Abstract This paper de nes local aversion to a risk index. I show that the local aversion to the Aumann and Serrano [2008] index of riskiness coincides with absolute risk aversion, and that the same applies for the Foster and Hart [2009] measure of riskiness. Using this\local consistency"I o er a new approach for axiomatizing the Aumann and Serrano index of riskiness and present other applications of this approach for relative risk aversion, \tardiness" (time preferences) and the appeal of information.
Keywords: ... JEL
Class: ...
AS
This paper is composed of two parts. In the rst part, I de ne for every index of riskiness Q the local aversion Q-riskinss, and show for two prominent indices of riskiness, the local aversion coincides with absolute risk aversion. I then show that for a vast class of indices, the local aversion must be ordinally equivalent to the absolute risk aversion. In the second part, I go in the other direction and present a set of conditions, importantly including the local consistency (with the absolute risk aversion coe cient), that are satis ed uniquely by R. The methodology is then applied to other settings of risk, as well as di erent settings such as time preferences (comparing the tardiness of di erent cashows) and information acquisition (comparing the appeal of di erent information transactions). The indices generated share several desirable properties.
Consider the following short paragraph: Alice is very averse to risk, and avoids risky actions whenever possible. Bernie is less averse to risk than Alice, and therefore sometimes takes rather risky actions. Although the previous paragraph does not sound too complex, pinning down the exact meaning of its components is a subtle task. Speci cally, three key issues need to be considered:
Thanks to Nageeb Ali, Robert Aumann, Dean Foster, Mira Frick, Drew Fudenberg, Sergiu Hart, Divya Kirti, Scott
Kominers, David Laibson, Jacob Leshno, Annie Liang, Eric Maskin, Moti Michaeli, Assaf Romm, Alvin Roth, Amnon
Schreiber and Tomasz Strzalecki. The author can be contacted at rshorrer@hbs.edu.
1

(i) What does\less averse to risk"mean?
(ii) What does\risky"mean?
(iii) Do less risk-averse people take riskier actions?
While it is very clear why items (i) and (ii) appear on the list, the need for item (iii) is less obvious. An a rmative to (iii) is however necessary for the validity of the inference\therefore."
1
The celebrated Arrow-Pratt index takes care of (i), as it allows us to compare the aversion to risk of any two agents with arbitrary wealth levels [Pratt, 1964, Arrow, 1965, 1971]. But this generality does not come without cost; as the index is \local" in the sense that it compares only aversions to in nitesimal risks. Pratt derives from the local index a characterization for being globally more risk averse; however, this yields a partial order that cannot compare many pairs of utility functions.
2
Recently, Aumann and Serrano [AS, 2008] addressed item (ii). They present an objective index of riskiness that can compare the riskiness of any two gambles. This index was derived from a small set of axioms, including centrally the \duality axiom," which ties riskiness to global (uniform) risk aversion. Roughly speaking, it asserts that (uniformly) less risk-averse individuals accept riskier gambles.
3
Subsequently Foster and Hart [FH, 2009] presented a di erent index of riskiness with an operational interpretation.Their index identi es for every gamble the critical wealth level below which it becomes \risky" to accept the gamble. Hart [2011] later demonstrated that both indices also arise from a comparison of acceptance and rejection of gambles.
4
Item (iii), however, remains partly untreated. While the consistency of riskiness and global aversion to risk lies at the foundations of the axioms of AS and is demonstrated by Hart [2010] and Hart
[2011],
5 the consistency of riskiness with local risk aversion was not established in any paper known to the author.That is, the fact that the Arrow-Pratt absolute risk aversion (ARA) coe cient (or the relative risk aversion coe cient for that matter) represent the\local AS riskiness aversion"or\local FH riskiness aversion"has not yet been established.
According to Aumann and Serrano [2008], the fact that Arrow and Pratt addressed risk aversion, a subjective concept that depends on a person’s utility function, but did not de ne the objective riskiness of gambles\..
. is like speaking about subjective time perception (‘this movie was too long’) without having an objective measure of time (‘3 hours’) or about heat or cold aversion (‘it’s too cold in here’) without an objective measure of temperature (‘20 degrees Fahrenheit’)." Using the above metaphor, proving the consistency of an index of riskiness with the Arrow-Pratt coe cient will exclude unintuitive conclusions such as a 3-hour movie being too long for a more patient agent, but a 6-hour movie is not for a less patient agent.
21 uis no less risk averse than vif and only if (8w) u
(w) v
(w), where u
(w) = u 00 (w) u 0 (w)
.Agent iuniformly no less risk-averse than agent jif whenever iaccepts a gamble at some wealth, jaccepts that gamble at any wealth.
5
3
Homm and Pigorsch [2012] supply an operational interpretation of the Aumann{Serrano index of riskiness.
4
Hart [2010] lemma 18 states that uis uniformly more risk averse than vif and only if inf w u 00 u
(w) sup w v u
( 0 (w) w
2
)
=
(w), where
(w) u
.
Recall that global risk aversion orders are very partial.

In Section 3, I establish this consistency. I show that the local aversion to AS riskiness is a complete order that coincides with the complete order of Arrow-Pratt local risk aversion. Simply put,
Alice is locally more averse to risk than Bernie in the Arrow-Pratt sense if and only if she is locally more averse to AS riskiness than he is. The same holds for the FH riskiness.
The fact that the AS and FH indices are consistent with the Arrow-Pratt coe cient is encouraging.
It indicates that they t well into the literature and, more importantly, that they\make sense." But this indication should not be regarded as de nitive evidence in favor of these indices; I show that there are in nitely many indices of riskiness which satisfy the above property. Moreover, some of these indices are unreasonable in the sense that they are not monotonic with respect stochastic dominance [Hanoch and Levy, 1969, Hadar and Russell, 1969, Rothschild and Stiglitz, 1970]. The section ends with conditions on indices of riskiness under which the local aversion to the riskiness index yields the same order as the Arrow-Pratt local risk aversion.
The results of Section 3 call for a new\dual"approach that takes local-consistency as a starting point. In Section 4 I derive the AS index from such axiomatization, using a novel general methodology which can be used to derive indices (of riskiness) from orders on utility and wealth pairs. I apply this new methodology to derive the index of relative riskiness of Schreiber [2012], using the Arrow-Pratt relative risk aversion order.
In section 5 I go back to Aumann and Serrano’s movie metaphor and use the new approach to derive an index of tardiness for investment cashows. Additionally, I show that the local appeal of two prominent indices of informativeness coincides with ARA, and derive the index presented in
Cabrales et al. [2012a] using the new approach. Importantly, just like AS is monotonic with respect to stochastic dominance, the index of tardiness is monotonic with respect to time dominance [B hren and Hansen, 1980, Ekern, 1981], and the index of informativeness is monotonic with respect to the partial order of Blackwell [1953]. These three concepts are analogous to one another in their respective spaces.
The last part of this paper discusses the results and some possible applications for empirical settings, as well for generating other objective indices. Additionallly, I show that in addition to being consistent with the Arrow-Pratt ARA, local AS (FH) aversion is slightly more general and can treat special functional forms, which cannot be treated with ARA.
A gamble gis a real-valued random variable with positive expectation and some negative values
(i.e., E[g] > 0 and P[g < 0] > 0); for simplicity, I assume that g takes nitely many values. Gis the collection of all such gambles. For any gamble g2G, L(g) and M(g) are respectively the maximal loss and gain from the gamble that occur with a positive probability. Formally, L(g) := maxsupp(g) and M(g) := maxsupp(g).G
is the class of gambles with support contained in an -ball around zero:
G
:= fg2G: maxfM(g);L(g)g g:
3

