Mixed Risk Apportionment andNon-financial Correlated Background Uncertainty∗
Takao Asano † Yusuke Osaki ‡This Version: September 7, 2023

Abstract
This paper considers a portfolio problem with one safe asset and one risky
asset in the presence of background risk. We assume that the background risk
is a non-financial variable and it is correlated to financial risk. The aim of
this paper is to investigate the eﬀect of correlation on portfolio choice. While
we find that an increase in correlation lowers (raises) the expected utility for
mixed correlation averse (seeking) individuals, contrary to intuition, it does notnecessarily decrease (increase) investment in the risky asset. We determine thatthe conditions needed to reduce (increase) investment can be instead related topreferences for the combination of good and bad. We also introduce ambiguity
into the correlation and investigate its eﬀects on the optimal portfolio.
JEL classification numbers: D81, D91, G11
Key Words: Ambiguity, Bivariate Utility Function, Linear Payoﬀ, Mixed
Correlation Aversion (Seekingness), Background Uncertainty, Portfolio Choice∗

We are grateful to Toshiki Honda, the seminar participants at Nagoya City University, andthe participants of the 2023 East Japan Section of the Japan Finance Association for their helpfulcomments. This research is financially supported by the JSPS KAKENHI Grant Numbers 20K01745
and 20H01519, 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

People need to make decisions in the face of multidimensional consequences thatare risky but also interdependent.1 Let us illustrate such a situation with two
examples. To start, imagine the situation where you are considering purchasing a
new health insurance policy. In this case, you need to consider not only your futurefinancial conditions, but also your future health status.2 Now imagine that you
are the general manager of a multinational firm in the East Asian region, where
you are concerned about the profits from other areas, such as the North American
region, which may aﬀect the management strategy of your region.3 These illustrative
examples highlight the need to incorporate multidimensional and interdependent
risks into a model and to investigate the eﬀects of their interdependence on decision
making.
The above cases are examples of the linear payoﬀ model in Dionne et al. (1993)
in the presence of non-financial background risk.4 Here, background risk is a typeof risk that is exogenous and cannot be controlled. For clarity, we specify thelinear payoﬀ model as a portfolio problem choosing between one safe asset and one
risky asset in the presence of a health risk.5 The reasons for choosing the above
specification are: (i) the portfolio choice problem is representative of the linear
payoﬀ problem, and (ii) empirical analysis is making progress on the properties of
1
This analysis focuses on bivariate consequences, but can also be applied to higher-order dimensions when all variables are exogenous except for two variables.
2
There is now consensus about the positive relationship between health status and income (e.g.,
Bloom and Canning, 2000), and the underlying mechanism is now explored in detail in, for example,
the importance of health status in childhood (Case et al., 2002) and the long-run eﬀects of healthimprovements (Kotschy, 2021).
3
He and Ng (1998) and Allayannis and Ihrig (2001) empirically find that correlations between
returns and exchange rates may be positive or negative, depending on the company’s characteristics.
He and Ng (1998) investigate Japanese multinational firms and demonstrate a positive correlation
between firm returns and exchange rates, while Allayannis and Ihrig (2001) analyze US manufacturing industries and find a negative correlation.
4
The former is like the analysis of the demand for insurance by Mossin (1968), where the bivariate
variables are wealth level and health status. The latter is like the analysis of production under pricerisk by Sandmo (1971), where the bivariate variables are the profits of one own region and anotherregion.
5This model is justifiable as the portfolio problem of a representative investor by empricalovserivations of a positive relationship between health status and income.

2
the bivariate utility function of wealth level and health status. The research questionof this paper is how the correlation between financial and health risks aﬀects portfolio
choice.
Background risk has been extensively studied in the literature under the assumption of independence from endogenous risks. For the portfolio choice problem,the eﬀects of background risk are analyzed in the univariate setting by Weil (1992)
and the bivariate setting by Crainich et al. (2017). Apart from specific contexts,
the analysis of background risk can be made using the derived utility function by
Kihlstrom et al. (1981) and Nachman (1982). This approach requires the assumptionof independence and cannot be applied to the case where nonfinancial backgroundrisk depends on endogenous financial risk.
However, independent background risk contradicts reality in many situations.
This leads to the need to relax the independence assumption and introduce dependent background risk. In the literature, the notion of stochastic dependenceby Lehmann (1966) is widely used to describe dependent background risk. This
stochastic dependence and its variants are stronger than correlation because thesenotions imply correlation (Li, 2011). Some interesting results are obtained under
dependent background risk. For example, investment is not undertaken, even withpositive excess returns. This relates to the stock market participation puzzle by
Mankiw and Zeldes (1991) and Haliassos and Bertaut (1995). However, because of
the analytical diﬃculty, we have obtained few results about dependent backgroundrisk when compared with independent background risk. We also note that previous
studies demonstrate the eﬀects of stochastic dependence for the comparison withindependence. Thus, more research is needed on the eﬀect of changes in stochastic
dependence on portfolio choices. In this paper, we adopt a diﬀerent description ofstochastic dependence and undertake further analyses.
This paper describes stochastic dependence following Doherty and Schlesinger(1983) that modelled the relationship between insurable and contract nonperformance risks. This can capture correlation through a single parameter. Moreover,3
in addition to facilitating the analysis, this description has several advantages: (i)the structure of mixed risk apportionment is embedded into the model to analyze
an increase in correlation,6 (ii) the eﬀect of correlation on portfolio choices can beinvestigated, and (iii) ambiguity can be incorporated into correlation.
This paper employs the term uncertainty in a broad sense, whereby uncertainty
is an umbrella term that includes the notions of risk and ambiguity.7 Whereas
ambiguity about asset returns in a univariate framework has been analyzed in theframework of portfolio choice,8 ambiguity about correlation in a bivariate framework
remains to be investigated.9 In general, it is diﬃcult to pin down the correlation
between two variables, for example, financial and health risks. We overcome thisdiﬃculty through ambiguous correlation and investigate its eﬀects on the optimal
portfolio in a bivariate framework. By adopting the smooth ambiguity model by
Klibanoﬀ et al. (2005), which is the primary ambiguity model used in past applications, we are able to diﬀerentiate between ambiguity and ambiguity attitude.10
The contribution of our paper is threefold. First, we provide a framework foranalyzing the eﬀects of correlation between financial and health background risks
on optimal portfolio choices under a bivariate setting. We use this to determine theconditions under which an increase in correlation monotonically aﬀects the optimal
portfolio choice. Second, we relate these conditions to mixed risk apportionment.
Finally, we introduce ambiguity in the correlation and show that ambiguous correlation decreases the optimal investment in the risky asset.
The organization of this paper is as follows. Section 2 provides a literature
review, Section 3 introduces a formal model, and Section 4 investigates the eﬀects
6

The notion of risk apportionment is introduced by Eeckhoudt and Schlesinger (2006), and ischaracterized by a specific pattern of the preference for a combination of “good” and “bad”. To
apply to bivariate utility functions, “mixed” is added to the beginning of the term.
7
Since seminal works by Gilboa and Schmeidler (1989) and Schmeidler (1989), the eﬀects ofambiguity on decision-making have been extensively investigated. For textbook explanations, see
Gilboa (2009).
8
See Guidolin and Rinaldi (2013) for a review.
9
Jiang et al. (2022) tackle ambiguous correlation in the portfolio problem. Using the meanvariance approach, they show that portfolio concentration occurs under the max–min expectedutility in Gilboa and Schmeidler (1989).
10
For applications of the smooth ambiguity model to finance, see Gollier (2011).

4of correlation on portfolio choices. Section 5 introduces ambiguity into correlationand investigates its eﬀects on portfolio choices. Section 6 relates the conditions
determined in Section 4 to mixed risk apportionment, which is a specific type ofpreference for the combination of good and bad. Section 7 presents higher-order
generalizations of the results in previous sections. Section 8 considers correlation
seeking and mixed correlation seeking individuals. Section 9 concludes.

2

Literature review

The extant literature has investigated the eﬀects of independent background risk.
For example, as early references for specific contexts, we refer to Eeckhoudt andKimball (1992) for insurance demand and Bryis et al. (1993) for hedging behavior. Apart from specific contexts, Gollier and Pratt (1996) introduce the notion ofrisk vulnerability under which the derived utility function is more risk averse thanthe underlying utility function. That is, background risk leads to more risk averse
behaviors under risk vulnerability. Malevergne and Rey (2009) apply the notionof risk vulnerability to the bivariate utility function and term it as cross risk vulnerability. Even though these previous works have obtained fruitful results under
independent background risk, as discussed earlier, the results cannot necessarily beobtained under dependent background risk.
There have been several attempts to investigate the eﬀects of dependent background risk. The main idea of dependent background risk draws on that of stochastic
dependence by Lehmann (1966) in the statistical literature and Wright (1987) for
economic applications.11 Tsetlin and Winkler (2005), Li (2011), and Chiu (2020)align with this description. Unlike independent background risk, these studies reveal
the possibility that investment is not made even with positive excess returns. Even
though their findings have some interest in particular problems, we face diﬃcultyin obtaining clear comparative static predictions within this framework. For exam11

We can find other ways to describe dependence. For example, bivariate stochastic dominanceby Denuit et al. (2011).

