*3.2. Discussion*

We analyze above the dominance condition between two specific portfolios, one with an IHB and one without an IHB. If indeed these are the two portfolios considered by all investors, assuming a positive cross derivative, we do not have dominance of the portfolio with the IHB over the portfolio with no IHB. Thus, a positive cross derivative is not a su fficient condition for BFSD. However, with a negative cross derivative, we may have dominance and, therefore, may obtain IHB rationalization by incorporating the peer e ffect. The existence of such IHB rationalization depends on the joint distribution of the investor's selected portfolio and the peer's portfolio. If all investors consider the same two portfolios (the market portfolio and the actual portfolio defined above), and if the condition *FA w*, *wp* ≤ *G M w*, *wp* holds, we have IHB rationalization, so long as the cross derivative is negative. However, generally there is no reason to assume that with actual data this condition is intact. Therefore, we generally do not have IHB rationalization with the above two portfolios, regardless of the sign of the cross derivative.

<sup>12</sup> Generally, if *<sup>F</sup>*(*x*) ≤ *<sup>G</sup>*(*x*) for all values x and there is at least one strict inequality, we say that *F* dominates G by first degree stochastic dominance (FSD).

In the above analysis, we consider two specific portfolios. In practice, not all investors hold these two portfolios, and each investor has her optimal univariate portfolio and her optimal bivariate portfolio. In this realistic case, only if we have for all investors the optimal domestic investment weight whereby the bivariate utility function is larger than the optimal domestic investment weight corresponding to the univariate framework, we have an IHB rationalization. As such case cannot be analyzed without the information on the preferences of all investors, we analyze below the peer e ffect on the IHB with the commonly employed KUJ preferences.

### **4. The IHB with Some Specific KUJ Preferences**

We proved above that unless the marginal distributions are kept unchanged with a positive cross derivative, we do not have BFSD, and with a negative cross derivative, the required condition is unlikely to hold; therefore, generally we do not have IHB rationalization, regardless of the sign of the cross derivative. However, it is possible that, despite not having dominance which corresponds to all possible bivariate preferences, for some important and commonly employed KUJ preferences (but not for all bivariate preferences, as we do not have BFSD), with various risk aversion parameters, the peer effect may induce an increase in the domestic investment weight relative to the univariate optimal domestic investment weight. As we shall see below, this is not the case and also the KUJ preferences empirically do no rationalize the IHB.

The procedure for testing the peer e ffect with KUJ preferences is as follows: We first solve with actual data for the optimum diversification with a univariate preference for various degrees of risk aversion and then solve for the optimum diversification with a KUJ preference with a similar form of the univariate preference but with the incorporation of the peer e ffect. Using this two-step analysis, we measure the marginal peer group e ffect on the optimal domestic investment weights. It is shown below empirically that, with the commonly employed KUJ preference with a positive cross derivative, adding the peer group e ffect does not rationalize the home bias phenomenon. Moreover, the optimal domestic investment weight with the KUJ preference (with a domestic peer group) is found to be even smaller than the optimal domestic investment weight with no peer group e ffect. Thus, despite the assumed positive cross derivative, which under some conditions implies CL, we find empirically that, counter intuitively, the IHB phenomenon is even enhanced when the peer e ffect is incorporated. Thus, shifting from univariate utility function to the bivariate utility function with peer e ffect does not rationalize the IHB and other explanations are called for. Let us elaborate.

### *4.1. The Optimal Diversification with a Univariate Utility*

The empirical analysis given in this section is done from the American investor's point of view. Thus, all the returns are in dollar terms. Similar analysis can be done from other points of view.

With a univariate CRRA utility function, we have:

$$\mathcal{U}(w) = [w^{1-\alpha}/(1-\alpha)]\_{\prime} \text{ with} \\ \alpha > 0. \tag{9}$$

With *n* available international assets, we solve for the vector of the investment proportions *p* (where 0 ≤ *pi* ≤ 1), which maximizes the following expected utility:

$$E[\left(p\_{lIS}y\_{lIS} + p\_2y\_2 + \dots .(1 - \sum\_{i=1}^{n-1} p\_n)y\_n\right)^{1-\alpha}/(1-\alpha)]\tag{10}$$

where *yUS* is the rate of return on the US index (a random variable), and *yi* is the return on foreign market *i*, where *i* = 2, 3, ..... *n*, and there are *n* stock indices, the US index plus *n* − 1 foreign indices. Let us specify the corresponding bivariate preferences employed in this study.

