The investment home bias with peer effect

Levy, Haim Article The investment home bias with peer effect Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Levy, Haim (2020) : The investment home bias with peer effect, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 5, pp. 1-19, https://doi.org/10.3390/jrfm13050094 This Version is available at: https://hdl.handle.net/10419/239182 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article The Investment Home Bias with Peer Effect Haim Levy Department of Finance, Hebrew University, Jerusalem 9190401, Israel; [email protected] Received: 26 February 2020; Accepted: 7 May 2020; Published: 11 May 2020   Abstract: Observed international diversification implies an investment home bias (IHB). Can bivariate preferences with a local domestic peer group rationalize the IHB? For example, it is argued that wishing to have a large correlation with the Standard and Poor’s 500 stock index (S&P 500 stock index) may induce an increase in the domestic investment weight by American investors and, hence, rationalize the IHB. While this argument is valid in the mean-variance framework, employing bivariate first-degree stochastic dominance (BFSD), we prove that this intuition is generally invalid. Counter intuitively, employing “keeping up with the Joneses” (KUJ) preference with actual international data even enhances the IHB phenomenon. Keywords: investment home bias (IHB); bivariate first-degree stochastic dominance (BFSD); keeping up with the Joneses (KUJ); correlation loving (CL) JEL Classification: D81; C91 1. Introduction The investment home bias (IHB) is well documented. For example, the US equity market accounts for about 35% of the world equity market, yet about 75% of Americans’ equity investment is allocated to the US market. Hence, the US equity IHB is in the magnitude of 40%. For most commonly employed utility functions, the univariate expected utility maximization also does not support the relatively large domestic investment weight; hence, the IHB puzzle emerges. It is advocated that the bivariate expected utility maximization rationalizes partially or fully the IHB. In this study, we employ the “keeping up with the Joneses” (KUJ) preference, where the investor’s wealth and the peer group’s wealth are the two attributes of this utility function, analyzing the peer effect on the empirically observed IHB phenomenon. The basic idea and the intuition of the KUJ argument for rationalizing the IHB phenomenon is as follows: suppose that you know your univariate utility function and that, for a given joint distribution of returns corresponding to the various international markets, you derive with this utility function the optimal investment weights in the domestic market as well as in the foreign markets under consideration. Furthermore, suppose that the optimal domestic investment weight is, say, p%. Now, suppose that you decide to consider, in addition to the joint distribution of returns, one more factor: you also want the performance of your portfolio to be as close as possible to the performance of a certain local stock index. For example, the American investor wants the return on her portfolio to be as close as possible to the return on the S&P 500 index, which, for simplicity of the discussion, is assumed to be the peer’s portfolio (the same analysis applies to any other local stock index). Thus, if the investor benefits from having a relatively large correlation with the S&P index, she may have an incentive to increase the domestic investment weight (which generally increases the correlation with the S&P stock index, and if the domestic investment weight is 100%, this correlation is +1) beyond J. Risk Financial Manag. 2020,13, 94; doi:10.3390/jrfm13050094 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 94 2 of 19 what is obtained by a maximization of a univariate expected utility function. Therefore, employing KUJ preferences may rationalize the IHB.1 Indeed, Lauterbach and Reisman (2004) use the KUJ preference prove that the IHB is rationalized by incorporating the peer effect. However, they use the mean-variance model with some approximations to achieve this result. We analyze in this paper the impact of incorporating the peer effect on the IHB in the most general bivariate expected utility case, not relying on the mean-variance framework and where no approximations of the various mathematical formulas are employed. We define the precise conditions, which guarantee the IHB rationalization, by adding the peer effect. We find that, in this unrestricted analysis, the appealing intuitive explanation of the IHB rationalization by the peer effect is generally wrong. Thus, we conclude that one should seek other economic explanations for the observed IHB phenomenon. For some interesting economic suggestions, see Coeurdacier and Rey (2013) and Berriel and Bhattarai (2013)2or other behavioral explanations. We employ in this study distribution-free bivariate first-degree stochastic dominance (BFSD), with no assumptions on the shape of the bivariate preferences and no approximations. We prove that, despite the above appealing intuition of the peer effect on the optimal domestic investment weight, using the bivariate preferences, the IHB may increase or decrease relative to the univariate optimal domestic investment weight. Moreover, we demonstrate with actual international data that adding the peer effect, counter intuitively, even intensifies the IHB from the American investor’s point of view. Hence, the IHB still exists. The structure of the rest of this paper is as follows. Section 2provides a brief literature review. Section 3presents bivariate first-degree stochastic dominance (BFSD) rule and the implied theoretical results. We analyze the various factors affecting the IHB and show that bivariate preferences rationalize the IHB phenomenon only in a limited and unrealistic case. Section 4is devoted to the commonly employed KUJ preferences, which is a specific set of all the bivariate preferences. We show empirically that the peer effect with KUJ preferences even enhances the IHB puzzle. Section 5concludes. 2. Literature Review Vanp é e and DeMoore (2012) show that the IHB exists in virtually all countries. The magnitude of the IHB phenomenon is relatively large, characterizing various periods, assets, and countries. While about three decades ago the American investment in the local market was more than 90%, implying a very large IHB, in recent years the IHB phenomenon has been mitigated, yet it is still about 40%. When it comes to fixed-income assets, the home bias is even larger. This phenomenon is not unique to the US and characterizes many capital markets (for a report on the IHB in various countries, regarding equity and fixed-income assets, see (Philips et al. 2012)). Actually, there is evidence that the home bias is even worse than reported (see Baxter and Jermann 1997). Researchers have analyzed various possible key explanations for the IHB. It is agreed that some portion of the domestic overinvestment may be induced by international trade barriers, foreign exchange risk, and regulation, as well as by a domestic peer group effect. However, with the increase in the rapid flow of information and market efficiency observed over the last few decades, the trade barriers, including possible asymmetrical information, have drastically declined. This may account for the observed slight decrease in the domestic overinvestment phenomenon. However, since 1998, the equity IHB of American investors has stabilized at about 40% (see Levy and Levy 2014). 1 Note, we analyze whether the peer effect increases the optimal domestic weight, which partially or fully rationalizes the IHB. The reason is that it is possible that the peer effect increases the optimal domestic weight by, say, 1%, but the IHB is, say, 40%, a case where other factors are needed to explain the observed IHB. In our study, we find empirically that the peer effect even enhanced the IHB; hence, the distinction between partial and full IHB rationalization is irrelevant. 2 They consider portfolio diversification when macroeconomic factors are incorporated into a two-country general equilibrium model, called the “Open Economy Financial Macroeconomics” model. They conclude that, with this equilibrium model, the home bias is less of a puzzle. Berriel and Bhattarai (2013) also suggest a macroeconomic model (related to the positive association between government spending and return on local stocks) to explain the home bias. J. Risk Financial Manag. 2020,13, 94 3 of 19 While most empirical studies analyze the IHB at the country level (see French and Poterba 1991; Tesar and Werner 1995), Kang and Stultz (1997), who study the IHB puzzle in Japan, analyze it at the individual firm level, showing that foreign investors hold disproportionally more Japanese shares of firms in the manufacturing industries, large firms, and firms with good accounting performance. Similarly, Dahlquist and Robertsson (2001) identify the characteristics of Swedish firms that attract foreign investors. Lewis (1999), who analyzes the effect of each economic factor that is considered as a barrier for efficient international diversification on the IHB, concludes that the trade barriers cannot explain the magnitude of the existing IHB. Therefore, the IHB puzzle is still an interesting research topic.3 Obviously, if the IHB does not incur economic loss, it does not constitute an economic puzzle. Indeed, the intensity of the IHB economic cost changes over time. Levy (2016) analyzes the trend in the IHB phenomenon over time. Moreover, he distinguishes between the economic home bias (EHB), which measures