5ple, previous studies have not obtained the conditions under which an increase instochastic dependence leads to more risk averse behaviors.
This paper adopts a diﬀerent notion of dependence as introduced by Doherty andSchlesinger (1983, 1990) to describe dependent background risk. This dependence
is widely adopted in various models, for example, index insurance (Clarke, 2016)
comorbidity risk (Bleichrodt et al., 2003; Fujii and Osaki, 2019), and precautionary savings (Courbage et al., 2020; Asano and Osaki, 2022). Using this notion ofstochastic dependence, the present study determines the conditions under which anincrease in correlation raises or lowers investment in risky assets. We also investigate
the eﬀect of ambiguous correlation on the optimal portfolio.
The signs of the cross derivatives of the bivariate utility functions play a crucialrole in determining the eﬀects of correlated background uncertainty. In a univariate
framework, Caballé and Pomansky (1996) provide the notion of mixed risk aversion(seekingness). Eeckhoudt and Schlesinger (2006) characterize the notion of mixedrisk aversion (seekingness) by the preference for the combination of good and bad
as constructed using two building blocks: a sure reduction and a zero-mean risk.
This idea is extended to the bivariate utility setting by Eeckhoudt et al. (2007) thatcharacterize the notions of correlation aversion and cross prudence. In a univariate setting, Eeckhdout et al. (2009) adopt a stochastic dominance relation to rank
good and bad, and characterize mixed risk aversion (seekingness). This framework
is extended to the bivariate setting by Jokung (2011). These works present usefulcharacterizations of individuals’ attitudes toward risk and correlation using the signsof the utility functions. A few studies also investigate the signs of the cross derivatives in experimental settings using the framework in Eeckhoudt et al. (2009), e.g.,
Attema et al., (2019).

3

The model

In this section, we provide a framework for analyzing the eﬀects of the correlation
between two risks, namely, financial risk and non-financial background risk, on port-

6
folio choice. As discussed earlier, for convenience, we interpret background risk asa health risk. We suppose that an individual faces a portfolio choice problem with
one safe asset and one risky asset in the presence of a health risk. We consider
a static model in which there are two dates, t = 0 and t = 1. The individual
makes a portfolio choice at t = 0, and its return is realized at t = 1. Utility isdetermined by the level of wealth and health status (w, h) ∈ W × H ⊆ R2+ . Let
u : W × H → R denote a bivariate utility function. We denote u(1,0) (w, h) as ∂u/∂w, u(0,1) (w, h) as ∂u/∂h, and u(1,1) (w, h) as ∂ 2 u(w, h)/∂w∂h. Similarly, u(i,j) (w, h)
stands for ∂ i+j u(w, h)/∂wi ∂hj . All higher-order partial and cross derivatives are assumed to exist if necessary for the analysis. We impose the standard assumptions on
the utility function for wealth level and health status, that is, u(w, h) is increasing
and concave in wealth level w and health status h. We consider the two cases ofthe signs of cross derivatives, u(1,1) (w, h) ≤ 0 and u(1,1) (w, h) ≥ 0, which captureindividuals’ correlation aversion and correlation seekingness, respectively.12 Because
the analysis of correlation seekingness is the same as correlation aversion, we focus
on the analysis of correlation aversion and present the results for correlation seekingness in Section 8. Note that we do not make any claims about the validity ofcorrelation aversion and seekingness. Indeed, both cases are possible. It then depends on situations whether correlation aversion or seekingness is compelling basedon observations from empirical studies.
The individual has initial wealth w0 to invest in either safe or risky assets. The
net return of the safe asset is rf . There are two possible returns of the risky asset:“good”, r̃G , and “bad”, r̃B with probability p and 1 − p. The excess returns aredenoted by x̃G = r̃G − rf and x̃B = r̃B − rf . We omit “excess” for the remainderof this analysis because only excess returns are considered. We assume that theserandom variables are defined over the same compact support [x, x] with x < 0 < x.
12

Eeckhoudt et al. (2007) show that the notions of correlation aversion and correlation seekingness
are captured by the signs of the cross derivatives of a bivariate utility function. See also Epstein andTanny (1980). In empirical studies, we obtain mixed evidence of correlation aversion and correlation
seekingness (for example, Edwards, 2008; Sloan et al., 1990; Viscusi and Evans, 1990). We notethat our analytical method can be applied to both cases of correlation aversion and correlation
seekingness.

7
The two types of risks are also involved in health status: “good”, h̃G , and “bad”,h̃B with probability q and 1 − q, which are defined over the same compact support[h, h] with h < h. We use the terms “good” and “bad” in the ranking of expectedutility; that is, good risk leads to higher expected utility than bad risk.
We introduce the stochastic dependence between risky return and health risks.
There are four combinations: two types of risky returns and two types of healthrisks,• x̃G and h̃G with probability kpq;
• x̃B and h̃G with probability (1 − kp)q;• x̃G and h̃B with probability p(1 − kq);
• x̃B and h̃B with probability 1 − p − q + kpq.
We calculate the probability of risky returns and health risks such that, for example,
the good risky return occurs with probability p = kpq+p(1−kq). Other probabilitiesof the risky returns and health risks can be calculated similarly. A value of k, taking
a positive value, is chosen so that all probabilities are strictly positive and the sumof all probabilities is unity. This k captures the stochastic dependence between therisky return and the health risk. When the value of k is unity, the risky returns
and health risks are independent. Indeed, if k = 1, then the probability that x̃Gand h̃G occur simultaneously is equal to pq. A value of k greater (less) than unity
indicates a positive (negative) correlation. Because the correlation increases in k,correlation and k have a one-to-one correspondence. Thus, we can treat k to indicate
correlation.
Under this setting, the individual determines how much to invest in the risky
and safe assets, which is denoted by (α, w0 − α). The value of this portfolio at t = 1
is written asα(1 + r̃) + (w0 − α)(1 + rf ) = w0 (1 + rf ) + α(r̃ − rf )= w1 + αx̃,8where r̃ denotes the random return of the risky asset, w1 = w0 (1+rf ), and x̃ = r̃−rf .
Then, the individual determines the investment of the risky asset α to maximize thefollowing expected utility:
V (α)= kpqE[u(w1 + αx̃G , h̃G )] + p(1 − kq)E[u(w1 + αx̃G , h̃B )]+ (1 − kp)qE[u(w1 + αx̃B , h̃G )] + (1 − p − q + kpq)E[u(w1 + αx̃B , h̃B )].
The first-order condition for (1) isV ′ (α∗ )= kpqE[x̃G u(1,0) (w1 + α∗ x̃G , h̃G )] + p(1 − kq)E[x̃G u(1,0) (w1 + α∗ x̃G , h̃B )]+ (1 − kp)qE[x̃B u(1,0) (w1 + α∗ x̃B , h̃G )] + (1 − p − q + kpq)E[x̃B u(1,0) (w1 + α∗ x̃B , h̃B )] = 0.
(1)
Because V (α) is concave by u(2,0) (w, h) ≤ 0, the second-order condition for a maximum is satisfied. For simplicity, we assume that the optimal investment in the risky
asset is interior, 0 < α∗ < w, and is unique throughout the paper. We sometimes
omit the asterisk when there is no possibility of confusion and use the notation α(k)to represent the optimal portfolio explicitly under the correlation of k.
Let us consider that the correlation increases by ∆k from k to k + ∆k. This
increases the probability of the combination of x̃G and h̃G and that of x̃B and h̃Bby ∆k × pq. However, it also decreases the probability of the combination of x̃G andh̃B and that of x̃B and h̃G by ∆k × pq. In other words, an increase in correlation
enhances the possibility of the combination of good and good,13 but reduces thepossibility of the combination of good and bad, which suggests that the structure ofrisk apportionment by Eeckhoudt and Schleisinger (2006) is incorporated into theeﬀects of correlation on portfolio choice.
13

Following the literature, we simply say the combination of good and good instead of bad andbad in the remainder of the paper. Similarly, we apply the same convention for good and bad byomitting bad and good.