[x
1 ; p
1
;x
2
;p
2
:::;x n
;p n
] represents a gamble which takes values x
1
;x
2
;:::;xwith respective probabilities of p
1
;p
2
;:::;p nn
+
. An index of riskiness is a function from the collection of gambles to the positive reals, Q: G!
R
AS
(g), the Aumann-Serrano index of riskiness of gamble g, is implicitly de ned by the equation
R E exp g R
AS
(g) = 1:
= 0:
(g), the Foster-Hart measure of riskiness of g,
. Note that an index of riskiness is objective, in the sense that its value depends only on the gamble and not on any agent-speci c attribute. An index of riskiness Qis homogeneous (of degree
1) if Q(tg) = t Q(g) for all t>0 and all gambles g2G.
(g)
FH and R
FH
E log 1
+
6 is implicitly de ned by the equation g R
FH
FH
(g
0
Note that both R are homogeneous. Additionally, these indices are monotone with respect to rst and second order stochastic dominance; namely, if gis stochastically dominated by g
0 then R
AS
(g) >R
AS
(g
0
) and also
R
FH
(g) >R) [Foster and Hart, 2009, Aumann and Serrano, 2008].
The image of R
AS and of R
FH is R
+ 1+ e c even when the domain is restricted to
G c e
; ;
1 1+ e c
] and g
0
00
(w) := u(w) u
0
(w) : u
00
(w) u
0
1 2
;
1+ c
; 1
2
] for which R
AS
(g) =
% u
1 c and R
FH
(g
0
) = 1 c for all >0. This is simply demonstrated by considering gambles of the form g= [ ;= [ ;. In this paper, a utility function is a von Neumann{Morgenstern utility function for money.
I assume that utility functions are strictly concave and twice continuously di erentiable with a positive rst derivative unless otherwise mentioned. The Arrow-Pratt absolute risk aversion (ARA) of uat wealth wis de ned
The Arrow-Pratt relative risk aversion (RRA) of uat wealth wis de ned
(w) := w u (w) :
Note that u
(w) and %(w) are personal attributes of the agent and that both u
(w) and % u
(w) yield a complete order on u utility-wealth pairs. That is, the risk aversion, as measured by (or %), of any two agents with two given wealth levels can be comapred.
A gamble gis accepted by uat wealth wif E[u(w+ g)] >u(w), and is rejected otherwise. Given an index of riskiness Q, a utility function uand a wealth level w: supfQ(g)jg2G
!0
De nition 1. R
Q
(u;w) := lim
+
!0
De nition 2.
S
Q
(u;w) := lim
+ inf fQ(g)jg2G

and gis accepted by uat wg and gis rejected by uat wg
6
I also refer to R
FH as an index of riskiness.
4

( g
R
Q
(u;w) is the limit of the Q-riskiness of the riskiest accepted gamble according to Q, taking the support of the gambles to f0gto capture the notion of \locality." S
7
Roughly speaking, R
QQ
(u;w) is the riskiest\local gambles"that uaccepts at w, and S
Q
(u;w) is the limit of the Q-riskiness of the safest rejected gamble according to Q, again taking the support of the gambles to f0g.(u;w) is the safest rejected\local gamble." f
Q i s
R
+ for all >0, R
Q
(u;w) S
Q
+
:
)j g
2
G
Fact 1. For any Qfor which the image of G (u;w) for every uand w.
and gis rejected by uat wg= R
Proof. By the properties of the supremum, since and gis accepted by uat wg[fQ(g)jg2G
8 The inverse of these numbers is a natural measure of the aversion to Q-riskiness.The local aversion of uto Q-riskiness at wis therefore de ned by the ordered pair
A
Q
(u;w) := 1 R
Q
(u;w) ; 1
S
Q
(u;w)
:
Finally, u at w is locally no less averse to Q-riskiness than v at w
0
Q
( u;w)
Q
( v;w
)
Q
( v;w
0
) if A
Q
(u;w) A
Q
(v;w
0
0
Lemma 1. For every index of riskiness Q, A
Q
0
0
7
Q(g)jg2G
coordinate-wise; that is, if both
5 u;w)
1 R
Q
( 1 R 0 and 1 S
0
0
1 S .
)
+
Q 0

At this point, locally no less averse to Qmight look like an arbitrary class of partial orders on (u;w) pairs. However, I claim that its members are natural candidates for de ning local risk aversion. The reason is that they re ne the following natural partial order [Yaari, 1969]: uat wis locally no less risk averse than vat w(written (u;w) m (v;w, if uaccepts gat wthen so does vat w)) if and only if there exists >0 such that for every g2G. An order Ore nes m if for all gand h, gm h =)gOh
). Then there exists
0 re nes the natural partial order. Proof. Assume that (u;w)m (v;w>0 such that for every g2G
0 if uaccepts gat wthen so does vat w. As in the de nition of Awe have -ball supports with !0
0 will not change the result. Note that for every < , disregarding all -balls for and gis rejected by vat w
0
fQ(g)jg2G and gis rejected by uat wg,
1 and 1= 0
1
.
8
The nested supports assure monotonicity which assures the existence of the limit in the wide sense. Any nested sequence of compact supports that contain 0 in their interior, and which shrinks to f0g, will give the same results.For our purposes, 0 = 1

and and gis accepted by uat wg Q(g)jg2G and gis accepted by vat w
0
:
Q
(u;w) (R
Q fQ(g)jg2G
This means that the in ma (suprima) in the de nition of S
Q
(v;w
0
) (R
Q
(v;w
0
)) are taken on a subset
(superset) of the corresponding sets in the de nition of S(u;w)), and therefore are higher. The proof is concluded using the well known result that weak inequalities are preserved in the limit.
The next step is to show that the local aversion to AS (FH) riskiness gives rise to a complete order and that this order coincides with the one implied by the Arrow-Pratt ARA coe cient.
Lemma 2 (3) shows exactly this.
u
AS
(u;w) = ( u
(w); u exists an interval I R such that u
(x) v
9
0
(x) >0 we have that u emma 2. For every utility function uand every w, A
8 >0 9 >0 s.t x2(w ;w+ ) )j
(w)). Proof. According to proposition 9 in Hart [2010], if uand vare two utility functions and there
(x) for every x2Ithen for every wealth level wand lottery gsuch that w+g I, if gis rejected by vat wit is also rejected by ufor the same wealth level.Put di erently, if gis accepted by uat wit is also accepted by vat the same wealth level. Keeping in mind that u(x) is continuous. Speci cally,
(x) u
(w)j< (3.0.1)
R
Recall that a CARA utility function with ARA of rejects all gambles with AS riskiness greater than
1 and accepts all gambles with AS riskiness smaller than
1
[Aumann and Serrano, 2008].
Using the CARA functions with ARA of u
(w) + and u
(w) as implied by Statement 3.0.1 together with Hart’s Proposition 9 completes the proof.
Lemma 3. For every utility function uand every w, A
R
FH
(u;w) = ( u
(w); u
(w)). Proof.
According to Statement 4 in Foster and Hart [2009]:
L(g) R
AS
(g) R
FH
(g) M(g): (3.0.2)
From Statement 3.0.3 one can deduce that R
R
FH
(u;w) = R
R
AS
(u;w) and S
R
FH
(u;w) = S
R
AS
(u;w).
Lemma 2 completes the proof.
FH
The result of Lemma 3 is not surprising in light of Lemma 2, as Foster and Hart [2009] already noted that the Taylor expansions around 0 of the functions that de ne Rand R
AS di er only
9
Actually, the statement of the proposition there is a bit weaker, however the proof su ces for this formulation as well.
6
Therefore, if g2G then: R
AS
(g) R
FH
(g) : (3.0.3)

