risk is unambiguously transferred from the right to the left in the support of the random variable.
The total amount of risk is left unchanged. MGT then show that this intuitive definition can be
characterized by conditions on the cumulative distribution functions, F(x) and G(x), representing
the two random alternatives. These conditions on the CDFs are given below and are used here as
the MGT definition of an increase in downside risk.
Definition 1: CDF G(x) has more downside risk than F(x) if and only if:
-
a)
b)
y x
a a
G s – F s ds dx 
for all y in [a, b] with equality holding at y = b and > 0
holding for some y in (a, b).
In this definition the supports of the random variables are assumed to be contained in a finite
interval denoted [a, b] and that assumption is maintained here. The first condition, part a),
requires the two random alternatives to have the same mean value. The second condition defines
a downside risk increase in that it requires the risk increases to unambiguously occur to the left
of the risk decreases. The condition that equality holds at y = b requires the risk changes in total
6
to add to zero; that is, the risk increases and decreases are equal in size. The profession has
accepted this definition and characterization of an increase in downside risk.
Following this characterization of an increase in downside risk, MGT demonstrate the
following theorem. This theorem provides one link between downside risk increases and
aversion to those increases by decision makers with utility functions displaying a positive third
derivative.
Theorem 1: F(x) is preferred or indifferent to G(x) for all decision makers with u(x) satisfying
u'''(x) > 0 if and only if G(x) has more downside risk than F(x).
Keenan and Snow (2009) demonstrate a general theorem for expected utility preserving
downside risk increases, which for the special case of u'(x) = 1, reduces to Theorem 2 given
below. This theorem also provides a link between downside risk increases and the third
derivative of utility.
Theorem 2: All increases in downside risk result in lower expected utility for u(x) if and only if
u'''(x)  0.
Keenan and Snow assume a weak rather than strong inequality on the third derivative of utility,
and also assume that u(x) is positively monotonic, a condition which MGT do not assume. This
assumption, that u'(x) > 0, is also made here so that division by zero is not an issue when
examining the measure of absolute risk aversion. These two theorems clearly establish a
connection between downside risk aversion and the sign of the third derivative of utility. As a
7
result the sign of the third derivative of utility has been an important feature of the extensions of
the work of MGT.
Keenan and Snow (K-S) (2009) note that u''(x)  0 characterizes risk aversion and that
Pratt (1964) has shown that ''(u)  0 when v(x) = (u(x)) characterizes increased risk aversion.
Using this, and the fact that u'''(x)  0 characterizes downside risk aversion, quite naturally K-S
suggest the following definition of increasing downside risk aversion.
Definition 2: A decision maker with utility function v(x) is more downside risk averse than a
decision maker with utility function u(x) if and only if v(x) = (u(x)) where the transformation
function (u) satisfies '''(u)  0.
Keenan and Snow go on to support this definition with extensive analysis. They also
expend considerable effort in attempting to obtain a measure of downside risk aversion that is
consistent with this definition of increasing downside risk aversion. They finally settle on a
measure denoted s(x), but find a uniformly larger s(x) is only a sufficient, not a necessary and
sufficient condition for a utility function to be more downside risk averse. K-S conclude that
“there is no measure that fully characterizes greater downside risk aversion.” p.1097).
Upon examining Definition 2 further, it becomes apparent that this definition does not
lead to a transitive or asymmetric partial order over utility functions. That is, if w(x) is more
downside risk averse than v(x) and v(x) is more downside risk averse than u(x) then it need not
be the case that w(x) is more downside risk averse than u(x). Furthermore, there exist distinct
utility functions such that each is more downside risk averse than the other.
8
To show that transitivity does not hold for Definition 2, one can verify that when v(x) =
 u x and w x = ψ v x then ψ◦)''' = ψ''''3+3ψ'''''+ ψ''''. Thus, ψ''' v  0 and '''(u)  0
cannot guarantee ψ◦)'''  0 in general. For example, let (u) = u1/2, ψ v = v3, and therefore
ψ◦)(u) = u3/2. Then ψ◦)'''(u) < 0 even though ''' u > and ψ''' v > . Additionally,
Definition 2 makes it possible for two distinct decision makers to each be more downside risk
averse than the other. That is, the partial order over risk preferences is not asymmetric. To see
this let u(x) = ecx and v(x) = edx. This implies that (u) takes the form (u) = ud/c. It can be
readily checked that '''(u) > 0 for both d/c = 3 and d/c = 1/3. Therefore the two distinct utility
functions e3x and ex are each strictly more downside risk averse than the other under Definition
2.3 This lack of transitivity and asymmetry are serious flaws for orders. Furthermore, because
the partial order is not transitive, there cannot be a numerical intensity measure for downside risk
aversion that is associated with Definition 2.
In order to add to the discussion of increased downside risk aversion, Definition 2 must
be replaced by one that is transitive. The next section uses more decreasing absolute risk averse
to define increased downside risk aversion. First, to justify this definition, a connection is
established between aversion to downside risk and decreasing absolute risk aversion.
3. DARA and Downside Risk Aversion
As mentioned in the introduction, the theorem that is presented next applies to all mean
preserving changes in random variables and is related to one demonstrated by Meyer and Meyer
(2010) and also to a stochastic dominance finding by Fishburn and Vickson (1978). Fishburn
and Vickson show that when the means of the random alternatives are the same, third degree
stochastic dominance (TSD) and DARA stochastic dominance are equivalent concepts. The
9
proof of the theorem presented here is brief and quite different from that provided by Fishburn
and Vickson.
In way of notation, assume that u'(x) > 0 represents any positive marginal utility function4
and let Au(x) =
- ''
denote the associated absolute risk aversion measure. The slope of
'
absolute risk aversion is given by Au'(x) =
- '''
'
''
. No assumption is made
'
concerning the sign of u''(x). With this notation, the following theorem is presented.
Theorem 3: For any H(x) such that

