Discussion Paper No. 609

PORTFOLIO INERTIA UNDER AMBIGUITY

Takao Asano

June 2004

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Portfolio Inertia under Ambiguity∗

Takao Asano †Institute of Social and Economic Research
Osaka University
Mihogaoka 6-1, Ibaraki, Osaka 567-0047
JAPANFirst Version: August 1, 2001
This version: December 1, 2003

Abstract
We consider individual’s portfolio selection problems. Introducing the conceptof ambiguity, we show the existence of portfolio inertia under the assumptions thatdecision maker’s beliefs are captured by an inner measure, and that her preferences
are represented by the Choquet integral with respect to the inner measure. Under the concept of ambiguity, it is considered that a σ-algebra is not necessarily
an appropriate collection of events to which a decision maker assigns probabilities.
Furthermore, we study the difference between ambiguity and uncertainty by considering investors’ behavior.
Journal of Economic Literature Classification Numbers: D81, G11
Key Words: Ambiguity, Knightian Uncertainty, Portfolio Inertia∗

I would like to thank Hiroyuki Ozaki for his discussions and comments on this work. I am gratefulto seminar participants at Keio University, Tokyo Institute of Technology, University of Tokyo, andYokohama National University as well as participants at the conference on “Knightian Uncertainty” heldat University of Tokyo. I am also grateful to Hiroyuki Kojima, Motonari Kurasawa, Daisuke Oyama,
and Satoru Takahashi as well as participants at the 2002 Annual Meeting of the Japanese Economic
Association in Otaru. Finally, I am greatly indebted to Akihiko Matsui, who patiently read throughearlier versions of the present paper, encouraged me through uncountable many meetings, and gave me
invaluable comments and suggestions. Needless to say, I am responsible for any remaining errors.
†
e-mail: tasano@iser.osaka-u.ac.jp

1

Introduction

In stock markets, we often observe portfolio inertia, i.e., a situation in which some stocks
are not traded or not priced for a few minutes or longer. This is neither an exceptionalsituation in which some stock price soars too high to be priced, nor the one where somestock price plummets too much to be traded. Now, a question is worth raising; what leads
to portfolio inertia?
Under the standard expected utility theory, Arrow (1965) shows that an investor
has the unique striking price, i.e., the price above which she is willing to sell, and below
which she is willing to buy. Portfolio inertia generically exists only if the striking price forselling and that for buying have a spread. The standard expected utility theory does notaccount for the existence of the spread, except for a transaction cost, which is too small
for institutional investors to justify the existence of portfolio inertia.
Dow and Werlang (1992) account for the existence of portfolio inertia under the
Choquet Expected Utility (henceforth CEU) with convex non-additive measures. In theliterature on CEU, decision makers’ (henceforth DM) attitude toward uncertainty is captured by the convexity of non-additive measures.1 However, the assumption of the convexity of non-additive measures is so strong that there exist possibilities that importanteconomic problems cannot be fully accommodated as an example in this paper explains.
In this paper, we weaken the convexity, and adopt a new concept, i.e., the concept of
ambiguity. In this alternative setting, DM’s attitude toward ambiguity, which is differentiated from risk and uncertainty, is captured by the super-additivity of non-additive
measures. The super-additivity is intuitively appealing compared with the convexity of
non-additive measures, it is weaker than the convexity, and it can accommodate the situations in which the convexity cannot. The purpose of this paper is to account for the
existence of portfolio inertia by adopting the concept of ambiguity.
In order to illustrate the point, suppose that an investor considers trading the stockof a multinational corporation that does business in two countries, Japan and the United
1

For the definitions of non-additive measures and convexity, see Section 2 in details.