the economic loss in terms of the differences in the certainty equivalent of two alternative international diversification strategies (with and without a home bias) and the IHB, which simply measures the deviations between the optimal international investment weights and the actual investment weights. He reports that, while the EHB was very large in the past, in the last 15 years, the EHB from the American investment point of view has become negligible, despite the existence of about 40% IHB. This reduction in the EHB is induced by the increasing trend in the international correlations. Thus, it seems that for the American investors the IHB is not a major economic puzzle. However, he also reports that for other countries, e.g., France, the EHB is still very large, and the economic puzzle exists. Moreover, in recent years, we have trend reversal in correlations, and a decrease in the average correlation between various markets has been recorded. As a result of this trend reversal, the EHB has recently increased, even for American investors. Thus, for most countries and with the recent trend reversal in correlation also for the US, the IHB still constitutes an economic puzzle that needs an explanation. The employment of the KUJ preference, namely incorporation of the peer effect, is considered as one of the promising paths in explaining the IHB puzzle. We employ in this paper a bivariate preference. Generally, with bivariate preference, the two variables can take many forms, e.g., wealth and health, climate and income, etc. Our study deals with investment choices. Hence, the two variables are the individual’s wealth and the peer group’s wealth. The peer group’s wealth can be the return on a certain domestic portfolio, and in our case, as mentioned above, we assume, for the simplicity of the discussion and without loss of generality, that it is the return on S&P 500 stock index. The common view is that the relevant bivariate utility function has a positive cross derivative (we will elaborate on this issue below) and that investors want, among other things, the performance of their portfolio to be as close as possible to the performance of the peer’s portfolio, i.e., a large correlation with the S&P stock index is desired. 4 Therefore, we focus our analysis on the positive cross-derivative case. Obviously, despite the desire for having a relatively large correlation with the S&P index, the investor will shift from a portfolio with a small correlation to a portfolio with a large correlation, only if the bivariate expected utility increases by such a shift. We turn to analyze the conditions under which indeed such shift takes place, namely that the IHB can be rationalized with the peer effect. 3 It is interesting to note that, even in a case in which there are no transparent trade barriers, there is a tendency to invest in firms that are geographically located close to the investor’s location. This phenomenon is well documented within the US (see Coval and Moskowitz 1999,2001;Huberman 2001). This indicates that the home bias is a complex phenomenon that is not easy to explain with conventional economic factors. 4 Tsetlin and Winkler (2009) advocate that correlation aversion prevails. However, in their model, the two attributes of the bivariate preference directly affect the utility of the decision maker, for example, income and quality of life. In our model, the two attributes are different: the individual’s wealth and the peer group’s wealth. As relative wealth may affect the individual’s utility, it is advocated in the literature that, when some conditions hold, correlation loving prevails. J. Risk Financial Manag. 2020,13, 94 4 of 19 3. Bivariate First-Degree Stochastic Dominance (BFSD) and the IHB We would like to stress at the outset that most of the mathematical formulas given in the first part of this section are not new and exist in the literature, albeit in different forms and in different connotations. However, we use these mathematical results, to the best of our knowledge for the first time to analyze the peer effect with KUJ preferences on the IHB phenomenon. 3.1. The Sufficient Conditions for BFSD Implying the IHB Rationalization Consider an individual with a bivariate preference U(w , wP) , where w denotes the return on the selected international portfolio by the investor under consideration, and wp denotes the return on the peer’s portfolio. We compare two bivariate investment portfolios, Fand G, where the domestic investment weight in portfolio Fis larger than the domestic investment weight in portfolio G(we will elaborate later on the selected portfolios, Fand G). Our aim is to examine the conditions under which F dominates Gby BFSD with the above bivariate utility function, where we first assume two assumptions on the preferences: ∂Uw,wp/∂w≡U1≥ 0 (monotonicity) and ∂2Uw,wp/∂w∂wp≡U12 ≥ 0 (later on we consider also U12 ≤ 0, a case usually not considered in KUJ economic research but emerges as important to our analysis). There is no constraint on the derivative ∂Uw,wp/∂wp≡U2 , which can be negative, zero, or positive. 