from the third term on. Roughly speaking, this means that for gambles with small supports R
AS and R
FH are close. Theorem 1 summarizes the consistency results of
Lemmata 1-3.
Theorem 1. (i) For any index of riskiness Q, A
Q re nes the natural partial order. (ii) For every utility function uand every w, A
R
AS
(u;w) = A
R
FH
(u;w) = ( u
(w); u
(w)).
Up until this point, I showed the consistency of the AS and FH riskiness indices with the standard de nition of local risk aversion. This means that we can now start with a small set of axioms, namely
Aumann and Serrano’s, and de ne a complete order of riskiness over gambles. We then can derive from it the local aversion of agents to this riskiness index and get the well known Arrow-Pratt coe cient. If we take seriously the cardinal interpretation of the AS riskiness index, we get a cardinal interpretation for u
(w); u
(w) =
1 2 v
(w
0
) if and only if uat wis willing to accept (small) gambles twice as risky as those that vis willing to accept at w
0
. Hence, Theorem 1 answers item (iii) from the introduction in the a rmative.
Theorem 1 might be interpreted as evidence that AS and FH were\well-chosen"in some sense.
However, Theorem 2 shows that while according to the above results the AS and FH indices are consistent with the standard notion of local risk aversion, there are other indices that will give rise to the same order of local aversion. Furthermore, some of these indices are not\reasonable"in the sense that they are not monotone with respect stochastic dominance. Informally, this mean that local consistency may be a necessary condition for an index to be reasonable, but it is not su cient.
AS
Theorem 2. (i) There exists a continuum of homogeneous riskiness indices for which the local aversion equals the local aversion to R-riskiness and R
FH10
-riskiness at all uand w(and coincides with the Arrow-Pratt coe cient).(ii) Moreover, some of these indices do not respect stochastic dominance.
Proof. (i) I rst show that for every a>0 any combination of the form Q a
(g) := R
FH
(g) + a R
FH to
R(g) R
FHAS
(g) is an index of riskiness for which the local aversion equals the local aversion. The reason is that for small supports, the second element in the de nition is vanishingly small by
Inequality 3.0.3, and so Q a and R
FH should be close. Fix a>0. First, note that
8g2G0 <R
FH
(g) R
FH
(g) + a R
FH AS
(g) ;
7
AS
R so Q a
(g) 2R
+
, R
FH
(g) R
FH
(g) + a FH (g) R
AS
(g) FH (g) + : (3.0.4)
AS
.
10
. Additionally, for every >0 there exists >0 small enough such that for every g2G
Omitting the homogeneity requirement would yield a trivial statement as, for example, an arbitrary change of the values of Rfor gambles taking values larger than some M>0 will result in a valid index. The requirement that that the local aversion to the index coincides with the Arrow-Pratt coe cient, and not just with the order it implies, is a normalization that rules out, for example, the use of positive multiples of R

Inequality 3.0.4 stems from the small support combined with Inequality 3.0.3. It tells us that the ordered pair that de nes the local aversion to Q a
-riskiness cannot be di erent from A
FH which equals A
R
AS according to Theorem 1. The proof of (i) is completed by recalling that R
RFHAS
FH
(g) = 1 and R(g) 1:26, hence Q
1
(g) <1:6. Now take g
0AS
= [1;1 ;1; ]. For small values of ,
R(g
0
) 0 but R
FH
(g
0
) > 1, so Q
1
(g
0
) > 1:6. Therefore, while g
0
rst order stochastically dominates g,
Q
1
(g) <Q
1
(g
0
). Finally, I present a set of su cient conditions on Qunder which the local aversion to
Q-riskiness yields the same order as the Arrow-Pratt local risk aversion. The result resembles the ndings of
Lemma 1, but it is necessary for the following section.
Axiom 1. Full-image. 8 >0 ImQ(G ) = R
+
: Axiom 1 says that even when the support of the gambles is limited, the image of Qis all of R
Q
. Axiom 2. Acceptance-cuto . 8u 8w 0 <R(u;w) S
Q
(u;w) <1:
Together with Axiom 1, Axiom 2 says that at a given wealth level, each agent has a unique
12
Q-riskiness cuto for acceptance and rejection of small gambles, and that it is proper (not all small gambles are accepted or rejected).
Axiom 3. Homogeneity. 8 >0 8g2G; Q( g) = Q(g). Axiom 3 embodies a cardinal interpretation of the index. It appears in the original axiomatic characterization of the AS index. Interestingly, Axioms 1-3 imply a cardinal interpretation for A
AS
; If cuto . This interpretation applies, for example, to Ror R v 0 QQ Q
FH0
0 and uu
(w) > ( ). Theorem 3. If Axioms 1, u
(w) > v
(w
0
)
Implies that (u;w) m (v;w
0
) , so Lemma 1 implies that A
Q
(u;w) A
Q
(v;w
0
). To see that A
Q
(u;w) 6=
R
Q
(v;w
AS
(g n
0 u
( w)+ v
( w
0
) 2
) = c. For a small >0 let h
1 n n1 n=1 n
1 n
= (1 + )g. It is easy to observe that for large values of n, g and n and h nn will be rejected by uat wand accepted by vat w
0
, so
S
Q
(v;w
0
) = R
Q
(v;w
0
) (1 + ) S
Q
(u;w) >S
Q
(u;w) = R
Q
(u;w);
11
An alternative proof could use indices of the form:(R
FH
)(R
AS 1
)
(u;w).
12
Recall Fact 1 which states that Axiom 1 implies S
Q
8
11 and that both indices are homogenous.
(ii) Follows from example 1.
1
+ e e ;
;
1
1
+ e
1 ]:R
AS
+
Example 1. Take Q
1
(h) := R
FH
(h) + R
FH
(h) R
AS
(h) and g= [1;

, 2(0;1). This form may prove to be useful in empirical work, since it enables some exibility in the estimation. In addition, it allows us to put some weight on the FH measure that \punishes" heavily for rare disasters [Barro, 2006].
(u;w) R
Q

such that k n
2G
1 n and Q(k n
) = c
0
, where (with a slight abuse of notation) c
0
:= where the strict inequality follows from the fact that S(u;w) >0 by Axiom 2. This proves that
A
Q
(u;w) >A
Q
(v;w
0
). In the other direction, if A
Q
(u;w) >A
Q
(v;w
0Q
) then there exists a sequence of gambles fk
QQ
( v;w( u;w)+ AAn g
0
) 21 n=1 1
. For a small >0 let l n
= (1 + )g n
. A similar argument shows that
S
R
AS
(v;w
0
) = R
R
AS
(v;w
0
) (1 + ) S
R
AS
(u;w) >S
R
AS
(u;w) = R
R
AS
(u;w); where the strict inequality follows from the fact that S
R
AS
(u;w) > 0 by Lemma 2. Using Lemma 2 again, this implies that u
(w) > v
(w
0
).
The ndings of Theorem 3 indicate that, under weak conditions, the only \reasonable" order of local aversion is the one induced by the Arrow-Pratt ARA. But according to Theorem 2, localconsistency is not enough to characterize a \reasonable" index of riskiness. These ndings call for new axioms for an index of riskiness, which take local-consistency as a starting point.
Q v) if infDe nition 3. Let Qbe an index of riskiness. uis globally no less averse to Q-riskiness than v
(written u% w
A
Q
(u;w) sup w
0
A
Q
(v;w
0Q v) if u%), where the in mum and supremum are taken coordinate-wise. uis globally more averse to Q-riskiness than v(written u
Q u.
Q vand not v%
De nition 3 is di erent from the AS de nition of uniformly more risk-averse. It is derived directly from the index Qand the utility function u. However, if Axioms 1-3 hold, the two de nitions are equivalent.
Q v, uaccepts gat w, and Q(g) >Q(h), then vaccepts hat w.Axiom 4. Global consistency. For every pair of utilities uand v, for every wand every gand h in G, if u
Theorem 4. R
AS is the unique index of riskiness that satis es Axioms 1-4, up to a multiplication by a positive number.
AS
Proof. The AS duality axiom states that if uis uniformly more averse to risk than v, uaccepts gat w, and Q(g) >Q(h), then vaccepts hat w. Using Theorem 3, Axioms 1-3 combined with Axiom 4 imply the duality axiom. But the only indices that satisfy Axiom 3 and the duality axiom are positive multiples of R[Aumann and Serrano, 2008]. Finally, observe that R
AS satis es the axioms.
hi s m
Note that Axioms 1 and 2 were used mainly to show that A
Q
9 e sa m e yi el d th or d er
.
T and u

d b e nc y co ul o ns ist e at lo ca l-c e a ns th as su su g g es or e m
4 h e e,
T h er ef or ts a g
.
T io m th es e ax pl ac e of m e d in

r d h fo e ni n p pr o ac d u al a e n er al g a f ris k av or d er
(o er si o n) o m a n n es s) fr
(o f ris ki n in d ex

