Arrow-Pratt’s risk aversion: 50 years later1
by
Louis Eeckhoudt
Iéseg School of Management (Lille) and CORE (Louvain)
The papers by Arrow (1965) and Pratt (1964) about the measurement of risk aversion
have had a huge impact both on the theoretical literature devoted to the economics of risk
and on fields of application that use concepts of risk (e.g. finance, agricultural economics or
health economics,…). A quick look at Google Scholar indicates besides that both papers had
a similar success: 4227 citations for Arrow (1965) and 5469 for Pratt (1964).
In this presentation I'll describe some theoretical developments that followed these
leading papers and, in some sense, contributed to their lasting success. Some of them will be
well known to many of you while others didn't receive – in my opinion – the attention they
deserved.
Let me also stress that, because of the longlife (biased) orientation of my research
activities, I'll limit myself to the theoretical developments inside the EU model2. This being
said, it should be obvious that the extension to non EU models does represent a very
valuable research topics, as I shall point out in the conclusion.
My presentation will be organized as follows. I'll start with a quick review of ArrowPratt's main results. In the following 3 sections I'll concentrate on the extensions and
"criticisms" that led for most of them to significant developments around the founding
papers.
The first development is due to M. Kimball (1990) who – by looking at precautionary
savings – defined the notion of absolute prudence and made the assumption that – as
absolute risk aversion – it should be decreasing in wealth. This observation was at the origin
of the "background risk" literature that culminated in the concept of risk vulnerability
(Gollier-Pratt (1996)) and gave rise also to the notion of absolute temperance.
The second development had – unfortunately – much less influence than the other
two ones. It started with the notion of "one-switch" utility functions by D.Bell (1988), which
represents a clever alternative (complement?) to the DARA assumption. In fact the idea of
"one-switch" utility puts forward two utility functions: the "sumex" utility and – when "oneswitch" is combined with DARA – the "linex" utility.
1
An alternative title might be "de Finetti and Arrow-Pratt's risk aversion: more than 50 years ago" keeping in
mind that de Finetti (1952) had already discussed in some depth the concept of risk aversion. For an update,
see Montesano (2009).
2
Survey papers either on the theoretical developments outside EU or on empirical applications would be
valuable complements to the present one. At this stage let me already mention three papers of which I am
aware that compare at the theoretical level the EU approach of risk aversion with non EU ones: M. Cohen
(1995), C. Hong, E. Karni, Z. Safra (1997) and M. Machina (1989). Of course a survey of a non EU approach of
risk aversion would probably reveal many more contributions.
1
Interestingly these two utility functions will appear again in other developments
discussed in the following sections. Indeed we show in section 3 that the sumex utility is
related to an important but (so far) little used paper by Caballé and Pomansky (1996) around
the notion of "mixed risk aversion" and its properties. The linex utility will appear again in
section 4 when we present the last developments around the Arrow-Pratt's contributions.
The last development discussed in section 4 resulted from two papers published by
Ross and Kihlstrom-Romer-Williams in the same year (1981) in the same journal,
Econometrica. Both papers deal essentially with the same question: how does an initial
random wealth (instead of a safe one as in Arrow-Pratt) affect the cost of a risk and its
properties? The paper by Ross led to many extensions and developments until recently and
we'll try to summarize them.
While the topics of all these sections are almost exclusively "historical" I'll end up in
the last section with a short statement about results recently obtained on related questions.
In the conclusion I'll indicate how in my opinion these developments might be useful
outside the expected utility model.
1. Arrow-Pratt's risk aversion: 1965-1964
Given my goal in this talk, it will be easier to follow Pratt's definition of more risk
aversion3.
For my purpose let me refer to his theorem 1 in section 5 "Comparative risk
aversion"4. Pratt presents 5 definitions of more risk aversion5, of which he shows that they
are equivalent.
A decision maker (DM) v is more risk averse than a decision maker u if:
(a) At each wealth level, the index of absolute risk aversion of v is at least equal (or
v ''( x)
u ''( x)
larger for at least one x ) to that of u 6: −
(1)
≥−
v '( x)
u '( x)
(b) For all pairs ( x, εɶ ) , the risk premium of v exceeds that of u :
π v ( x, εɶ ) ≥ π u ( x, εɶ )
where, as is well known π ( x, εɶ ) is defined by
u ( x + E (εɶ ) − π u ( x, εɶ )) = E[u ( x + εɶ )]
3
(2)
While Pratt's paper "dominates" that of Arrow for the definition of increased risk aversion, Arrow's
contribution to its implications is clearly richer than that of Pratt. Arrow's 1963 paper on medical care is a good
example.
4
His paper which is 15 pages long is made of 13 sections, most of them very concise and intense.
5
Most of these definitions are well known but I repeat them for the sake of completeness. Others are less well
known and give rise to comments that will be useful later.
6
Unless indicated otherwise we use a standard notation for successive derivatives of the utility function. For
instance the third derivative of u might be denoted either u ''' or u (3) .
2
(c) The third condition is much less known and is related to the "probability premium"7.
Let's consider a binary zero-mean lottery.
½
½
The probability premium p u associated with this lottery for DM u given initial
wealth x is defined by:
1

1

u ( x) =  − p u  u ( x − ε ) +  + pu  u ( x + ε )
2

2

(3)
Of course if v is more risk averse than u , p v > p u .
At this stage let's make an observation that will be useful at the end of my talk.