'
only if

'
,
for all u'''(x) > 0 if and
for all Au'(x) < 0.
Proof: First the well known part. u'''(x) > 0 is a necessary condition for Au'(x) < 0, therefore
when

'
the other direction, assume

'

'
for all u'''(x) > 0 then
for all Au'(x) < 0. To show
for all Au'(x) < 0 and let u'(x) be any marginal
utility function with u'''(x) > 0. Consider the family of marginal utility functions (u'(x) + k) for
any k  0. For this family of marginal utility functions,
for all k  0. This follows because
'
=
'
. The slope of the absolute risk aversion
measure for marginal utility (u'(x) + k) is:
- '''
'
+
''
'
. For k large enough, the sign
of this slope is the opposite of the sign of u'''(x). Thus, if u'''(x) > 0, then the sign of this slope is
negative and
'
 0 for this marginal utility (u'(x) + k), and therefore for u'(x)
10
as well. A magnitude for k exceeding the value of:
''
'''
- u'(x) for all x is sufficiently large
for the signs of Au'(x) and u'''(x) to be opposite of one another.
QED
As in the Fishburn and Vickson analysis, the key assumption is that
,
which when H(x) = G(x) - F(x), implies that the random alternatives with CDFs F(x) and G(x)
have the same mean. Of course condition a) in the MGT definition of an increase in downside
risk requires those increases to be mean preserving. Theorem 4 given below is obtained directly
from Theorem 1 and Theorem 3.
Theorem 4: For distributions with the same mean, F(x) is preferred or indifferent to G(x) for all
decision makers with u(x) such that Au'(x) < 0 if and only if G(x) has more downside risk than
F(x).
Theorem 4 establishes a link between aversion to increases in downside risk and the slope
of the absolute risk aversion function. This connection suggests using this slope to define what it
means for one decision maker to be more downside risk averse than another. This is carried out
next.
Definition 3: A decision maker with utility function v(x) is more downside risk averse than a
decision maker with utility function u(x) if and only if -Av'(x) > -Au'(x) for all x in [a, b], where
Av'(x) and Au'(x) are the slopes of the absolute risk aversion functions for v(x) and u(x),
respectively.
11
To show that Definition 3 is transitive is straightforward. If -Aw'(x) > -Av'(x), and
-Av'(x) > -Au'(x), then -Aw'(x) > -Au'(x). Quite obviously, Definition 3 does lead to an
asymmetric partial order over risk preferences as well. In addition, Definition 3 is also very easy
to check. That is, given utility functions u(x) and v(x), it is easy to determine whether one is
more downside risk averse than the other. This is in contrast to verifying that '''(u)  0 which,
as Keenan and Snow point out, may be difficult. In fact, K-S attempt to characterize Definition 2
using downside risk aversion measure namely s(x) =
'''
'
-
''
and succeed in
'
showing that sv(x)  su(x) is a sufficient, but not necessary condition for v(x) = (u(x)) with
'''(u)  0. For the example given earlier to illustrate the lack of asymmetry, the s(x) associated
with e3x is - 4.5 for all x while that associated with ex is - .5. Thus, the sufficient condition
sv(x)  su(x) for all x indicates that ex is more downside risk averse than e3x by Definition 2, but
fails to indicate that e3x is also more downside risk averse than ex.
Associated with Definition 3 is a measure of downside risk aversion namely the negative
of the slope of absolute risk aversion, -Au'(x). This may or may not be a good measure of the
intensity of downside risk aversion. It is, however, related to two of the more prominent
suggested measures of downside risk aversion. To see this note that -Au'(x) =
'''
-(
'
''
'
)2 .
Thus, this measure of downside risk aversion lies between the measure proposed by Modica and
Scarsini who suggest d(x) =
'''
'
, and that given by K-S, s(x) =
'''
'
-
''
'
.5
While Definition 3 is a simple and natural way to define when one decision maker is
more downside risk averse than another, a different necessary and sufficient condition proves to
12
be more useful when demonstrating propositions based on Definition 3. This necessary and
sufficient condition applies to the marginal utility functions for u(x) and v(x), rather than the
utility functions themselves. Of course, since utility is unique to a positive linear transformation,
marginal utility completely characterizes risk preferences. Meyer (2010) discusses this in more
detail.
Let marginal utility v'(x) = '(x)·u'(x); that is, let v'(x) be written as the product of two
functions, one being marginal utility u'(x), and another function '(x). Of course, for any given
u'(x) and v'(x), '(x) =
'
'