on utility-wealth pairs. The rest of this section presents an application of this approach using the
Arrow-Pratt relative risk aversion (RRA) index % u13
(w).
De ne U:= fu: R
g2GjR
FH
+
!R j% u p >1 ()E[log(1 + g)] >0:
Fact 2. R
FH
+
(1 + g
FH
:
(w) >18w>0g, the set of (twice continuously di erentiable) utility functions with relative risk aversion higher than the logarithmic utility function. Additionally, let H:=(g) <1 be the set of gambles with
FH riskiness smaller than 1. The following is a result of FH: i
) i
(g) <1 () QThe interpretation of R
14
(g) <1 is that gambles of the form wgare accepted by a logarithmic utility function at wealth w. I will consider multiplicative gambles, so that now uaccepts gat wif u(w+ gw) >u(w), and rejects gotherwise.
Adjusting the previous axioms to the current setting yields the following axioms for an index of riskiness Q: H!R
Axiom 5. Scaling. 8 >0 8g2H; Q((1 + g) 1) = Q(g):
As before, Axiom 5 embodies a cardinal interpretation.
1
5
Axiom 6. Local consistency. There exists a strictly increasing function : R !R such that A
Q
(u;w) =
( % u
(w); % u
(w)), for every uin Uand every w>0.
16
In the interest of brevity and concreteness, Axiom 6 is stated in a less primitive way. It could be derived from Axiom 2, a weaker version of Axiom 1 and Axiom 5.Importantly, it states that A
Q
(u;w) is ordinally equivalent to % u
(w).
Q v, uaccepts gat w, and Q(g) >Q(h), then vaccepts hat w.Axiom 7. Global consistency. For every uand vin U, for every w>0 and every gand hin H, if u Theorem 5. (i) For any g2Hthere is a unique positive number S(g) such that E h (1 + g)
1 S(g) i =1. (ii) Sis the unique index of riskiness that satis es Axioms 5-7 up to a multiplication by a positive number.
Lemma 4. g2H()log(1 + g) 2G.
Proof. In one direction, g2H)g2Gand R
FH
(g) <1. Since R
FHFH
(g) L(g) we know log(1+g) is well-de ned. As g2G, it assumes a negative value with positive probability and therefore so does log(1 + g):Since R(g) <1, we have E[log(1 + g)] >0. Hence, log(1 + g) 2G. In the other direction, if log(1 + g) 2Gwe have that log(1 + g) assumes a negative value withpositive probability and therefore so does g. In addition, we have Pp i log(1 + g i
) >0. Hence, by Fact 2, g2H.
13
This section is inspired by Schreiber [2012].
14 gcan be interpreted as the return on some risky asset.
15
Importantly, note that for every >0 if g2Hthen (1 + g)
16 If we wanted to assume the weakest axioms, we could also replace Axiom 1 with a requirement of a dense image.
10
1 2Hby fact 2.

Lemma 5. For every g2Hthe equation E h (1 + g) 1
S
Proof. Note that for every g2Hand S>0, we have E h (1 + g)
AS
(log(1 + g)).
1
S i = E h e
1 S( g) u
1
. Now observe that:
1 i = 1 has a unique positive solution.
log (1+ g)
S i .
Consequen-
1
(1 + g) tially, Lemma 4 and Theorem A in AS imply that the unique positive solution for the equation is S(g) = R
Claim 1. For all g 2H, If u 2Uhas a constant RRA then % (w) 1 <
E[u(w+ wg)] >u(w) ()E w
()E h e
(1 ) log (1+ g) i <1 ()R
AS
1
>w
1
()E (1 + g)
(log(1 + g)) < 1 1 () 1 < 1
S(g)
:
<1 () if and only if
E[u(w+ wg)] >u(w) 8w>0.
Proof. As positive a ne transformations of the utility function do not change acceptance and rejection, it is enough to treat functions of the form u(w) = w
(x
0 x
Lemma 6. For every u;v2U, if inf% u
(x) sup x
0
% v
) then for every w, if uaccepts gat wso does v.
0 t logv
0
(tw) = logv
0
(tw) logv
0
(w) =
= t 1 sw v
00
(sw) s v
0
(sw) ds
1 t 1 logu
0
(tw)
1 u
00
1
0
(sw) ds= s
1 s w
Proof. Without loss of generality, assume that v(w) = u(w) = 0 and that v (w) = 1. For every t>1
@logv
0
(sw)
@s ds= t w v
00
(sw) v
0
(sw) ds= ds= t s w v
00
( w s
) ds=
1 w s logv
0
( wt ) = logv
0
( wt ) logv
0
(w) = t
(
@logv
0
) @s
1 w 2 0 s (
) v
0
(w) = u

w sw
0
( s
)
= t
1
1 ds t
ws
w
0 00
(
0
( s
)
) v v
00
( 1 s
1 s
This means that for every t>0: t
0
( w s u) u
0 ds= logu t )
(sw)ds wv 1
11 v(tw) = v(tw) v(w) = t wu
0
(sw)ds= u(tw)
1

And so, if E[u(w+ wg)] >u(w) = 0 then necessarily E[v(w+ wg)] >v(w) = 0 as E[v(w+ wg)] E[u(w+ wg)].
Lemma 7. For every u 2Uand every w > 0, the local aversion of u to S at w equals (% u
(w) 1;% u
(w)
1).
The proof of Lemma 7 is analogous to the proof of Lemma 2 and is therefore omitted. Recalling that the CRRA utility function with parameter is often expressed as w
1
= w
( 1)
; t
( ) seems particularly natural.
h i S((1 + g)1) = R
AS
(log(1 + g)) = R
AS
( log(1 + g)) = R
AS
(log(1 + g)) = S(g), so Ssatis es Axiom 5.
By Lemma 7, Ssatis es Axiom 6.
w t
% v
To see that Ssatis es Axiom 7, observe that the fact that Ssatis es Axiom 6 implies that if r w
0
% u
(w
0
). Therefore, by Lemma 6 if vaccepts g at wso does an agent with a CRRA utility function with RRA equals . Furthermore, by Claim 1, a n i o t a f o r m f o n
% u
For uniqueness, assume that ^ Qsatis es the axioms. By ^ P(g) ^
Lemma 4 := Q(e g
^
^ P: G!R
+
. For every >0, we have ^ P( g) =1) = ^
Q(e g
P(g), so
^ Q(e
g
1) = ^ Q((1 + e g
AS
^ P and R
AS
AS
^ Q= S, for some >0.
1)^ Qorder
12
1) is an index of riskiness^ Psatis es Axiom 3. From Axioms 7 and 6 we know that Sand1) = lotteries in the same manner (using CRRA functions). Hence, ^ Pand Ralso agree on the order of lotteries. Since bothare homogenous, we have that ^ P = Rfor some >0. This in turn, implies that
In the previous sections, I used the concept of local aversion (to riskiness), to present a new dual

approach for deriving indices (of riskiness). For the case of additive gambles, I showed that there is a unique\reasonable"order of local aversion to riskiness (for a vast class of indices of riskiness). With this result at hand, I showed that the only index that can satisfy the property of global consistency is the AS index of riskiness.
To demonstrate the generality of the new approach, I provided another application in the realm of risk. I showed that the index of relative riskiness suggested by Schreiber [2012] could be derived from the Arrow-Pratt relative risk aversion order, using the same technique.
This section shows that the new dual approach could be used for deriving\objective"indices in other settings.

5.1.1 Preliminaries Consider a capital budgeting setting. In this setting, an investment cashow is a sequence of outows (investment) followed by inows (return), and a sequence of times when they are conducted.
Denote by (x i
;t i
)
N i=1 such a cashow that pays (requires) x i at time tand assume, without loss of generality, that t
1
<t
2
<:::<t
N
. Further, assume that x
1it; be the collection of cashows with t
1
<0 and
Px i
>0. Let Cdenote the collection of such cashows, and C t t
N
, and t
N t
+1
< . An index of tardiness is a function T: (C;t) !Rfrom the product of time and the collection of cashows to the positive reals. A cashow cis said to be T-tardier then c
0 relative to tif T(c;t) >T(c
0
;t). The second entry, t, represents the time at which the cashow is considered. T is homogenous of degree 0 in payo s if T (x i
;t i
)
N i=1
;t =
T ( x i
;t i
)
N i=1
;t for any >0and any t. This means that doubling all the sums of money (both investment and return) does not change the T-tardiness of the investment cashow. T is homogenous of degree 1 in dates relative to tif T (x i
;t+ (t iN i=1 t));t = T (x i
;t i
)
N i=1
;t for any >0 and any t. This means that doubling the periods between the ows of money while preserving their relative distance from tdoubles the T-tardiness. Tis homogenous of degree 1 in dates, if it is homogenous of degree 1 in dates relative to tfor all t. Tis translation invariant if T (x i
;t i
+ ))
N i=1
;t+ = T (x i
;t i
)
N i=1
;t for all >0 and all t. This property means that tardiness is a time expression, like\in a week"or \a year before,"and it does not depend on the current date.
I consider a setting in which agent idiscounts using a smooth schedule of positive instantaneous discounting rates, r i17 ;18
(t).In this case agents must accept or reject suggested shifts to their cashows only according to their net present value (NPV).
5.1.2 Axiomatic Characterization I now apply the dual approach from the previous section to present axioms for an index of \tariaccepts c;c2C !0
t; + diness." Using the framework described above, I de ne agent i
0 sT-tardiness aversion at tas the inverse of limsupT(c;t), and denote it by TA
T
(i;t). Roughly speaking, this number is the inverse of the T-tardiness of the latest \local cashow" with respect to tthat is accepted
0
(denoted by j J
T;t o t t
0
TA
13
17
18
19 y i. Say that iis uniformly more T-tardiness averse than j after t o i) if inf
T
(i;t) >supTA
T19
(j;t).
t t o
Axiom 8. Acceptance-cuto . For all tand i, i’s T-tardiness aversion at tis positive and nite, and equal to the inverse of liminfT(c;t).
!0
+ irejects c;c2C t;
An alternative interpretation may be a social planner with such time preferences[Foster and Mitra, 2003].
For a discussion of this condition see B hren and Hansen [1980] and references provided there.
Although in some settings it will make no sense to assume that di erent agents face di erent paths of interest rates
[Debreu, 1972], the comparison is reasonable in many reasonable settings [B hren and Hansen, 1980, Ekern, 1981].
It is also possible to make inter-temporal comparisons for the same agent by shifting tin one side of the formula.