Observe that in (b) and (c) a risk εɶ is involved along with a sure wealth x . If the risk εɶ is
zero mean this corresponds to a mean preserving increase in risk. Relative to the
absence of risk, this "second order" deterioration is compared in (b) and (c) with "first
order" changes: either a sure loss in wealth in (b) or an improvement in the probability of
the good outcome in (c). As we indicate in section 4, this observation is exploited in e.g.
Machina-Neilson (1987) and in Liu-Meyer (2013).
(d) The utility function of v is an increasing and concave transformation of u .
v = k (u )
with k ' > 0 and k '' < 0 .
(4)
(e) For any pair of points x and y with x < y , v is more risk averse than u if8:
v '( x) v '( y )
>
u '( x) u '( y )
(5)
7
More information about this definition and its properties as well as extensions can be found in Jindapon
(2010).
8
Eq. (5) is not found as such in Pratt. It appears in Kihlstrom, Romer and Williams (1981).
3
The graphical interpretation given in figure 1 offers the corresponding intuition.
u
u
v
x
y
Figure 1
wealth
As condition (c), condition (e) is almost never used. However we'll see that it is
interesting when we discuss Ross' notion of "stronger risk aversion".
After the formal definition of risk aversion (and its increase) Pratt presents in section
7 the assumption of decreasing (global or local) risk aversion that will play for decades a
central role in the analysis of risky choices.
While the first 12 sections of Pratt's paper concentrate on theoretical aspects of the
notion of risk aversion, the last section, nicely called "related work of Arrow" indicates
how to use these notions, along with the assumption of decreasing absolute risk aversion
(DARA) to analyze risky choices (more precisely the optimal composition of a portfolio
made of one safe and one risky asset).
In preparing this talk I noticed a point to which I had never paid attention before:
Pratt's paper has no bibliography9. This anecdotal observation will have some impact on
the development of research in the field since for a long time no attention will be paid to
Friedman-Savage's paper of 1948 where they had defined the interesting notion of
"utility premium" besides that of the risk premium10 (using however a different
terminology).
The notion of "utility premium" received almost no attention for decades. To the best
of my knowledge the only exception is a paper by Hanson-Menezes (1973) under the
very appropriate title "On a neglected aspect of the theory of risk aversion". But again,
this paper had no influence in the profession until developments were made rather
9
The only citation is in a footnote at the beginning of section 13 where Arrow's class notes for Economics 285
at Stanford University are mentioned!
10
While Friedman-Savage notion of utility premium was neglected for decades, their assumption of a utility
function with a concave part and a convex one has retained regular attention.
4
recently around the concept of "higher-order" risk attitudes (see Eeckhoudt-Schlesinger
(2009) and Eeckhoudt (2012) for a survey).
2. Kimball's prudence (1991)
Kimball's paper published more than 25 years after those of Arrow-Pratt – in the
middle of our review period – contributed much to the lasting success of the original
contribution.
While the paper is famous for its link with the behavior of precautionary savings, it
can be equivalently interpreted in a different way. Since marginal utility (and not total utility)
is the relevant concept for the analysis of decisions, it might be interesting to characterize
the cost of risk through its impact on marginal (and not total) utility. Instead of computing a
risk premium that keeps total utility constant, Kimball suggests the prudence premium (ψ )
that keeps marginal utility constant. Formally one has:
u '( x + E (εɶ ) −ψ u ( x, εɶ )) = E[u '( x + εɶ )]
(6)
It is then easy to show that:
ψ u ( x, εɶ ) ≅
where −
σ ε2ɶ  u '''( x) 
−

2  u ''( x) 
(7)
u '''( x)
is the index of absolute prudence.
u ''( x)
In the same way as Arrow-Pratt had shown that more risk aversion induces decisionmakers to hold portfolios with a larger share of safe assets, Kimball proves that more
prudence induces more precautionary savings.
Pursuing the relationship with Arrow-Pratt, Kimball makes then the assumption that
the index of absolute prudence – as that of absolute risk aversion – is decreasing in wealth
(DAP).
Quite interestingly the joint assumption of DARA and DAP became the starting point
of a very intense research activity around the topics of "background risks", a topics quite
specific to the expected utility model11. Intuition suggests that a decision-maker who has to
face (independent) background risks that he doesn't control (and doesn't like) will behave in
a more risk averse way towards the risk he can manage. The question then is: which
property should the utility function satisfy in order to generate the intuitive behaviour? This
search – to which Pratt contributed much – gave rise to the new concepts of "standardness"
(Kimball (1993)), properness (Pratt-Zeckhauser (1987)) and risk vulnerability (Gollier-Pratt
(1996)) which are more restrictive than the DARA one and which are summarized in Gollier
(2001), especially chapters 8 and 9. Notice that all these developments contributed to the
11
To the best of my knowledge, the first papers on these topics were published by Doherty-Schlesinger (1983)
and Mayers-Smith (1983) without using the notion of prudence that was developed later.
5
popularity of a third index of absolute risk attitude, i.e. the index of absolute temperance
u ''''( x)
.
defined by −
u '''( x)
It is fair to say that the theoretical developments in this direction, at least in the
expected utility model, have come to maturity and I do not think many new concepts will be
elaborated. However there is still much room for a rich empirical research around the impact
of background risk. This line of research was initiated by L. Guiso and his co-authors on
Italian data12 and pursued by Arrondel-Prado-Olivier (2010) in France. While these papers
deal with unidimensional utility functions, a theoretical and empirical literature also
develops around the impact of background risks in a multidimensional framework (see e.g.