### *4.2. The KUJ Preferences (for* α > 1*)*

We discuss below various bivariate functions with a positive cross-derivative, which are natural extensions of the myopic univariate reference. We first employ the bivariate utility suggested by Abel (1990). As this function has a positive cross derivative only for α > 1, we also employ a slightly different preference that conforms to the KUJ requirement for the whole range of the risk aversion parameter (α > <sup>0</sup>). Abel suggests a general bivariate formula that, under some conditions for the various parameters, reduces to:

$$\mathcal{L}(w, w\_P) = \left[ w / w\_P^\circ \right]^{1-\alpha} / (1-\alpha) \text{ where} \alpha > 0 \text{ and} \mathbf{y} \ge 0. \tag{11}$$

With γ = 0 this function is reduced to the univariate CRRA function. Hence, this is a neutral extension to the univariate myopic function. Thus, this bivariate preference is a natural extension of the univariate CRRA function, *<sup>U</sup>*(*w*)=[*w*1−<sup>α</sup>/(<sup>1</sup> <sup>−</sup><sup>α</sup>)], which is commonly employed in economic research (see (Merton and Samuelson 1974)) to the bivariate case, in which the peer group effect is incorporated into the analysis. Note also that, when *wp* is constant, Equation (11) collapses to Equation (9). As we wish to have monotonicity with regard to *w* and a positive cross derivative (because with positive cross derivative there is an appealing intuition for an increase in the domestic investment weight), let us examine these two derivatives. Monotonicity always exists with this function, because:

$$\mathcal{U}\_1 = \left(w/w\_P^\vee\right)^{-\alpha} w\_P^{-\gamma} = w^{-\alpha} w\_P^{-\gamma(1-\alpha)} \ge 0.$$

The cross derivative, which is given by:

$$\mathcal{U}\_{12} = -\gamma (1 - \alpha) w^{-\alpha} w\_P^{-\gamma (1 - \alpha) - 1} \mathcal{I}$$

is positive only for α > 1. Thus, we have a KUJ preference with a positive cross-correlation for α > 1. It is worth mentioning that with this function we also have jealousy, namely, *U*2 = <sup>∂</sup>*U*/∂*wp* < 0.

Generally, in economics, the range of the risk aversion parameter is α ≥ 0. To avoid the above constraint on α (namely α > 1), we also employ the following function:

$$\mathcal{U}(w, w\_P) = [w^{1-\alpha}/(1-\alpha)][w\_P^{1-\beta}/(1-\beta)] \text{ with} \beta < 1. \tag{12}$$

With this preference, the cross derivative is positive for all values α > 0, as required by the myopic preference. Once again, when *wp* is constant, this function collapses to the univariate myopic preference (multiplied by a positive constant). The partial derivative with respect to *w* of this bivariate function, given by ∂*U*/∂*w* ≡ *U*1 = *w*<sup>−</sup>α*w*1−<sup>β</sup> *P* /(1 − β) ≥ 0, is positive only for β ≤ 1 and for any value α. As we assume monotonicity in the numerical solution to the optimum investment, we take the β ≤ 1 constraint into account.

The cross-derivative is positive for the whole range of relevant parameters:

$$
\partial^2 \mathcal{U} / \partial w \partial w\_P \equiv \mathcal{U}\_{12} = w^{-\alpha} w\_P^{-\beta} > 0.
$$

It is easy to see that, with this function, *U*2 may be negative, zero, or positive, depending on whether α is larger than, equal to, or smaller than 1. Therefore, with this function, we allow for jealousy as well as altruism.