cuto l evel of
T-ta rdin ess for acc epta nce and ocal
"dec isio ns, ther e exis ts a his
T axio m say s that for\l reje ctio n.
eit y.
F or e n o g all t,
T(
;t
9.
H o m
A xi o m

d e gr e n o us of e
1 in g e m o h o
) is d at es
.
the cas how is con duct ed twic e as late, then the entir e cas how noti on that if eac h pay men t in his
T axio m repr ese nts the

ri a in va ns lat io n nc e.
T
1
0.
Tr a
A xi o m is tr n in va ri a ns lat io a nt
.
A
F or all
> m a g e.
1.
F ull
-i xi o m
1 is twic e as tard y.

)
=
R
+
.
T(
C
;t
;t all t,
I m
0 a n d
Axiom 12. Global consistency. Let j J
T;t o i, c= (x i
;t i
)
N i=1 and
1 t
; i
= x
0
;t i N
0 i= and t o c
0
0 i 1 i 0 be an investment cashow. If r k
(s) <r j
(s) for all s2[t
1
;t
)
X t i e
r k
( s) ds x i i i
Lemma 8. Let c= (x
Proof.
Denote by i i i
= Xe t i
r k
( s) ds x i i>i
+ Xe t i
r k
( s) ds x i i i
= X i
Xe t i
r k
( s) ds x t i jx i i>i j+
X t i jx i j 0 ()e t i
r k
( s) ds
0
@ e
r k
( s) ds i 0 implies that P
i e
r k
( s) ds i t i
r j
( s) ds x <0 .
<0. Then t i k
( s) ds e r jx i>i j+ Xe t i k
( s) ds t i
X e
r k
( s) ds i jx j; jx ii>i j+ Xe t i
r k
( s) ds jx i j 1A 0:
r i i s sma ller than t, t
1 and t
0 i
A x i . Then, if R(c;t) <R(c
N i=1
;t), and iaccepts c
0
, then jaccepts c.
N i

e] th t r e a t a
1
2 d a n
8 s o m h
.
o a c p r a p l u a d e w n h e t f o x r u c h e

) a
, e
.2
)
B ut n d i i
(5
.1
(5
.1
.1
es t in d ex wi th x g h e hi e n
Pt h i i r k d s
)
( s
0
@ t i i>i e t i t i
X e
r k
( s) ds i i i>i t i t i jx i j+ Xe t i
r k
( s) ds jx i j 1A = Xe i i t i t i t i r k
( s) ds i i>i
X e r ( s) ds jx nl y m ult ipl s ar e o as p os iti ve j+ Xe t i r ( s) ds jx j j i jx i j+ X j t i r k
( s) ds jx j> i

s a n d n e g u m b er m all er n ie d by s ati ve er
(p os iti by gr e at ult ipl ie d s ar e m m b er s.
n u ve
)
.
9 a
I
L e m m f c=
(x i
;t ii
P e rt i x i

)
N u m b er os iti ve n q u e p s a u ni w th e n nt ca sh o th er e ex ist ve st m e is a n in or e, if
~r ur th er m at
=
0.
F rs uc h th i=1
14
1
;t
N

e n th e t2
[t]
, th
N
P
V
) fo r all
(t)
>r
>^ r(t of ci n d is p
, a g
~r e g s n ati ve us in os iti ve us in g
^r.

i
X e rt i x i
= 0 ()e rti
Xe rt i x i i
= 0 () X e r( t i t) x i i
Axiom 10 is also satis ed as Xe rt i x i
= 0 ()e rti
Xe rt i x i i
= 0 () X
= 0 8t:
To see that 11 is satis ed, simply consider cashows of the form: e r
( t i t) x i
= 0 8t8 >0: k
(1;t; e;t+ ):
For Axiom 8, I use the smoothness of r ii
( ) to deduce that for every small >0 there exists >0 such that if s2(t ;t+ ) then r(t) <r i
(s) <r i
(t) + . This fact, together with Lemmata 8 and 9, implies that Axiom
8 is satis ed.
Finally, to see that Axiom 12 is satis ed, consider an agent that discounts at the constant rate , with sup
1 r j
( t) 1
inf
1 r i
( t)
, where the supremum and in mum are taken on the relevant domain. Label this agent . Lemma 8 implies that accepts any gamble accepted be j, Lemma 9 implies that he also accepts cashows with higher IRR, and another application of Lemma 8 implies that iaccepts these cashows.
I now turn to show that the only indices that satisfy Axioms 812 are positive multiples of D. This is done in two steps. In the rst step, I show that indices that satisfy the axioms agree with the order induced by D. Then, I show that they are also multiples of this index.
For the rst step, assume by way of contradiction that there exists another index, Q, that satis es the axioms but does not agree with Don the ordering of two cashows at some given time points. There are three possibilities:
1. Q(c;t) >Q(c
0
;t
0
) and D(c;t) <D(c
0
;t
0
) for time points tand t
0 and cashows cand c
0 a
0
0 and cashows cand c
0
;t
0
) for time points tand t
0 n and cashows cand c
0
;t
0
) for time points tand t
0
1 d r
2
) and D(c;t) = D(c
0
;t
0
) and D(c;t) <D(c
0
D(c;t) < 1r
2
< 1r
1
<D(c
0
;t
0
);
1
;t
0
. 2. Q(c;t)
>Q(c. 3. Q(c;t) = Q(c. Using Axiom 9, there is no loss of generality in treating just the rst case, since using the rst degree homogeneity ccan be shifted slightly in a way that would preserve the strict inequality, but break the equality in the right direction, leading to the rst case. To obtain a contradiction, choose rsuch that and consider two agents that discount with the constant and r
2 rates r
1 and r
2
1 (with a slight abuse of notation). Using Lemma 9 both r
1 r
2 and r
2 accept cand rejects c
0 does not
Q;t o
, and are labeled accordingly r. Together with Axioms 9 and 10, Theorem 6 implies that rJfor all t. But this means that Qviolates Axiom 12, as r
2o
, the impatient agent, accepts c, the Q-tardy cashow, but r
1
16