Evans-Viscusi (1991), Eeckhoudt-Hammitt (2001) and the references therein).
At this stage it is to be mentioned that Pratt's 1964 paper and Kimball's one share an
unexpected feature. As already mentioned Pratt pays mostly attention to the intensity of risk
aversion and disregards the concept of utility premium of Friedman-Savage (1948) (i.e. "a
direction"). The same is to some extent true for Kimball's paper where the notion of
– i.e. an intensity – plays a central role while the paper of
absolute prudence −u '''
u ''
Menezes-Geiss-Tressler (1980) that deals with the positive sign of u ''' (i.e. a direction) is not
even quoted13. Menezes-Geiss-Tressler (1980) termed u ''' > 0 "downside risk aversion" and
related it to the concept of "mean variance preserving transformations"14 which stress the
role of skewness in the evaluation of risk and have implications for the analysis of selfprotection (see Briys-Schlesinger (1990) for the initial contribution and Eeckhoudt-Gollier
(2005) for an update).
(
)
Finally, to anticipate future developments mainly in section 5 a last comment on
Kimball's contribution is in order.
As is well known, Kimball's prudence is defined from the analysis of a decision
problem related to the optimal amount of precautionary savings. This presentation has an
advantage and…a cost. The advantage is obvious: it nicely motivates the use of the concept
and it opens the way for many developments around the application, here the theoretical or
empirical analysis of savings decisions. However, without surprise, there is a cost related to
the specificity of the decision problem and one might search for a more general presentation
of the concept linked exclusively to the preferences of the decision-maker. It is worth
pointing out at this stage that Menezes-Wang (2004) and Chiu (2005) gave a justification of
prudence based exclusively on the analysis of the utility function of the decision-maker in
relationship with statistical transformations of the risks he faces (for a generalization and an
extension to the nth order see Denuit-Eeckhoudt (2010a)
12
See e.g. "Earnings uncertainty and precautionary savings" by Guiso, Jappelli, Terlizzese (1992), "Background
uncertainty and the demand for insurance against insurable risks" by Guiso, Jappelli (1998) and "Risk aversion,
wealth and background risk" by Guiso, Paiella (2008).
13
Interestingly another paper published in the Journal of Finance (Scott-Horvath (1980)) also pays attention to
the sign of u ''' and relates it to an attitude towards skewness.
14
At the 2nd order u '' < 0 is related to "mean preserving transformations" in the sense of Rothschild-Stiglitz
(1970, 1971).
6
3. "One switch" utility and mixed risk aversion
To justify the idea of a "one-switch" utility, D. Bell (1988) compares two risky
situations denoted A and B15. Given your current wealth you prefer A but you also know that
if your initial wealth were at least 25000€ higher you would prefer B (you would "switch"). If
you obey the "one switch rule" it means that if your wealth increases by more than 25000€
you'll never return to A and that for any wealth below your current one you'll always prefer
A16.
Before characterizing "one switch utility functions", D. Bell first observes that there
exist only two zero-switch utility functions: the linear and the exponential utilities (which are
also the only utility functions that have constant absolute risk aversion (CARA) everywhere).
Intuition then suggests that the DARA assumption and the one-switch rule are very
close concepts. However, this is not quite true because:
•
•
•
Some DARA utility functions (e.g. the log utility) are not one switch and Bell
illustrates with an example for which a logarithmic utility generates two
switches.
IARA (increasing absolute risk aversion) utility functions may also generate the
one-switch rule.
Finally, and maybe more importantly, DARA applies to a comparison between
risk and certainty while the one-switch is defined for two risky situations. In
this respect, it is interesting to observe that Bell (1988) ignores Ross's
contribution (1981) where also two risky situations are compared.
In fact, in his proposition 2, D. Bell (1988) shows that a utility function satisfies the
one switch rule if and only if it belongs to one of the following families17:
a)
b)
c)
d)
The quadratic
The sumex u ( w) = aebx + ce dx (with all constants negative)
The linex u ( w) = ax + becx (with a > 0 and b and c < 0 )
The linear times exponential
If one adds two conditions:
-
DARA
At extremely high wealth levels one approaches risk neutrality
then the only feasible utility is the linex one (proposition 3 in Bell (1988))18.
15
Notice the difference with Arrow-Pratt where risk is compared to certainty.
nd
Although he doesn't mention it, D. Bell makes the implicit assumption that there is no 2 order stochastic
dominance relationship between A and B which would yield the "zero switch rule". This "zero switch rule" also
occurs with linear or exponential utility functions.
17
Quite interestingly the proof of proposition 2 (which is the central result) implicitly uses the never mentioned
concept of the utility premium proposed by Friedman-Savage (1948) and defined here in appendix 1 (eq.
(A.1.3)). (See D. Bell (1988) page 1418).
18
Although only part of Bell's initial contribution is presented here, it should be mentioned for the interested
reader that he proposed further developments in related papers (Bell (1995) and Bell and Fishburn (2001)).
16
7
Despite its interest Bell's paper didn't receive much attention. However – as we will
see – some of its propositions were developed later on, using a different methodology and
without reference to the original paper. Maybe this can be explained by the fact that at the
time of Bell's publication – end of 80's in Management Science – this journal was not yet
popular in the economics profession.