. Also in Keenan and Snow’s formulation, '(x) = '(u(x)). Recall
that u'(x), v'(x) and hence '(x) are all assumed to be positive. When marginal utilities are related
this way, v'(x) = '(x)·u'(x), simple calculation shows that the absolute risk aversion measures
associated with these three marginal functions satisfy Av(x) = Au(x) + A(x). That is, the
absolute risk aversion measure associated with a marginal utility function v'(x) is the sum of the
absolute risk aversion measures associated with u'(x) and '(x).
This relationship between absolute risk aversion measures makes it very easy to specify
when v'(x) represents more decreasingly absolute risk averse risk preferences than u'(x). What is
required is that the slope of A(x) be less than zero. In fact, -Av'(x) > -Au'(x) for all x in [a, b] if
and only if A'(x) < 0 for all x in [a, b]. Thus, the definition of increased downside risk aversion
can be restated as: marginal utility v'(x) displays more downside risk aversion than u'(x) if and
only if '(x) displays decreasing absolute risk aversion. This working definition of increased
downside risk aversion is used often in the various demonstrations presented here.6
4. Expected Utility Preserving Increases in Downside Risk
13
Definition 3 flows quite naturally from Theorem 4 which states the fact that for mean
preserving changes DARA is a necessary and sufficient condition on risk preferences for
aversion to the downside risk increases defined by MGT. Associated with this new definition of
increasing downside risk aversion is an alternate definition of an expected utility preserving
increase in downside risk. This new definition is both an extension of the definition of an
increase in downside risk given by MGT, and is also expected utility preserving for an arbitrary
reference decision maker. Moreover, this alternative definition of expected utility preserving
downside risk increases is paired with Definition 3 in that when an expected utility preserving
increase in downside risk for a reference decision maker occurs, all those more downside risk
averse than this person are averse to the change.
Definition 4: For a decision maker with marginal utility u'(x), G(x) is an expected utility
preserving increase in downside risk from F(x) if and only if:
a)
b)
-
'
'
–

for all y in [a, b] with equality holding at y = b and
> 0 holding for some y in (a, b).
When Definition 4 is compared with Definition 1 presented by MGT, the only difference
is the u'(x) in the expressions. The original MGT definition of an increase in downside risk
treats the risk neutral person (u'(x) = 1) as a reference decision maker. Thus, it is confirmed that
Definition 4 is indeed an extension of the definition of an increase in downside risk given by
MGT. In addition, since condition a) requires that expected utility be the same for F(x) and G(x)
for the reference decision maker, this definition is an extension to the expected utility preserving
14
case. Thus, the extension proposed here is an alternative to that given and used by Keenan and
Snow, who also extend the MGT definition to the expected utility preserving case. Before
comparing these two alternative extensions, a theorem connecting Definition 3 with Definition 4,
the two new DARA based definitions, is presented and demonstrated.
Theorem 5: For expected utility preserving changes, F(x) is preferred or indifferent to G(x) for
all decision makers v(x) who are more downside risk averse (Definition 3) than u(x) if and only
if G(x) is an expected utility preserving increase in downside risk (Definition 4) from F(x) for
u(x).
Proof: First note that Theorem 4 is a special case of Theorem 5 when the reference decision
maker, represented by u(x) in Theorem 5, is risk-neutral or u'(x) = 1. Since Theorem 4 is
obtained directly from Theorems 1 and 3, the strategy here is to first generalize Theorem 1 to the
case of expected utility preserving downside risk increases, which is Lemma 1 stated and proven
in the Appendix, and then use Lemma 1 and Theorem 3 to prove Theorem 5.
As previously pointed out, for any two utility functions u(x) and v(x) related by v'(x) =
'(x)·u'(x), A'(x) < 0 is necessary and sufficient for v(x) to be more downside risk averse than
u(x). So the “if” part of Theorem 5 is immediately obtained from Lemma 1 by noting that
A'(x) < 0 implies '''(x) > 0.
For the “only if” part, suppose a change from F(x) to G(x) is expected utility preserving
for u(x) and reduces expected utility for all v(x) who are more downside risk averse than u(x).
That is,
'
-
15
and for all v(x) with v'(x) = '(x)·u'(x) and A'(x) < 0,