5 If such dominance exists, then all investors, regardless of the precise shape of the bivariate preference, will switch from Gto F. Hence, the optimal domestic investment weight increases, and therefore the peer effect rationalizes the IHB phenomenon. Note that the main ingredient of the KUJ preference is that the cross derivative (U12) is positive, implying that the individual’s marginal utility increases with an increase in the peer group wealth (see Ljungqvist and Uhlig 2000). 6 Therefore, as explained before, it seems that the investor with a positive cross derivative would incline to overinvest domestically, as she prefers her wealth to be positively correlated with the peer’s wealth. While the above intuitive explanation is appealing, in the following proposition, it is formally shown that generally only under some specific conditions, indeed a positive cross derivative is tantamount to correlation loving, where correlation loving implies that, by increasing the domestic investment weight, the bivariate expected utility increases. Namely, if the conditions required in the proposition are intact, the investor increases her bivariate expected utility by overinvesting domestically (relative to the optimal univariate expected utility maximization optimal domestic investment weight), and by doing so, the correlation increases. Thus, if the proposition required conditions hold in practice, we have by the KUJ preferences a rationalization of the IHB, and the IHB puzzle may vanish. As we explain below, in practice, the required conditions for IHB rationalization are not intact. Before stating the proposition, we need the following definition: Definition 1. Definition of correlation loving (CL): The investor is CL if and only if, by increasing the correlation between her portfolio and the peer’s portfolio, the expected bivariate utility increases. Hence, CR investors who maximize the bivariate expected utility would increase the domestic investment weight relative to the optimal univariate expected utility weight. 5Note that a negative sign implies jealousy, and a positive sign implies altruism (see Dupor and Liu 2003). 6 Numerous studies suggest replacing the univariate expected utility analysis with the expected bivariate utility analysis with various definitions of the two variables: past and present consumption, consumption of the individual, and consumption of the peer group, the wealth obtained by the individual and the opponent in an ultimatum game, and so forth. For studies that assume that the utility is derived not from the absolute wealth (or consumption) of the individual but from the relative wealth (or consumption), in which the wealth’s position relative to the peer group plays an important role, as well as for other factors that do not affect the classic univariate expected utility but affect the bivariate expected utility, see, for example, Abel (1990), Constantinides (1990), Bolton (1991), Rabin (1993,1998), Gal í (1994), Campbell and Cochrane (1999), Bolton and Ockenfels (2000), Dupor and Liu (2003), Zizzo (2003), and Demarzo et al. (2008). J. Risk Financial Manag. 2020,13, 94 5 of 19 Proposition 1. Suppose that the investor faces two alternate bivariate prospects, F(w , wp) and G(w , wp) , where w as well as wp can take only two different outcomes. As there are only two outcomes, they can be rearranged to have either a correlation of +1 or a correlation of − 1. Diversification between w and wp is not allowed, implying that the marginal distributions are identical (namely, Fw=Gw and FwP=GwP , regardless of the outcomes arrangement; see, for example, Table 1). Under these specific conditions, the investor with a bivariate preference is CL if and only if the cross derivative is positive, namely U12 ≥ 0. Specifically, under the conditions of the proposition, with CL, the prospect with a correlation of +1 yields a higher bivariate expected utility than any other possible prospect. (For proof, with some other notation, see (Eeckhoudt et al. 2007)). Thus, if the conditions of the proposition were intact, the American investor who likes her investment performance to be as close as possible to the S&P index would have a higher expected utility by increasing the domestic investment weight. Actually, under the conditions of the proposition, having a correlation of +1 with the S&P index is optimal, implying that investing 100% domestically is optimal, which creates a negative IHB puzzle (because in practice less than 100% is invested domestically). In short, if the conditions of Proposition 1 are intact, we have: CL ⇔U12 >0 (1) Note