KIER DISCUSSION PAPER SERIESKYOTO INSTITUTEOFECONOMIC RESEARCH
Discussion Paper No.1071“Precautionary Saving against Correlation
under Risk and Ambiguity”

Takao Asano

Yusuke Osaki

January 2022

KYOTO UNIVERSITY
KYOTO, JAPANPrecautionary Saving against Correlation
under Risk and Ambiguity∗
Takao Asano † Yusuke Osaki ‡This Version: January 11, 2022

Abstract
This paper considers precautionary saving against the correlation between
two risky attributes (wealth and health) and investigates how the correlation affects optimal savings under multivariate preferences. The signs of higher-ordercross derivatives play a key role in determining the direction of precautionarysaving against such correlation. Mixed correlation averse (seeking) individuals
increase (decrease) savings in response to increases in correlation. Furthermore,
we introduce ambiguity to the correlation and investigate how ambiguity aﬀects
the amount of optimal savings. The analyses enable us to deepen our understanding of saving behavior under multivariate preferences in the presence ofcorrelation.

Key Words: Mixed correlation aversion (seekingness), Multivariate
preferences, Smooth ambiguity model, Stochastic dominance∗
We are grateful to Chiaki Hara, Nobuyuki Hanaki, Takashi Kunimoto, Takehito Masuda, Hitoshi Sadakane, Tadashi Sekiguchi and thank participants in seminars held at Kyoto and Osaka
Universities for their comments and suggestions. Of course, all remaining errors are our sole responsibility. This research is financially supported by JSPS KAKENHI Grant Numbers 20K01745,
17K03806, 20H01507, 20H01519, 21K01393, and the Joint Research Program of KIER.
†
Faculty of Economics, Okayama University, 3-1-1 Tsushimanaka, Kita-ku, Okayama 700-8530,
Japan. e-mail: takaoasano@okayama-u.ac.jp‡
Faculty of Commerce, Waseda University, 1-6-1 Nishiwaseda, Shinjyku-ku, Tokyo 169-8050,
Japan. e-mail: osakiy@waseda.jp

1

1

Introduction

How much to save and consume are essential household financial decisions. Uncertainty is one reason why households save, and this is commonly referred to as
the precautionary saving motive.1 In the literature on precautionary saving, onedimensional risk has mainly been investigated. However, saving decisions often facemultidimensional risks. For example, both the riskiness of the wealth levels andhealth conditions of individuals aﬀect saving decisions because both are importantin our daily lives.
The analyses of multivariate preferences provide clues for understanding our saving behaviors in the presence of multidimensional risks. Moreover, it is plausible toassume that the level of wealth and the condition of health are closely related. Thisimplies that the role of correlation between the wealth level and the health condition
should be incorporated into analyses of precautionary saving decisions. Motivatedby the above recognition, the purpose of this paper is to analyze precautionarysaving against the correlation between two risky attributes.
In most cases, it is more diﬃcult to quantify the correlation between two risky
attributes than the risk of an individual attribute due to the lack of experience andinsuﬃcient observations. We acknowledge this diﬃculty by introducing ambiguity
about the correlation between two risky attributes. One of the purposes of thispaper is to extend the analyses of precautionary saving to include correlation with
ambiguity and to investigate how correlation with ambiguity aﬀects the amount ofsavings.
Since the seminal paper by Ellsberg (1961), the importance of ambiguity has
been recognized in the literature. The developments of Choquet expected utility
and maxmin expected utility have enabled us to investigate behaviors and decisions under ambiguity (see Gilboa and Schmeidler (1989) and Schmeidler (1989)
for axiomatizations). This paper adopts the smooth ambiguity model in Klibanoﬀ
1

Uncertainty is used as an umbrella term that includes both risk and ambiguity. Risk is thesituation where uncertainty can be measured by a unique probability and ambiguity is the situation
where it cannot.

2
et al. (2005, 2009). The model can diﬀerentiate the decision makers’ attitudes
toward ambiguity from their perception of ambiguity.2 Furthermore, the model ismore tractable than most of the models analyzing ambiguity. The usefulness of thesmooth ambiguity model has been widely recognized, for example, an application to
the portfolio problem by Gollier (2011).
Precautionary saving has been investigated extensively in the literature. It dates
back to Leland (1968) that compares future labor income under risk with that undercertainty. Since Leland (1968), numerous studies have used theoretical and empiricalapproaches.3 Kimball (1990) is a milestone in research on precautionary saving andcharacterizes the notion of prudence that is related to the convexity of marginalutility. Most studies on precautionary saving have focused on investigating situations
where households face only risky future incomes. This paper advances this research
field by examining the eﬀects of correlation between two risky attributes on optimalsavings in a framework of multivariate preferences. In addition, we analyze thesituation in which there are multiple possible realizations of correlation by applying
the notion of ambiguity.
This paper makes the following three contributions to the literature on precautionary saving and multivariate preferences. First, we provide a framework for analyzing the eﬀects of correlation between two risky attributes on precautionary saving
under multivariate preferences. Second, we demonstrate how the signs of successive
cross derivatives of a utility function aﬀect precautionary saving toward correlation.
The condition is called mixed correlation aversion (seekingness) which is a bivariate
extension of mixed risk aversion (seekingness) by Caballé and Pomansky (1996) ina univariate framework. Mixed correlation aversion (seekingness) generalizes correlation aversion (seekingness) with a positive (negative) sign of a cross derivative.
Mixed correlation averse (seeking) individuals increase (decrease) their precaution2
Many decision models designed to capture ambiguity aversion are proposed with axiomatic
foundations and are applied to various problems that appear in economics and finance. Machina
and Siniscalchi (2014) provide a review of canonical models.
3
See Baidrdi et al. (2020) for a review of the theoretical studies and Liguilde et al. (2019) forthe empirical studies.