-
'
and '(x) above as H(x) and u'(x) in Theorem 3, respectively, we
-
Treating '
.
have, according to Theorem 3,

-
'
for all '''(x) > 0. This, applying Lemma 1 once again, implies G(x) is an expected utility
preserving increase in downside risk from F(x) for u(x).
QED
A corollary to this result which extends Theorem 2 is:
Corollary 1: All expected utility preserving increases in downside risk (Definition 4) with u(x)
as the reference decision maker result in lower expected utility for any decision maker who is
more downside risk averse than u(x) (Definition 3).
Definition 4 gives an alternative to the K-S definition of an expected utility preserving
downside risk increase. Theorem 5 relates this definition to the DARA based definition for an
increase in downside risk aversion. For expected utility preserving changes, this alternative
provides a necessary and sufficient condition for those who are more downside risk averse by the
DARA definition to not prefer an expected utility preserving downside risk increase.
While both our definition and that of K-S extend the definition of MGT, and both
preserve expected utility for a reference decision maker with utility u(x) or marginal utility u'(x),
the two definitions differ. The K-S definition requires that
'
'
–

for all y in [a, b]
16
while the definition given here, Definition 4, requires that
'

–
for all y in [a, b] instead.
These two restrictions are not in general the same as one another.
Each definition is for an expected utility preserving downside increase in risk, and is
constructed so that a theorem concerning the effect of these risk increases on those more
downside risk averse than a reference person can be determined. That is, each is defined to
match or be paired with an existing definition of what an increase in downside risk aversion is.
Since the partial order over risk preferences associated with the definition of an increase in
downside risk aversion associated with '''(u(x))  0 is not transitive, the matching Keenan and
Snow definition of an expected utility preserving downside risk increase has a weak partner. On
the other hand, Definition 4 given here corresponds to the transitive DARA based definition of
increased downside risk aversion, Definition 3.
The Keenan and Snow definition of an expected utility preserving increase in downside
risk ensures that the utility distributions associated with F(x) and G(x) for the reference person
are not only equal in mean, but also satisfy the original downside risk increase definition of
MGT. Thus, the variance of the utility distribution is preserved, and the risk increases occur to
the left of the risk decreases and are equal in size in utility space.
The condition that
'
–

for all y in [a, b] in Definition 4
has quite a different interpretation. Since there is no transformation of the utility function of the
reference person involved in the definition of more downside risk averse, the utility distributions
associated with F(x) and G(x) do not play a role. Instead, the focus shifts to the properties of risk
adjusted probability distributions associated with F(x) and G(x). Consider the expression
17
u'(x)(G(x) – F( x)), this expression behaves much like the difference between two cumulative
distribution functions, with adjustments that are determined by the marginal utility function of
the reference decision maker. u'(x)(G(x) – F( x))] changes sign at exactly the same points as
[G(x) – F(x)]. With proper normalization u'(x)(G(x) – F( x)) can be used to define the
difference between two CDFs. The normalization is required to ensure that the total increase and
decrease are less than or equal to one. For some t > 0, (x) – (x) = t[u'(x)(G(x) – F( x))]
defines the difference between two CDFs. An example and further discussion of this is given in
the Appendix.
5. Prudence and Downside Risk Aversion
Prudence, defined as u''' > 0, is shown by Leland (1968) to characterize the motive to
save for future uncertainty. Kimball (1990) and others have used P(x) =
- '''
''
to measure the
intensity of prudence. Kimball shows that the precautionary saving is larger the more prudent an
individual is. Since MGT connect prudence to downside risk aversion, the role of the prudence
measure in downside risk aversion has also been explored. Chiu (2005) shows that, under
certain conditions, an individual dislikes any mean-preserving and variance-decreasing stochastic
change that preserves the expected utility for another individual if and only if the former has a
larger prudence than the latter everywhere.7 Jindapon and Neilson (2007) demonstrate that an
individual is willing to pay a higher utility cost to move to a payoff distribution with a lower
downside risk if and only if the individual is uniformly more prudent. More recently, Keenan
and Snow (2010) explore the relationship between greater prudence and their greater downside
18
risk aversion according to Definition 2. They find that greater prudence implies greater
downside risk aversion only under some strong restrictions on utility functions.
The analysis in this section starts with a definition of increased downside risk aversion
that is based on the prudence measure. It then establishes a close relationship between this
version of increased downside risk aversion and a corresponding concept of expected utility
preserving downside risk increases. The labeling of the definitions and theorems is meant to
correspond to the similar DARA related definitions and theorems.
Definition 3': A decision maker with utility function v(x) is more downside risk averse than a
decision maker with utility function u(x) if and only if Pv(x) > Pu(x) for all x in [a, b], where
Pv(x) and Pu(x) are the prudence measures for v(x) and u(x), respectively.
This definition yields a transitive and asymmetric partial order over utility functions.
Associated with this is a definition of an expected utility preserving downside risk increase.
When
and
are either both risk averse or both risk loving in [a, b], an assumption we
will maintain in this section, define (u) such that '
 '
. The following lemma relates
the sign of the second order derivative of the transformation function with prudence-based more
downside risk aversion:
Lemma 2: v(x) is more downside risk averse than u(x) in [a, b] if and only if ''(u' u'' < 0 in
[a, b].
Proof: From '  ' , we have v'' = 'u'' and v''' = ''u''2 + 'u''', which in turn yield '