accept c
0 which is less Q-tardy. Thus, Qand Dmust agree on the ordering of any two cashows at any given time point.
For the second step, choose an arbitrary cashow c
0
= (x i
;t i
)
N i=1
, a point in time t o
D (x i
;t o
+ (t i t o
))
N i=1
(x;t oi
;t o
+ (t i t o
))
N
;t i=1
(x o
;t o
0
;t o = T(c;t). The rst step implies that+ (t
0
;t o i t o
))
N i=1
;t o
= D(c
0
;t o i
;t o i t o
N i=1
;t
0
;t ). Altogether this means that T( c;t)
T( c)
= D( c;t)
D( c)
, and an index that satis es the axioms, T. For any cash ow cand time t, there exists a positive number >0 such that T = D(c;t). But D io
), and also T (xfor every c.+ (t)) = T(c o
5.1.3 Properties of the Index D Just like the AS-riskiness of a gamble depends \on its distribution only|and not on any other parameters, such as the utility function of the decision maker or his wealth,"[Aumann and Serrano,
2008] Ddepends solely on the cashow, and not on any agent speci c properties. In this sense, D is an\objective"measure of tardiness. A related property of Dis that it is also independent of the date when the cashow is considered.
Dis homogenous of degree 0 in payo s and unit free. This means, for example, that the Dtardiness of two cashows denominated in di erent currencies may be compared without knowing the exchange rate.
Another property that Dand R
AS share is monotonicity. R
AS
17
R
AS is monotonic with respect to rst and second order stochastic dominance. The analogous property for cashows is time-dominance
[B hren and Hansen, 1980, Ekern, 1981]. B hren and Hansen [1980] show that if a rst-order time dominates bthen every agent with positive time preferences prefers ato b. Positive time preferences mean that the agent prefers a dollar at time sto a dollar at time s+ for all >0, or that the agent’s discounting function is decreasing. They also show that if asecond-order time dominates b then every agent with a decreasing and convex discounting function prefers ato b. Their proposition 3 states that Dis monotonic with respect to time-dominance of any order.
is much more sensitive to the loss side of gambles than it is to gains. Analogously Dis more sensitive to early ows than it is to later ones. This follows from the properties of the exponential function in the de nition of the IRR.
5.1.4 Further Remarks This section demonstrates the relation between indices of riskiness and the
IRR. The IRR is a counterpart of the rate of return over cost suggested by Fisher [1930] as a criterion for project selection almost a century ago. Later, some economists dismissed this criterion, arguing that the
NPV was superior in comparing pairs of cashows. Yet, others mentioned that this criterion has the bene t of objectivity, in that it does not require the value judgment of setting the future discounting rates [Turvey, 1963]. For example, Stalin and Nixon would agree on the IRR of an investment even though they might not agree on its NPV. This resembles the point made by Hart [2011] that in

general there are many pairs of agents and pairs of gambles such that each agent accepts a di erent gamble and rejects the other - our axioms just compare\strong"rejections.
There are other similarities between the AS index and the index D. Value at Risk (VaR) is a family of indices commonly used in the nancial industry [Aumann and Serrano, 2008]. VaR indices depend on a parameter called a con dence level. For example, the VaR of a gamble at the
95 percent con dence level is the largest loss that occurs with probability greater than 5 percent.
Unlike the AS index, VaR is una ected by tail events or rare-disasters, extremely negative outcomes that occur with low probability. In the context of project selection, Turvey [1963]
The set of Investment opportunities B = b2R k k j Pp k b k
0 , consists of all no arbitrageassets. In particular it includes the option of inaction. The de nition of no arbitrage investment opportunities assumes that p k20 is the price of an Arrow-Debreu security that pays only in state k. Hence, pplays a dual role in this setting.
and state kis realized, his wealth becomes w+ b k
When an agent with initial wealth wchooses investment b2B. Before choosing his mentions that \the Pay-o Period, the number of years which it will take until the undiscounted sum of the gains realized from the investment equals its capital cost," was used by practitioners in the
West and in Russia. He adds that \[p]ractical men in industries with long-lived assets have perforce been made aware of the de ciencies of this criterion and have sought to bring in the time representing the information that aentails. To be more precise, is given by a nite set of signals
S and probability distributions k
2 (S ) for every k. When the state of nature is k, the probability that the signal sis observed equals (s). Thus, the information structure may be represented by a stochastic matrix M
k
, with Krows and jSjcolumns, and the total probability of the signals is p p M . Without loss of generality, assume that p(s) >0 for all s. Further, denote by q s k the probability the agent assigns to state kconditional on observing the signal s, using Bayes’ law.
Note that although my notation does not indicate it, (q s k
)
K k=1
= q s
2 (K) depends on .
:= element." The pay-o period criterion su ers from de ciencies similar to those of VaR. For example, shifting early or late payo s does not change its value.
investment, the agent has an opportunity to engage in an information transaction a = ( ; ), where > 0 is the cost of the transactions, and is the information structure
5.2.1 Preliminaries This section follows closely Cabrales et al. [2012a]. In this setting, I consider agents with concave and twice continuously di erentiable utility functions who have some initial wealth and face uncertainty about the state of nature. There are K2N states of nature, over which the agents have the prior p2 (K) which is assumed to have a full support.
ais said to be excluding if for every sthere exists some ksuch that q s k 18
20
Cabrales et al. [2012a] treat the more general case as well.

= 0. This means that

for every signal the agent receives, he knows that some states will not be realized. Throughout, I will assume that information transactions are not excluding.
Agents are assumed to choose the optimal investment opportunity in B given their belief, q.
Therefore, the expected utility of an agent with utility uand initial wealth wwith beliefs qis b2B
V(u;w;q) = sup k
Xq k u(w+ b k
): s p (s)V(u;w ;q s
Accordingly, an agents accepts an information transaction if
X) V(u;w;p) = u(w) and rejects it otherwise. Denote by Athe class of information transactions described above.
Additionally, denote by Athe sub-class these information transactions such that kpq s k
1 +
:Finally, the index of appeal Asuggested by Cabrales et al. [2012a] is de ned by< for all s. An index of appeal of information transactions is a function from the class of information transactions to the positive reals
G: A!R
A(a) = 1 ln s
Xp (s)exp(d(pjjq s
)) !; where d(pjjq) = k
Xp k ln pq kk is the Kulback-Leibler divergence [Kullback and Leibler, 1951]. Finally, de ne the local appeal of
Q-informativeness for an agent uwith wealth w, as the the appeal of the most Q-appealing transaction that is accepted, and provides just a little information.
More precisely, it is de ned by limsup
21
Q(a).
!0
+ a2A ;aisaccepted
5.2.2 Local Consistency Theorem 8 and Corollary 1 are the analogous of Theorem 1 in the current context.
Theorem 8. The local appeal of Afor uwith wealth wis equal to (w). Furthermore, it is equal to liminfA(a).
u
!0
+ a2A ;aisrejected
Proof. The proof is similar to the proof of Theorem 1. First, Lemma 2 of Cabrales et al. [2012a] proves that as approaches 0, so does the scale of the optimal investment kb k k. Therefore, for small enough, w + bis in a environment of wfor all k, if a= ( ; ) 2Ais accepted. For the second step, observe that u u
(w) is continuous, and so for every >0 there exists a >0 small enough such that x2(w ;w+ ) implies j (x) u
(w)j<.
19
21
Note that in this setting the index is not independent of the prior p.

For the nal step, choose a small positive number , and consider the CARA agents with absolute risk aversion coe cients u
(w) + and u
(w) . For a small enough environment of w, I, u
(w) inf x2I u
(x) sup x2I u
(x) u
(w) + :
This, in turn, implies, using Theorem 3 of Cabrales et al. [2012a] and a slightly modi ed version of their Theorem 2, that the local appeal of Afor uwith wealth wis equal to u
(w).
Cabrales et al. [2012b] suggested the entropy reduction as a measure of informativeness of an information structure for investors. It is de ned by
I es
( ) = H(p) Xp (s) H(q s
); where, H(q) = k2K
Xq k ln(q k
):
In the current context, consider the index J e
, the cost adjusted entropy reduction de ned by
J e
( ; ) = I e
( ) : for uwith wealth wis equal to (w). Furthermore, it is equal to liminfJ(a).
u a l a o c l p p e a h e
T
.
1 y a r l l r o
C o l f o
J e
!0
+ a2A ;aisrejected e
Proof. I use rst order approximations for Aand I e suggested by Cabrales et al. [2012a] to show that when priors and posteriors are close, so are Aand J. The result follows directly. For a=