9
We can confirm this structure with the marginal expected utility for k. By
diﬀerentiating (1) with respect to k, it follows that
∂V (α)∂k= pq{E[u(w1 + αx̃G , h̃G )] + E[u(w1 + αx̃B , h̃B )] − (E[u(w1 + αx̃G , h̃B )] + E[u(w1 + αx̃B , h̃G )])}.
The eﬀects of an increase in correlation can be classified into two parts: the former
two terms represent gains in utility, and the latter two terms represent losses inutility or disutility. The former is the combination of good and good (and bad andbad), and the latter is the combination of good and bad (bad and good).

4

Correlation and portfolio choice

We suppose that good and bad risks are ranked via first-order stochastic dominance(FSD). In the remainder of this paper, the cumulative distribution functions ofgood and bad risks are denoted by G and B, respectively. Distribution function Gx̃
dominates Bx̃ in the sense of FSD if Gx̃ (x) ≤ Bx̃ (x) for all x ∈ [x, x]. The subscriptx̃ represents the cumulative distribution functions of random variable x̃. We also
say that random variable x̃G dominates random variable x̃B in the sense of FSD
when x̃G and x̃B have the distribution functions Gx̃ and Bx̃ . For all α ∈ (0, w) andh ∈ [h, h],E[u(w + αx̃G , h)] ≥ E[u(w + αx̃B , h)]for all utility function satisfying u(1,0) (w, h) ≥ 0 when x̃G dominates x̃B in the senseof FSD. Because good risky returns lead to higher expected utility than do badrisky returns for individuals with u(1,0) (w, h) ≥ 0, FSD can identify the good andbad risky returns. Similarly, good health risk h̃G dominates bad health risk h̃Bin the sense of FSD. Individuals with u(0,1) (w, h) ≥ 0 then obtain higher expectedutility from a good health risk than a bad health risk.
Before starting the analysis, we prepare the following lemma. The proof can be10found in Appendix.
Lemma 1. Let π(w, h) be the payoﬀ function and define
E[π(w̃, h̃)] = kpqE[π(w̃G , h̃G )] + (1 − kpq)E[π(w̃G , h̃B )]+ p(1 − kq)E[π(w̃B , h̃G )] + (1 − p − q + kpq)E[π(w̃B , h̃B )].
Suppose that• w̃G dominates w̃B in the sense of FSD;
• h̃G dominates h̃B in the sense of FSD.
The following two conditions are also equivalent:(i) π(1,1) (w, h) ≤ 0.
(ii) E[π(w̃, h̃)] is decreasing in k.
By replacing π with u in Lemma 1, we ascertain that the expected utility forcorrelation averse individuals decreases in correlation, that is, ∂V (α)/∂k ≤ 0 for allα ∈ (0, w). The result is summarized as the following proposition.
Proposition 1. The following two conditions are equivalent:(i) Individuals are correlation averse.
(ii) An increase in correlation lowers expected utility.
Proposition 1 is a result consistent with the notion of correlation aversion. By
reducing investment in the risky asset, individuals can mitigate the eﬀect of correlation because the safe asset does not correlate with background health risk. In thecase of correlation aversion, we expect that an increase in correlation reduces investment in the risky asset, ∂α(k)/∂k ≤ 0, because an increase in correlation lowers
expected utility.
This intuition is only correct in the limited case where correlation averse individuals are cross prudent neutral, which is defined by u(2,1) (w, h) = 0. We have

11
that
∂V ′ (α)≤0∂k⇔ E[x̃G (u(1,0) (w + αx̃G , h̃G ) − u(1,0) (w + αx̃G , h̃B ))]≤ E[x̃B (u(1,0) (w + αx̃B , h̃G ) − u(1,0) (w + αx̃B , h̃B ))].

(2)

Because x̃G dominates x̃B in the sense of FSD, (2) holds under the following condition:E[x(u(1,0) (w + αx, h̃G ) − u(1,0) (w + αx, h̃B ))]is decreasing in x, that is,∂E[x(u(1,0) (w + αx, h̃G ) − u(1,0) (w + αx, h̃B ))] ≤ 0∂x⇔ E[u(1,0) (w + αx, h̃G )] + E[αxu(2,0) (w + αx, h̃G )]≤ E[u(1,0) (w + αx, h̃B )] + E[αxu(2,0) (w + αx, h̃B )].
By u(2,1) (w, h) = 0, we have thatE[αxu(2,0) (w + αx, h̃G )] = E[αxu(2,0) (w + αx, h̃B )].
This leads to the following:E[u(1,0) (w + αx, h̃G )] ≤ E[u(1,0) (w + αx, h̃B )] ⇔ u(1,1) (w, h) ≤ 0.
We have confirmed that an increase in correlation reduces investment in therisky asset for correlation averse individuals who are also cross prudent neutral.
From the above analysis, we know that correlation aversion is not enough to obtain
the intuitive comparative static results, and we can guess that the sign of u(2,1) (w, h)
plays a crucial role in obtaining the intuitive results. We say that individuals arecross prudent if u(2,1) (w, h) ≥ 0, and are cross imprudent if u(2,1) (w, h) ≤ 0.14
14

In Eeckhoudt et al. (2007), individuals with u(2,1) (w, h) ≥ 0 is are referred to as cross prudent

12
We introduce the two intensity measures of cross prudence to obtain definitive
comparative static results without assuming cross prudent neutrality.
Definition 1. We call the following two measures the degree of partial cross prudence
and the degree of relative cross prudence, respectively.
• the degree of partial cross prudence:−

xu(2,1) (w + x, h);u(1,1) (w + x, h)• the degree of relative cross prudence:−

wu(2,1) (w, h).
u(1,1) (w, h)

The terms “partial” and “relative” are based on the terminology in a univariate
setting, partial risk aversion (Menezes and Hanson, 1970) and relative risk aversion(Pratt, 1964).15
Because it follows from (1) that V ′ (α) = E[π(w̃, h̃)] by setting π(w, h) = xu(1,0) (w+αx, h) and applying Lemma 1, we obtain that V ′ (α) decreases in k if and only if
π(1,1) (w, h) ≤ 0 ⇔ −α(k)xu(2,1) (w + α(k)x, h)≤ 1.
u(1,1) (w + α(k)x, h)(3)

Using the terminology of Definition 1, the condition can be said that the partial crossprudence is less than unity. Because V ′ (α) decreases in k, we know that the optimal
investment in the risky asset decreases in k, which is obtained by the following:
V ′ (α(k); k) = 0 ⇒ V ′ (α(k); k ′ ) ≤ 0
for k ≥ k ′ . The value k after the semicolon means that the correlation is equal to k.
in health to distinguish them from individuals with u(1,2) (w, h) ≥ 0, who are cross prudent in
wealth. Because u(2,1) (w, h) only appears in this paper, we can omit “health” without confusion.
15
These intensity measures are used in the context of the consumption–saving model by Courbage
et al. (2022).

13
Assuming cross prudence, u(2,1) (w, h) ≥ 0, a suﬃcient condition for (3) is that−wu(2,1) (w, h)≤ 1.
u(1,1) (w, h)(4)This condition means that the degree of relative cross prudence is less than unity.
Applying a similar argument, we can prove that (4) is a suﬃcient condition
under which an increase in correlation decreases the optimal portfolio for correlation
seeking and cross imprudent individuals. We summarize the above argument as thefollowing proposition and provide interpretations of these conditions in the next
section.
Proposition 2. Suppose that individuals are correlation averse. Then,(i) the degree of partial cross prudence is less than unity if and only if an increasein correlation reduces the investment in the risky asset, and(ii) if individuals are cross prudent and the degree of relative cross prudence is lessthan unity, then an increase in correlation reduces the investment in the risky asset.
As discussed in Section 3, an increase in correlation has a greater likelihood ofproviding the combination of good and good than good and bad. The combinationof good and good is undesirable for correlation averse individuals because an increasein correlation lowers the expected utility as in Proposition 1.16 There are two eﬀects
concerning the FSD shift of risky returns: the substitution eﬀect and the wealtheﬀects. In a univariate setting, it is known that the wealth eﬀect dominates thesubstitution eﬀect when the degree of relative risk aversion is less than unity (e.g.,
Fishburn and Porter, 1976). Correlation averse individuals should then reduce theirinvestment in the risky asset in response to an increase in correlation to avoid theundesirable combination when the wealth eﬀect dominates the substitution eﬀect. To
determine the condition, we need to incorporate the eﬀect of health risks. Combining
these two eﬀects, the condition is determined by comparing the degrees of relative
or partial cross prudence with the threshold value being equal to unity.
16

We relate the preference for combining good with bad (good) to the signs of the cross derivativesof the utility function in the following section.