v   u  
 
      u  . Obviously, Pv(x) > Pu(x) if and only if ''(u' u'' < 0.
v   u  

QED
19
Definition 4': For a decision maker u(x), G(x) is an expected utility preserving increase in
downside risk from F(x) if and only if:
-
a)
b)
''
–
for all y in [a, b] with equality holding at y = b
and < holding for some y in (a, b).
Two observations can be immediately made about Definition 4'. First, Definition 1 is a
special case of Definition 4' when ''
'
-
- . Second, a) and b) imply that
. So the change from F(x) to G(x) is indeed expected utility
preserving for u(x) and is an extension of the definition of a downside risk increase given by
MGT.
The following proposition, which is more general than Theorem 1, establishes a close
relationship between Definition 3' and Definition 4'. The proof is in the Appendix.
Theorem 5': F(x) is preferred or indifferent to G(x) for all decision makers v(x) who are more
downside risk averse (Definition 3') than u(x) if and only if G(x) is an expected utility preserving
increase in downside risk from F(x) ( Definition 4') for u(x).
The following proposition, which is more general than Theorem 2, also establishes a
close relationship between Definition 3' and Definition 4'. Note that it is “if” in Corollary 1, but
is “if and only if” in Corollary 1'. The proof is in the Appendix.
20
Corollary 1': All expected utility preserving increases in downside risk (Definition 4') with u(x)
as the reference decision maker result in lower expected utility for a decision maker v(x) if and
only if v(x) is more downside risk averse than u(x) (Definition 3').
6. Discussion and Conclusion
This paper explores the role of decreasing absolute risk aversion (DARA) and prudence
in defining and analyzing increased downside risk aversion and expected utility preserving
downside risk increases. It begins by closing a gap in the work of Menezes, Geiss and Tressler
(1980) and establishing a close relationship between DARA and downside risk aversion.
Specifically, the analysis here shows that, for two random distributions with the same mean, one
is preferred or indifferent to another for all decision makers displaying DARA if and only if the
latter is an increase in downside risk from the former (Theorem 4).
From there, alternative DARA-based definitions have been given for two important
concepts, increases in downside risk aversion and expected utility preserving increases in
downside risk. The first replaces a definition that has serious flaws in that the partial order over
utility functions that it yields is not transitive or asymmetric. The second is defined to be paired
with the first. Theorem 5 verifies the relationship between the two new definitions.
These alternative DARA based definitions are compared to the existing ones due to
Keenan and Snow (2009). Keenan and Snow emphasize the fact that their definitions build upon
and extend to downside risk and downside risk aversion many of the same things that were first
presented for risk and risk aversion. The same is true for the two definitions given here. To
emphasize this, a paragraph from Keenan and Snow’s introduction (2009, p 1093) is repeated
21
and modified. Their discussion concerning v(x) = (u(x)) is replaced with similar discussion
involving v'(x) = '(x)·u'(x). Although their language is borrowed, Keenan and Snow are not
quoted directly, so quotation marks are omitted. Using their exact language as much as possible
is intentional in order to make the point that the DARA based definitions given here are also
natural extensions of the analysis leading to the Arrow-Pratt measure of risk aversion and R-S
definition of increasing risk. The paragraph is the following:
We provide an analysis of downside risk aversion that mirrors the standard
results for risk aversion. Specifically, a marginal utility function u' is risk
averse if u'' < 0 and v' = '·u' is more risk averse than u' if the function ' is,
itself, risk averse, that is, A(x) > 0. In like manner, u' is downside risk
averse if Au' < 0, and we say that v' is more downside risk averse than u' if
A' < 0. Given arbitrary functions u and v, verifying these properties of the
function ' is an easy and a straightforward exercise. For the case of risk
aversion, the condition A(x) > 0 is equivalent to Av(x) > Au(x). For
downside risk aversion, A'(x) < 0 is equivalent to v' being more downside
risk averse than u'.
Because the DARA based definition of an increase in downside risk aversion leads to a
measure of downside risk aversion intensity that has weaknesses (Footnote 5), a prudence based
definition of greater downside risk aversion is given as well. This definition also yields a
transitive and asymmetric partial order over utility functions. This paper demonstrates global
22
properties of the prudence based order by linking it to yet another definition of an expected
utility preserving downside risk increase, Theorem 5' and Corollary 1'.
23
Appendix
A1. Lemma 1
Lemma 1: F(x) is preferred or indifferent to G(x) for all decision makers v'(x) = '(x)·u'(x) with
'''(x) > 0 if and only if G(x) is an expected utility preserving increase in downside risk from
F(x) for u(x).
Proof: The proof follows the similar steps in MGT’s proof of Theorem 1. Using v' x =
'(x)·u'(x),
-
'
EFv(x) - EGv(x) =