that investing more intensively domestically, hence increasing the correlation between the investor’s portfolio and the peer’s portfolio, generally does not imply CR as defined above. The reason is that, with investment in practice, by increasing the domestic investment weight, although the correlation increases, generally, other parameters of the portfolio may also change, the marginal distributions may change (hence, the conditions of the proposition are violated), and the bivariate expected utility may decrease. Therefore, the American investor may decide not to decrease the domestic investment weight, despite the desire to have large correlation with the S&P index. However, by the above definition, the investor is CL only if, after considering all effects, the bivariate expected utility increases. However, note that, by Proposition 1, the marginal distributions are kept unchanged, and the correlation can take only the extreme values of either +1 or − 1. This is because in Eeckhoudt et al. (2007) original proposition, each variable can get only two possible values. Hence, by reordering these values, the marginal distributions are kept unchanged. Also, diversification between wand wp is not allowed, because if it is allowed, the marginal distribution of the individual’s wealth, w, generally will not be kept constant. Thus, the statement given in Proposition 1 is suitable to some choices, where the variables are, for example, wealth and health, with only two outcomes (say, bad and good health, high and low income, etc.). As we shall see below, with international diversification, we have more than two outcomes corresponding to each prospect, and diversification is allowed. Hence, the marginal distributions generally change when the selected diversification changes. Therefore, a positive cross derivative in our analysis does not necessarily imply CL. As a result, we may even obtain an IHB phenomenon enhanced with bivariate preferences relative to the univariate IHB, despite the fact that a positive cross derivative is assumed. Let us turn now to the conditions for BFSD of the distribution of returns of the portfolio with the IHB over the distribution of returns with no IHB. The two portfolios that we compare, Fand G, have bivariate density functions, denoted by f(w , wP) and g(w , wP) , respectively. As we focus on the possible IHB rationalization, it is assumed, as explained before, that Fstands for a portfolio with an IHB, that is, the domestic weight in this portfolio is larger than the corresponding weight in G. Thus, if the domestic investment weight in Gis equal to the optimal theoretical univariate expected utility maximization domestic weight (say, the international market portfolio), the BFSD of Fover G implies that the peer effect rationalizes the IHB phenomenon, as all investors would prefer Fover J. Risk Financial Manag. 2020,13, 94 6 of 19 G. 7 Assuming that ∂2w, wp/∂w∂wp)≡U12 > 0 with KUJ preference 8 to explain various observed economic phenomena is very common. As seen in Proposition 1, this assumption is an important ingredient also needed to rationalize the IHB phenomenon, so long as the conditions of Proposition 1 hold. Therefore, we examine the role of the cross derivative on the BFSD relation. To examine possible rationalization of the observed IHB with KUJ preferences, we extend the expected utility univariate analysis to the bivariate expected utility analysis by adding the peer effect. The expected bivariate utility of portfolios Fand Gis given by: EFU=Rw wRwp wpU(w,wp)fw,wpdwdwp EGU=Rw wRwp wpU(w,wp)gw,wpdwdwp (2) where w and w denote the minimal and maximal values of w (which can be −∞ and ∞ ); similarly, wP and wPdenote the minimal and maximal values of wP. Thus, ∆i≡EFU−EGU=Zw wZwp wp U(w,wp)hfw,wp−gw,wpidwdwp Integrating by parts the above equation with respect to both variables yields: ∆i≡EFU−EGU=Rwp wpRw wU12[F(w,wp)−G(w,wp)]dwdwp+Rw wU1[G(w)− F(w)]dw +Rwp wpU2[G(wp)−F(wp)]dwp ≡A+B+C (3) where ∆i denotes the expected utility difference corresponding to the ith investor, F(w , wP) and G(w , wP) are the two bivariate cumulative distributions, F(w)=Fw,wp is the marginal cumulative distribution function of w , Fwp=Fw,wp is the marginal cumulative distribution function of wP , and U1 , U2 , and U12 denote the partial derivatives: U1≡∂U/∂w , U2≡∂U/∂wP , and U12 ≡∂2U/∂w∂wP , respectively. For the derivation of Equation (3) with slightly different notations, see Levy and Paroush (1974, p. 131) and Atkinson and Bourguignon (1982, pp. 185–86). 