3
ary savings in response to an increase in correlation. Third, we consider a situation
where there are many possible correlations because of the diﬃculty of quantifying
the relationship between multiple attributes compared with one-dimensional uncertainty itself. This situation is captured by the notion of ambiguity. We determine
the conditions under which ambiguous correlations make decision makers increase
their precautionary savings by more than risky correlations. These conditions are
determined by the combination of risk and ambiguity attitudes.
The organization of this paper is as follows. Section 2 presents the literature
review. Section 3 presents a basic framework for analyzing the eﬀects of correlation
between two risky attributes on optimal savings under risk. Section 4 introduces thenotion of stochastic dominance. Section 5 analyzes how correlation aﬀects optimalsavings under risk. Section 6 discusses the notion of being mixed correlation averse(seeking). Section 7 introduces ambiguity into the correlation between two risky
attributes and shows that whether optimal savings increase or not depends on attitudes toward both risk and ambiguity. Section 8 relates the experimental results andempirical observations in the existing literature to our theoretical results. Section 9
concludes the paper.

2

Literature review

This paper lies at the intersection of the literature on multivariate preferences andprecautionary saving. As in Eeckhoudt et al. (2007), we restrict our analyses tobivariate preferences by fixing all but two of the attributes. The first attribute is
the level of wealth, and the second attribute is the condition of health. Even though
we could use other variables as the second attribute, for example, other’s level of
wealth or the condition of the environment, we adopt the condition of health becauseof its importance in household financial decisions and its use in many studies.
It is diﬃcult to interpret the signs of higher-order cross derivatives. Eeckhoudt
et al. (2007) provide their systematic characterization by applying a combination of

4
two bads, sure reduction and zero-mean risk.4 They relate preferences for specificcombinations of the two bads to the signs of successive higher-order cross derivatives of a utility function.5 Jokung (2011) applies stochastic dominance relations
to express a preference for the good attribute and the bad one, and shows that thesigns of cross derivatives of a utility function can be related to the preference for the
combination of good and bad. The signs also play a significant role in our analyses.
Most studies on precautionary saving have focused on settings in which individuals face risky future incomes in a one-dimensional situation. This suggests two
directions for advancing research on precautionary saving. One direction is to compare one risky situation with another risky situation. Another direction is to shedlight on precautionary saving motives in multidimensional settings. Eeckhoudt andSchlesinger (2008) are a seminal paper on the first direction and demonstrate how
risky shifts in future income aﬀect optimal savings based on the notion of changes
in higher-order stochastic dominance. We compare two diﬀerent risky situations as
in Eeckhoudt and Schlesinger (2008), but the diﬀerence between their paper andours is our inclusion of correlation between two risky attributes and multivariate
preferences.
Courbage and Rey (2007) analyze precautionary saving in a multidimensionalsetting and investigate conditions under which individuals increase their savings
by more when there is risk in the second attribute. Whereas Courbage and Rey(2007) compare two diﬀerent situations in which health conditions are either certain
or risky, we compare two diﬀerent situations in which health conditions are bothrisky but diﬀer in correlation against income risks. Even though Courbage and Rey(2007) consider a situation in which future income and health risks are correlated,
4This characterization is an extension of Eeckhdout and Schlesinger (2006) in one dimension.
Even though the arguments are closely related between univariate and multivariate preferences, weonly provide a review by Eeckhoudt and Schlesinger (2014) instead of referring to individual studies
of univariate preferences.
5
Eeckhoudt et al. (2007) show that (i) an individual is correlation averse if and only ifu(1,1) (x, y) ≤ 0 for all x, y, (ii) an individual is cross-prudent in health if and only if u(2,1) (x, y) ≥ 0
for all x, y, (iii) an individual is cross-prudent in wealth if and only if u(1,2) (x, y) ≥ 0 for all x, y,and (iv) an individual is cross-temperate if and only if u(2,2) (x, y) ≤ 0 for all x, y, where u(x, y) isa bivariate utility function of wealth x and health y, where u(i,j) (x, y) stands for ∂ i+j u/∂xi ∂y j .

5this paper adopts a simpler representation of the correlation based on Doherty andSchlesinger (1990). This representation enables us to examine precautionary saving
against the correlation. In addition, we analyze the situation in which there are
multiple possible realizations of correlation by applying the notion of ambiguity.
Some studies investigate how ambiguity in future income aﬀects saving decisions
in a univariate framework. Gierlinger and Gollier (2017) and Osaki and Schlesinger(2014) determine conditions under which individuals increase their savings to protect
against ambiguity in future income compared with the case of risk.6 Following Osaki
and Schlesinger (2014), ambiguity aversion distorts the relative weights on worse
priors and the preference for consumption timing. This distortion also occurs in
our analyses. Unlike Osaki and Schlesinger (2014), we introduce ambiguity into thecorrelation between the two risky attributes, not future income.

3

Optimal savings in the presence of correlation

In this section, we provide a basic framework for analyzing the eﬀects of correlation
between two attributes on optimal savings under risk. Let us consider a simple
dynamic model with two dates, t = 0 and t = 1. An individual enjoys lifetime
time-separable utility from two attributes (x, y) ∈ X × Y ⊆ R2+ . The first attributeis a financial variable and the second is a nonfinancial variable. For the sake of exposition, we assume that the financial variable is wealth level and the nonfinancial
variable is health condition. Other examples of nonfinancial variables are other’s
wealth level and environmental conditions. The analysis can be applied to nonfinancial variables, which can be measured numerically.
Let us denote a bivariate utility function u : X × Y → R. We denote u(1,0) (x, y)
as ∂u/∂x, u(0,1) (x, y) as ∂u/∂y and u(1,1) (x, y) as ∂ 2 u/∂x∂y. The same notation
can be used for the function u(i,j) (x, y), which stands for ∂ i+j u/∂xi ∂y j . We assume
that all higher-order partial and cross derivatives exist if necessary for the analysis.
6

Gierlinger and Gollier (2017) consider the term structure of interest rates under ambiguous
consumption growth. As in Berger (2014), their results can be translated in terms of precautionarysaving against ambiguity.

