**2. Literature Review**

Vanpée 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, ye<sup>t</sup> 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 e ffect. However, with the increase in the rapid flow of information and market e fficiency 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).

<sup>1</sup> 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.

<sup>2</sup> 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 sugges<sup>t</sup> a macroeconomic model (related to the positive association between governmen<sup>t</sup> spending and return on local stocks) to explain the home bias.

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 e ffect of each economic factor that is considered as a barrier for e fficient 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.<sup>3</sup>

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 di fferences 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 e ffect, 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.<sup>4</sup> 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 e ffect.

<sup>3</sup> 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.

<sup>4</sup> 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.

### **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 Su*ffi*cient Conditions for BFSD Implying the IHB Rationalization*

Consider an individual with a bivariate preference *<sup>U</sup>*(*<sup>w</sup>*, *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, *F* and *G*, where the domestic investment weight in portfolio *F* is larger than the domestic investment weight in portfolio *G* (we will elaborate later on the selected portfolios, *F* and *G*). Our aim is to examine the conditions under which *F* dominates *G* by BFSD with the above bivariate utility function, where we first assume two assumptions on the preferences: <sup>∂</sup>*U<sup>w</sup>*, *wp*/∂*w* ≡ *U*1 ≥ 0 (monotonicity) and <sup>∂</sup><sup>2</sup>*U<sup>w</sup>*, *wp*/∂*w*∂*wp* ≡ *U*12 ≥ 0 (later on we consider also *U*12 ≤ 0, a case usually not considered in KUJ economic research but emerges as important to our analysis). There is no constraint on the derivative <sup>∂</sup>*U<sup>w</sup>*, *wp*/∂*wp* ≡ *U*2, which can be negative, zero, or positive.<sup>5</sup> If such dominance exists, then all investors, regardless of the precise shape of the bivariate preference, will switch from *G* to *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 (*<sup>U</sup>*12) is positive, implying that the individual's marginal utility increases with an increase in the peer group wealth (see Ljungqvist and Uhlig 2000).<sup>6</sup> 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.

<sup>5</sup> Note that a negative sign implies jealousy, and a positive sign implies altruism (see Dupor and Liu 2003).

<sup>6</sup> Numerous studies sugges<sup>t</sup> 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), Galí (1994), Campbell and Cochrane (1999), Bolton and Ockenfels (2000), Dupor and Liu (2003), Zizzo (2003), and Demarzo et al. (2008).

**Proposition 1.** *Suppose that the investor faces two alternate bivariate prospects, <sup>F</sup>*(*<sup>w</sup>*, *wp*) *and <sup>G</sup>*(*<sup>w</sup>*, *wp*)*, where w as well as wp can take only two di*ff*erent 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 U*12 ≥ 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:

$$\mathbb{CL} \Leftrightarrow \mathbb{U}\_{12} > 0 \tag{1}$$

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 ge<sup>t</sup> only two possible values. Hence, by reordering these values, the marginal distributions are kept unchanged. Also, diversification between *w* and *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, *F* and *G*, have bivariate density functions, denoted by *f*(*<sup>w</sup>*, *wP*) and *g*(*<sup>w</sup>*, *wP*), respectively. As we focus on the possible IHB rationalization, it is assumed, as explained before, that *F* stands 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 *G* is equal to the optimal theoretical univariate expected utility maximization domestic weight (say, the international market portfolio), the BFSD of *F* over *G* implies that the peer effect rationalizes the IHB phenomenon, as all investors would prefer *F* over *G*.<sup>7</sup> Assuming that <sup>∂</sup><sup>2</sup>w, *wp*/∂w∂*wp*) ≡ *U*12 > 0 with KUJ preference<sup>8</sup> 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 *F* and *G* is given by:

$$\begin{aligned} E\_F \mathcal{U} &= \int\_{\underline{\text{uv}}}^{\overline{\text{uv}}} \int\_{\underline{\text{uv}}\_p}^{\overline{\text{uv}}\_p} \mathcal{U}(w, w\_p) f(w, w\_p) dudw\_p \\ E\_G \mathcal{U} &= \int\_{\underline{\text{uv}}}^{\overline{\text{uv}}} \int\_{\underline{\text{uv}}\_p}^{\overline{\text{uv}}\_p} \mathcal{U}(w, w\_p) g(w, w\_p) dudw\_p \end{aligned} \tag{2}$$

where *w* and *w* denote the minimal and maximal values of *w* (which can be −∞ and ∞); similarly, *wP* and *wP* denote the minimal and maximal values of *wP*. Thus,

$$\Delta\_l \equiv \, \_FL \\ \mathcal{I} - E\_G \\ \mathcal{U} = \int\_{\underline{w}}^{\overline{w}} \int\_{\underline{w}\_p}^{\overline{w}\_p} \mathcal{U}(w, w\_p) [f(w, w\_p) - g(w, w\_p)] dw dw\_p$$