14
Example 1. As a concrete example, we consider the following Cobb–Douglas utility
function:u(w, h) =(wψ h1−ψ )1−γ.
1−γHere, ψ takes a value in [0, 1] for u(1,0) (w, h), u(0,1) (w, h) ≥ 0 and u(2,0) (w, h), u(0,2) (w, h) ≤0. The sign of u(1,1) (w, h) depends on the parameter value of γ. When γ = 1, we
have that u(1,1) (w, h) = 0, that is, γ = 1 is the threshold to determine the signsof u(1,1) (w, h). We have that u(1,1) (w, h) > (<)0 for γ < (>)1, which represents
correlation seekingness (aversion). For γ < 1, it holds that−wu(2,1) (w, h)= 1 − (1 − γ)ψ ≤ 1.
u(1,1) (w, h)Thus, from Proposition 8 in Section 8, we conclude that an increase in correlation
raises the investment in the risky asset for investors who have the Cobb–Douglas
utility function with γ < 1. For γ > 1, it holds that−wu(2,1) (w, h)= 1 − (1 − γ)ψ ≥ 1.
u(1,1) (w, h)(5)For correlation averse individuals, we cannot obtain clear comparative statics results.
We note that an increase in correlation lowers the investment in the risky asset eventhough (5) holds but (3) is satisfied.
It is known that the sign of the investment in the risky asset is the same asthat of the risky asset return in a univariate setting, that is, individuals invest in apositive amount of risky assets when their expected returns are positive. This result
can be extended to the portfolio problem in the presence of independent backgroundrisk. For our setting, k = 1 corresponds to independent background risk. We havethat
V ′ (0) = E[ỹu(1,0) (w1 , z̃)]= E[ỹ]E[u(1,0) (w1 , z̃)],

(6)

15
where ỹ = px̃G + (1 − p)x̃B and z̃ = q h̃G + (1 − q)h̃B .
Let us consider a risky asset ỹ whose expected return is equal to zero, E[ỹ] =pE[x̃G ]+(1−p)E[x̃B ] = 0. When k is more than unity, correlation averse individuals
make negative investments in the risky asset when (4) holds by Proposition 2 andshort selling is allowed. When short selling is not allowed, the negative investment
means that individuals do not hold the risky asset and hold only the safe asset.
By a continuity argument, this means that correlation averse individuals may makenegative investments in risky assets with positive expectations. This observation canbe related to the stock market participation puzzle by Mankiw and Zeldes (1991)
and Haliassos and Bertaut (1995). The puzzle involves empirical observations thatmany households do not own any equities and hold only safe assets despite positive
excess returns.

5

Mixed risk apportionment

Eeckhoudt and Schlesinger (2006) introduce a notion of risk apportionment as particular types of preferences between two lotteries. This preference can be related
to the signs of higher-order derivatives of a utility function in a univariate setting
within an expected utility framework. A sure reduction and noise risk are two building blocks used to construct the pairs of lotteries to characterize risk apportionment.
By comparing zero, these two building blocks are bad for u′ (w) ≥ 0 and u′′ (w) ≤ 0.17
The two lotteries are constructed by the combination of good and bad: combining
good with bad and good with good.Risk apportionment can be classified into two
types: preferences for combining good with bad and good with good. Eeckhoudt
and Schlesinger (2006) show that the former is related to (−1)n+1 un (w) ≥ 018 andCrainich et al. (2013) show that the latter is related to un (w) ≥ 0. In this section,
17
A sure reduction −k with k > 0 lowers utility for every increasing utility function becauseu(w − k) ≤ u(w) given intimal wealth w. Now recall that the term of good and bad is related to theranking of expected utility. Comparing [−k] with [0], we can say that [−k] is bad and [0] is good.
Similarly, we can say that a noise risk [ε̃] is bad for every concave utility function by comparing [0].
18
We denote un (w) = ∂ n u(w)/∂wn .

16
we show that all the second conditions in Proposition 2 are related to the particular
type of preference for the combination of good and bad in a bivariate setting. In thepresent study, this type of preference is called mixed risk apportionment, following
the term in a univariate setting. As in risk apportionment, mixed risk apportionment can be classified into two types: the preference for combining good with bad orgood with good. Mixed risk apportionment can be considered a consistent preference
because it exhibits the preference for the same pattern of the lottery preference.
First, we consider the two conditions in Proposition 2: correlation aversion andcross prudence. Let (w, h) ∈ R2+ denote a nonnegative vector of wealth level andhealth status, and let [A; B] denote a lottery where outcomes A and B occur withprobability one-half. For positive constants k and c with k < w and c < h, we define
two lotteries: A(1,1) = [(w, h); (w − k, h − c)] and B(1,1) = [(w − k, h); (w, h − c)].19
Similarly, for a positive constant c and a noise risk ϵ̃, we define two lotteries: A(2,1) =[(w, h); (w + ϵ̃, h − c)] and B(2,1) = [(w + ϵ̃, h); (w, h − c)]. The lotteries have thesame pattern for the combination of good and bad. Lotteries A(1,1) and A(2,1) areconstructed by combining good with good (and bad with bad). Conversely, lotteries
B(1,1) and B(2,1) are constructed by combining good with bad (and bad with good).
Following Eeckhoudt et al. (2007), particular types of lottery preferences between
the above pairs can be related to the signs of the cross derivatives of a bivariate
utility function.
Result 1 (Eeckhoudt et al. (2007)). Eeckhoudt et al. (2007) show the following:(i) Individuals are correlation averse, u(1,1) (w, h) ≤ 0 if and only if they prefer B(1,1)
over A(1,1) .
(ii) Individuals are cross prudent, u(2,1) (w, h) ≥ 0 if and only if they prefer B(2,1)
over A(2,1) .
Next, we consider another condition in Proposition 2: the degree of relative
cross prudence is less than unity. We introduce a multiplicative version of lot19

Because this pair of lotteries is used to characterize the sign of u(1,1) (w, h), the subscript of, forexample, A(1,1) , corresponds to the subscript of utility function u(1,1) (w, h). Similar notations areapplied to other lottery pairs.

17tery pairs. For 0 < k < 1 and 0 < r < 1, we define the two lotteries: A(2,1) =[(w, h); (w(1 − k)(1 − r), h)] and B(2,1) = [(w(1 − k), h); (w(1 − r), h)]. The lotteries
are the same as Eeckhoudt et al. (2009a), except that health status is included.
Next, we consider preferences between two lotteries for marginal utility in health,u(0,1) (w, h), to incorporate the eﬀect of health risks into a bivariate utility function.
This idea is in the same spirit as Kimball (1990), where the precautionary premium is defined for marginal utility. In other words, precautionary eﬀects against
health risks are incorporated into the lottery preferences. Applying an argument of
Eeckhoudt et al. (2009a), we obtain the following proposition.
Proposition 3. When individuals are correlation averse, the following conditions
are equivalent:(i) The degree of relative cross prudence is less than unity.
(ii) Lottery B(2,1) is preferred to A(2,1) .
Proof. See Appendix.
In Proposition 2, there are three conditions for obtaining intuitive comparative
static results:• correlation aversion, cross prudence, and the degree of relative cross prudence
being less than unity;
These conditions are related to mixed risk apportionment that corresponds to thepreference for combining good with bad. In other words, the second conditions in
Proposition 2 are consistent with the preference for the combination of good andbad.

6

Ambiguous correlation

In general, understanding the relationship of stochastic dependence between financial and health risks is a more diﬃcult task when compared with grasping the financial and health risks themselves. Applying to our context, it is more diﬃcult

18
to determine the correlation between risky returns and health risks compared with
figuring out risky returns and health risks themselves. To overcome this diﬃculty,
we introduce ambiguity into correlation through the parameter k. We assume thata 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 employs the smooth ambiguity model in Klibanoﬀ et al. (2005).
Define a concave and strictly increasing function ϕ whose variable takes expectedutility. The function ϕ is assumed to be twice diﬀerentiable. The concavity of ϕ captures ambiguity aversion. The convexity and linearity of ϕ correspond to ambiguity
seeking and ambiguity neutral attitudes, respectively.
Given α, the objective function is written as

V (α) =

Θ∑qθ ϕ(U (α, kθ )).

(7)

θ=1

Here, U (α, k) is the expected utility given α and k, that is,
U (α, kθ ) = kpqE[u(w1 + αx̃G , h̃G )] + p(1 − kq)E[u(w1 + αx̃G , h̃B )]+ (1 − kp)qE[u(w1 + αx̃B , h̃G )] + (1 − p − q + kpq)E[u(w1 + αx̃B , h̃B )]. for θ = 1, 2, . . . , n.
As in the previous sections, good and bad are ranked via FSD, that is, x̃G dominatesx̃G in the sense of FSD, and h̃G dominates h̃G in the sense of FSD,
The first-order condition for (7) isV ′ (α∗ ) =Θ∑qθ ϕ′ (U (α∗ , kθ ))Uα (α∗ , k) = 0.