6
The utility function u(x, y) is increasing and concave in both wealth and health,u(1,0) (x, y) ≥ 0, u(0,1) (x, y) ≥ 0, and u(2,0) (x, y) ≤ 0, u(0,2) (x, y) ≤ 0. The concavity
means risk aversion for both wealth and health. We do not impose any restriction on
the sign of u(1,1) (x, y). The signs of the higher-order cross derivatives play a crucialrole in our analyses.
The individual faces a future income risk and future health risk. Two types ofrisks, which are called “good” and “bad,” are involved in both the future income andhealth risks. Regarding the future income risk, the random variables ϵ̃B and ϵ̃G occur
with probability p and 1 − p. Regarding the future health risk, the random variablesδ̃B and δ̃G occur with probability q and 1 − q. We assume that all future risks, ϵ̃G ,ϵ̃B , δ̃G , and δ̃B , are mutually independent. The terms “good” and “bad” indicatethat both good wealth and health risks are preferred, that is, E[u(x + ϵ̃G , y)] ≥E[u(x + ϵ̃B , y)] and E[u(x, y + δ̃G )] ≥ E[u(x, y + δ̃B )] for all x, y. As in the next
section, we rank good and bad by stochastic dominance relations.
There are four possible combinations of future income and health risks. The
following are the combinations and their probabilities:• ϵ̃B and δ̃B with probability kpq;
• ϵ̃G and δ̃B with probability (1 − kp)q;• ϵ̃B and δ̃G with probability p(1 − kq);• ϵ̃G and δ̃G with probability 1 − p − q + kpq.
We can calculate the probability of the future income and health risks, for example,
the probability of the simultaneous occurrence of bad future income and health risk
is kpq. Other probabilities of the future income and health risks can be calculated
similarly. A positive value of k is chosen so that all probabilities are nonnegativeand less than unity. This value of k captures the correlation between the future
income risk and the health risk. When the value of k is unity, the future income risk
and the health risk are independent. Indeed, if k = 1, the probability occuring ϵ̃B7and δ̃B is equal to pq, which is the multiplication of each probability. A value of k
greater (less) than unity indicates a positive (negative) correlation. The correlation
increases in k.
The individual earns the sure income w and is endowed with the sure healthcondition h in both t = 0 and t = 1. In addition to the sure income and health, theindividual faces the income and health risks at t = 1. The individual must decidea level of saving at t = 0 to maximize the lifetime utility. The savings are invested
at the risk-free rate of interest. Negative savings are the amount of borrowing
from future income for current consumption. We assume that the risk-free rate ofinterest is zero and the individual is not impatient, the latter of which means notime-discounting. Because of this simplified setting, we can focus on the eﬀect ofrisk on saving decisions. The individual determines the level of saving to maximize
lifetime time-separable utility from wealth and health:max U (s)s= u(w − s, h) + v(s, k)= u(w − s, h) + kpqE[u(w + s + ϵ̃B , h + δ̃B )] + (1 − kp)qE[u(w + s + ϵ̃G , h + δ̃B )]+ p(1 − kq)E[u(w + s + ϵ̃B , h + δ̃G )] + (1 − p − q + kpq)E[u(w + s + ϵ̃G , h + δ̃G )].
(1)Here, v(s, k) denotes the expected utility at t = 1 given s and k.
The first-order condition for (1) is
U ′ (s)= − u′ (w − s∗ , h) + vs (x∗ , k)
= − u′ (w − s∗ , h) + kpqE[u(1,0) (w + s∗ + ϵ̃B , h + δ̃B )] + (1 − kp)qE[u(1,0) (w + s∗ + ϵ̃G , h + δ̃B )]+ p(1 − kq)E[u(1,0) (w + s∗ + ϵ̃B , h + δ̃G )] + (1 − p − q + kpq)E[u(1,0) (w + s∗ + ϵ̃G , h + δ̃G )] = 0.
(2)
Because U (s) is concave by u(2,0) ≤ 0, the second-order condition for a maximum is8satisfied. For simplicity, we assume that the optimal amount of savings is interior,−w ≤ s∗ ≤ w, and is unique throughout the paper.

4

Stochastic dominanceIn this section, we introduce the notion of stochastic dominance to represent the“good” or “bad” future income risk and health risk. Stochastic dominance is apartial order to compare two random variables. Let us consider two random variablesx̃ and ỹ with the cumulative distribution functions F and G, which are defined overa bounded support [a, b]. We note that the notation of x̃ and ỹ is used for the
exposition of stochastic dominance in this section.
The distribution function F dominates the distribution function G in the senseof first-order stochastic dominance (FSD) if F (z) ≤ G(z) for all z ∈ [a, b]. If therandom variables x̃ and ỹ have the distribution functions F and G, and F dominates
G in the sense of FSD, it is said that a random variable x̃ dominates ỹ in the senseof FSD. The same goes for other notions of stochastic dominance. Applying FSD tothe future income risk, the individual with u(1,0) ≥ 0 prefers the good future income
risk ϵ̃G to the bad one ϵ̃B . Formally, the following two conditions are equivalent:• ϵ̃G dominates ϵ̃B in the sense of FSD;
• E[u(w + ϵ̃G , h)] ≥ E[u(w + ϵ̃B , h)] for u(1,0) ≥ 0.
The same argument can be applied to the future health risk. The individual withu(0,1) ≥ 0 prefers the good future health risk δ̃G to the bad one δ̃B , that is, E[u(w, h+
δ̃G )] ≥ E[u(x, h + δ̃B )] for u(0,1) ≥ 0 when δ̃G dominates δ̃B in the sense of FSD.
For the distribution functions F and G on [a, b], let us define F1 (z) = F (z) and∫z∫z
G1 (z) = G(z), and define Fn (z) = a Fn−1 (t)dt and Gn (z) = a Gn−1 (t)dt for allz ∈ [a, b] and for all n = 2, 3, . . . , N . Following Jean (1980) and Ingersoll (1987),
the distribution function F dominates the distribution function G in the sense of
N th-order stochastic dominance (NSD) if FN (z) ≤ GN (z) for all z ∈ [a, b] and9Fn (b) ≤ Gn (b) for all n = 1, 2, . . . , N − 1. The following result is known in theliterature, for example, see Ingersoll (1987).
• ϵ̃G dominates ϵ̃B in the sense of NSD;
• E[u(w+ϵ̃G , h)] ≥ E[u(w+ϵ̃B , h)] for any function u such that (−1)n+1 u(n,0) ≥ 0
for all n = 1, 2, . . . , N .
The individual prefers the future good income risk to the bad one, which are ranked
according to second-order stochastic dominance because we assume that u(1,0) ≥0 and u(2,0) ≤ 0. For the third-order stochastic dominance, we need to assumeu(3,0) ≥ 0 in addition to u(1,0) ≥ 0 and u(2,0) ≤ 0 so that the individual prefers thegood future income risk to the bad one. Because the positive third-order derivative
represents prudence in the univariate utility framework, we refer to u(3,0) ≥ 0 as
prudence for wealth. The same argument can be applied to the future health risk.
The following two conditions are equivalent:• δ̃G dominates δ̃B in the sense of NSD;
• E[u(w, h+ δ̃G )] ≥ E[u(w, h+ δ̃B )] for any function u such that (−1)n+1 u(0,n) ≥0 for all n = 1, 2, . . . , N .
Following the terminology coined by Caballé and Pomansky (1996), the second condition represents mixed risk aversion in wealth and health, respectively. In otherwords, individuals are called mixed risk averse if the signs of successive derivativesof utility functions have alternate signs, with all positive odd derivatives and all
negative even derivatives. As shown by Brockett and Golden (1987), the utility
functions commonly adopted in economics have the property of having all positive
odd derivatives and all negative even derivatives.7
A real-valued function u(x) on (0, ∞) is complete monotone if its derivatives un (x) of all orders
exist and (−1)n un (x) ≥ 0 for all x > 0 and all n = 0, 1, 2, . . .. A real-valued, continuous utility
function u defined on [0, ∞) exhibits mixed risk aversion if it has a completely monotone first
derivative on (0, ∞) (i.e., (−1)n un+1 (x) ≥ 0 for all x > 0 and for all n = 0, 1, 2, . . .) and u(0) = 0.
As also pointed out by Pratt and Zeckhauser (1987), the majority of utility functions analyzed inapplied work have completely monotone first derivatives. For example, if a class of utility functions
u is the class of hyperbolic absolute risk aversion with −u′′ (x)/u′ (x) = 1/(a + bx) for a > 0 andb > 0, then they are mixed risk averse. See Caballé and Pomansky (1996, p.490) for details.
7