Integrating by parts the above equation with respect to both variables yields:

$$\begin{aligned} \Delta\_i &\equiv E\_F \mathcal{U} - E\_G \mathcal{U} = \int\_{\underline{w}\_P}^{\overline{w}\_p} \underline{\int}^{\overline{w}} \mathcal{U}\_{12} [F(w, w\_p) - G(w, w\_p)] dw dw\_p + \int\_{\underline{w}}^{\overline{w}} \mathcal{U}\_1 [G(w) - G(w, w\_p)] dw dw \\ F(w) [dw + \int\_{\underline{w}\_P}^{\overline{w}\_p} \mathcal{U}\_2 [G(w\_p) - F(w\_P)] dw\_p \\ &\equiv A + B + C \end{aligned} \tag{3}$$

where Δ*i* denotes the expected utility difference corresponding to the *i*th investor, *<sup>F</sup>*(*<sup>w</sup>*, *wP*) and *<sup>G</sup>*(*<sup>w</sup>*, *wP*) are the two bivariate cumulative distributions, *<sup>F</sup>*(*w*) = *<sup>F</sup><sup>w</sup>*, *wp* is the marginal cumulative distribution function of *w*, *<sup>F</sup>wp* = *<sup>F</sup><sup>w</sup>*, *wp* is the marginal cumulative distribution function of *wP*, and *U*1, *U*2, and *U*12 denote the partial derivatives: *U*1 ≡ ∂*U*/∂*<sup>w</sup>*, *U*2 ≡ ∂*U*/∂*wP*, and *U*12 ≡ ∂<sup>2</sup>*U*/∂*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).<sup>9</sup> 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 *<sup>G</sup>wp* = *<sup>F</sup>wp*, therefore term *C* in Equation (3) is equal to zero. Thus, the rest of the paper relate only to terms *A* and *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 *w* and *wp* is not allowed, the marginal distributions are equal (namely, *<sup>G</sup>*(*w*) = *<sup>F</sup>*(*w*), see also the example given in

<sup>7</sup> 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.

<sup>8</sup> The KUJ and CUJ literature is very extensive; hence, we mention here only a few of these studies. Abel (1990) and Galí (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.

<sup>9</sup> Note that Equation (2) is reduced to the well-known univariate formula employed to derive the FSD rule, where *U*12 = *U*2 = 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).

Table 1. Hence, in this specific case also term *B* is equal to zero, and we are left with term *A*. If *F* represents the +1 correlation and *G* the −1 correlation, we must have with the two outcomes case that *<sup>F</sup><sup>w</sup>*, *wp* ≥ *<sup>G</sup><sup>w</sup>*, *wp* (see example 1 in Table 1. Hence, in this case by Equation (3), with *B* = *C* = 0, the condition *U*12 ≥ 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)).<sup>10</sup> It is easy to verify that, in this specific case, *U*12 ≥ 0 is a necessary and sufficient condition for dominance.<sup>11</sup> 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 *B* is not necessarily equal to zero. As we shall see in this general and realistic case, *U*12 ≥ 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 *F* and *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 *U*2 is irrelevant. Namely, term *C* = 0 and there is no need to assume jealousy (*<sup>U</sup>*2 < 0) or altruism (*<sup>U</sup>*2 > 0) to obtain our results corresponding to the portfolio investment case. Thus, as for the analysis of the IHB, term *C* is equal to zero, and Equation (3) is reduced to:

$$
\Delta\_{\bar{l}} = A + B.\tag{4}
$$

However, note that generally we cannot assume that also term *B* is 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 *B* of 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 *F* and *G* under 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 *FA* and *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%.*

<sup>10</sup> 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 Δ*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.

<sup>11</sup> If *U*12 < 0, in some range, one can always find a bivariate preference, such that outside this range the cross derivative is close to zero; hence, Δ*i* is negative. Therefore, to guarantee that Δ*i* is non-negative, the cross derivative cannot be negative.

Assuming that term *C* is equal to zero, and rewriting Equation (4) in terms of the above two portfolios, we obtain:

$$\begin{aligned} \Delta\_i &\equiv E\_{F\_A} \mathcal{U} - E\_{G\_M} \mathcal{U} = \int\_{\underline{w}}^{\overline{w}\_p} \int\_{\underline{w}}^{\overline{w}} \mathcal{U}\_{12} [F\_A(w, w\_p) - G\_M(w, w\_p)] dw dw\_p \\ &+ \int\_{\underline{w}}^{\overline{w}} \mathcal{U}\_1 [G\_M(w) - F\_A(w)] dw \equiv A + B \end{aligned} \tag{5}$$

Suppose that without the peer e ffect the market portfolio is optimal. If Δ*i* > 0, the *i*th investor under consideration who considers also the peer e ffect 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 Δ*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 *U*12 ≥ 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 *U*12 ≥ 0, term *A* of Equation (5) is positive only if *FA w*, *wp* ≥ *G M w*, *wp* , but this implies that *FA*(*<sup>w</sup>*, ∞) = *FA*(*w*) ≥ *<sup>G</sup>*(*<sup>w</sup>*, ∞) = *<sup>G</sup>*(*w*), and therefore, term *B* is negative. The sum *A* + *B* may be negative, implying that there is no BFSD.