The first development in the spirit of Bell's contribution is linked to the sumex utility
function19. It was made by Caballé-Pomansky (1996) who discussed the concept of "mixed
risk aversion".
Their paper starts with the observation that "most utility functions used to construct
examples of choice under uncertainty share the property of having odd derivatives positive
and all even derivatives negative". The purpose of the paper is then "to characterize the
class of utility functions exhibiting this property" and they call them "mixed risk averse".
Using Bernstein's theorem they then show that such functions are mixtures of exponential
utilities20, as the sumex of D. Bell.
Although the paper is pretty technical (at least from my point of view) it also contains
very intuitive developments. One of them refers to a comparison of simple lotteries. These
lotteries, termed even or odd ones, are generated by throwing n times a balanced coin. In
the first sequence of lotteries – the even ones – the individual's payoff is kh €21 when the
number of heads is even ( k ) and it is zero when the number of heads is odd. In the second
sequence of lotteries – the odd ones – the reverse occurs: you get kh if k is odd and zero
otherwise.
Besides, as shown by Denuit, Eeckhoudt and Schlesinger (2013) the linex utility is implied by the assumptions of
DARA and DAP in Ross' sense (see section 4).
19
The second one – based on the linex utility – is developed in section 4.
20
Caballé-Pomansky show that utility functions such as the power and logarithmic ones (with very minor
modifications) are in fact a mixture of exponentials even if they don't look like that at first glance.
21
h is any strictly positive number.
8
Let's illustrate for n = 2 , n = 3 , n = 4
Even lotteries
n=2
3
Odd lotteries
0
1
4
xɶB
xɶ A
1
n=3
0
2
1
4
2h
0
5
2
1
8
1h
0
2
3
xɶB '
xɶ A '
3
n=4
1
8
8
8
2h
3h
0
9
0
1
16
6
xɶ A ''
1
16
16
1h
2
1
xɶB ''
2h
1
4h
4
1h
4
3h
Figure 2
What do we observe?
First, for any n , the first (n − 1) moments of the even and odd lotteries are equal.
Besides when n is even the nth moment of the even lottery is larger than that of the odd
one. When n is odd the reverse occurs22.
It follows from these properties that these lotteries correspond to what Ekern (1980)
termed " nth degree increase in risk"23 which is a very interesting and rich special case of nth
degree stochastic dominance.
At n = 2 , E ( xɶ A2 ) > E ( xɶ B2 ) and at n = 4 , E ( xɶ A4 " ) > E ( xɶ B4 " ) while at n = 3 , E ( xɶ 3A ' ) < E ( xɶ B3 ' ) .
A similar (but not identical) situation can be obtained from alternative simple stories. See e.g. "risk
apportionement" (Eeckhoudt-Schlesinger (2006)) or "combining good with bad" (Eeckhoudt-Schlesinger-Tsetlin
(2009))
22
23
9
Then, using (again without mentioning it) the concept of utility premium, CaballéPomansky prove that the odd lotteries will always be preferred to the even ones if the utility
function is mixed risk averse (see their proposition 3.1), i.e. when it is a mixture of
exponential utilities for which the successive derivatives of u alternate is sign.
The paper by Caballé-Pomansky also stresses another important feature of "mixed
risk aversion". These authors extend to higher orders the coefficient of absolute risk aversion
u ( n +1) ( x)
denoted An ( x) 24.
and absolute prudence by considering expressions such as − n
u ( x)
They then show (see their proposition (3.3)) that u is mixed risk averse if and only if
An ( x) is non-increasing in x for all n = 1, 2, 3...
Besides these theoretical developments Caballé-Pomansky also describe interesting
properties of mixed risk aversion for topics such as stochastic dominance and portfolio
selection25.
I think that globally Caballé-Pomansky's paper didn't receive so far the attention it
deserved but things might change because of developments in financial economics. Papers
by Corrado and Su (1997), Harvey-Siddique (2000), Dittmar (2002), Chabi-Yo (2012), Dionne
et al. (2012) and Duan-Zhang (2014) all stress the importance of skewness and/or kurtosis
for the evaluation of risks in financial markets while the paper by Caballé-Pomansky is
among the first ones in economics to create interesting links between these moments and
properties of the utility function.
Progress in this direction is also made in Chiu (2005) and in Denuit-Eeckhoudt
(2010a). In his paper, Chiu considers an individual u who is indifferent between lotteries A
and B where A and B have the same mean while A has a larger variance and a larger
u ''' 
 v '''
skewness. Then any decision maker v who is more prudent than u  −
> −  will
u '' 
 v ''
prefer A (since this more prudent decision maker is relatively more sensitive to an increase
in skewness than an increase in variance).
u (n)
u ( n −1)
and the associated couple of moments. For instance, consider temperance. If mr u is
indifferent between  and B̂ that have the same mean and variance while  has a higher
The paper by Denuit-Eeckhoudt (2010a) extends this result to any ratio such as
We'll meet again such expressions in this section and in section 4 for n > 3 . When Caballé-Pomansky wrote
their paper, such expressions didn't yet exist and they interestingly note in 1996 that "we lack an economic
interpretation of An ( x) for values of n greater than 2" (page 495). Recent developments described in this
section and in section 4 have the objective of giving such an economic interpretation. Notice also that AumannKurtz (1977) and Foncel-Treich (2005) have extended −u ''
in the other direction ( u ' ) by defining the
u'
u
notion of fear of ruin.