1States. We suppose that the company’s profit is affected by the economic conditionsof both countries. We also assume that the state of the world is described by one of
the four elements in S = {gg, bg, gb, bb}, where the first and second letters stand forthe economic conditions of Japan and the US, respectively. We assume that she assigns
probabilities to the events that contain the state where the economic conditions of thetwo countries are good. However, we assume that she does not assign probabilities tothe events where the conditions of the two countries are both good, and so forth. We
make these assumptions in order to describe the situation in which investors do not haveenough information to pin down whether the economic conditions of the two countries
are in good, or in bad, or in opposite condition. Then the collection A of the events towhich probabilities are assigned is A = {S, ∅, {gg, gb}, {gb, bb}, {bg, bb}, {gg, bg}}.2 In thestandard model, probabilities are assumed to be defined on a family of subsets of a state
space, called a σ-algebra, while A is not.3 We call all subsets of S outside A ambiguous
events. As we explain in Subsection 2.2, there exists a situation in which the existence of
portfolio inertia cannot be accounted for by the convexity of non-additive measures thatis considered as capturing uncertainty aversion in the literature on non-expected utility
theories. Before we provide the definition of ambiguity, let us review some of the existing
concepts.
It is Ellsberg (1961) who first casts doubt on the validity of the Subjective Expected
Utility (henceforth SEU) theory axiomatized by Savage (1954), which is widely applied
to economic problems.4 Since then, a number of generalizations have been developed toovercome its shortcomings.5
In the course of development of non-expected utility theories, Knight’s approach(1921) is reconsidered as one way to account for Ellsberg-type situations. According
to Knight (1921), the risk situation is a situation where a probability measure can be
2

See Subsection 2.2 in details.
Note that A is not closed with respect to intersections.
4
SEU states that if a certain set of axioms is satisfied, then DM’s beliefs are captured by a uniqueprobability measure and her preferences are represented by the expected utility.
5
Using an easy-to-understand example, Ellsberg shows that people often violate Savage’s crucial axiom,
the Sure Thing Principle.
32

assigned, and the uncertainty situation is “all other situations.” In the literature on nonexpected utility theories, Knight’s risk situation is interpreted as a situation where DM’s
beliefs are captured by a unique probability measure, while the uncertainty situation isinterpreted as a situation where DM’s beliefs are not captured by a unique probability.
We call an approach based on the latter interpretation the uncertainty approach. In thelate 1990s, another approach based on a different interpretation of “all other situations”
is proposed. This approach focuses on domains of probability measures, in other words,
this considers that a σ-algebra is not necessarily an appropriate collection of events towhich a DM assigns probabilities, while the other two focus on set functions by which
DM’s beliefs are captured.6 In the present paper, we call this approach the ambiguity
approach.
The uncertainty approach is axiomatized and analyzed in the framework of theMaxmin Expected Utility (henceforth MMEU) theory and CEU.7 MMEU states that if acertain set of axioms is satisfied, then DM’s beliefs are captured by a set of finitely additive
measures and her preferences are represented by the minimum of expected utilities overthe set of these measures. On the other hand, CEU states that if a certain set of axioms
is satisfied, then DM’s beliefs are captured by a non-additive measure and her preferences
are represented by Choquet integrals. Unlike MMEU, CEU does not contain any axiom
that captures DM’s attitude toward uncertainty.8 Showing that CEU with a convex nonadditive measure is equivalent to a special case of MMEU, Schmeidler (1989) endows themeaning of uncertainty aversion with CEU. This enables us to analyze DM’s behavior
under uncertainty through the convexity of non-additive measures.9 Our understanding
6

See Epstein (1999) and Zhang (1999)
MMEU is axiomatized by Gilboa and Schmeidler (1989) in the Anscombe and Aumunn (henceforthAA) framework and axiomatized by Casadesus-Masanell, Klibanoff and Ozdenoren (2000) in the Savage
framework. CEU is axiomatized by Schmeidler (1989) in the AA framework and axiomatized by Gilboa(1987) in the Savage framework.
8
See Axiom 5 (Uncertainty Aversion) in Gilboa and Schmeidler (1989).
9Let µ be a convex non-additive measure, M be the set of finitely additive measures on (S, 2S ), andlet B(S, R) denote the space of bounded functions from S into R. ThenZZ
X(s)µ(ds) = min
X(s)P (ds) | P ∈ core(µ) ,7where X ∈ B(S, R) and the core of µ is the set of finitely additive measures on (S, 2S ) that dominate3