10
We introduce another stochastic dominance relation referred to as an increasein N -th degree risk. As a special case of NSD, Ekern (1980) proposes that the
distribution function G has more N th-degree risk than the distribution function F
if FN (z) ≤ GN (z) for all z ∈ [a, b] and Fn (b) = Gn (b) for all n = 1, 2, . . . , N . This
indicates that the first (N − 1)th-moments of F and G coincide. Ekern (1980) showsthat the following conditions are equivalent:• δ̃B has more N th-degree risk than δ̃G ;• E[u(w + ϵ̃G , h)] ≥ E[u(w + ϵ̃B , h)] for (−1)N +1 u(N,0) ≥ 0.
The same argument can be applied to the case of health risk. We provide someexamples from the literature. An increase in risk by Rothschild and Stiglitz (1970)
corresponds to a second-degree increase in risk. Rothschild and Stiglitz (1970) showthat any increase in risk can be obtained by a sequence of mean-preserving spreads.
This implies that two random variables with the same means are ranked by anincrease in risk. Risk averse individuals dislike any increases in risk. An increasein downside risk introduced by Menezes et al. (1980) corresponds to a third-degreeincrease in risk. This implies that two random variables with the same means andthe same variances are ranked by an increase in downside risk. Prudent individuals
dislike any increases in downside risk.

5

Precautionary saving against correlation under risk

In this section, we examine how correlation aﬀects optimal savings under risk. In
our setup, this means that we examine how optimal savings change in k, which isthe parameter representing correlation. Before stating the main result, we present
the following lemma.

11Lemma 1. Let f (ϵ, δ) be the payoﬀ function and define
E[f (ϵ̃, δ̃)] = kpqE[f (ϵ̃B , δ̃B )] + (1 − kp)qE[f (ϵ̃G , δ̃B )]+ p(1 − kq)E[f (ϵ̃B , δ̃G )] + (1 − p − q + kpq)E[f (ϵ̃G , δ̃G )].
Suppose that
• ϵ̃G dominates ϵ̃B in the sense of NSD;
• δ̃G dominates δ̃B in the sense of MSD.8
If (−1)n+m f(n,m) ≥ (≤)0 for all n = 1, 2, . . . , N and m = 1, 2, . . . , M , then E[f (ϵ̃, δ̃)]
increases (decreases) in k.
Proof

We will prove the case that E[f (ϵ̃, δ̃)] increases in k, because the oppo-

site case can be proven in a similar way.
By a simple calculation, we have the following:()∂E[f (ϵ̃, δ̃)]= pq E[f (ϵ̃B , δ̃B )] − E[f (ϵ̃G , δ̃B )] − E[f (ϵ̃B , δ̃G )] + E[f (ϵ̃G , δ̃G )] .
∂kThis leads to the following:(sgn∂E[f (ϵ̃, δ̃)]∂k)≥ 0 ⇔ E[f (ϵ̃G , δ̃G )] − E[f (ϵ̃G , δ̃B )] ≥ E[f (ϵ̃B , δ̃G )] − E[f (ϵ̃B , δ̃B )].
(3)

When
(−1)n+1∂ n {E[f (ϵ, δ̃G )] − E[f (ϵ, δ̃B ])}≥ 0 for all n = 1, 2, . . . , N ,
∂ϵn

(4)

we have that
E[f (ϵ̃G , δ̃G )] − E[f (ϵ̃G , δ̃B )] ≥ E[f (ϵ̃B , δ̃G )] − E[f (ϵ̃B , δ̃B )]8

Recall that NSD and MSD denote N th-order stochastic dominance and M th-order stochastic
dominance, respectively.

12
because ϵ̃G dominates ϵ̃B in the sense of NSD. We can prove (3) by determining thecondition in which (4) holds. The condition (4) can be rewritten as follows:(−1)n+1 E[f(n,0) (ϵ, δ̃G )] ≥ (−1)n+1 E[f(n,0) (ϵ, δ̃B )] for all n = 1, 2, . . . , N .

(5)Inequality (5) holds for
m+1 ∂

(−1)

m {(−1)n+1 f(n,0) (ϵ, δ)}= (−1)n+m+2 f(n,m) (ϵ, δ) = (−1)n+m f(n,m) (ϵ, δ) ≥ 0∂δ mfor all n = 1, 2, . . . , N and m = 1, 2, . . . , M , because δ̃G dominates δ̃B in the senseof MSD. Thus, the proof is complete.