(8)

θ=1

The second-order condition is easily verified by the concavity of u and ϕ.
∑
We define kO by kO = θ qθ kθ and find i ∈ {1, 2, . . . , n} such that ki ≤ kO ≤ki+1 . The level of the optimal portfolio is denoted by αO under kO . Because linear

19
ϕ corresponds to expected utility, we haveUα (αO , kO ) =n∑qθ U (αO , kθ ) = 0.

θ=1

We assume that u exhibits correlation aversion. From Proposition 1, U (α, k)decreases in k. This leads to the following:• U (αO , kθ ) ≤ U (αO , kO ) ⇔ ϕ′ (U (αO , kθ )) ≤ ϕ′ (U (αO , kO )) for θ = 1, . . . , i, and• U (αO , kθ ) ≥ U (αO , kO ) ⇔ ϕ′ (U (αO , kθ )) ≥ ϕ′ (U (αO , kO )) for θ = i + 1, . . . , n.
From Proposition 2, the optimal portfolio decreases in k when the individual iscorrelation averse, cross prudent, and−wu(2,1) (w, h)≤ 1.
u(1,1) (w, h)This leads to the following:• Uα (αO , kθ ) ≤ 0 for θ = 1, . . . , i, and• Uα (αO , kθ ) ≥ 0 for θ = i + 1, . . . n.
From the above two relations, we have the following:• ϕ′ (U (αO , kθ ))Uα (αO , kθ ) ≤ ϕ′ (U (αO , kO ))Uα (αO , kO ) ≤ 0 for θ = 1, . . . , i, and• 0 ≤ ϕ′ (U (αO , kθ ))Uα (αO , kθ ) ≤ ϕ′ (U (αO , kO ))Uα (αO , kO ) for θ = i + 1, . . . n.
Finally, we obtain the following:
V ′ (α∗ ) =

i
∑
θ=1

≤

i
∑

n
∑

qθ ϕ′ (U (αO , kθ ))Uα (αO , kθ ) +qθ ϕ′ (U (αO , kθ ))Uα (αO , kθ )θ=i+1

qθ ϕ′ (U (αO , kO ))Uα (αO , kO ) +θ=1= ϕ′ (U (αO , kO )

n
∑
θ=1

n
∑
θ=1

Uα (αO , kθ ) = 0.

qθ ϕ′ (U (αO , kO ))Uα (αO , kO )

(9)

20
Inequality (9) shows that ambiguous correlation decreases the optimal portfolio. We
summarize the above argument into the following proposition.
Proposition 4. Suppose that risky return and health risk are ranked by FSD. If u
exhibits correlation aversion, cross prudence, and−wu(2,1) (w, h)≤ 1,u(1,1) (w, h)then ambiguous correlation reduces the investment in the risky asset.
Ambiguity aversion puts more weight on worse expected utilities. For correlation
averse individuals, worse expected utilities correspond to higher correlation. From
Proposition 2, the conditions on u guarantee that the demand for the risky asset
is lower when individuals face higher correlation. Combining the above arguments,
ambiguous correlation reduces the optimal portfolio.

7

Higher-order generalization

This section considers higher-order stochastic dominance relations to rank good andbad risks. The basic idea can be shared with the case of FSD, and the analysis can beextended from FSD to higher-order stochastic dominance relations. We consider two
types of higher-order stochastic dominance, N -th order stochastic dominance (NSD,
e.g., Ingersoll, 1987) and an N -th degree increase in risk (Ekern, 1980). Let Gw̃
and Bw̃ denote two cumulative distribution functions of good and bad wealth risks,w̃G and w̃B , that are defined on the same compact support, [w, w]. Let us define∫w n1
G1w̃ (w) = Gw̃ (w) and Gn+1w̃ (w) = w Gw̃ (z)dz for n = 1, 2, . . . , N and Bw̃ (w) =∫w nn+1(z)dz for n = 1, 2, . . . , N . Distribution function Gw̃
Bw̃ (w) and Bw̃(w) = w Bw̃
Ndominates distribution function Bw̃ in the sense of NSD if GNw̃ (w) ≤ Bw̃ (w) forn (w) for n = 1, 2, . . . , N − 1. As in Section 4, we
all w ∈ [w, w] and Gnw̃ (w) ≤ Bw̃

also say that w̃G dominances w̃B in the sense of NSD when random variables w̃G
and w̃B have the distribution functions Gw̃ and Bw̃ , respectively. Similarly, we can

21
define NSD for health risks h̃G and h̃B whose distribution functions are Gh̃ and Bh̃ ,respectively.
We assume that the good risky return x̃G dominates the bad risky return x̃Bin the sense of NSD, and that the good health risk h̃G dominates the bad healthrisk h̃B in the sense of MSD. We also assume that individuals are mixed risk averse
in both wealth and health, that is, (−1)i+1 u(i,0) (w, h) ≥ 0 for i = 1, 2, . . . , N and(−1)j+1 u(0,j) (w, h) ≥ 0 for j = 1, 2, . . . , M .20 By this assumption, stochastic dominance relations of wealth and health risks can be related to the ranking of expectedutility. Thus, good risky return (health risk) leads to higher expected utility thandoes bad risky return (health risk). We say that individuals are mixed correlation
averse if(−1)i+j+1 u(i,j) (w, h) ≥ 0 for i = 1, 2, . . . , N and j = 1, 2, . . . , M .
We define the two intensity measures for mixed correlation aversion as follows.
Definition 2. We call the following two measures the degree of partial mixed correlation aversion and the degree of relative mixed correlation aversion, respectively.
• the degree of partial mixed correlation aversion in (n, m):−xu(n+1,m) (w + x, h).
u(n,m) (w + x, h)• the degree of relative mixed correlation aversion in (n, m):−wu(n+1,m) (w, h).
u(n,m) (w, h)

These measures are higher-order versions of the degrees of partial and relative
cross prudence. From the above preparation, we provide the following results about
correlation and portfolio choices.
20

The term for mixed risk aversion is borrowed from Caballé and Pomansky (1996) in a univariate
setting.

22
Proposition 5. Suppose that good risky return dominates bad risky return in thesense of NSD and good health risk dominates bad health risk in the sense of MSD,
and individuals are mixed correlation averse. Then,(i) the degree of partial mixed correlation aversion in (n, m) is less than n for n =1, 2, . . . , N and m = 1, 2, . . . , M if and only if an increase in correlation reduces theinvestment in the risky asset, and(ii) if the degree of relative mixed correlation aversion in (n, m) is less than n for
n = 1, 2, . . . , N and m = 1, 2, . . . , M , then an increase in correlation reduces theinvestment in the risky asset.
Proof. See Appendix.
Next, we consider whether N -th and M -th degree increases in risk introduced by
Ekern (1980) can be related to the signs of the higher-order cross derivatives of theutility function. Random variable w̃G is an N -th degree riskier than random variableNnnw̃G if GNw̃ (w) ≤ Bw̃ (w) for all w ∈ [w, w] and Gw̃ (w) = Bw̃ (w) for n = 1, 2, . . . , N −1.

The latter condition means that the first N − 1 moments have the same values.
Individuals with (−1)N +1 u(N,0) (w, h) ≥ 0 prefer w̃G over w̃B . Similarly, we candefine good and bad health risks in which h̃G is an M -th degree riskier than h̃G . For
example, N = 2 corresponds to the increase in risk introduced by Rothschild andStiglitz (1970). It is constructed using a series of mean-preserving spreads. Goodrisky return (health risk) leads to higher expected utility than does bad risky return(health risk) when individuals are averse to wealth (health) risk.21
From Proposition 5, we can obtain the following corollary.
Corollary 1. Suppose that bad risky return is an N -th degree riskier than good risky
return and bad health risk is an M -th degree riskier than good health risk, and that(−1)N +M +1 u(N,M ) (w, h) ≥ 0. Then,21

N = 3 corresponds to the increase in downside risk in Menezes et al. (1980) that is ranked
by skewness with the same means and variances. The preference for an increase in downside risk
can be related to the positive third derivative of the utility function and is called prudence. N = 4
corresponds to the increase in outer risk by Menezes and Wang (2005) that is ranked by kurtosis
with the same means, variances, and skewnesses. The preference for an increase in outer risk canbe related to the negative fourth derivative of the utility function and is called temperance.