of DM’s behavior under uncertainty has been deepened by these new theories themselves
and their applications to game theory (Dow and Werlang (1994), Lo (1996, 1998, 1999),
and Marinacci (2000)), finance (Dow and Werlang (1992), and Epstein and Wang (1994,
1995)) or agency theory (Ghirardato (1994)).
Recently, Epstein (1999) casts doubt on the usage of the convexity of non-additive
measures as uncertainty aversion, and introduces the concept of ambiguity. Epstein (1999)
defines unambiguous events as those events to which a DM can assign probabilities, andambiguous events as the ones to which she cannot. This corresponds to the second interpretation of Knight. Thus, we consider Epstein’s approach as the ambiguity approach.
One of the purposes of this paper is to seek the possibility of applications of the ambiguity
approach to economic problems.
Zhang (2002) proposes a new axiomatization of CEU. This states that if a certain
set of axioms is satisfied, then DM’s beliefs are captured by an inner measure and her
preferences are represented by the Choquet integral with respect to the inner measure.10
Zhang’s axiomatization is more restrictive than Schmeidler’s one. However, adopting
Zhang’s approach, we can incorporate the concept of ambiguity into CEU, and shed lighton the possibility of its applications to other economic problems.
Based on Zhang (2002), we show that there exists portfolio inertia under CEU
with an inner measure, which is neither additive nor convex. This approach is one of
the starkest contrasts to the one that depends on the convexity of non-additive measures.
µ for all E ∈ 2S , i.e., core(µ) = P ∈ M |(∀E ∈ 2S ) P (E) ≥ µ(E) . This result implies that it is theconvexity of non-additive measures that serves the bridge between MMEU and CEU, and enables us toanalyze DM’s behavior under uncertainty through the Choquet integral. Mathematically, analyses of the
Choquet integral are easier than those of the minimization of expected utilities over a set of measures in
the framework of MMEU. For example, see Ghirardato (1997), and Ghirardato, Klibanoff and Marinacci(1998). The reason that the convexity of non-additive measures has been used in the literature on CEU
and MMEU is this mathematical tractability of the Choquet integral and the meaning of uncertainty
aversion through MMEU.
10
If a certain set of axioms is satisfied, her preferences are represented as follows: there exists a
nonconstant function u : X → R and a premeasure p on (S, A) such thatZZ(∀f, g ∈ F) f  g ⇔u ◦ f dpA ≥u ◦ g dpA ,where X is some set of outcomes, F is some set of acts, and pA is the inner measure corresponding to(S, A, p) and integrals are in the sense of Choquet integrals. For the definition of Choquet integrals, see
Subsection 2.1.

4

Furthermore, we study the reason that different investors have different degrees of portfolio
inertia.
The organization of this paper is as follows. Section 2 provides an example, somemathematical tools including the concept of the Choquet integral, inner and outer measures, and λ-systems. Section 3 presents main results of the present paper. Section 4
concludes this paper. Proofs and mathematical results are relegated to Appendices.

2

Preliminaries

2.1

Choquet Integrals

Let S be a set and let 2S be the power set of S. A set function µ : 2S → [0, 1] is a
non-additive measure (or capacity) if (a) µ(∅) = 0 and (b) E, F ∈ 2S and E ⊂ F implyµ(E) ≤ µ(F ), where ∅ denotes the empty set. A non-additive measure µ is convex ifµ(E ∪ F ) + µ(E ∩ F ) ≥ µ(E) + µ(F ) for all E, F ∈ 2S .
Let B(S, R) denote the space of bounded functions from S into R and let X ∈B(S, R). The integral of X with respect to a non-additive measure µ is called the Choquet
integral, and is defined byZ=

X(s) µ(ds)Z ∞µ ({s ∈ S| X(s) ≥ α}) dα +Z 0[µ ({s ∈ S| X(s) ≥ α}) − 1] dα,−∞0where integrals on the right hand side are in the sense of Riemann integrals. Throughout
this paper, all the integrals are in the sense of Choquet integrals.