=
-
'
.
(The “only if” part)
Suppose EFv(x)
EGv(x) for all '''(x) > 0.
Let 1 x = θx3 /3 + x and 2 x = θx3 /3 – x where θ > . Since 1'''(x) = 2'''(x) > 0, we
have


b
a
b
a
( x 2  1)u ( x)(G( x)  F ( x))dx  0
( x 2  1)u ( x)(G( x)  F ( x))dx  0
which continues to hold as θ approaches zero. So
'
-
''
'
= 0, which is a) in
Definition 4.
'
Note that when
EFv(x) - EGv(x) =

'
-
= 0,
.= -
-
.
Now let 3 x = θx3 /3 + x2/2 and 4 x = θx3 /3 – x2/2 where θ > . Since 3'''(x) = 4'''(x) > 0,
we have
24
b
x
a
a
b
x
a
a
  (2 x  1)  u ( s)(G ( s)  F ( s))dsdx  0
  (2 x  1)  u ( s)(G ( s)  F ( s))dsdx  0
which continue to hold as θ approaches zero. So
–
'
, which is
the equality part of b) in Definition 4.
-
'
With
'''
EFv(x) - EGv(x) =
–
'
= 0 and
–
'
,
.
We prove the weak inequality of b) by contradiction. Assume
–
'
[α, β], such that
for some y0 in [a,b]. Then there exists an interval in [a, b],
–
'
 x3 / 6,
 5 ( x)   3
 x / 6,
for all y in [α, β]. Therefore, for
x  [ ,  ]
otherwise
,
EFv(x) - EGv x < for sufficiently small θ. Because 5'''(x) > 0, that contradicts the original
assumption that EFv(x)
Moreover,
EGv(x) for all '''(x) > 0.
–
'
for some y in (a, b) as long as G(x) and
F(x) are not identical.
(The “if” part)
Suppose G(x) is an expected utility preserving increase in downside risk from F(x) for
u(x). Then,
EFv(x) - EGv(x) =
QED
'''
'
–
 0 for all '''
> 0.
25
A2. Risk Adjusted Cumulative Distribution Functions
For continuously differentiable CDFs, note that the derivative of u'(x)(G(x) – F( x)) is
u'(g – f) + u''(G – F), where f and g are the densities associated with CDFs F and G. First assume
u''  0. Then u'·g + u''·G  0 and u'·f + u''·F  0. These two expressions can be density functions
if the probabilities sum to one. Integrate each over [a, b] to find the total value for each is u'(b)
and thus the normalizing factor is t = 1/u'(b). Next assume u'' < 0 and consider u'·g + u''·F  0
and u'·f + u''·G  0. Again these can be density functions if the probabilities add to one.
Because
'
'' - ' - ''
,
'
''
=
'
''
= 1/t is the
normalizing factor.
The probability distributions for these random variables have been “adjusted” by the
reference decision maker with marginal utility u'(x) so that the means of (x) and (x) are the
same. That is, they are risk adjusted so that for this person the choice between them depends
only on their mean values. It is also the case that the conditions in Definition 4 imply that (x)
is a downside increase in risk from (x) by the original MGT definition. Thus, the function
u'(x)(G(x) – F( x)), properly normalized, adjusts the random alternatives associated with G(x)
and F(x) so that he or she can act as risk neutral person and choose between them based only on