(Q.E.D.)

A similar result can be obtained for N th-degree risk.
Lemma 2. For the payoﬀ function f (ϵ, δ), define
E[f (ϵ̃, δ̃)] = kpqE[f (ϵ̃B , δ̃B )] + (1 − kp)qE[f (ϵ̃G , δ̃B )]+ p(1 − kq)E[f (ϵ̃B , δ̃G )] + (1 − p − q + kpq)E[f (ϵ̃G , δ̃G )].
Suppose that
• ϵ̃B has more N th-degree risk than ϵ̃G ;• δ̃B has more M th-degree risk than δ̃G .
If (−1)N +M f(N,M ) ≥ (≤)0, then E[f (ϵ̃, δ̃)] is increasing (decreasing) in k.
Assume that future income risk is ranked by NSD and future health risk is ranked
by MSD, that is,• ϵ̃G dominates ϵ̃B in the sense of NSD;
• δ̃G dominates δ̃B in the sense of MSD.

13
It follows from Lemma 1 thatvs (s, k)= kpqE[u(1,0) (w + s + ϵ̃B , h + δ̃B )] + (1 − kp)qE[u(1,0) (w + s + ϵ̃G , h + δ̃B )]+ p(1 − kq)E[u(1,0) (w + s + ϵ̃B , h + δ̃G )] + (1 − p − q + kpq)E[u(1,0) (w + s + ϵ̃G , h + δ̃G )].
(6)
increases (decreases) in k for (−1)n+m u(n+1,m) (x, y) ≥ (≤)0 for all n = 1, 2, . . . , N
and m = 1, 2, . . . , M . Here, recall that v represents the expected utility at t = 1
given s and k.
Let us consider an individual whose correlation between future income risk andhealth risk is represented by k. We consider two cases where the correlation is low orhigh, denoted by kL and kH with kL ≤ kH and all other things equal. Note that thecorrelation increases in k. We denote sL and sH as optimal savings under kL andkH , respectively. Suppose that (−1)n+m u(n+1,m) (x, y) ≥ (≤)0 for all n = 1, 2, . . . , N
and m = 1, 2, . . . , M . We obtain the following inequality:
Vs (sL , kL ) = −u′ (w − sL , h) + vs (sL , kL ) = 0≤ (≥) − u′ (w − sL , h) + vs (sL , kH ) = Vs (sL , kH )(7)

⇔ sL ≤ (≥)sH
The inequality follows from Lemma 1 where f (ϵ, δ) is set to u(1,0) (w + s + ϵ, h + δ).
Now, we can summarize the above argument into the following proposition. We referto individuals as being mixed correlation averse (seeking) if the following holds:(−1)n+m u(n+1,m) (x, y) ≥ (≤)0 for n = 1, 2, . . . , N and m = 1, 2, . . . , M.
Proposition 1. Suppose that the future income risk is ranked by NSD and the future
health risk is ranked by MSD, that is,• ϵ̃G dominates ϵ̃B in the sense of NSD;

14
• δ̃G dominates δ̃B in the sense of MSD.
If the following condition holds,(−1)n+m u(n+1,m) (x, y) ≥ (≤)0 for n = 1, 2, . . . , N and m = 1, 2, . . . , M,
then optimal savings increase (decrease) in k.
Let us consider the special case of N = M = 1. In this case, if u(2,1) (x, y) ≥ (≤)0,then optimal savings increase in k. Following the terminology by Eeckhoudt et al.
(2007), individuals are called cross prudent (imprudent) if u(2,1) (x, y) ≥ (≤)0. We
provide an interpretation of the condition on the sign of the cross derivatives andprovide the intuition of this proposition in the next section.

6

Signs of cross derivatives

Following Eeckhoudt et al. (2007), we provide the intuition of the result in Proposition 1 by relating the signs of the cross derivatives to the precautionary saving
against correlation. In the following analyses, we investigate individual’s preferences for a 50–50 lottery. Let [A, B] denote a lottery that pays either A or B witha probability of one-half. Cross prudence (imprudence) is characterized as a typeof lottery preference.9 An individual is cross prudent (imprudent) if the lottery[(w + ϵ̃, h), (w, h − c)] is preferred to the lottery [(w, h), (w + ϵ̃, h − c)] (the lottery[(w, h), (w + ϵ̃, h − c)] is preferred to the lottery [(w + ϵ̃, h), (w, h − c)]) such thath − c > 0. Each outcome of the lottery occurs with equal probability. The firstoutcome is wealth level and the second outcome is health status. ϵ̃ is zero-mean risk
in wealth and −c is sure loss in health. ϵ̃ is considered to be “bad” compared with 0
9Strictly speaking, this definition corresponds to cross prudence (imprudence) in health. Cross
prudence (imprudence) in wealth can be defined in a similar way. An individual is cross prudent(imprudent) in wealth if the lottery [(w, h + δ̃), (w − k, h)] is preferred to the lottery [(w, h), (w −k, h + δ̃)] (the lottery [(w, h), (w − k, h + δ̃)] is preferred to the lottery [(w, h + δ̃), (w − k, h)]) suchthat w − k > 0. Eeckhoudt et al. (2007, p.120, Proposition 1) show that an individual is crossprudent (imprudent) in health if and only if u(2,1) ≥ (≤)0 and that an individual is cross prudent(imprudent) in wealth if and only if u(1,2) ≥ (≤)0. Because only the results about cross prudence(imprudence) in health appear in the following analyses, we omit the phrase “in health.”