2.2

An Example

Suppose that an investor considers trading the stock of a multinational corporation that
does business in two countries, Japan and the United States. We suppose that the company’s profit is affected by the economic conditions of both countries. We also assume
that the state of the world is described by one of the four elements in S = {gg, bg, gb, bb},
where the first and second letters stand for the economic conditions of Japan and the US,5respectively. For example, gb stands for the state where the economic condition of Japanis good and the economic condition of the US is bad. Let X : S → R+ denote the stock
price of the company at t = 1. Let X(gg) = a1 , X(bg) = a2 , X(gb) = a3 , and X(bb) = a4 .
We assume that the company’s profit is more likely to be affected by the economic condition of the US than that of Japan. So, we suppose that a1 > a2 > a3 > a4 . It is also
assumed that she assigns probabilities to the events that contain the state in which theeconomic conditions of the two countries are good. However, we assume that she does not
assign probabilities to the events in which the conditions of the two countries are both
good, and so forth. We make these assumptions in order to describe the situation in whichinvestors do not have enough information to pin down whether the economic conditionsof the two countries are in good, or in bad, or in opposite condition. An interpretationis that she is unwilling to, even subjectively, judge the correlation between the two countries’ economic conditions. We assume that she assigns p1 and p2 to {gg, gb} and {gg, bg},
respectively. It is also assumed that 1 − p1 and 1 − p2 are assigned to {gg, gb}c = {bg, bb}
and {gg, bg}c = {gb, bb}, respectively.11 Thus, the collection A of the events to which
probabilities are assigned is as follows:
A = {S, ∅, {gg, gb}, {gb, bb}, {bg, bb}, {gg, bg}} .
The assigned values are given byp(S) = 1, p(∅) = 0, p ({gg, gb}) = p1 ,
p ({gg, bg}) = p2 , p ({bg, bb}) = 1 − p1 , p ({gb, bb}) = 1 − p2 .
We assume that she dislikes “ambiguity,” in other words, she is very cautious, and that
she uses a set function pA : 2S → [0, 1] given bypA (B) = sup{p(A) | A ∈ A, A ⊂ B},
for every subset B ∈ 2S , not necessarily in A.12 This function assigns a number to every
subset B, which is approximated from the inside by an “unambiguous” event A ∈ A. Also
11
12

We assume that p1 and p2 are in (0, 1).
pA is a non-additive measure. See Lemma 1.

6

assume that her preferences are represented by Choquet integrals with respect to pA , andthat she is a CEU maximizer.
In the following situations, suppose that p2 > p1 .13 First, consider a situation
where she will buy the stock. Since the expected value of buying the stock isZ

X(s) pA (ds) = (a2 − a4 )p2 + a4 ,R
the expected return from buying it is X(s)pA (ds) − q, where q is the current price ofRthe stock. Thus, if q is less than X(s)pA (ds), she will buy the stock. Next, consider asituation where she will sell the stock. Since the expected value of selling the stock isZ−X(s) pA (ds) = (a2 − a1 )p1 + (a3 − a2 )p2 − a3 ,R
the expected return from selling it is q + −X(s)pA (ds). Thus, if q is more thanR
− −X(s)pA (ds), she will sell the stock. Note that the next inequality holds:−

Z−X(s) pA (ds) >Z

X(s) pA (ds).

Thus, if the current price q is between these two numbers, she will not consider trading
the stock.14 If this investor’s beliefs are captured by a unique probability, in other words,
if she does not care about ambiguity, then the above inequality is replaced by equality.
Thus, under the standard model like Arrow (1965), the existence of portfolio inertia is
not accounted for.
More observations are in order. A collection of the events to which probabilities
are assigned is typically assumed to be a σ-algebra. In our example, however, since{gg, gb} ∩ {gb, bb} = {gb} ∈/ A, A is not a σ-algebra. Moreover, sincepA ({gg, gb}) + pA ({gg, bg}) > pA ({gg, gb, bg}) + pA ({gg}) ,13

For this inequality, we need assume that she believes that {gg, bg} is more likely to happen than{gg, gb}. Even if the reverse inequality is assumed, portfolio inertia still exists.
14This phenomenon occurs when the expected values from buying and selling the stock are both negative, i.e.,ZZ
X(s)pA (ds) − q < 0 and q +−X(s)pA (ds) < 0.

7

pA is not convex.
In summary, we observe three points. First, a collection of the events to which thisinvestor assigns probabilities is not a σ-algebra. Second, a set function that captures her
attitude toward ambiguity is not a convex non-additive measure. Thus, such a situation
as this example cannot be accounted for in the framework of Dow and Werlang (1992).
Finally, even if the set function is not convex, portfolio inertia still exists.