25
For an application to self-protection see Dachraoui et al (2004) who also define the notion of "more mixed
risk aversion".
24
10
 v (4)
u (4) 
skewness and a higher kurtosis then if v is more temperant than u  − (3) > − (3)  he will
u 
 v
prefer  .
4. Stronger measures of risk aversion
During the same year (1981) in the same journal (Econometrica) S. Ross and
Kihlstrom-Romer-Williams (KRS in short) raised a very important question.
As is obvious from equations (2) and (3), Arrow-Pratt had always compared a risky
situation to a risk free one. The question raised by Ross (1981) and KRS (1981) can then be
stated as follows: are all the nice properties obtained for the "risk-risk free" case maintained
when one compares 2 risky situations?
Before looking at the developments around this question, its importance should first
be stressed. Indeed for many decades, hundreds (thousands?) of mostly empirical papers
dealt with the following problem: what is the willingness to pay (WTP) for a reduction in the
probability of an adverse event (car accident, illness, death,…)26? Quite obviously such
questions necessarily involve a comparison of risky situations. Nevertheless many empirical
papers on these topics explicitly or implicitly refer to the Arrow-Pratt concepts. This is an
illustration of a situation where the empirical developments have preceded the theoretical
ones.
To develop their analysis, Ross and KRW use a similar structure: an individual faces
two sources of risks ( xɶ and εɶ ) and is willing to pay to get rid of one of them. To compare
the two papers, we use a simple representation linked to that adopted by Ross
KRW
Ross
1
2
−ε
1
1
2
p
+ε
1
1
1− p
1− p
x2
−ε
x1
x1
p
2
1
2
2
+ε
−ε
x2
0
1
Figure 3(a)
2
+ε
Figure 3(b)
26
An even partial review of this literature is out of the topics of the present paper. Let me just mention that –
to the best of my knowledge – the first paper on this issue dealt with the value of a statistical life and was
published by Drèze (1962) in a French journal of operations research.
11
In both papers one discusses the willingness to pay to get rid of εɶ 27 while keeping
the initial risk xɶ . The difference between the two papers is that for KRW the two risks are
independent which is not the case for Ross where the condition E[εɶ / xɶ = x] = 0 for all x
prevails.
Because KRW's analysis is limited to independent random variables (so that a small
number of risky situations are compared than in Ross) the condition imposed on the utility
function to maintain the AP properties are not very strict. In KRW's model the results are
obtained if at least one of the utility function exhibits non increasing absolute risk aversion, a
not so demanding condition.
Matters are very different in Ross' framework. To give the intuition behind his result,
Ross approximates the risk premium the individual is willing to pay to get rid of εɶ in the
lottery described in figure 3a (left side) while retaining the foreground risk on xɶ . He easily
obtains28:
1
pu ''( x1 )ε 2
2
π u (εɶ ) ≅
pu '( x1 ) + (1 − p )u '( x2 )
(8)
Then, using the exponential utility function as an illustration, he shows that even if
the decision maker v is more risk averse than u in the Arrow-Pratt' sense, it may be that
π v (εɶ ) < π u (εɶ )
and he explains why this counter intuitive result occurs29. It is essentially due to the
fact that the benefit of the risk reduction is obtained in one state of the world ( x1 ) while the
premium is paid in both states and this observation will play an important role soon.
To avoid such a situation Ross needs the concept of a stronger measure of risk
aversion that appears in the following condition:
v is strongly more risk averse than u if and only if for any pair ( x1 , x2 ) there exists a
positive constant λ such that:
v ''( x1 )
v '( x2 )
≥λ ≥
u ''( x1 )
u '( x2 )
Then in his theorem 3 Ross shows the equivalence between (9) and
π v (εɶ ) ≥ π u (εɶ )
27
In figure 3a, it is assumed that a degenerated zero mean risk is attached to x2 . However any zero mean risk
(including εɶ itself) might be attached to x2 .
28
Since Ross's paper was published 10 years before Kimball's one he couldn't observe that the denominator of
(8) is linked to the…prudence premium. More on this in the last section of the present paper.
29
An alternative and interesting explanation can also be found in Pratt (1990).
12
(9)
Besides Ross adds a 3rd equivalent condition:
There exists a non increasing and concave function G such that:
v = λu + G
(10)
At this point, it is worth stressing that the economic interpretation of
condition (9) is very much linked to the notion of the utility premium30 (see Appendix 1).
Indeed when one shifts from u to v :
- the left hand side of (9) represents the relative increase in the pain inflicted by a
zero mean risk
- the right hand side of (9) is the relative increase in the pain inflicted by the sure loss
of a monetary unit of wealth.
Because the pains can be inflicted at different points in the interval of wealth that is
considered (see above), condition (9) must be satisfied for all pairs of x1 and x2 if one wants
that π v is at least equal to π u .
In order to show that his definition of strongly more risk averse is not vacuous, Ross
gives an example on page 630: it is the linex utility that D. Bell suggested seven years later as
one of the "one-switch" utilities. We'll come back to this "co-incidence".
For a long period Ross' contribution raised little interest in the profession despite two
very important suggestions made by Machina-Neilson (1987) in their "strengthening and
extension of Ross' characterization of more risk aversion". In that paper indeed, besides
other interesting points Machina-Neilson:
•
•
suggest to view the risk premium as a marginal rate of substitution (MRS) "between
risk and premium payments". MRS is a basic concept in models of choice under
certainty and – surprisingly – it remained widely unused in models of choice under
risk even after Machina-Neilson's paper31.
while in Arrow-Pratt's framework the risk premium is paid in the state of certainty, it
may be paid in only some or in all states of the world when two risky situations are
compared. This observation by Machina-Neilson is very important and it plays in fact
an important role in some recent developments.