15
because ϵ̃ is preferred to 0 by u(2,0) ≤ 0. Cross-prudent (imprudent) individuals prefer to accept noise risk δ̃ to wealth in good (bad) health outcomes. Cross prudence(imprudence) displays a type of preference to receive one of the two bads for certain
rather than both two bads or nothing (both two bads or nothing rather than one of
the two bads for certain). This is referred to as a type of preference for combining
good with bad (good) following Eeckhoudt and Schlesinger (2006) and Eeckhoudt
et al. (2007).
Next, we see the intuition as to why cross-prudent individuals increase theirsavings as k increases. Cross-prudent individuals dislike occurring bad future income
and health risks simultaneously. Because the simultaneous occurrence increases in k,
cross-prudent individuals increase their savings to protect against the simultaneous
occurrence of the two bad future risks. This leads to positive precautionary saving
against correlation for cross-prudent individuals. The opposite intuition can beapplied for cross-imprudent individuals.
Epstein and Tanny (1980) propose the notion of correlation aversion. An individual is correlation averse (seeking) if the lottery [(w − k, h), (w, h − c)] is preferredto the lottery [(w, h), (w − k, h − c)] (the lottery [(w, h), (w − k, h − c)] is preferredto the lottery [(w − k, h), (w, h − c)]). Here, sure losses −k and −c are considered tobe “bad.” Eeckhoudt et al. (2007, p.120, Proposition 1) show that an individual iscorrelation averse (seeking) if and only if u(1,1) ≤ (≥)0.10 Correlation aversion (seekingness) is a type of preference for combining good with bad (good with good). We
can characterize higher-order versions of correlation aversion (seekingness), which
are called mixed correlation aversion (seekingness). Cross prudence (imprudence)
can be viewed as the third-order version of correlation aversion (seekingness).
Let us consider two risks ϵ̃G and ϵ̃B in wealth and assume that ϵ̃G dominates ϵ̃Bin the sense of NSD. Then, individuals with (−1)n+1 u(n,0) ≥ 0 for n = 1, 2, . . . , N
prefer ϵ̃G to ϵ̃B , that is, E[u(w + ϵ̃G , h)] ≥ E[u(w + ϵ̃B , h)] for all w and h. The two
random variables ϵ̃G and ϵ̃B can be viewed as good and bad, respectively. Let us
10

See also Richard (1975, p.14, Theorem 1) and Epstein and Tanny (1980, p.20).

16
also consider two risks δ̃G and δ̃B in health and that δ̃G dominates δ̃B in the senseof MSD. Then, individuals with (−1)m+1 u(0,m) ≥ 0 for m = 1, 2, . . . , M prefer δ̃G toδ̃B , that is, E[u(w, h + δ̃G )] ≥ E[u(w, h + δ̃B )] for all w and h. The two variables δ̃Gand δ̃B can be viewed as good and bad, respectively.
An individual is called mixed correlation averse (seeking) if11(−1)n+m+1 u(n,m) (x, y) ≥ (≤)0 for n = 1, 2, . . . , N and m = 1, 2, . . . , M.
Jokung (2011, p.449, Theorem 3) shows that if individuals are mixed correlation
averse (seeking), then the lottery [(w+ϵ̃G , h+ δ̃B ), (w+ϵ̃B , h+ δ̃G )] is preferred to thelottery [(w+ϵ̃G , h+ δ̃G ), (w+ϵ̃B , h+ δ̃B )] (the lottery [(w+ϵ̃G , h+ δ̃G ), (w+ϵ̃B , h+ δ̃B )]is preferred to the lottery [(w + ϵ̃G , h + δ̃B ), (w + ϵ̃B , h + δ̃G )]).12
As in correlation aversion (seekingness), mixed correlation aversion (seekingness)
is a type of preference for combining good with bad (good). The condition in our
proposition is mixed correlation aversion (seekingness) excluding correlation aversion(seekingness). We obtain the intuition of the result by the analogy of the case
N = M = 1. Because the simultaneous occurrence of the two bad future risks
increases in k, mixed correlation-averse individuals increase their savings to protect
against the simultaneous occurrence. The opposite argument can be applied tomixed correlation-seeking individuals.

7

Ambiguous correlation

In sections above, we considered the case of risk, in the sense that the correlation is uniquely identified. In this section, we introduce ambiguity into correlation
11

Although this property was first proposed by Jokung (2011), in this paper, it might be better
called “mixed correlation averse (seeking)” based on the notion of being “mixed risk averse” by
Caballé and Pomansky (1996).
12
By introducing the notion of increasing concave order, Jokung (2011, p.449, Theorem 3) extends
Eeckhoudt et al. (2009, p.997, Theorem 3) to the bivariate case. Eeckhoudt et al. (2009, p.997,
Theorem 3) show that if X̃ dominates Ỹ in the sense of NSD and Z̃ dominates T̃ in the sense of
MSD, then [X̃ + T̃ , Ỹ + Z̃] dominates [X̃ + Z̃, Ỹ + T̃ ] in the sense of (N + M )th-order stochastic
dominance.

17
through the parameter k. We assume that a plausible set of possible k is the set{k1 , k2 , . . . , kΘ } . Without loss of generality, kθ is arranged in ascending order,
k1 < k2 < . . . < kΘ . The individual attaches subjective probability qθ to the parameter of kθ for θ = 1, 2, . . . , Θ. We assume that the individual follows the recursive
version of the smooth ambiguity model by Klibanoﬀ et al. (2005, 2009). Define anincreasing and concave second-order utility function ϕ whose variable takes expectedutility. The function ϕ is assumed to be thrice diﬀerentiable. The concavity of ϕcaptures ambiguity aversion. Given s, the objective function is written asΘ∑V (s) = u(w − s, h) + ϕ−1 (qθ ϕ(v(s, k)).

(8)

θ=1

Recall that v(s, k) is the expected utility at t = 1 given s and k, that is,v(s, k)= kpqE[u(w + s + ϵ̃B , h + δ̃B )] + (1 − kp)qE[u(w + s + ϵ̃G , h + δ̃B )]+ p(1 − kq)E[u(w + s + ϵ̃B , h + δ̃G )] + (1 − p − q + kpq)E[u(w + s + ϵ̃G , h + δ̃G )].
The first-order condition for (8) is
′

∗

′

∗

V (s ) = −u (w − s , h) +

Θ
∑
θ=1

qθ

ϕ′ (v(s∗ , kθ ))∑vs (s∗ , kθ ) = 0.
ϕ′ (ϕ−1 ( θ qθ ϕ(v(s∗ , kθ ))))

(9)The second-order condition is easily verified by the concavity of u and ϕ.
∑
We define kO by kO = θ qθ kθ . The level of optimal savings is denoted sO under
kO . To examine the eﬀect of ambiguous correlation on optimal savings, we evaluate