2.3

λ-systems, Inner Measures

This subsection defines and discusses the concept of ambiguity, and provides mathematicalresults related to the concept. First of all, the definition of λ-systems is provided.
Definition 1. A non-empty class of subsets A ⊂ 2S is a λ-system if(λ.1) S ∈ A,(λ.2) A ∈ A implies Ac ∈ A, and(λ.3) hAi i∞
i=1 ⊂ A and Ai ∩ Aj = ∅ for i 6= j implyS∞

n=1 An ∈ A.

In the standard model, a probability measure is defined on a σ-algebra, while in
the present paper, a probability measure is defined on a λ-system.
Definition 2. Let S be a set and A be a λ-system on S. Consider a set function p : A →[0, 1] and let p satisfy the following property:P∞S∞(p.1) If hAi i∞
i=1 is a sequence of disjoint sets in A, then p ( i=1 Ai ) =i=1 p(Ai ).

Then p is called a premeasure on (S, A).15

If a premeasure p on (S, A) satisfies p(S) = 1, then (S, A, p) is called a λ-systemprobability space. Throughout this paper, (S, A, p) denotes a λ-system probability space.
Some comments on Definitions 1 and 2 are in order. Conditions (λ.1) and p(S) = 1
imply that a DM knows what the whole space is. Conditions (λ.2) and (p.1) imply that
if she knows what A is, she knows what the complement is. To put it differently, if
she can assign probabilities to an event, she can assign probabilities to its complement.
15

Formally speaking, a premeasure is defined on an algebra, not on a λ-system. For example, see
Folland (1999).

8

Conditions (λ.3) and (p.1) imply that if she knows what A1 and A2 are, she knows what
the disjoint union of them is. In other words, if she can assign probabilities to disjoint sets
A1 , A2 , respectively, she can assign probabilities to its disjoint union. On the other hand,as we see in the previous subsection, she could not assign probabilities to intersectionsof any sets in A. Thus, a λ-system can be considered as a collection of subsets that she
could assign probabilities to “unambiguously,” in other words, can be considered as acollection of “unambiguous events.” Following Epstein (1999), this paper uses a λ-system
A for a collection of unambiguous events. Every subset of S outside A is called ambiguous
events.16 Next, we consider a set function to capture DM’s attitude toward ambiguity.
Definition 3. A set function pA : 2S → [0, 1] defined by(∀B ∈ 2S ) pA (B) ≡ sup {p(A) | A ∈ A, A ⊂ B}
is the inner measure corresponding to (S, A, p).
Since ∅ ∈ A, pA is well-defined. Note that a DM whose beliefs are captured bypA is very cautious. Similarly, the outer measure corresponding to (S, A, p) is defined asfollows : A set function pA : 2S → [0, 1] defined by(∀B ∈ 2S ) pA (B) ≡ inf {p(A) | A ∈ A, B ⊂ A}
is the outer measure corresponding to (S, A, p). Since S ∈ A, pA is also well-defined. Now,
we must check what conditions the inner and outer measures satisfy.
Lemma 1. The inner and outer measures, pA and pA are non-additive measures.
Proof. See Appendix.
Note that pA is not necessarily a convex non-additive measure. Since the convexity of nonadditive measures has been assumed in order to capture DM’s attitude toward uncertainty,
its non-convexity of non-additive measures is one of the starkest contrasts to approaches
adopted in the literature on CEU and its applications. However, the inner measure has
useful results.
16

Epstein and Zhang (2001) define ambiguous events in terms of DM’s preferences.

9Lemma 2. Let pA and pA be the inner and outer measures corresponding to (S, A, p).
(a) pA is the conjugate of pA , that is, for all B ∈ 2S , pA (B) = 1 − pA (B c ).
(b) pA is superadditive, that is, for all B1 , B2 ∈ 2S such that B1 ∩ B2 = ∅, pA (B1 ∪ B2 ) ≥pA (B1 ) + pA (B2 ).
Proof. See Appendix.
Recall that the inner and outer measures are non-additive measures. The non-additivity
provides the next lemma.
Lemma 3. Let pA and pA be the inner and outer measures corresponding to (S, A, p).
Then for all X ∈ B(S, R),Z

X(s) pA(ds) ≤Z

X(s) pA (ds).

Proof. See Appendix.