After two decades with no further developments around Ross' approach, three papers in
the period 2005-2008 revived the interest for his contribution. A very interesting paper by
Modica-Scarsini (2005) extends Ross' development to the 3rd order. As Ross lotteries
depicted in figure 2 clearly indicate, Ross is interested in 2nd order risk increases in the sense
30
In fact I discovered the economic interpretation of (9) after I found back the notion of the utility premium
while working on the paper Eeckhoudt-Schlesinger (2006).
31
See also the concept of first order and second order risk aversion by Segal-Spivak (1990) which also use
implicitly the MRS approach. Similarly in the two states diagram for the representation of binary risks, which is
very popular in insurance economics, the notion of MRS also appears.
13
of Rothschild-Stiglitz (1970-1971) i.e. mean preserving increases in risk. However, when risky
situations are compared many other cases can arise and one may have to compare lotteries
that correspond to mean variance preserving increases in risk. Then the equivalent of
condition (9) becomes condition (11)
v '''( x1 )
v '( x2 )
≥λ ≥
(11)
u '''( x1 )
u '( x2 )
to always generate an increase in the risk premium for a partial risk reduction when
one shifts from u to v .
A by-product of this condition is that, for x1 = x2 , u '''
is an appropriate measure of
u'
the intensity of downside risk aversion, which is an alternative to the intensity of prudence
proposed by Kimball − u '''
(see Crainich-Eeckhoudt (2008))32.
u ''
(
)
Once it was shown how to go from the second to the third order, the nth order
extension was rather immediate and it was discussed in Li (2009) and in Denuit-Eeckhoudt
(n)
n −1 u
that can be compared with those
(2010b). It gave rise to coefficients such as (−1)
u'
u(n)
resulting from the idea of mixed risk aversion − ( n −1) as in Caballé-Pomansky (1996)33.
u
The third paper of the 2005-2008 period is that of Jindapon-Neilson (2007). It is
especially interesting because in a sense it generalizes the very important contribution of
Ehrlich-Becker (1972) about self-protection and its relationship with market or selfinsurance.
Remember that Ehrlich-Becker discuss how much should be spent to reduce the
probability of a bad event in a binary lottery. In other words efforts are made to increase the
likelihood of a good event which corresponds to a (specific) improvement in the sense of 1st
order stochastic dominance. Quite naturally Jindapon-Neilson raise the same question when
the effort increases the likelihood of a better event in the sense of an nth order reduction in
risk a la Ekern (1980). They show that either the Arrow-Pratt-Kimball's approach or that of
Ross can be generated by considering the way in which the cost of the effort is expressed
u(n)
u (n)
and they obtain expressions such as − ( n −1) or (−1)n −1
to characterize risky choices.
u
u'
The discussion around Caballé-Pomansky (1996) and the more recent extensions of
Ross' contribution shows a sequence of partial progresses on the topics of "higher-order risk
32
For another definition of downside risk aversion see Keenan-Snow (2002).
Quite interestingly the term u ''' seems to appear in Harvey and Siddique (2000), a paper of the Journal of
u'
Finance devoted to the role of conditional skewness in the asset pricing model. Notice also that in a recent
research paper, Huang (2012) reviews alternative measures of downside risk aversion that were proposed in
the literature.
33
14
attitudes" and obviously a synthesis was necessary34. Fortunately for us this synthesis was
provided a few months ago by Liu-Meyer (2013).
These authors return to the fundamental idea of introducing the concept of marginal
rate of substitution in risk theory (see also Machina-Neilson 1987)). More precisely they
define a general rate of substitution of one stochastic change (of order n ) to another of
order m with n > m ≥ 1 .
To illustrate their approach consider 3 cumulative distributions F ( x) , G ( x) , H ( x)
where:
G ( x) represents an nth degree increase in risk of F ( x) (in Ekern's (1981) sense)
H ( x) represents an mth degree increase in risk of F ( x) again in Ekern's (1981) sense
Then for a utility function u , the marginal rate of substitution between these
changes, denoted Tu ( F ( x), G ( x), H ( x)) is equal to
b
∫ u ( x) d ( F − G )
∫ u ( x)d ( F − H )
a
b
(12)
a
given that xɶ is in the interval [a, b] 35.
At this stage it is worth stressing (again) the importance of the concept of utility
premium in equation (12) since the numerator and the denominator of (12) are each a utility
premium as defined in (A.1.3). Notice further that the Arrow-Pratt premium is a special case
of (12). Indeed let n = 2 and m = 1 and consider that F and H are degenerate distributions
(at x for F and at k smaller than x for H ) while G is a mean preserving increase of F . In
that case
Tu =
u ( x) − E[u ( x + εɶ )]
u ( x) − u (k )
(13)
with k < x .
When k = x − π one recovers Arrow-Pratt's risk premium and Tu is equal to 1: the
change in utility in the numerator and that in the denominator are said to be "an equivalent
variation" of each other (see footnote 8 in Liu-Meyer).