23(i) the degree of partial mixed correlation aversion in (N, M ) is less than N if andonly if an increase in correlation reduces the investment in the risky asset, and(ii) if the degree of relative mixed correlation aversion in (N, M ) is less than N ,then an increase in correlation reduces the investment in the risky asset.
All the conditions in Proposition 2 can be related to mixed risk apportionment
and the specific type of lottery preferences for the combination of good and bad.
Following Jokung (2011), we relate the signs of the higher-order cross derivatives
to mixed risk apportionment. Suppose that w̃G dominates w̃B in the sense of
NSD and h̃G dominates h̃B in the sense of MSD. Let us consider two lotteries:
A(N,M ) = [(w̃G , h̃G ); (w̃B , w̃B )] and B(N,M ) = [(w̃G , h̃B ); (w̃B , w̃G )]. The lottery
A(N,M ) is constructed by combining good with bad, and the lottery B(N,M ) is constructed by combining good with good. Using the lottery preference, mixed correlation averse individuals prefer B(N,M ) to A(N,M ) . The following result is a naturalextension of Eeckhdout et al. (2009) from the univariate to bivariate setting.
Result 2 (Jokung (2011)). Individuals are mixed correlation averse if and only if(−1)i+j+1 u(i,j) (w, h) ≥ 0 for i = 1, 2, . . . , N and j = 1, 2, . . . , M .
For N = M = 1, the condition degenerates into correlation aversion.
Finally, we consider the condition on the degree of relative mixed risk aversion.
We now apply the lottery preference to u(0,m) (w, h) for m = 1, 2, . . . , M to incorporate the eﬀect of changes in health risks. Let us consider two lotteries that are
multiplicative versions of the above lotteries: A(N,M ) = [(w(1+ϵ̃G )(1+δ̃B ), h); (w(1+ϵ̃B )(1 + δ̃G ), h)] and B(N,M ) = [(w(1 + ϵ̃G )(1 + δ̃G ), h); (w(1 + ϵ̃B )(1 + δ̃B ), h)]. Here,random variables ϵ̃G and ϵ̃B are defined over the same compact support [−1, ϵ], andrandom variables δ̃G and δ̃B are defined over the same compact support [−1, δ]. ϵ̃G
and ϵ̃B are ranked via NSD, and δ̃G and δ̃B are ranked via FSD. The four risks(ϵ̃G , ϵ̃B , δ̃G and δ̃B )in the lotteries A(N,M ) and B(N,M ) are mutually independent.
The following characterizations can be viewed as an extension of Eeckhdout et al.
(2009) from a univariate setting with additive lotteries to a bivariate setting with

24
multiplicative lotteries.22 The proof can be found in Appendix.
Proposition 6. When individuals are mixed correlation averse, the following two
conditions are equivalent:(i) The degree of relative mixed correlation aversion of (n, m) is less than n for
n = 1, 2, . . . , N and m = 1, 2, . . . , M .
(ii) Lottery B(N,M ) is preferred to A(N,M ) .

8

Correlation seekingness

Up until now, we focused on the analysis of the correlation averse or mixed correlation averse individuals. This section provides the results for correlation seeking
individuals parallel to those of correlation averse individuals. Therefore, we only
present the results and omit their proofs and interpretations.
Note that correlation seeking individuals and cross imprudence individuals arecaptured by u(1,1) (w, h) ≥ 0 and u(2,1) (w, h) ≥ 0, respectively. In this section, goodand bad are ranked via FSD, that is, x̃G dominates x̃B in the sense of FSD, andh̃G dominates h̃B in the sense of FSD. To obtain the first two results, we use thefollowing lemma, which is a symmetric version of Lemma 1.
Lemma 2. Let consider π(w, h) in Lemma 1. Suppose that• w̃G dominates w̃B in the sense of FSD;
• h̃G dominates h̃B in the sense of FSD.
The following two conditions are also equivalent:(i) π(1,1) (w, h) ≥ 0.
(ii) E[π(w̃, h̃)] is increasing in k.
Applying Lemma 2, Proposition 7 is obtained by setting π(x, h) = u(w + αx, h)and Proposition 8 is obtained by setting π(x, h) = xu(w + αx, h).
22Wang and Li (2010) provide a higher-order generalization of Eeckhoudt et al. (2009a) following
Eeckhoudt and Schlesinger (2006). They construct the lottery pairs using the combination of a sure
reduction and independent risks.

25
Proposition 7. The following two conditions are also equivalent:(i) Individuals are correlation seeking.
(ii) An increase in correlation raises expected utility.
Proposition 8. Suppose that individuals are correlation seeking. Then,(i) the degree of partial cross prudence is less than unity if and only if an increasein correlation increases the investment in the risky asset, and(ii) if individuals are cross imprudent and the degree of relative cross prudence isless than unity, then an increase in correlation increases the investment in the risky
asset.
Note that the conditions in Proposition 2 are the same as those in Proposition
8. This is because the signs of u(1,1) (w, h) diﬀer between correlation aversion andcorrelation seekingness. Because the sign is reversed twice, comparing correlation
aversion and correlation seekingness results in the same conditions appearing in
Propositions 2 and 8.
Applying a similar argument in Section 5, we have the following proposition.
Proposition 9. Suppose that the risky return and health risk are ranked by FSD.
If u exhibits correlation seekingness and cross imprudence, and−wu(2,1) (w, h)≤ 1,u(1,1) (w, h)then ambiguous correlation decreases the investment in the risky asset.
The conditions in Proposition 8 can be related to the lottery preference.
Result 3 (Eeckhoudt et al. (2007)). Eeckhoudt et al. (2007) show the following:(i) Individuals are correlation seeking, u(1,1) (w, h) ≥ 0, if and only if they preferA(1,1) over B(1,1) .
(ii) Individuals are cross imprudent, u(2,1) (w, h) ≤ 0, if and only if they prefer A(2,1)
over B(2,1) .
The following proposition corresponds to Proposition 3.

26
Proposition 10. When individuals are correlation seeking, the following conditions
are equivalent:(i) The degree of relative cross prudence is less than unity.
(ii) Lottery A(2,1) is preferred to B(2,1) .
We note that the signs of u(1,1) (w, h) diﬀer between correlation aversion andseekingness, and that the first condition in Proposition 3 and 10 is the same, thatis, the degree of relative cross prudence is less than unity.
In Proposition 8, there are three conditions for obtaining intuitive comparative
static results:• correlation seekingness, cross imprudence, and the degree of relative cross prudence being less than unity.
Combining Result 3 and Proposition 10, these conditions are related to mixed risk
apportionment, which corresponds to the preference for combining good with good.
In other words, the conditions in Proposition 8 are consistent with the preference
for the combination of good and good.
As in FSD, we can obtain the corresponding results of higher-order stochastic
dominance for mixed correlation seeking individuals. Thus, we omit these to avoid
repetition.

9

Conclusion

This paper considered the portfolio choice problem with one risk-free asset and one
risky asset in the presence of background risk. We investigated the eﬀects of correlation on the optimal portfolio. While increases in correlation reduce (raise) expectedutility for mixed correlation averse (seeking) individuals, contrary to intuition, they
do not necessarily reduce (raise) the optimal portfolio. We determined the conditions
for a reduction (rise) in the optimal portfolio that can be related to the preference
for the combination of good and bad. Furthermore, we introduced ambiguity into
correlation and investigated its eﬀects on the optimal portfolio.

27
There are several directions for future research that extend this study. First, because the linear payoﬀ problem has versability, this study can be applied to specificproblems, such as the demand for insurance and a firm’s production under price risk,
by incorporating characteristics specific to each problem. Second, it is interesting
to consider health risk as controllable. In this study, health risk is treated as background risk. However, it may be controlled by introducing, for example, insurance
and prevention activities. Third, experimental studies are worth investigating. Onlya few studies have experimentally investigated higher-order risk preferences basedon bivariate utility functions (e.g., Attema et al., 2019). These studies are contextfree and use the framework of Eeckhoudt et al. (2007). However, to the best of ourknowledge, there is no experimental investigation of these under specific contexts.
In addition to our results, experimental studies would illuminate the importanceof higher-order risk preferences and the eﬀects of correlation in multidimensionalframeworks.