18(9) at sO as follows:∑ϕ′ (v(sO , kθ ))∑vs (sO , kθ )ϕ′ (ϕ−1 ( θ qθ ϕ(v(sO , kθ ))))
θ∑∑qθ ϕ′ (v(sO , kθ ))′O= −u (w − s , h) + ′ −1 θ∑qθ vs (sO , kθ )ϕ (ϕ ( θ qθ ϕ(v(sO , kθ ))))V ′ (sO ) = −u′ (w − sO , h) +

qθ

(10)

θ

+

Cov(ϕ′ (v(sO , kθ ), vs (sO , kθ ))∑ϕ′ (ϕ−1 ( θ qθ ϕ(v(sO , kθ ))))The second equality is obtained by applying Cov(x̃, ỹ) = E[x̃ỹ] − E[x̃]E[ỹ] for all
random variables x̃ and ỹ.
The second term of (10) is called the timing of uncertainty eﬀect by Osaki andSchlesinger (2014). Because the probability is linear in expected utility theory,∑qθ vs (sO , kθ ) = vs (sO , kO )(= −u′ (w − sO , h)).

θ

When

∑

O

β(s ) =qθ ϕ′ (v(sO , kθ ))θ∑≥ (<)1,ϕ′ (ϕ−1 ( θ qθ ϕ(v(sO , kθ ))))

the eﬀect implies that savings should increase (decrease) because the individual
places more (less) weight on the future.
Let λ(z) ≡ −ϕ′′ (z)/ϕ′ (z) denote the coeﬃcient of absolute ambiguity aversion
defined by Klibanoﬀ et al. (2005). If λ(z) is a decreasing function in z, then thefunction ϕ exhibits decreasing absolute ambiguity aversion (DAAA). Following Osaki
and Schlesinger (2014), it holds that β(sO ) ≥ 1 if and only if ϕ exhibits DAAA.13 The
proof can be found in Appendix. We need to determine the sign of the covariancein the third term of (10) to see whether ambiguous correlation increases the amountof savings.
We assume for now that the future income risk is ordered by NSD and the future
health risk is ordered by MSD. Applying Lemma 1 to v(s, k) and vs (s, k), we have
Note that β(sO ) < 1 if and only if ϕ exhibits strictly increasing absolute ambiguity aversion,
which defines that λ(z) is a strictly increasing function in z.
13

19
the following Lemma.
Lemma 3. Suppose that
• ϵ̃G dominates ϵ̃B in the sense of NSD;
• δ̃G dominates δ̃B in the sense of MSD.
When the individual is mixed correlation averse (seeking), that is, (−1)n+m u(n,m) ≥(≤)0 for all n = 1, 2, . . . , N and m = 1, 2, . . . , M , then v(s, k) is increasing (decreasing) in k and vs (s, k) is decreasing (increasing) in k.
Assume that u exhibits mixed correlation aversion or seeking. From Lemma 3,
the signs of ∂v(sO , k)/∂k and ∂vs (sO , k)/∂k are diﬀerent. Because ϕ′ (·) is a decreasing function, the covariance is positive when u exhibits mixed correlation aversion
or seeking. Combining the above arguments, we have the following proposition.
Proposition 2. Suppose that the future income risk is ranked by NSD and the future
health risk is ranked by MSD, that is,• ϵ̃G dominates ϵ̃B in the sense of NSD;
• δ̃G dominates δ̃B in the sense of MSD.
If ϕ exhibits DAAA and u exhibits mixed correlation aversion or seeking, then ambiguous correlation raises the optimal amount of savings.
Even though decreasing absolute risk aversion is a reasonable property, there
are few experimental or empirical studies that examine the property of absoluteambiguity aversion to our knowledge. Berger and Bosetti (2020) are an exception
and found experimental evidence on DAAA. We need additional evidence to conclude
that absolute ambiguity aversion is either decreasing or increasing.
We provide the intuition for Proposition 2 under DAAA. Assume that v(sO , kθ )is decreasing in θ without loss of generality. Ambiguity averse individuals place moreweight on higher θ, which is a worse correlation in the sense that expected utility

20
is lower. When individuals increase their precautionary savings in the face of worse
correlation, vs (sO , kθ ) is increasing in θ. Combining the above argument, ambiguous
correlation raises savings to protect against the worse correlation. This case holds
for mixed correlation aversion. The opposite case where v(sO , kθ ) is increasing in θand vs (sO , kθ ) is decreasing in θ holds for mixed correlation seeking.
The same result can be obtained for the case where the future income risk andthe future health risk are replaced with N th-degree risk and M th-degree risk. First,
we show the following lemma corresponding to Lemma 3.
Lemma 4. Suppose that
• ϵ̃B increases ϵ̃G in N th-degree risk;• δ̃B increases δ̃G in M th-degree risk.
When (−1)N +M u(N,M ) ≥ (≤)0, v(s, k) is increasing (decreasing) in k and vs (s, k)
is decreasing (increasing) in k.
The same argument leads to the following proposition corresponding to Proposition 2.
Proposition 3. Suppose that the future income risk is ranked by N th-degree risk
and the future health risk is ranked by M th-degree risk, that is,• ϵ̃B increases ϵ̃G in N th-degree risk;• δ̃B increases δ̃G in M th-degree risk.
If ϕ exhibits DAAA and u satisfies (−1)N +M u(N,M ) ≥ (≤)0 and (−1)N +M u(N +1,M ) ≥(≤)0, then ambiguous correlation raises the optimal amount of savings.
We note that mixed correlation aversion (seeking) is a suﬃcient condition for(−1)N +M u(N,M ) ≥ (≤)0 and (−1)N +M u(N +1,M ) ≥ (≤)0 to hold when N + M is odd(even). The intuition of Proposition 2 can be applied to Proposition 3.