34
A rather similar sequence applied to the impact of mean preserving increases in risk on risk taking. The
question was initially raised by Rothschild-Stiglitz in 1971 and it then gave rise to a sequence of partial
developments for about 25 years. All this ended up with Gollier (1995) contribution. For a survey of this
literature see Eeckhoudt-Gollier (2000).
35
As any MRS, Tu is a ratio of marginal utilities since both the numerator and denominator represent changes
in utility. Besides Liu-Meyer show that there exists a link between their MRS and the probability premium.
15
It is also easy to show that for m > n ≥ 1 , Tu is the solution of:
∫ u( x)dG( x) = (1 − T ) ∫ u ( x)dF ( x) +T ∫ u ( x)dH ( x)
u
u
This presentation is very close to that of Jindapon-Neilson, with a (small) advantage
for that of Liu-Meyer since it applies to preferences in general and is not linked to a specific
choice problem.
Once Tu is defined, Liu and Meyer present a Theorem that extends to any n and m
the equivalent theorem in Ross for n = 2 and m = 1 described on page 12 and they show
n
that if Mr v is more ( ) risk averse than Mr u in the sense of Ross then Tv > Tu . From their
m
th
theorem they also produce for utility u the n
degree of absolute risk attitude denoted
m
( )
A( x) n =
m
(−1) n −1 u ( n ) ( x)
(−1)m −1 u ( m ) ( x)
which includes as special cases all the indexes we presented earlier (absolute risk
aversion, prudence, downside risk aversion, temperance,…).
5. Some additional results
In this last section I present some complementary results around the developments
induced by Ross's paper.
First notice that Ross (1981) and KRW (1981) both use one of the various
transformations that induce a mean preserving spread (namely the addition of an
independent zero-mean risk). However the problem can be stated in more general terms by
asking what the decision maker is willing to pay to transform a zero mean risk Yɶ into
another one Xɶ that is less risky. To get intuition about the problem, let us represent the
initial situation on a line of increasing utility as in figure 4.
Yɶ
π YX
πX
Xɶ
0
increasing utility
πY
Figure 4
Of course the utility of 0 with certainty exceeds that of Xɶ which itself exceeds that
of Yɶ . The arrow from Yɶ to Xɶ represents the willingness to pay to reduce risk ( π YX ) and it is
easily seen that π YX is related to the difference between two Arrow-Pratt's risk premia: π Y
and π X . Although π YX seems related to π Y and π X by:
16
π YX = π Y − π X
(14)
this relationship is not exact . The reason for this is related to the notion of prudence and
its link with the idea of risk apportionment (Eeckhoudt-Schlesinger (2006)). When π YX is
being paid to reduce the risk Yɶ the decision maker becomes poorer so that – under u ''' > 0
– the risk Xɶ loses some attractivity and this slows down the expansion of π YX . This intuition
36
is confirmed by first formally defining π YX which is an amount of money such that:
E[u ( w + Yɶ )] = E[u ( w + Xɶ − π YX )]
(15)
As in Arrow-Pratt, a second order approximation is applied to the left hand side of
(15) and a first order one to its right hand side, so that:
σ Y2
u ''( w) ≅ E[u ( w + Xɶ )] − π YX E[u '( w + Xɶ )]
(16)
2
Applying now a second order approximation to E[u ( w + Xɶ )] and simplifying we are
left with37:
u ( w) +
σ Y2 − σ X2
π YX =
(−u ''( w))
2
u '( w −ψ Xɶ )
(17)
where we have used the definition of prudence (ψ Xɶ ) as in Kimball to replace E[u '( w + Xɶ )] .
To get further insight, (17) can also be written:
π YX =
σ Y2 − σ X2 (−u ''( w))
2
u '( w)
u '( w) u '( w −ψ Xɶ )
(18)
If one were to apply the approximation to (14) one would be left with the first two
terms on the right hand side of (18) i.e. half the difference between the variances multiplied
by the degree of absolute risk aversion. It is then obvious that this approximation is
improved if one multiplies the first two terms by a correction factor u '( w)
which
u '( w −ψ Xɶ )
is smaller than unity under risk aversion and prudence. This correction factor has an
important feature: it jointly represents risk aversion and prudence. Indeed prudence
explicitly appears in the denominator and risk aversion is implicitly present because the
correction factor is a ratio of marginal utilities at two different points when prudence is
strictly different from zero38.
This result also confirms the intuition developed earlier: prudence slows down the
increase in π YX . As a result when one deals with partial risk reductions, the absolute risk
aversion – that tells the full story for a risk elimination – has to be combined with prudence
36
The true relationship is presented in appendix 2 following an argument developed in Doherty-LoubergéSchlesinger (1987).
37
Compare with Ross (1981) especially his equation (1) in which the denominator is an expected marginal
utility, hence involving prudence, a concept that was to be defined 10 years later.
38
If u is quadratic, prudence is zero and the correction factor equals unity. In that case π YX = π Y − π X .
17
in order to determine the value of π YX . With respect to 2nd order partial risk reductions, risk
aversion and prudence appear to be jointly relevant.
This joint effect of risk aversion and prudence on π YX has implications for the
notion of increased risk aversion as defined by Pratt, i.e. the concavification of the utility
function (eq. (4)). As shown in appendix 3 this concavification has an ambiguous impact on
the degree of prudence so that it is impossible to claim that the concavification of u will
raise π YX . In fact to obtain such a result one has to consider as in Eeckhoudt-Schlesinger
(1994) and Liu-Meyer (2013) the following transformation:
v( w) = aw + u ( w)
(19)
which is a generalization of the linex utility of D. Bell (1988).