( ; ),A(a) 1 s
Xp
k
(s) X e p(k)(ln(p(k) lnq s k
) and, I e s
( ) Xp
k
(s) Xp(k)(1 + lnq s k
)(ln(p(k) lnq s k
) so J e
X (s) Xp(k)(1 + lnq s k
)(ln(p(k) lnq s k
): p k
(a) 1 s
Hence, for an arbitrary >0, for transactions involving su ciently small amounts of information, jJ e
( ) A(a)j<
1
1 s
Pp (s) P k;s
max k p(k)(lnqj(lnq s ks kk;s
)j max)(ln(p(k) lnqj(ln(p(k) lnq s ks k
) +
2
)j+ 2
!0 in the de nition of local appeal, and the fact that p k
Using the fact that kpq s k
1
20
>0 for all k, the continuity of the logarithmic function on the positive reals implies that for transactions

involving su ciently small amounts of information, jJ e
( ) A(a)j< :
As was arbitrary, the proof is complete.
5.2.3 Axiomatic Characterization For an index Q, say that Q-informativeness is globally more appealing to agent ithan to agent j
(written j/
Q i) if the in mum over wof the local appeal of Q-informativeness for iis greater than the supremum of the local appeal of Q-informativeness for jover w.
u
Axiom 13. Local consistency. For every uand w, The local appeal of Q-informativeness for u with wealth wis equal to (w), for some strictly increasing function : R !R .
Axiom 14. Global consistency. For any wand a;b2A, if j/
Q i, A(a) <A(b) and jaccepts aat w, then iaccepts bat w.
Axiom 15. Homogeneity. For every information transaction a= ( ; ) and every >0;Q( ; ) =
1
Q(a).
This axiom states that Qis homogenous of degree -1 in transaction prices. This axiom entails the cardinal content of the index. It is particularly interesting if the units of the index are interpreted as\information per dollar."
Theorem 9. Ais the unique index that satis es Axioms 13-15, up to a multiplication by a positive number.
Proof. Axiom 13 implies that if j/
Q ithen iis uniformly more risk averse than j. Combined with this fact,
Axioms 14 and 15 imply the two axioms that are uniquely satis ed by positive multiples of A, according to Theorem 4 in Cabrales et al. [2012a]. That Asatis es Axiom 13 follows from
Theorem 8. That the other Axioms are satis ed is shown in Cabrales et al. [2012a].
For the sake of simplicity, Axiom 13 was stated in a less primitive way. To complete the analysis,
I provide conditions which are satis ed by Aunder which the axiom holds.
Axiom 16. Full-image. For every >0, ImQ(A ) = R
+
.
Axiom 17. Acceptance-cuto . For all uand w, the local appeal of Q-informativeness to uat wis positive and nite. Further more, it is smaller or equal than liminfQ(a).
!0
+ a2A ;aisrejected
Lemma 10. If Qsatis es Axioms 15, 16 and 17, then it satis es Axiom 13.
21

Proof. The proof uses the same techniques used above. If u
(w) > v
(w
0
) then there exists some >0 such that u
(w) >(1 + ) v
(w
0 then uaccepts ((1 +
2
). Following the arguments used before, for >0 small enough, if vaccepts a= ( ; ) 2A) ; ). Together with Axioms 15 and 17 this implies that the local appeal of Q-informativeness to uat wis greater than the local appeal of Q-informativeness to vat w
0
.
In the other direction, assume u
(w) = v
(w
0
), and by way of contradiction assume that the local appeal of Q-informativeness to uat wis not equal to the local appeal of Q-informativeness to vat w
0
.
Without loss of generality, assume that the local appeal of Q-informativeness to uat wis greater than the local appeal of Q-informativeness to vat w
0n1 n=1
. This means that there exists a sequence fagof information transactions, such that for every n, a n
= ( n
; ) satis es (i) a n
2A
1 n
;(ii) For some small >0, n nn 0n
17, and so the local appeal of Q-informativeness to uat wis equal to the local appeal of
Q-informativeness to vat w
0
.
Corollary 2. Ais the unique index that satis es Axioms 14-17, up to a multiplication by a positive number.
5.2.4 Properties of the Index A The setting presented here is based on Cabrales et al. [2012a] and is related to Cabrales et al.
[2012b]. It is somewhat di erent than other settings that are discussed in this paper, in that the index depends on the prior, and is therefore not completely \objective." Cabrales et al. [2012b] showed that this problem cannot be avoided. The fact that in the setting presented here the prior and the prices (which are observable) coincide is comforting in this regard.
An important property of the index A, is that it is monotonic with respect to Blackwell’s[1953] partial ordering of information structures [Cabrales et al., 2012a]. According to Blackwell’s order, an information structure is more informative than another if the later is a garbling of the prior. Blackwell
[1953] proved that one information structure is more informative than another according to this partial ordering if an only if every decision maker prefers it to the other. Cabrales et al. [2012a] show that if is more informative than in the sense of Blackwell, then A( ; ) >A( ; ) for every >0. As
Blackwell’s ordering is an analog to stochastic-dominance and time-dominance, this property is analogous to the properties of the indices presented before.
(a) Rating agencies and simple decision rules. Let us interpret the gambles as risky assets and the riskiness index as an objective rating, similar to the ones that rating agencies produce. The consistency results of Section 3 imply that for small investments individuals are able to form a simple investment strategy, based only on the objective rating
22

of the risky asset, their wealth and their utility function, that will result in very similar decisions to the ones an expected utility maximizer produces.
In the context of additive gambles, the rating could be R
AS and a possible strategy strategy will reject gambles riskier than
1
+ and accept gambles that are less risky than
1 u
( w)0 u
( w)
. This result is especially interesting for environments in which attaining and interpreting information about the risky asset is costly [for example Dellavigna and
Pollet, 2009].
(b) Cardinal interpretation. If we take seriously the cardinal content of an index of riskiness Q, then the local aversion to the Q-riskiness also carries some cardinal content. For example, from the AS riskiness index, we get a cardinal interpretation for (w); u
(w) =
1 2 v
(w
0u
) if and only if uat wis willing to accept (small) gambles twice as risky as those that vis willing to accept at w
0
. The same applies to the other indeices presented here.
(c) An incomplete order of local aversion to riskiness. In the previous sections I assumed that utility functions were twice continuously di erentiable with a positive rst derivative. In some cases, however, one might be interested in utility functions that are less well behaved. In these cases, the de nition of the local aversion of uto Qat wmight still work well, as the following example illustrates: w u(w) = minfe
2 w
; 1 e 2 g;v(w) = minfe w
4 w
; 3 e4 g:
22
While vis no less risk averse than uat 0 according to Yaari’s natural partial order, we cannot apply the Arrow-Pratt de nition in order to make the comparison.In contrast, one can show that vis locally no less averse to AS riskiness than u.
(d) Complete orders of local aversion to riskiness. While Ayields a partial order on utility-wealth pairs, each of its coordinates yields a complete order. As they both re ne A
QQAS
;by Lemma 1 they automatically re ne the natural partial order. Lemmas 2 and 3 show that for Rand R
FH
1 3
, in the class of utility functions treated in this paper the two orders coincide. This does not have to be the case for all utility functions and all risk indices, as the following example illustrates: u(w) := jwj c e
sgn(w), where sgn( ) is the sign function. uat wealth 0 accepts any gamble of the form g= [ ; c
1+ e
; ;
1 1+ e c
], for which R
AS
(g) = c, hence
1 R
R AS
( u;0)
= 0. On the other hand, for small >0, uat wealth 0 rejects gambles of the form g
0
= [2 ;
1 3
+ ; ;
2 323
].R
AS
(g
0
) !
!0
0, and therefore
1 S
R AS
( u;0)
= 1. Comparing uwith any CARA utility function completes the example.
23
22
In this example we can use the pair of one-sided derivatives.
23
does not depend on .