3

Portfolio Selection Problems

In this section, we derive our main results. Under CEU with the inner measure pA , weshow that there exists a range of prices, in which investors neither buy nor sell.
First, we consider the portfolio selection problem à la Arrow (1965) and provide
one of the main results of this paper. Let W ∈ R+ be the wealth at t = 0, N ∈ R be theamount of money invested at t = 0, q > 0 be the price of a risky asset at t = 0, and X be
the random payoff of the asset at t = 1, where X ∈ B(S, R). Moreover, let u : R → R bea monotonic increasing concave function.17
An investor is assumed to invest in two assets, a riskless asset and a risky asset.
The problem is to choose N so as to maximizeZu (W − N + (N/q) X(s)) pA (ds).
In other words, she chooses the amount of money N to invest the risky asset in order tomaximize her non-expected utility of the terminal wealth.
17

In this paper, we capture investor’s attitude toward risk by u.

10

Theorem 1. Let pA and pA be the inner and outer measures corresponding to (S, A, p).
Then, this investor will neither buy nor sell the risky asset, if q satisfies:
Z
Z
X(s) pA (ds) < q <X(s) pA(ds).
Proof. See Appendix.
This theorem states that the range she will neither buy nor sell the risky asset does notdepend on her attitude toward risk measured by u. Moreover, the range does not dependon her attitude toward uncertainty captured by the convexity of non-additive measures.
In our setting, the range is determined by a set of unambiguous events A, and the innerand outer measures pA and pA corresponding to (S, A, p).
Before we investigate a possibility of the further application of the ambiguity approach, we provide an example.
Example 3.1. Suppose that two investors consider trading the stock of GE whose current
price is $35 per share. On the one hand, one investor might want to sell it if the stock
price is over $40, and might want to buy it if the stock price is below $30. On the other
hand, the other investor might want to sell it if the stock price is over $38, and might wantto buy it if the stock price is below $32. In this case, they will never consider changing
their positions. Moreover, ceteris paribus, suppose that the stock’s current price is $39.
In this case, the first investor does not consider trading the stock. However, the second
one does sell it.
While such a situation is not actually observable, it would cause no trouble tosuppose that different investors behave in a different way. To put it differently, it is natural
to suppose that a spread between the striking price for selling and that for buying is notunique among different investors. What brings the differences of the spreads between the
striking price for selling and that for buying among investors? This phenomenon as wellas the existence of portfolio inertia cannot be accounted for under the standard expectedutility theory. However, adopting the ambiguity approach enables us to obtain a solution
to the problem. Before we present the last theorem of this paper, some mathematical11

results are in order. Let (S, A1 , p1 ) and (S, A2 , p2 ) be λ-system probability spaces, whereA1 ⊂ A2 , and p1 = p2 |A1 , i.e., p1 is the restriction of p2 to A1 . Now, we are in a position
to present two results, which play a crucial part in obtaining the last result of this paper.
Lemma 4. Let (S, A1 , p1 ) and (S, A2 , p2 ) be λ-system probability spaces, where A1 ⊂ A2 ,and p1 = p2 |A1 . Then, for all B ∈ 2S , pA1 (B) ≤ pA2 (B) and pA2 (B) ≤ pA1 (B).
For the inner measure, Lemma 4 states that a fine λ-system yields a fine assessment
to all the subsets outside A by approximating them from the inside. For the outer measure,Lemma 4 states that a fine λ-system yields a fine assessment to all the subsets outside A
by approximating them from the outside. From above observations, pA can be considered
as a set function to capture a form of aversion to ambiguity through A.
Theorem 2. Let (S, A1 , p1 ) and (S, A2 , p2 ) be λ-system probability spaces, where A1 ⊂A2 , p1 = p2 |A1 , and X ∈ B(S, R). Then
Z
Z
Z
Z
A2
X(s) pA1 (ds) ≤ X(s) pA2 (ds) and X(s) p (ds) ≤ X(s) pA1 (ds).
Proof. See Appendix.
This theorem states that the Choquet integral with respect to pA is increasing in A,
and that the Choquet integral with respect to pA is decreasing in A. In other words,
the monotonicity of the Choquet integral with respect to A provides a fine evaluation ofX(s).
Theorems 1 and 2 provide the last theorem of this paper.
Theorem 3. Let (S, A1 , p1 ) and (S, A2 , p2 ) be λ-system probability spaces, where A1 ⊂A2 , and p1 = p2 |A1 , and let q be the price of a risky asset at t = 0, and let X ∈ B(S, R) be
the asset price at t = 1. Suppose that one investor’s preferences are represented by CEU
with the inner measure pA1 corresponding to (S, A1 , p1 ), and the other investor’s preferences are represented by CEU with the inner measure pA2 corresponding to (S, A2 , p2 ).
Then both investors neither buy nor sell the asset if q satisfies:
Z
Z
X(s) pA2 (ds) < q < X(s) pA2 (ds).
12