9 Note that, as the marginal utility of the peer’s portfolio is identical under the various investment strategies (with and without intensive domestic investment). Namely, we have Gwp=Fwp , therefore term Cin Equation (3) is equal to zero. Thus, the rest of the paper relate only to terms Aand B. Let us first analyze the relation between Equation (3) (with C=0) and the conditions given in Proposition 1. If each of the two random variables, w and wp , has only two possible different outcomes, the correlation is either +1 or − 1. Also, when diversification between wand wp is not allowed, the marginal distributions are equal (namely, G(w)=F(w) , see also the example given in 7 Obviously, we have a different optimum portfolio for each utility function, but, as we shall see below, the analysis is intact, independent of the assumed preference. 8 The KUJ and CUJ literature is very extensive; hence, we mention here only a few of these studies. Abel (1990) and Gal í (1994) use this bivariate framework to explain optimal choices. Ljungqvist and Uhlig (2000) examine the role of tax policies in economics with CUJ utility functions. Campbell and Cochrane (1999) assume that the preference is a function of the relative consumption, when the individual’s consumption is measured relative to the weighted average of the past consumption of all individuals. In these models, when the peer group’s variable (e.g., consumption) is a lagged variable, the model is commonly called the CUJ model, and when the individual’s variable and the peer group variable relate to the same time period (e.g., return on investment), it is commonly called the KUJ model. In this paper, we analyze the optimal portfolio investment decision in the KUJ set-up. 9 Note that Equation (2) is reduced to the well-known univariate formula employed to derive the FSD rule, where U12 =U2= 0. For more details, see Hadar and Russell (1969) and Hanoch and Levy (1969). Although we focus in this paper on FSD, one can assume risk aversion and employ stronger investment rules; for example, see Rothschild and Stiglitz (1970) and Levy (2015). J. Risk Financial Manag. 2020,13, 94 7 of 19 Table 1. Hence, in this specific case also term Bis equal to zero, and we are left with term A. If F represents the +1 correlation and Gthe − 1 correlation, we must have with the two outcomes case that Fw,wp≥Gw,wp (see example 1 in Table 1. Hence, in this case by Equation (3), with B=C=0, the condition U12 ≥ 0 is a sufficient condition for dominance of the joint distribution with the +1 correlation over the joint distribution with the − 1 correlation (see Equation (3)). 10 It is easy to verify that, in this specific case, U12 ≥ 0 is a necessary and sufficient condition for dominance. 11 Thus, Equation (3) is perfectly consistent with Proposition 1, so long as the conditions given in the proposition are intact. However, Equation (3) corresponds to the general case, as it covers the more realistic scenarios where more than two outcomes are possible. Diversification is allowed, and the marginal distributions are not necessarily equal, hence term Bis not necessarily equal to zero. As we shall see in this general and realistic case, U12 ≥ 0 is neither a necessary nor a sufficient condition for dominance. We turn now to analyze the possible dominance of the portfolio with the IHB over a portfolio with no IHB in the most general case. To examine possible rationalization of the IHB phenomenon with bivariate preferences, let us first take a deeper look at the marginal distributions corresponding to the international diversification issue analyzed in this paper. Returning to Equation (3), note that, as mentioned above, it is reasonable to assume that the third term on the right-hand side of Equation (3), term C, is equal to zero, as the investor in the capital market generally cannot affect the peer group investment decision; hence, the peer’s group marginal distribution is identical under Fand G. This condition conforms to the requirement in Proposition 1, even in the case where more than two outcomes exist. This is a reasonable assumption with the investment choices that we analyze in this study but not with ultimatum games in which the individual decision affects the opponent’s outcome. Moreover, in the portfolio investment case, this term is equal to zero, regardless of whether the peer group portfolio is domestic or international. Thus, regarding the issue that we investigate in this paper (investment with a particular stock index as the peer group’s portfolio), as advocated above, the sign of the derivative U2 is irrelevant. Namely, term C=0 and there is no need to assume jealousy (U2<0) or altruism (U2> 0 ) to obtain our results corresponding to the portfolio investment case. Thus, as for the analysis of the IHB, term Cis equal to zero, and Equation (3) is reduced to: ∆i=A+B. (4) However, note that generally we cannot assume that also term Bis equal to zero, as by changing the diversification strategy, we change the marginal distribution of the individual’s wealth. Thus, with international portfolio diversification, the condition of equal marginal distributions (see term Bof Equation (3)) required by Proposition 1 does not hold. To be able to determine whether the peer effect induces an increase in the optimal domestic investment relative to the univariate expected utility optimal domestic investment weight, we need to be more specific regarding the definitions of portfolios Fand Gunder consideration. We examine here the possible existence of BFSD by considering the two specific portfolios with direct implication to the IHB issue analyzed in this paper. These two portfolios are denoted by FAand GM, as defined below. Definition 2. GM is the portfolio with the international market weights. If the American investor holds this market portfolio, she would invest 35% (which is the weight of the American market in the world market) domestically; hence, the IHB does not exist. Distribution FA stands for the actual aggregate portfolio held by the American investors. Namely, the actual domestic weight held by the American investor is 75%; hence, holding this portfolio implies an IHB of 40%. 10 Actually, it is required to have at least one strict inequality with the distribution functions as well as with the cross derivative to avoid the trivial case of having ∆i= 0. In the rest of the paper, when we write such inequalities, we always mean that there is at least one strict inequality, but to avoid a complex writing, we will not write it down everywhere. 11 If U12 < 0, in some range, one can always find a bivariate preference, such that outside this range the cross derivative is close to zero; hence, ∆iis negative. Therefore, to guarantee that ∆iis non-negative, the cross derivative cannot be negative. J. Risk Financial Manag. 2020,13, 94 8 of 19 Assuming that term Cis equal to zero, and rewriting Equation (4) in terms of the above two portfolios, we obtain: ∆i≡EFAU−EGMU=Rwp wpRw wU12hFAw,wp−GMw,wpidwdwp +Rw wU1[GM(w)−FA(w)]dw ≡A+B (5) Suppose that without the peer effect the market portfolio is optimal. If ∆i> 0, the ith investor under consideration who considers also the peer effect prefers the actual portfolio to the market portfolio; hence, the investor increases the bivariate expected utility by increasing the domestic investment weight. However, to have BFSD and IHB rationalization, we need to have that ∆i> 0 for all investors i=1, 2, . . . n, regardless of the precise shape of their preferences. A few conclusions, some of them in contradiction to the common view regarding the role of the cross derivative, can be drawn from Equation (5). First, if U12 ≥ 0 is assumed (as needed in Proposition 1 to justify the rationalization of the IHB), we find that there is no BFSD; hence, in this setting there is no IHB rationalization. The reason is that if U12 ≥ 0, term Aof Equation (5) is positive only if FAw,wp≥GMw,wp , but this implies that FA(w,∞)=FA(w)≥G(w,∞)=G(w) , and therefore, term Bis negative. The sum A+Bmay be negative, implying that there is no BFSD. Surprisingly, in contrast to Proposition 1, the condition U12 ≤ 0 may allow BFSD; hence, it may allow IHB rationalization. We have BFSD and IHB rationalization with U12 ≤ 0 if the following two conditions hold: FAw,wp≤GMw,wp(6a) FA(w)≤GM(w)(6b) But as condition (a) implies condition (b), unlike the positive cross-derivative case, these two conditions can simultaneously hold. Therefore, if condition (a) on the joint distribution holds, both terms Aand Bare positive (with U12 <0)and therefore ∆i≥0 for i=1, 2, . . . n. Example 1. The marginal distributions and the BFSD. In this example, we demonstrate the relation between the BFSD and the positive cross derivative in the case where the conditions of Proposition 1 are intact, and then we demonstrate the more realistic case, where the marginal distributions are not kept constant; hence, BFSD does not exist, despite the positive cross-derivative assumption. Suppose that the S&P index return wp is equal to 3 or 4, each outcome with an equal probability of 0.5. We consider investing in either portfolio For portfolio G, both yielding return w of either 2 or 5 with equal probability of 0.5. However, Fhas a correlation of +1 with the S&P index (with joint returns of (2, 3) with a probability of 0.5 and joint returns of (5, 4) with a probability of 0.5). Ghas a negative correlation of −1 with the S&P index with joint returns of (2, 4) with a probability of 0.5 and joint returns (5, 3) with a probability of 0.5 (see Table 1). All other joint probabilities are equal to zero. Denoting the joint distribution corresponding to the correlation +1 by FA and the joint distribution corresponding to correlation − 1 by GM , we have with the above example with the joint probabilities the following relationship: FAw,wp≥GMw,wpfor all values w,wp(7) with at least one strict inequality (see lower part of Table 1Part a), e.g., FA(2, 3)=0.5 >GM(2, 3)=0, J. Risk Financial Manag. 