24
(e) Risk lovers. In their concluding remarks Aumann and Serrano [2008] suggested extending their approach to gambles with negative expectation, which will apply to risk lovers. The current results highlight this need, since the local aversion to Qas it is currently de ned is unable to accommodate risk loving behavior.
Similarly, the index of relative riskiness Swas constructed here using the class of gambles with FH-riskiness lower than 1. It is easy to see that the same criticism applies to it, as it cannot handle multiplicative gambles with geometric mean smaller than 1, and hence it cannot accommodate utility functions with relative risk aversion lower than that of the logarithmic utility. Finally, the index Dalso su ers from the same aws. Dcannot accomodate investments cashow in which the return in smaller than the investments.
AS
(f) Monotonicity. A common property of all the indices for which axioms were provided in this paper is that they respect monotonicity with respect to some intuitive partial order. In the case of Rit is stochastic-dominance, while in the other cases it is timedominance, or Blackwell’s partial order.
(e) Empirical Applications. Recently, attempts are bring made to apply the AS and FH indices in the empirical seeting. The most prominent example is Kadan and Liu [2012] who observe that tail events and rare disasters are unaccounted for by traditional performance evaluation measures
[Barro, 2006]. They propose a reinterpretation of the AS and FH indices as performance measures, and illustrate the applicability of these measures by using them to evaluate popular anomalies and investment strategies, and by applying them to the selection of mutual funds.
The ndings of this paper may turn out to be useful in the empirical setting as well. First, the index
Sof relative risk aversion, seems to be a more natural choice for portfolio selection. Moreover, similar to the FH measure it ampli ed the weight of rare disasters, but unlike the FH measure it is continuous in its domain, thus avoiding numerous di caulties in estimation. Second, the paper suggested a way to generate indices for di erent environments. In cases where some property must be measured, but no acceptable index exists, this approach may be useful for generating an index as required.
are positive by de nition.
24
24
To see this, note that both coordinates of A
Q

Pratt [1964] explains that u
(x) can be interpreted as\a measure of local risk aversion (risk aversion in the small)." Moreover, he states that while he does not introduce a simple measure to compare
\risk aversion in the large," global risks are considered. Speci cally, he shows that uis globally more averse to risk than vif and only if uis locally more averse to risk than vat all wealth levels.
In contrast, Aumann and Serrano [2008] start by de ning some global properties of utility functions. They de ne two partial orders that represent global risk aversion; \no less risk averse,"
Pratt’s order, and \uniformly no less risk averse," a stronger property. Their duality axiom links between global risk aversion and riskiness.
The current paper takes a di erent approach. For an index of riskiness Q, I start by introducing the concept of local aversion to Q, so riskiness is de ned prior to the de nition of its (local) aversion. I then show that for two prominent riskiness indices the concept coincides with the Arrow-Pratt local aversion to risk. However, I nd that this local consistency is not satis ed uniquely by these indices, and provide examples for\unreasonable"indices with this property.
Using the concept of local aversion to a riskiness index, I present a unique approach for deriving indices (of riskiness). For the case of additive gambles, I show that there is a unique \reasonable" order of local aversion (still, for a general class of indices of riskiness). With this result at hand, I show that the only index that satis es the property of global consistency and homogeneity is the AS index of riskiness.
To demonstrate the generality of this methodology, I provide other applications. Still in the contect of risk, I show that the index of relative riskiness suggested by Schreiber [2012] could be derived from the Arrow-Pratt relative risk aversion order, using the exact same technique. In di ereent contexts, I use the same methodology to derive an index of tardiness-aversion, and the index of appeal of information transactions [Cabrales et al., 2012a].
Future work should use this methodology for deriving other \objective" indices. A particualar setting that seems promising in this regard is the measurment of inequality, which has many similarities to the setting of riskiness [Atkinson, 1970].
Kenneth J. Arrow. Aspects of the Theory of Risk-Bearing. Helsinki: Yrj o Jahnssonin S a atio,
1965.
Kenneth J. Arrow. Essays in the Theory of Risk Bearing. Chicago: Markham, 1971.
Kenneth J. Arrow and David Levhari. Uniqueness of the internal rate of return with variable life of investment. The Economic Journal, 79(315):pp. 560{566, 1969. ISSN 00130133. URL http://www.jstor.org/stable/2230382.
Anthony B. Atkinson. On the measurement of inequality. Journal of economic theory, 2(3):244{263,
1970.
Robert J. Aumann and Roberto Serrano. An economic index of riskiness. Jour-
25

nal of Political Economy, 116(5):pp. 810{836, 2008. ISSN 00223808. URL http://www.jstor.org/stable/10.1086/591947.
Robert J. Barro. Rare disasters and asset markets in the twentieth century. The Quarterly Journal of
Economics, 121(3):823{866, 2006. doi: 10.1162/qjec.121.3.823. URL http://qje.oxfordjournals.org/content/121/3/823.abstract.
David Blackwell. Equivalent comparisons of experiments. The Annals of Mathematical Statistics,
24(2):pp. 265{272, 1953. ISSN 00034851. URL http://www.jstor.org/stable/2236332.
yvind B hren and Terje Hansen. Capital budgeting with unspeci ed discount rates. The
Scandinavian Journal of Economics, 82(1):pp. 45{58, 1980. ISSN 03470520. URL http://www.jstor.org/stable/3439551.
Antonio Cabrales, Oliver Gossner, and Roberto Serrano. The appeal of information transactions.
2012a.
Antonio Cabrales, Oliver Gossner, and Roberto Serrano. Entropy and the value of information for investors. 2012b.
Gerard Debreu. Theory of value: An axiomatic analysis of economic equilibrium, volume 17. Yale
University Press, 1972.
Stefano Dellavigna and Joshua M. Pollet. Investor inattention and friday earnings announcements.
The Journal of Finance, 64(2):709{749, 2009. doi: 10.1111/j.1540-6261.2009.01447.x.
Steinar Ekern. Time dominance e ciency analysis. The Journal of Finance, 36(5):1023{1033, 1981.
Irving Fisher. The theory of interest. New York: Kelley, Reprint of the 1930 Edition, 1930.
Dean P. Foster and Sergiu Hart. An operational measure of riskiness. Journal of Political Economy,
117(5):pp. 785{814, 2009. ISSN 00223808. URL http://www.jstor.org/stable/10.1086/644840.
James E. Foster and Tapan Mitra. Ranking investment projects. Economic Theory, 22:469{494,
2003. ISSN 0938-2259. doi: 10.1007/s00199-002-0339-y. URL http://dx.doi.org/10.1007/s00199-002-0339-y.
Josef Hadar and William R. Russell. Rules for ordering uncertain prospects. The American
Economic Review, 59(1):pp. 25{34, 1969. ISSN 00028282. URL http://www.jstor.org/stable/1811090.
Giora Hanoch and Haim Levy. The e ciency analysis of choices involving risk. The Review of
Economic Studies, 36(3):pp. 335{346, 1969. ISSN 00346527. URL http://www.jstor.org/stable/2296431.
26

Sergiu Hart. Comparing risks by acceptance and rejection. Discussion Paper no. 531, Hebrew Univ.
Jerusalem, 119(4), 2010.
Sergiu Hart. Comparing risks by acceptance and rejection. Journal of Political Economy, 119(4): pp.
617{638, 2011. ISSN 00223808. URL http://www.jstor.org/stable/10.1086/662222.
Ulrich Homm and Christian Pigorsch. An operational interpretation and existence of the
Aumann-Serrano index of riskiness. Economics Letters, 114(3): 265 { 267, 2012. ISSN 0165-1765. doi: 10.1016/j.econlet.2011.10.030. URL http://www.sciencedirect.com/science/article/pii/S0165176511004101.
Ohad Kadan and Fang Liu. Performance evaluation with high moments and disaster risk. Available at SSRN: http://ssrn.com/abstract=2020724 or http://dx.doi.org/10.2139/ssrn.2020724, 2012.
S. Kullback and R.A. Leibler. On information and su ciency. The Annals of Mathematical Statistics,
22(1):79{86, 1951.
Carl J. Norstr m. A su cient condition for a unique nonnegative internal rate of return. Journal of
Financial and Quantitative Analysis, 7(03):1835{1839, 1972.
John W. Pratt. Risk aversion in the small and in the large. Econometrica, 32(1/2):pp. 122{136, 1964.
ISSN 00129682. URL http://www.jstor.org/stable/1913738.
John W. Pratt and John S. Hammond. Evaluating and comparing projects: Simple detection of false alarms. The Journal of Finance, 34(5):1231{1242, 1979. ISSN 1540-6261. doi:
10.1111/j.15406261.1979.tb00068.x. URL http://dx.doi.org/10.1111/j.1540-6261.1979.tb00068.x.
Michael Rothschild and Joseph E. Stiglitz. Increasing risk: I. a de nition. Journal of Economic
Theory, 2(3):225{243, September 1970. URL http://ideas.repec.org/a/eee/jetheo/v2y1970i3p225-243.html.
Amnon Schreiber. An economic index of relative riskiness. Discussion Paper no. 597, Hebrew Univ.
Jerusalem, 2012.
Ralph Turvey. Present value versus internal rate of return-an essay in the theory of the third best.
The Economic Journal, 73(289):93{98, 1963.
Menahem E. Yaari. Some remarks on measures of risk aversion and on their uses. Journal of
Economic Theory, 1(3):315{329, 1969. URL http://econpapers.repec.org/RePEc:eee:jetheo:v:1:y:1969:i:3:p:315-329.
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