Furthemore, for such an asset price, the spread between the striking price for selling andthat for buying with respect to A1 is wider than that with respect to A2 , i.e.,
Z
Z
Z
Z
A2
X(s) pA1 (ds) ≤ X(s) pA2 (ds) < q < X(s) p (ds) ≤ X(s) pA1 (ds).
The latter part of this theorm provides a solution to the problem: what leads to thedifferences of the spreads between the striking price for selling and that for buying among
different investors? If an investor has more events to which she assigns probabilities,
then the spread shrinks compared with the case in which she has less events to which
she assigns probabilities. This interpretation is made possible by capturing ambiguity
through λ-systems.18

4Conclusion

This paper adopts the ambiguity approach to show that there exists portfolio inertia
under CEU with an inner measure. A related work is Dow and Werlang (1992) who
adopt the uncertainty approach to obtain a result similar to ours. However, there are twodifferences between their paper and this one. First, their model is based on CEU withevery convex non-additive measure. On the other hand, we adopt CEU with the innermeasure pA , which is neither additive nor convex. As the example in Subection 2.2 shows,there exists some situation where the convexity of non-additive measures does not accountfor investor’s behavior. This implies that such a situation cannot be accounted for within
the framework of theirs, and that the convexity assumption is not crucial for the existenceof portfolio inertia. Second, we introduce the concept of ambiguity to portfolio selection
problems by letting λ-systems serve as domains of probabilities. Adopting the conceptof ambiguity enables us to account for a problem that cannot be explained by Dow andWerlang (1992). To the best of my knowledge, analyses based on the ambiguity approach
have not been done in the fields of game theory or finance. Our results provide a new view
18

A different approach appears in Dow and Werlang (1992). Let µ, µ1 and µ2 be non-additive measureson (S, 2S ). They define uncertainty aversion, by c(µ, A) = 1 − µ(A) − µ(Ac ), measured by µ, for every
set A in 2S . Furthermore, they say that µ2 is more uncertainty averse than µ1 if c(µ2 , A) ≥ c(µ1 , A) for
every set A in 2S . Based on this definition, they show that if µ2 is more uncertainty averse than µ1 , aspread between the striking price for selling and that for buying expands.

13

point by illuminating the difference between the uncertainty approach and the ambiguity
approach. Furthemore, our results imply that the convexity of non-additive measures is
not a panacea for analyzing DM’s behavior under uncertainty. If this paper becomes afirst step toward the further research on the ambiguity approach, the goal of this paper
will be achieved.

14

Appendix
A. Mathematical Results
In this subsection, let µ be a non-additive measure on (S, 2S ).
Fact 1.
(∀X, Y ∈ B(S, R)) X ≥ Y ⇒Z

X(s) µ (ds) ≥Z

Y (s) µ (ds).

Fact 2.
(∀X ∈ B(S, R))(∀a ∈ R)(∀b ∈ R+ )Z

(a + bX(s))µ (ds) = a + b

ZX(s)µ (ds).

Fact 3.
(∀X ∈ B(S, R))Z

X(s) µ(ds) = −Z−X(s) µ′ (ds),where µ′ is the conjugate of µ.
Lemma 5. Let u : R → R. If u is concave on (a, b) and t0 ∈ (a, b), then there exists
β ∈ R such thatu(t) ≤ u(t0 ) + β(t − t0 ) for all t ∈ (a, b).
For such a β, the next inequlities hold:u(t) − u(t0 )u(t) − u(t0 )≡ u′+ (t0 ) ≤ β ≤ lim≡ u′− (t0 ).
t↑t0
t↓t0
t − t0t − t0lim

Furtheremore, if u is increasing on (a, b), then 0 ≤ u′+ (t0 ).
Proof of Lemma 5. For the first claim, see Folland (1999, p.109), or Royden (1987, p.115).
The second and third ones are immediate.