2020,13, 94 15 of 19 We employ the bivariate first-degree stochastic dominance (BFSD) rule and prove theoretically that bivariate preferences with a positive cross derivative rationalizes the observed IHB, only in the unrealistic case in which the marginal distributions of all possible portfolio under consideration are identical. Of course, this does not hold in practice, as not all international markets are identical, and therefore also the marginal distributions of various selected diversified portfolios are not identical. Thus, even with peer effect, overinvesting domestically may be an inferior investment strategy, hence the IHB cannot be explained by the peer effect. With actual empirical international stock market data (obviously, with unequal empirical marginal distributions), we find that the commonly employed KUJ preference with a positive cross derivative, which intuitively implies a desire to increase the correlation by overinvesting domestically, decreases rather than increases the domestic investment weight, hence the peer effect even enhances the IHB puzzle. Moreover, once again counter intuitively, we find that, with a bivariate preference with a negative cross derivative, the optimal domestic investment increases. Thus, a positive cross derivative is neither necessary nor sufficient for IHB rationalization. In sum, employing a general bivariate utility function with peer effect, with no constraints on the preference employed, generally cannot rationalize the empirically observed IHB. Employing the commonly employed specific KUJ bivariate preferences also does not rationalize the IHB. As the IHB is an empirical fact, to rationalize this phenomenon, one needs to seek other explanations and other research strands, as the intuitive explanation of the peer effect for rationalizing the IHB phenomenon is misleading. Funding: This research received no external funding. Acknowledgments: I would like to thank three anonymous referees of this Journal for their helpful comments which greatly improved the paper. Conflicts of Interest: The authors declare no conflict of interest. J. Risk Financial Manag. 2020,13, 94 16 of 19 Appendix A Table A1. The annual rates of return 1988–2012. Year USA Canada Germany France The Netherlands Norway Sweden UK Australia Japan Emerging Markets 1988 0.16 0.18 0.21 0.39 0.16 0.43 0.49 0.06 0.38 0.36 0.40 1989 0.31 0.25 0.47 0.37 0.37 0.46 0.33 0.22 0.11 0.02 0.65 1990 −0.02 −0.12 −0.09 −0.13 −0.02 0.01 −0.20 0.10 −0.16 −0.36 −0.11 1991 0.31 0.12 0.09 0.19 0.19 −0.15 0.15 0.16 0.36 0.09 0.60 1992 0.07 −0.11 −0.10 0.03 0.03 −0.22 −0.14 −0.04 −0.10 −0.21 0.11 1993 0.10 0.18 0.36 0.22 0.37 0.43 0.38 0.24 0.37 0.26 0.75 1994 0.02 −0.02 0.05 −0.05 0.13 0.24 0.19 −0.02 0.06 0.22 −0.07 1995 0.38 0.19 0.17 0.15 0.29 0.07 0.34 0.21 0.12 0.01 −0.05 1996 0.24 0.29 0.14 0.22 0.29 0.29 0.38 0.27 0.18 −0.15 0.06 1997 0.34 0.13 0.25 0.12 0.25 0.07 0.13 0.23 −0.10 −0.24 −0.12 1998 0.31 −0.06 0.30 0.42 0.24 −0.30 0.15 0.18 0.07 0.05 −0.25 1999 0.22 0.54 0.21 0.30 0.07 0.32 0.81 0.12 0.19 0.62 0.66 2000 −0.13 0.06 −0.15 −0.04 −0.04 0.00 −0.21 −0.12 −0.09 −0.28 −0.31 2001 −0.12 −0.20 −0.22 −0.22 −0.22 −0.12 −0.27 −0.14 0.03 −0.29 −0.02 2002 −0.23 −0.13 −0.33 −0.21 −0.20 −0.07 −0.30 −0.15 0.00 −0.10 −0.06 2003 0.29 0.55 0.65 0.41 0.29 0.50 0.66 0.32 0.51 0.36 0.56 2004 0.11 0.23 0.17 0.19 0.13 0.54 0.37 0.20 0.32 0.16 0.26 2005 0.06 0.29 0.11 0.11 0.15 0.26 0.11 0.07 0.18 0.26 0.35 2006 0.15 0.18 0.37 0.35 0.32 0.46 0.45 0.31 0.33 0.06 0.33 2007 0.06 0.30 0.36 0.14 0.21 0.32 0.01 0.08 0.30 −0.04 0.40 2008 −0.37 −0.45 −0.45 −0.43 −0.48 −0.64 −0.49 −0.48 −0.50 −0.29 −0.53 2009 0.27 0.57 0.27 0.33 0.43 0.89 0.66 0.43 0.77 0.06 0.79 2010 0.15 0.21 0.09 −0.03 0.02 0.12 0.35 0.09 0.15 0.16 0.19 2011 0.02 −0.12 −0.17 −0.16 −0.12 −0.09 −0.15 −0.03 −0.11 −0.14 −0.18 2012 0.16 0.10 0.32 0.23 0.21 0.20 0.23 0.15 0.22 0.08 0.19 J. Risk Financial Manag. 2020,13, 94 17 of 19 Appendix B Table A2. The correlation matrix for the period 1988–2012. USA Canada Germany France The Netherlands Norway Sweden UK Australia Japan Emerging Markets USA 1 0.68 0.79 0.83 0.85 0.48 0.76 0.86 0.58 0.44 0.52 Canada 0.68 1 0.77 0.76 0.75 0.84 0.88 0.79 0.81 0.67 0.78 Germany 0.79 0.77 1 0.90 0.89 0.70 0.80 0.84 0.72 0.59 0.68 France 0.83 0.76 0.90 1 0.88 0.66 0.84 0.83 0.75 0.62 0.67 The Netherlands 0.85 0.75 0.89 0.88 1 0.73 0.77 0.93 0.75 0.45 0.66 Norway 0.48 0.84 0.70 0.66 0.73 1 0.79 0.75 0.83 0.55 0.76 Sweden 0.76 0.88 0.80 0.84 0.77 0.79 1 0.80 0.79 0.80 0.74 UK 0.86 0.79 0.84 0.83 0.93 0.75 0.80 1 0.78 0.42 0.65 Australia 0.58 0.81 0.72 0.75 0.75 0.83 0.79 0.78 1 0.63 0.83 Japan 0.44 0.67 0.59 0.62 0.45 0.55 0.80 0.42 0.63 1 0.67 Emerging Markets 0.52 0.78 0.68 0.67 0.66 0.76 0.74 0.65 0.83 0.67 1 J. Risk Financial Manag. 2020,13, 94 18 of 19 References Abel, Andrew B. 1990. 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