the mean values. This leads to an interpretation of Theorem 5. This theorem indicates that for
those who are more downside risk averse than the person doing this risk adjustment, that is, those
who are more decreasingly absolute risk averse than u'(x), this risk adjustment is not sufficient
and (x) is preferred or indifferent to (x).
Consider the following example for F(x) and G(x) defined on [0, 1].
26
Let G(x) =
and F(x) =
0
.4x
.4x + .6
1
x<0
0  x < .75
.75  x < 1
x1
0
.8x
.6
1
x < .25
.25  x < .75
.75  x < 1
x1
This implies that
0
G(x) – F(x) = .4x
-.4x
.4x
0
x<0
0  x < .25
.25  x < .75
.75  x < 1
x1
Consider utility function u(x) = ln x so that u'(x) = 1/x. This implies that
u'(x)(G(x) – F(x)) =
0
.4
-.4
.4
0
x<0
0  x < .25
.25  x < .75
.75  x < 1
x1
This sum of the increases (or decreases) for u'(x)(G(x) – F(x)) is 1.2 which is greater than 1 so a
scaling factor less than or equal to 1/1.2 is needed to allow u'(x)(G(x) – F(x)) to represent the
difference the difference between two CDFs. These rescaled CDFs are given below and can be
viewed as the difference between two risk adjusted CDFs where person with utility u(x) = ln x
has adjusted CDFs F(x) and G(x) so that they have equal mean values. For that person the
ranking of the two CDFs is the same, but for those more decreasing absolute risk averse, the
adjustment is not sufficient, and (x) is preferred or indifferent to (x).
27
0
1/3
-1/3
1/3
0
(x) – (x) =
x<0
0  x < .25
.25  x < .75
.75  x < 1
x1
A3. Proof of Theorem 5'
Proof: The “if” part Suppose that G(x) is an expected utility preserving increase in downside
risk from F(x) for u(x). Then for any utility function v(x),
 ''
=
=
–
''
EFv(x) - EGv(x) =

–
''
–
''
.
where the first equality is due to a) and the third equality due to b) in Definition 4'. Also from b)
in Definition 4',
–
''
downside risk averse than u(x) – 
for all x. Therefore, v(x) being more
''
< 0 according to Lemma 2 – implies EFv(x) - EGv(x)
.
The “only if” part Suppose u(x) is risk averse. The case in which u(x) is risk loving
can be similarly dealt with. Now suppose that F(x) is preferred or indifferent to G(x) for all risk
averse decision makers v(x) who are more downside risk averse than u(x). That is EFv(x)
EGv(x) =
'
–
 '
–
for all  such that 
28
Let  ' be θ '
2
+1 and θ ' 2 -1, respectively, where θ > . That the above inequality
holds for both  ' and for all positive θ implies
-
= 0, which is a) in
Definition 4'.
-
With

= 0, we have EFv(x) - EGv(x)
 '
. Let  ' be θ ' 2 + ' and θ '
–
–
2
- ', respectively,
where θ > . Again, that this inequality holds for these  ' and for all positive θ implies
–
-
With
- EGv(x) =
, which is the equality in b in Definition 4’.