Surprisingly, in contrast to Proposition 1, the condition *U*12 ≤ 0 may allow BFSD; hence, it may allow IHB rationalization. We have BFSD and IHB rationalization with *U*12 ≤ 0 if the following two conditions hold:

$$F\_{\mathcal{A}}(w, w\_p) \le G\_{\mathcal{M}}(w, w\_p) \tag{6a}$$

$$F\_A(w) \le G\_M(w) \tag{69}$$

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 *A* and *B* are positive (with *U*12 < 0) and therefore Δ*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 *F* or portfolio *G*, both yielding return *w* of either 2 or 5 with equal probability of 0.5. However, *F* has 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). *G* has 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 *G M*, we have with the above example with the joint probabilities the following relationship:

$$F\_A(w, w\_{\mathcal{P}}) \ge G\_M(w, w\_{\mathcal{P}}) \text{ for all values } \{w, w\_{\mathcal{P}}\} \tag{7}$$

with at least one strict inequality (see lower part of Table 1 Part a), e.g.,

$$F\_A(\mathbf{2}, \mathbf{3}) = 0.5 > G\_M(\mathbf{2}, \mathbf{3}) = 0,$$

and it is easy to verify that with the marginal distributions, we have:

$$G\_M(w) = F\_A(w)$$

and

$$G\_M(w\_p) = F\_A(w\_p)$$

for all possible values(all marginal cumulative distributions ge<sup>t</sup> the values of 0.5 at the lower outcome and 1 at the larger outcome). Therefore, the conditions of Proposition 1 are intact and terms *B* and *C* of Equation (3) are equal to zero and we are left only with term *A*; hence, if the cross derivative is positive, the joint distribution yielding (2, 3) and (5, 4) dominates the joint distribution (2, 4) and (5, 3) for all bivariate preferences with a positive cross derivative. Thus, term *A* of Equation (3) is positive, and term *B* is equal to zero; hence, we have BFSD of the joint distribution with correlation +1 over the joint distribution with correlation −1. So far, this example conforms to the conditions given in Proposition 1 and provides the IHB rationalization by adding the peer effect, so long as the cross derivative is positive. Let's turn to another example, where one of the conditions of Proposition 1 is violated.



Make now the following change in the previous example: the outcomes with the −1 correlation are 2 and 10 with equal probability of 0.5 rather than 2 and 5 as we have in the previous example. Thus, we simply replaced the outcome 5 by outcome 10 (see Table 1 Part b). All the other outcomes are kept unchanged. Thus, under the choice with a correlation of −1, we have the joint outcomes of (2, 4) or (10, 3) with an equal probability of 0.5, and with the choice corresponding to correlation +1, we have as before the joint outcomes (2, 3) or the outcomes (5, 4) with an equal probability of 0.5. It is easy to verify that as before, also with this change we have:

$$F\_A(w, w\_p) \ge G\_M(w, w\_p) \tag{8}$$

with at least one strict inequality, e.g.,

$$F\_A(\mathfrak{2}, \mathfrak{3}) = 0.5 > G\_M(\mathfrak{2}, \mathfrak{3}) = 0.1$$

Hence, if the cross derivative is positive, term *A* of Equation (5) is also positive. However, with this change, the marginal distribution of the individual wealth also changes, and as cumulative distribution of (2, 10) is located to the right of the cumulative distribution of (2, 5), namely, *G M*(*w*) ≤ *FA*(*w*) for all values w, and there is at least one strict inequality, e.g., *G M*(*<sup>w</sup>* = 5) = 0.5 > *FA*(*w* = 5) = 1, see Table 1 Part b (this means that the univariate prospect *A* dominates prospect M by FSD12); hence, term *B* of Equation (5) is negative. Therefore, *A* + *B* may be positive, zero, or negative, and we do not have BFSD, and as a result, the IHB cannot be rationalized in this case, despite the assumed positive cross derivative. Actually, it is easy to find a specific bivariate preference with *U*12 → 0 and *U*1 very large such that *A* + *B* < 0, implying that *F* does not dominate *G*.

Finally, even if we stick to the original set of numbers by diversification of say, 0.5 in each asset (*w* and *wp*), the marginal distribution of the individual's wealth becomes either 2.5 = 0.5 × 2 + 0.5 × 3 or 4.5 = 0.5 × 5 + 0.5 × 4 (with equal probability) in the case of a correlation of +1, and either 3 = 0.5 × 2 + 0.5 × 4 or 4 = 0.5 × 5 + 0.5 × 3 (with an equal probability) in the case of correlation of −1 (see Table 1 Part a). Thus, allowing diversification between the two variables the marginal distribution of w is not kept constant, and the conditions required by Proposition 1 are violated, and once again, we do not have BSFD, even where the cross derivative is positive. Moreover, by changing the investment weights, we change the marginal distribution of *w*. Hence, we do not have BFSD of the +1 correlation joint distribution over the −1 correlation joint distribution.

In sum, with the first example, we have IHB rationalization if diversification is not allowed. In the second example, we do not have IHB, even if diversification is not allowed, let alone if it is allowed. We show below that the case given in Table 1 Part b, where the marginal distributions are not kept constant, conforms to actual stock market data; therefore, KUJ with a positive cross derivative does not rationalize empirically the IHB phenomenon.