It is easily shown that with such a transformation of u the introduction of a strict
positive a reduces risk aversion and also π YX because the risk neutral component of v is
reinforced.
Before concluding a last remark is in order. The results presented so far can be easily
extended to an nth order increase in risk as defined by Ekern, of which the mean preserving
change in risk is a special case ( n = 2 ).
If Yɶ and Xɶ are zero mean risks with Yɶ ≺ Xɶ in the sense of Ekern (1980) one has:
n
π YX =
E (Yɶ n ) − E ( Xɶ n )  u n ( w)  u '( w)
−

n!
 u '( w)  u '( w −ψ Xɶ )
(20)
6. Conclusion
Arrow and Pratt showed in the mid-sixties that the notion of absolute risk aversion
was important and useful to characterize attitudes towards risk and understand risky
choices.
The literature since then has confirmed this opinion but it has also shown that:
•
•
absolute risk aversion is not the only relevant risk attitude. Other attitudes
such as e.g. prudence or temperance add to our understanding of risk
attitudes and are even necessary to analyse some specific risky choices.
while absolute risk aversion in the Arrow-Pratt's sense is necessary to discuss
risk elimination it has to be strengthened to value risk reductions.
Our review also shows that the concepts developed by Arrow and Pratt and their
extensions have gained in importance thanks to their close relationship with the stochastic
dominance literature introduced in economics by Hadar and Russel (1969) and developed
since then from the contributions by Rothschild-Stiglitz (1970, 1971) and Ekern (1980) and
extensions thereof.
18
As indicated in the introduction, my review is limited to theoretical results obtained
under the expected utility model. Hence one may wonder if they can be relevant outside this
specific model of choice.
I am definitely unable to fully answer this question. However one point seems pretty
obvious (at least to me). Results obtained under expected utility can be seen as a "reference
point" for alternative models of choices under risk (and ambiguity?).
Indeed if the analysis of a risky decision requires e.g. prudence in the expected utility
model it is very likely that a similar notion might be relevant in a non-expected utility model.
This suggests that it may be worth considering in non-expected utility models the equivalent
of higher order risk attitudes as developed inside expected utility and this question might
induce valuable research in the future.
Appendix 1
In Friedman-Savage (1948), the utility premium ( R ) measures the loss of utility due
to the presence of the zero mean risk εɶ .
Formally
R2 = u ( x) − E[u ( x + εɶ )]
(A.1.1)
Of course u '' < 0 implies R2 > 0 and a second order approximation yields:
R2 ≅
σ ε2ɶ
2
(−u ''( x))
which is the numerator of Arrow-Pratt's approximation39.
The idea of the "utility premium" defined here at the 2nd order can be used at any
order. For the first-order one has:
R1 = u ( x) − u ( x − k )
(A.1.2)
which measures the pain associated with a sure loss k .
At the nth order one has:
Rn = E[u ( x + εɶ )] − E[u ( x + θɶ )]
(A.1.3)
where θɶ is a nth order increase in risk of εɶ (see Ekern (1980)).
In this case Rn measures the pain induced by a deterioration of the zero mean risk εɶ
into the riskier zero mean one θɶ .
39
To obtain the approximated risk premium one divides R2 by u '( x) the marginal utility of wealth. This is the
standard procedure in micro-economics to obtain the monetary equivalent of a non-monetary damage.
19
Appendix 2
One observes from figure 4 that to go from Yɶ to 0 , two ways can be followed:
- either one goes directly from Yɶ to 0 and π Y is paid so that one ends up with
w − π Y obtained with certainty.
- or one proceeds in two steps. First one pays π YX to go from Yɶ to Xɶ . This
intermediate situation is made of a sure component of wealth equal to w − π XY plus a zero
mean risk Xɶ . From there one pays a sure amount of money to reach 0 . This amount
corresponds to the Arrow-Pratt risk premium to eliminate Xɶ when one avails upon sure
wealth equal to w − π YX ( w) .
The equivalence between the two terms implies:
E[u ( w − π Y ( w))] = u ( w − π YX ( w) − π X ( w − π YX ( w)))
(A.2.1)
where π Y ( w) is the risk premium attached to Yɶ with initial wealth w 40 while
π X ( w − π YX ( w)) is the risk premium paid to eliminate Xɶ with an initial wealth w − π YX ( w) .
The equality (A.2.1) implies:
π Y ( w) = π YX ( w) + π X ( w − π YX ( w))
(A.2.2)
or
π YX ( w) = π Y ( w) − π X ( w − π YX ( w))
which is a non linear equation in one unknown π YX ( w)
(A.2.3)
Appendix 3
If one transforms u into v = k (u ) with k ' > 0 and k '' < 0 as in Arrow-Pratt it can be
shown by straightforward differentiation that the degree of absolute prudence attached to
−v '''
v (i.e.
) can be larger or smaller than that of u .
v ''
However if one uses the transformation suggested by Eeckhoudt-Schlesinger (1994)
and Liu-Meyer (2013), i.e.
v( w) = aw + u ( w)
where a > 0 implies that v is less risk averse than u , one obtains:
Initially this expression was written as π Y . We specify here the initial wealth level at which the premium is
evaluated.
40
20
−v '''( w) −u '''( w)
=
v ''( w)
u ''( w)
Hence with such a transformation changes in risk aversion induce no change in
prudence (or in any higher order risk attitude).
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