28
Appendix
Proof of Lemma 1. We show that (i) implies (ii). By straightforward calculation,
we have(sgn)∂
E[π(w̃, h̃)] ≤ 0
∂k

(10)

⇔ E[π(w̃G , h̃G )] − E[π(w̃G , h̃B )] ≤ E[π(w̃B , h̃G )] − E[π(w̃B , h̃B )].
Because w̃G dominates w̃B in the sense of FSD, (10) holds if E[π(w, h̃G )]−E[π(w, h̃B )]decreases in w, which is equivalent to E[π(1,0) (w, h̃G )] ≤ E[π(1,0) (w, h̃B )]. Again applying the property of FSD, we haveE[π(1,0) (w, h̃G )] ≤ E[π(1,0) (w, h̃B )] ⇔ π(1,1) (w, h) ≤ 0.
We conclude the proof that (i) implies (ii).
Next, we show by contradiction that (ii) implies (i). Assume that π(1,1) (wo , ho ) >0 for the neighborhood of wo and ho . Let us consider that w̃G and w̃B that are
identical except for the neighborhood of wo and ho . In this case, we have thatE[π(w̃, h̃)] increases in k. Because this is a contradiction, we complete the proof.
Proof of Proposition 3. We only provide a proof that (ii) implies (i). The opposite
can be easily proven by contradiction, and it can be omitted.
Because we assume that B(2,1) ⪰ A(2,1) , we have
1
1
1
1
u
(w(1 − r), h) + u(0,1) (w(1 − k), h) ≥ u(0,1) (w, h) + u(0,1) (w(1 − r)(1 − k), h)
2 (0,1)2
2
2⇔ u(0,1) (w(1 − k), h) − u(0,1) (w, h) ≥ u(0,1) (w(1 − r)(1 − k), h) − u(0,1) (w(1 − r), h). (11)From (11), it holds that u(0,1) (w(1 − r)(1 − k), h) − u(0,1) (w(1 − r), h) decreases

29
in r, that is,∂u(w(1 − r)(1 − k), h) − u(0,1) (w(1 − r), h) ≤ 0∂r (0,1)⇔ (1 − k)u(1,1) (w(1 − k)(1 − r), h) ≥ u(1,1) (w(1 − r), h).

(12)

From (12), it holds that u(1,1) (w(1 − r)(1 − k), h) increases in k, that is,∂u(w(1 − r)(1 − k), h) ≥ 0∂k (1,1)⇔ −u(1,1) (w(1 − k)(1 − r), h) − w(1 − k)(1 − r)u(2,1) (w(1 − k)(1 − r), h) ≥ 0⇔ −w(1 − k)(1 − r)u(2,1) (w(1 − k)(1 − r), h)≤ 1.
u(1,1) (w(1 − k)(1 − r), h)(13)This completes the proof. The proof is almost the same as Eeckhoudt et al. (2009a),
but the inequality is reversed in the last step of (13) because of u(1,1) ≤ 0 bycorrelation aversion.
To show Proposition 5, we need the following lemma, which is a generalizationof Lemma 1.
Lemma 3. Suppose that• w̃G dominates w̃B in the sense of NSD;
• h̃G dominates h̃B in the sense of MSD.
The following two conditions are equivalent:(i) (−1)n+m π(1,1) (w, h) ≤ 0.
(ii) E[π(w̃, h̃)] is decreasing in k.
Proof of Lemma 3. To show that (i) implies (ii), as in the proof of Lemma 1, we
need to prove thatE[π(w̃G , h̃G )] − E[π(w̃G , h̃B )] ≤ E[π(w̃B , h̃G )] − E[π(w̃B , h̃B )].

(14)

30
Because w̃G dominates w̃B in the sense of NSD, (14) holds if}∂n {E[π(w,h̃)]−E[π(w,
h̃
)]
≤0
G
B
∂wn
⇔ (−1)n+1 E[π(n,0) (w, h̃G )] ≤ (−1)n+1 E[π(n,0) (w, h̃B )].
(−1)n+1

Applying the property of MSD, we have}∂m {(−1)n+1 π(n,0) (w, h) ≤ 0
m
∂h
(n+m)
⇔ (−1)
π(n,m) (w, h) ≤ 0.
(−1)m+1

Similar to the proof of Lemma 1, we can show that (ii) implies (i).
Proof of Proposition 5. Applying Lemma 3 to π(x, h) = xu(1,0) (w + αx, h), we canobtain the condition. There are two cases where n + m is an even or odd number.
Note that π(n,m) (w, h) = nαn−1 u(n,m) (w + αx, h) + αn xu(n+1,m) (w + αx, h). For thecase of n + m being even numbers, we have the following:
π(n,m) (x, h) ≤ 0⇔ nαn−1 u(n,m) (w + αx, h) + αn xu(n+1,m) (w + αx, h) ≤ 0⇔ nu(n,m) (w + αx, h) + αxu(n+1,m) (w + αx, h) ≤ 0⇔ −αxu(n+1,m) (w + αx, h)≤ n,u(n,m) (w + αx, h)

where the last equivalence holds because (−1)n+m+1 u(n,m) is positive by mixed correlation aversion, and thus u(n,m) is negative in the case of n+m being even numbers.
Because u(n+1,m) (w, h) ≥ 0 for n + m being an even number, we have−

αxu(n+1,m) (w + αx, h)(w + αx)u(n+1,m) (w + αx, h)≤−.
u(n,m) (w + αx, h)u(n,m) (w + αx, h)Thus, if the right-hand-side of the inequality is less than n, that is, the degree ofrelative mixed correlation aversion in (n, m) is less than n, then the degree of partial

31
mixed correlation aversion in (n, m) is less than n. Thus, an increase in correlation
reduces the investment in the risky asset. We complete the proof for the case wheren + m is an even number. For the case of (n + m) being odd numbers, we can apply
a similar argument.
Proof of Proposition 6. As in the proof of Proposition 3, we only provide a proof
that (ii) implies (i). Because B(n,m) ⪰ A(n,m) , it follows that11
E[u(0,m) (w(1 + ϵ̃G )(1 + δ̃B ), h)] + E[u(0,m) (w(1 + ϵ̃B )(1 + δ̃G ), h)]22
11≥ E[u(0,m) (w(1 + ϵ̃G )(1 + δ̃G ), h)] + E[u(0,m) (w(1 + ϵ̃B )(1 + δ̃B ), h)]22⇔ E[u(0,m) (w(1 + ϵ̃G )(1 + δ̃B ), h)] − E[u(0,m) (w(1 + ϵ̃G )(1 + δ̃G ), h)]≥ E[u(0,m) (w(1 + ϵ̃B )(1 + δ̃B ), h)] − E[u(0,m) (w(1 + ϵ̃B )(1 + δ̃G ), h)]. (15)By NSD, (15) holds when∂n{E[u(0,m) (w(1 + ϵ)(1 + δ̃B ), h)] − E[u(0,m) (w(1 + ϵ)(1 + δ̃G ), h)]} ≥ 0∂ϵn⇔ E[wn (1 + δ̃G )n u(n,m) (w(1 + ϵ)(1 + δ̃G ), h)] ≥ E[wn (1 + δ̃B )n u(n,m) (w(1 + ϵ)(1 + δ̃B ), h)].
(16)
By FSD, (16) is equivalent to the following:∂{wn (1 + δ)n u(n,m) (w(1 + ϵ)(1 + δ), h)] ≥ 0∂δ⇔ nwn (1 + δ n−1 )u(n,m) (w(1 + ϵ)(1 + δ), h) − wn+1 (1 + δ n )(1 + ϵ)u(n+1,m) (w(1 + ϵ)(1 + δ), h)] ≥ 0(17)(17) is equivalent to−w(1 + ϵ)(1 + δ)u(n+1,m) (w(1 + ϵ)(1 + δ), h)≤ n.
u(n,m) (w(1 + ϵ)(1 + δ), h)This completes the proof.

32

References[1] Allayannis, G. and J. Ihrig (2001): “Exposure and Markups,” Review of Financial Studies 14, 805-835.
[2] Asano, T. and Y. Osaki (2022): “Precautionary Saving against Correlation
under Risk and Ambiguity,” KIER Discussion Paper, 1071.
[3] Attema, A. E., O. l’Haridon, and G. van de Kuilen (2019): “Measuring Multivariate Risk Preferences in the Health Domain,” Journal of Health Economics
64, 15-24.
[4] Ayyagari, P. and D. He (2017): “The Role of Medical Expenditure Risk in
Portfolio Allocation Decisions,” Health Economics 26, 1447-1458.
[5] Bleichrodt, H., D. Crainich, and L. Eeckhoudt (2003): “The Eﬀect of Comorbidities on Treatment Decisions,” Journal of Health Economics 22, 805-820.
[6] Bloom, D. E. and D. Canning (2000): “The Health and Wealth of Nations,”
Science 287(5456), 1207-1209.
[7] Briys, E., M. Crouhy, and H. Schlesinger (1993): “Optimal Hedging in a Futures
Market with Background Noise and Basis Risk,” European Economic Review
37, 949-960.
[8] Caballé, J. and A. Pomansky (1996): “Mixed Risk Aversion,” Journal of Economic Theory 71, 485-513.
[9] Case, A., D. Lubotsky, and C. Paxson (2002): “Economic Status and Healthin Childhood: The Origins of the Gradient,” American Economic Review 92,
1308-1334.
[10] Chiu, W. H. (2020): “Financial Risk Taking in the Presence of Correlated Nonfinancial Background Risk,” Journal of Mathematical Economics 88, 167-179.
[11] Clarke, D. J. (2016): “A Theory of Rational Demand for Index Insurance,”
American Economic Journal: Microeconomics 8, 283-306.
[12] Courbage, C., R. Peter, and B. Rey (2022). “Incentive and Welfare Eﬀects ofCorrelated Returns,” Journal of Risk and Insurance 89, 5-34.
[13] Crainich, D., Eeckhoudt, L., and Le Courtois, O. (2017): “Health and Portfolio
Choices: A Diﬃdence Approach,” European Journal of Operational Research
259, 273-279.
[14] Crainich, C., L. Eeckhoudt, and A. Trannoy (2013): “Even (Mixed) Risk Lovers
are Prudent,” American Economic Review 103, 1529-1535.