''
Moreover,
, we have EFv(x)
. For all  >0, which implies
–
''
–
''
–
= 0 and
, the weak inequality of b) in Definition 4'.
–
''
for some y in (a, b) as long as G(x) and
F(x) are not identical.
QED
A4. Proof of Corollary 1'
Proof: Suppose v(x) is more downside risk averse than u(x). Then for any expected utility
preserving increase in downside risk from F(x) to G(x) with u(x) as the reference decision maker,
=
=
–
''
EFv(x) - EGv(x) =
''
'''
–
''
''
–
.
where the first equality is due to a) and the third equality due to b) in Definition 4'.
29
If we further assume that v''(x) u''(x) > 0 in [a,b], then v(x) is more downside risk averse
than u(x) implies '''(y)
y
a
''
x
a
0. Therefore EFv(x) - EGv(x)
G s – F s ds dx
since
according to b) in Definition 4'.
Conversely, if EFv(x) - EGv(x)
for all expected utility preserving downside risk
increases with u(x) as the reference decision maker, then it has to be the case that '''(y)
and
v(x) is more downside risk averse than u(x) (with the additional assumption that v''(x) u''(x) > 0
in [a, b]). The proof of this is by contradiction with arguments parallel to those in Keenan and
Snow (2009). Suppose '''(y) > 0 at some point y0 in [a, b]. Then there would exist a
surrounding interval within [a, b] on which '''(y)
, which implies by choosing G(x) - F(x) to
be concentrated on the same interval, one would obtain EFv(x) - EGv(x)
initial assumption that EFv(x) - EGv(x)
.
, contradicting the
QED
30
References
W.H. Chiu, Skewness preferences, risk aversion, and the precedence relationships on stochastic
changes, Manage. Sci. 51 (2005) 1816-1828.
D. Crainich, L. Eeckhoudt, On the intensity of downside risk aversion, J. Risk Uncertainty 36
(2008) 267-276.
M. Denuit, L. Eeckhoudt, A general index of absolute risk attitude, Manage. Sci. 56 (2010) 712715.
L. Eeckhoudt, H. Schlesinger, Putting risk in its proper place, Amer. Econ. Rev. 96 (2006) 280289.
P. Fishburn, R. Vickson, Theoretical foundations of stochastic dominance, in G. Whitmore and
M. Findlay (eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk (1978),
Lexington Books, D.C. Heath and Company: Lexington, Massachusetts.
P. Jindapon, W. Neilson, Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A
comparative statics approach, J. Econ. Theory 136 (2007) 719-728.
D. Keenan, A. Snow, Greater downside risk aversion, J. Risk Uncertainty 24 (2002) 267-277.
D. Keenan, A. Snow, Greater downside risk aversion in the large, J. Econ. Theory 144 (2009)
1092-1101.
D. Keenan, A. Snow, Greater prudence and greater downside risk aversion, J. Econ. Theory 145
(2010) 2018-2026.
M. Kimball, Precautionary saving in the small and in the large, Econometrica 58 (1990) 53-73.
H. Leland, Saving and uncertainty: the precautionary demand for saving, Quart. J. Econ. 82
(1968) 465-473.
C. Menezes, C. Geiss, J. Tressler, Increasing downside risk, Amer. Econ. Rev. 70 (1980) 921932.
S. Modica, M. Scarsini, A note on comparative downside risk aversion, J. Econ. Theory 122
(2005) 267-271.
D. Meyer, J. Meyer, A Diamond-Stiglitz approach to the demand for self-protection, J. Risk
Uncertainty 42 (2010) 45-60.
31
J. Meyer, Representing risk preferences in expected utility based decision models, Annals of
Operations Research, ," Annals of Operations Research 176 (2010) 179-190.
J. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964) 122-136.
M. Rothschild, J. Stiglitz, Increasing risk I: A definition, J. Econ. Theory 2 (1970) 225-243.
*
The authors thank Louis Eeckhoudt and Harris Schlesinger for helpful comments. Support from the Private
Enterprise Research Center at Texas A&M University is gratefully acknowledged.
1
Menezes, Geiss and Tressler say that “It is natural to expect an individual to be averse to downside risk if he is
decreasingly risk averse.”
An example provided later in the paper also shows that the '''(u)  0 based definition allows distinct u(x) and v(x)
to each be more downside risk averse than the other. Thus, the definition is not asymmetric either.
2
-
3
If risk averse functions are desired, v(x) = -e dx, u(x) = -e
example.
-cx
and (u) = -(-u)
d/c
provides an almost identical
4
Because not all functions have integrals of closed form, it may not be possible to determine the u(x) associated
with marginal utility u'(x).
5
All these measures of the intensity of downside risk aversion have some desirable as well as undesirable properties.
See Crainich and Eeckhoudt (2008) for the desirable properties of d(x), Keenan and Snow (2002, 2009) for the
desirable properties of s(x). An undesirable property shared by s(x) and -Au'(x) here is that the intensity measure of
downside risk aversion may be inconsistent with the direction of downside risk aversion. For example, quadratic
utility functions are downside risk neutral (because u'''=0), but s(x) < 0 and -Au'(x) < 0. In the case of constant
absolute risk averse utility functions, the direction of downside risk aversion is positive (meaning decision makers
with these utility functions are downside risk averse) while the intensity measure is zero for -Au'(x) and negative for
s(x).
6
Notice that this same use of marginal utility can be made when discussing when one decision maker is more risk
averse than another. It is the case that v'(x) is more risk averse than u'(x) if and only if A (x)  0. This provides an
additional characterization of the Arrow-Pratt definition of more risk averse.
7
Denuit and Eeckhoudt 2 1
extend Chiu’s analysis to higher order risk aversion.