21

8

Linkage between actual observations and theoreticalresults

There are two types of actual observations that can be related to our theoreticalresults:• experimental observations about the signs of cross derivatives of multivariate
preferences;• empirical observations of household financial decisions in the presence of multivariate risks.
We consider the case of risk in the sense that correlation is uniquely identified.
Based on the framework of Eeckhoudt and Schlesinger (2005), numerous studies
have conducted an experimental analysis to test higher-order risk aversion in a onedimension framework since Deck and Schlesinger (2010).14 However, few studies test
higher-order risk attitudes of multivariate preferences. Even though it is diﬃcult todraw their concluding evidence, Attema et al. (2019) assume that the first attributeis the level of wealth and the second is the length of longevity which is viewed as aproxy of the condition of health. We apply the framework of Attema et al. (2019) in
our analysis.15 For ease of exposition, we consider a special case of N = M = 1, thatis, both future income and health risks are ranked by the FSD. Proposition 1 in thispaper states that the amount of optimal savings increases (decreases) in correlation
if an individual is cross prudent (imprudent). Attema et al. (2019) observe bothcross prudence and cross imprudence. From their experimental observations, we canconclude that the optimal amount of savings can increase or decrease in correlation.
As a realistic case, let us assume that the wealth level and health status are positively
correlated, k > 1. In this case, the optimal amount of savings is higher (lower) than
14
Instead of referring to individual studies, we cite the review by Trautmann and van de Kuilen(2015).
15
Ebert and van de Kuilen (2015) are another experimental study on this topic. They take thefirst attribute as the level of current wealth, and the second as time, the level of future wealth andthe other’s wealth

22
in the case of independence, k = 1, when an individual is cross prudent (imprudent).
Next, we consider the case of ambiguity in the sense that multiple correlations
are perceived. As in the previous section, we assume that an individual is ambiguity
averse and exhibits decreasing absolute ambiguity aversion. As in the case of risk,
we assume that both the future income and health risks are ranked by FSD. From
our Proposition 2, ambiguous correlation raises the optimal amount of savings if
an individual is correlation averse and cross prudent or correlation seeking andcross imprudent in wealth and health. Attema et al. (2019) observe that mostsubjects exhibit correlation aversion in the gain domain and correlation seekingness
in the loss domain. However, Attema et al. (2019) observe both cross prudence andimprudence in both the gain and loss domains. For example, when an individual
exhibits correlation aversion and cross prudence, ambiguous correlation raises the
optimal amount of savings. Following the experimental evidence of Attema et al.
(2019), when health status is considered in the gain domain, ambiguous correlation
raises the optimal amount of savings if an individual is cross prudent. However, forcross-imprudent individuals, we cannot obtain a definitive prediction, and ambiguous
correlation might lower the optimal amount of savings.16
Finally, we discuss the literature that empirically investigates the eﬀect of healthconditions on household financial decisions. Rosen and Wu (2004) observe that poor
health status leads to safer investment choices. These observations are consistent
with cross prudence, which means positive precautionary savings against correlation. The subject of this paper, precautionary savings against correlation, is closely
related to the literature on the role of the public health system in household financial
decisions because it aﬀects the correlation between wealth and health. The coveragerate of medical expenses depends on the public health system and its expansion canseparate the riskiness of wealth and health to some extent, which means a decreasein correlation in our context. Atella et al. (2012) investigate the influence of current
health conditions and future health risk on households’ investments in risky assets
Note that Cov(ϕ′ (v(sO , kθ ), vs (sO , kθ )) is negative for correlation-averse and cross-imprudent
individuals.
16

23
and empirically find that households are more willing to invest in risky assets whenhealth risk is mitigated by a highly protective national health care system. Naranjo
and Gameren (2016) find that the population of Mexicans aged from 50 to 75 years
accumulate precautionary savings for fear that the future of the social security system is uncertain. Ayyagarari and He (2017) find that the reduction in prescription
drug spending by the introduction of Medicare Part D in the US in 2006 increased
risky investments. As a recent study related to Ayyagarari and He (2017), Christelis
et al. (2020) empirically show that Medicare eligibility has quantitatively and statistically significant eﬀects on stockholding for households with college degrees. It
is observed that the public health system aﬀects precautionary saving. Chou et al.
(2003) observe that the introduction of public health insurance reduced the amountof precautionary savings in Taiwan. Jappelli et al. (2007) find that the amount of
precautionary savings is higher in districts with lower health care quality in Italy.
The former corresponds to a decrease in correlation between wealth and health,and the latter corresponds to an increase in correlation. These empirical results are
consistent with cross prudence in our theoretical results, which is also observed inhouseholds’ portfolio choices in the literature.

9

Conclusion

This paper provided a framework for analyzing the eﬀect of correlation among risky
attributes in multivariate preferences with higher-order risk changes on the optimal
amount of savings. We investigated not only the situation in which a unique correlation is identified but also the situation in which multiple correlations are perceived.
Under the risky situation, we showed that whether the optimal amount of savings
increases or not depends on the signs of the derivatives of the utility functions. Furthermore, in the ambiguous situation, we showed that whether the optimal amountof savings increases or not depends on not only the signs of the derivatives of theutility functions (being mixed correlation averse (seeking)) but also the coeﬃcientof absolute ambiguity aversion.

24
Future research should address several issues. As we mentioned above, it shouldbe emphasized that experimental evidence is insuﬃcient, and more studies are necessary to obtain firm observations on the properties of higher-order risk attitudes
of multivariate preferences. From a theoretical perspective, an extension to multiperiod models is worth investigating. The results can be embedded into multipleperiod models in a straightforward manner under the same settings. For example,
we can investigate the situation where the value and the uncertainty of correlation
change over time in multiperiod frameworks.

25
Appendix
We present the proof that shows that decreasing absolute ambiguity aversion
is equivalent to β(s) ≥ 1. A complete proof of this claim can be found in Osaki and Schlesinger (2014). We define the ambiguity premium πA and ambiguity
precautionary premium ψA , which are analogous to risk, as follows:∑qθ ϕ(v(s, kθ )) = ϕ[v(s, kO ) − πA ],θ∑qθ ϕ′ (v(s, kθ )) = ϕ′ [v(s, kO ) − ψA ].

θ

We can therefore rewrite β(s) asβ(s) =

ϕ′ [v(s, kO ) − ψA ].
ϕ′ [v(s, kO ) − πA ]Here, we note that ϕ′ [ϕ−1 ϕ[v(s, kO ) − πA ]] = ϕ′ [v(s, kO ) − πA ]. Because ϕ′ (·) is adecreasing function because of ambiguity aversion, we haveβ(s) ≥ 1 ⇔ ψA ≥ πA .
Thus, we obtain that decreasing absolute ambiguity aversion is equivalent to ψA ≥πA , which is an analogous property of decreasing absolute risk aversion in expectedutility.

26

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