15

B. Proofs
Proof of Lemma 1. For all B1 , B2 ∈ 2S , let B1 ⊂ B2 . Then {p(A)| A ∈ A, A ⊂ B1 } ⊂{p(A)| A ∈ A, A ⊂ B2 }, which implies pA (B1 ) ≤ pA (B2 ). The proof that pA (∅) = 0 is
trivial.
Proof of Lemma 2. (a)(∀B ∈ 2S ) pA (B) = sup {p(A) | A ∈ A, A ⊂ B}
= − inf {−p(A) | A ∈ A, A ⊂ B}
= − inf {p(Ac ) − 1 | Ac ∈ A, B c ⊂ Ac }= 1 − inf {p(Ac ) | Ac ∈ A, B c ⊂ Ac }= 1 − pA (B c ).
(b) If ε > 0, there exist A1 , A2 ∈ A, A1 ⊂ B1 , A2 ⊂ B2 , B1 ∩ B2 = ∅ such that pA (B1 ) −ε ≤ p(A1 ) and pA (B2 ) − ε ≤ p(A2 ). So
pA (B1 ) + pA (B2 ) − 2ε ≤ p(A1 ) + p(A2 )= p(A1 ∪ A2 )≤ pA (B1 ∪ B2 ).
The first equality follows from the facts that B1 ∩ B2 = ∅ ⇒ A1 ∩ A2 = ∅, A1 ∪ A2 ∈ A
and p is a premeasure on A. The last inequality holds by the definition of pA . Since ε isarbitrary, the proof is completed.
Proof of Lemma 3. This follows from Fact 1 and the fact that pA (B) ≤ pA (B) for all
B ∈ 2S .

16

Theorem 4 (Jensen’s Inequality). Let (S, 2S , µ) be a measure space with µ(S) = 1,
where µ is a non-additive measure. Let X : S → (a, b) be in B(S, R), and u be a concave,increasing and real-valued function on (a, b). Then for all non-additive measures µ,ZZu(X(s)) µ(ds) ≤ uX(s) µ(ds) .
Proof. Let t0 =R

X(s) µ(ds) and t = u(s). Then t0 ∈ (a, b) and t ∈ (a, b). By Lemma 5,

there exists β ∈ [u′+ (t0 ), u′− (t0 )] ⊂ R+ such that
Z



Z
u (X(s)) ≤ uX(s) µ(ds) + β X(s) −X(s) µ(ds) for all X(s) ∈ (a, b).
Integrating both sides with respect to µ and Fact 1 implyZu (X(s)) µ(ds)Z  ZZ≤u
X(s) µ(ds) + β X(s) −X(s) µ(ds)µ(ds).
ThenZ

u (X(s)) µ(ds)
Z


Z 
Z
≤ u
X(s) µ(ds) + β
X(s) −
X(s) µ(ds) µ(ds)
Z

Z

Z
= u
X(s) µ(ds) + β
X(s) µ(ds) −X(s) µ(ds)Z= u
X(s) µ(ds) .
The inequality and the first equality follow from Fact 2.
Proof of Theorem 1. Assume
Z

X(s) pA (ds) < q.

Then(∀N ≥ 0)Z

u (W − N + (N/q) X(s)) pA (ds)Z≤ u(W − N + (N/q) X(s)) pA(ds)
17



Z
= u W − N + (N/q)
X(s)pA (ds)< u (W − N + (N/q) q)= u(W ).
The first inequality follows from Theorem 4. The first equality holds by Fact 2. The
second inequality follows from the assumption.
Next, assumeq<ZX(s) pA (ds).

Similarly,(∀N ≤ 0)Z

u (W − N + (N/q) X(s)) pA (ds)Z≤ u(W − N + (N/q) X(s)) pA (ds)Z
= u W − N − (N/q) (−X(s)) pA (ds)

 Z

A
= u W − N − (N/q) −X(s) p (ds)< u (W − N − (N/q) (−q))= u(W ).

The first inequality holds by Theorem 4. The first equality holds by Fact 2. The second
equality holds by Fact 3. The last inequality follows from the assumption.
Proof of Theorem 2. The first and second inequalities follow from the boundedness of X,
the definition of the Choquet integral, and Lemmas 3 and 4.

18

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