33[15] Deck, C. and H. Schlesinger (2014): “Consistency of Higher Order Risk Preferences,” Econometrica 82, 1913-1943.
[16] Denuit, M. M., L. Eeckhoudt, and M. Menegatti (2011): “Correlated Risks,
Bivariate Utility and Optimal Choices,” Economic Theory 46, 39-54.
[17] Dionne, G., L. Eeckhoudt, and C. Gollier (1993): “Increases in Risk and Linear
Payoﬀs,” International Economic Review 34, 309-319.
[18] Doherty, N. A. and H. Schlesinger (1983): “Optimal Insurance in Incomplete
Markets,” Journal of Political Economy 91, 1045-1054.
[19] Doherty, N. A. and H. Schlesinger (1990): “Rational Insurance Purchasing:Consideration of Contract Nonperformance,” Quarterly Journal of Economics
105, 243-253.
[20] Edwards, R. D. (2008). “Health Risk and Portfolio Choice,” Journal of Businessand Economic Statistics 26, 472-485.
[21] Eeckhoudt, L., J. Etner, and F. Schroyen (2009a): “The Values of Relative Risk
Aversion and Prudence: A Context-free Interpretation,” Mathematical Social
Sciences 58, 1-7.
[22] Eeckhoudt, L. and M. Kimball (1992): “Background Risk, Prudence, and theDemand for Insurance,” in Contributions to Insurance Economics, ed. by G.
Dionne. Boston: Kluwer, 239-254.
[23] Eeckhoudt, L., B. Rey, and H. Schlesinger (2007): “A Good Sign for Multivariate Risk Taking,” Management Science 53, 117-124.
[24] Eeckhoudt, L. and H. Schlesinger (2006): “Putting Risk in Its Proper Place,”
American Economic Review , 1329-1336.
[25] Eeckhoudt, L., H. Schlesinger, and I. Tsetlin (2009): “Apportioning of Risks
via Stochastic Dominance,” Journal of Economic Theory 144, 994-1003.
[26] Ekern, S. (1980): “Increasing N th Degree Risk,” Economics Letters 6, 329-333.
[27] Epstein, L. G. and S. M. Tanny (1980): “Increasing Generalized Correlation: A
Definition and Some Economic Consequences,” Canadian Journal of Economics
13, 16-34.
[28] Fishburn, P. C. and R. B. Porter (1976): “Optimal Portfolios with One Safe and
One Risky Asset: Eﬀects of Changes in Rate of Return and Risk,” Management
Science 22, 1064-1073.
[29] Fujii, Y. and Y. Osaki (2019): “The Willingness to Pay for Health Improvement
under Ambiguity,” Journal of Health Economics 66, 91-100.

34[30] Gilboa, I. (2009): Theory of Decision under Uncertainty, Cambridge University
Press.
[31] Gilboa, I. and D. Schmeidler (1989): “Maxmin Expected Utility with NonUnique Priors,” Journal of Mathematical Economics 18, 141-153.
[32] Gollier, C. (2011): “Portfolio Choice and Asset Prices: The Comparative Statics
of Ambiguity Aversion,” Review of Economic Studies 78, 1329-1344.
[33] Gollier, C. and J. W. Pratt (1996): “Risk Vulnerability and the Tempering
Eﬀect of Background Risk,” Econometrica 64, 1109-1123.
[34] Guidolin, M. and F. Rinaldi (2013): “Ambiguity in Asset Pricing and Portfolio
Choice: A Review of the Literature,” Theory and Decision 74, 183-217.
[35] He, J. and L. K. Ng (1998): “The Foreign Exchange Exposure of JapaneseMultinational Corportations,” Journal of Finance 53, 733-753.
[36] Haliassos, M. and C. C. Bertaut (1995): “Why Do So Few Hold Stocks?”,
Economie Journal 105, 1110-1129.
[37] Ingersoll, J. (1987): Theory of Financial Decision Making, Rowman & Littlefield.
[38] Jiang, J. J. Liu, W. Tian, and X. Zeng (2022): “Portfolio Concentration, Portfolio Inertia, and Ambiguous Correlation,” Journal of Economic Theory, 203,
105463.
[39] Jokung, O. (2011): “Risk Apportionment via Bivariate Srochastic Dominance,”
Journal of Mathematical Economics 47, 448-452.
[40] Kihlstrom, R. E., D. Romer, and S. Williams (1981): “Risk Aversion with
Random Initial Wealth,” Econometrica 49, 911-920.
[41] Kimball, M. (1990): “Precautionary Savings in the Small and the Large,”
Econometrica 58, 53-73.
[42] Klibanoﬀ, P., M. Marinacci, and S. Mukerji (2005): “A Smooth Model of Decision Making under Ambiguity,” Econometrica 73, 1849-1892.
[43] Kotschy, R. (2021). “Health Dynamics Shape lLife-cycle Incomes,” Journal of
Health Economics, 75, 102398.
[44] Lehmann, E. L. (1966): “Some Concepts of Dependence,” The Annals of Mathematical Statistics 37, 1137-1153.
[45] Li, J. (2011): “The Demand for a Risky Asset in the Presence of a BackgroundRisk,” Journal of Economic Theory 146, 372-391.

35[46] Mankiw, G. and S. Zeldes (1991): “The Consumption of Stockholders and Nonstockholders”, Journal of Financial Economics 29, 97-112.
[47] Malevergne, Y. and B. Rey (2009): “On Cross-risk Vulnerability,” Insurance:
Mathematics and Economics 45, 224-229.
[48] Menezes, C., C. Geiss, and J. Tressler (1980): “Increasing Downside Risk,”
American Economic Review 70, 921-932.
[49] Menezes, C. F., and H. Wang (2005): “Increasing Outer Risk,” Journal of
Mathematical Economics 41, 875-886.
[50] Mossin, J. (1968): “Aspects of Rational Insurance Purchasing,” Journal of
Political Economy 76, 553-568.
[51] Nachman, D. C. (1982): “Preservation of “More Risk Averse” under Expectations,” Journal of Economic Theory 28, 361-368.
[52] Rothschild, M. and J. E. Stiglitz (1970): “Increasing Risk: I. A Definition,”
Journal of Economic Theory 2, 225-243.
[53] Sandmo, A. (1971): “On the Theory of the Competitive Firm under Price
Uncertainty,” American Economic Review 61, 65-73.
[54] Schmeidler, D. (1989): “Subjective Probability and Expected Utility without
Additivity,” Econometrica 57, 571-587.
[55] Sloan, F. A., W. K. Viscusi, H. W. Chesson, C. J. Conover, and K. WhettenGoldstein (1998): “Alternative Approaches to Valuing Intangible Health Losses:
the Evidence for Multiple Sclerosis,” Journal of Health Economics 17, 475-497.
[56] Tsetlin, I. and R. L. Winkler (2005): “Risky Choices and Correlated Background Risk,” Management Science 51, 1336-1345.
[57] Viscusi, W. K., and W. N. Evans (1990): “Utility Functions that Depend on
Health status: Estimates and Economic Implications,” American Economic
Review 80, 353-374.
[58] Wang, J. and J. Li (2010): “Multiplicative Risk Apportionment,” Mathematical
Social Sciences 60, 79-81.
[59] Weil, P. (1992): “Equilibrium Asset Prices with Undiversifiable Labor Income
Risk,” Journal of Economic Dynamics and Control 16, 769-790.
[60] Wright, R. (1987): “Expectation Dependence of Random Variables, with an
Application in Portfolio Theory,” Theory and Decision 22, 111-124.