Let us write down the equations employed in the derivation of the optimal investment weights with the various preferences. With *n* available international assets, we solve for the vector *p* (where 0 ≤ *pi* ≤ 1), which maximizes the expected utility. For the preference suggested by Abel (see Equation (11)), we have:

$$E(p\_{us}y\_{us} + p\_2y\_2 + \dots .(1 - \sum\_{i=1}^{n-1} p\_n)y\_n] / (w\_p^\vee) \Big|^{1-\alpha} / (1-\alpha) \Big|\Big) \tag{13}$$

and with the KUJ preference given by Equation (12) we have:

$$E[\left[\left(p\_{\rm hs}y\_{\rm hs} + p\_2y\_2 + \ldots \right)1 - \sum\_{i=1}^{n-1} p\_n\right) y\_n]^{1-a} / \left(1 - a\right) [\left[y\_{\rm hs}\right]^{1-\beta} / \left(1 - \beta\right)] \},\tag{14}$$

where *yUS* is the return on the US index (a random variable), and *yi* is the return on foreign market *i*, where *i* = 2, 3, ..... *n*, and there are *n* stock indices, the US index plus *n* − 1 foreign indices.

### *4.3. Data and Results*

The purpose of the empirical analysis given below is to examine the peer e ffect with some specific preferences on the IHB phenomena with actual data in the case where BFSD does not necessarily exist. It is not intended to find the optimal investment portfolio for investment, a case where statistical adjustment should be made in the empirical distribution, which is very noisy

We employ the 11 countries' actual *ex-post* joint distribution to analyze the peer group e ffect on the domestic investment weight from the American investor's point of view.<sup>13</sup> Obviously, the empirical distributions of return of all countries are not identical, and the marginal distribution of the various diversified portfolios under consideration are not identical; hence, the conditions of Proposition 1 are not intact; hence, the BFSD does not exist with positive cross derivative. Yet, it is possible that, with some specific bivariate preferences, the domestic investment weight increases. Thus, we examine whether the above two specific KUJ preferences with a positive cross derivative explain the IHB, despite the fact that we do not have BFSD. We use in the empirical study annual rates of return taken from the Bloomberg MSCI stock indices.

We report the optimum investment weights corresponding to Abel's bivariate preference function (see Equation (13)). The results corresponding to the bivariate function given by Equation (14) are very similar, so for brevity sake they are not reported here in detail. Both forms of the bivariate functions reveal the same surprising results: the commonly employed KUJ preference with a positive cross derivative induces a decrease rather than an increase in the US investment weight. Thus, the adding the peer e ffect does not rationalize the IHB.

Table 2 reports the results obtained using Abel's preference (see Equation (13)). The table employs the actual 25-year historical data (see Appendix A for the annual rates of return of the 11 countries for the years 1988–2012 and Appendix B for the correlation matrix). Recalling that, for γ = 0, the bivariate function is reduced to the univariate CRRA function, we see that, by adding the KUJ peer group e ffect with the domestic peer group, the US investment weight decreases quite sharply.


**Table 2.** The US optimal investment weights in the US market with Abel's bivariate preferences with empirical data for the period 1988–2012.

\* Only for α > 1 do we have a KUJ preference with *U*12 > 0. We also have with this function *U*1 > 0 (monotonicity) and *U*2 < 0 (jealousy).

<sup>13</sup> We are aware of the large potential statistical errors involved in the derivation of the optimal investment weight with historical data (see Britten-Jones 1999; Levy and Roll 2010). However, the goal of this empirical analysis is not to derive the optimal investment weights for *ex-ante* investment purposes but rather to demonstrate that, with empirical data covering 11 international markets and 25 years, it is possible that, with some commonly employed KUJ preferences with a positive cross derivative, the peer group effect induces a decrease rather than an increase in the domestic investment weight, which is in contradiction to the equal marginal distribution case and to the economic intuition.

Specifically, with the 11 countries for α = 2, we find with no peer group e ffect (a univariate utility function) that the optimal investment weight in the US is 0.49. Employing Equation (13) (Abel's preference), we find that the domestic investment in the US decreases rather than increases. For example, for α = 2 and γ = 1, the US domestic weight decreases from 0.49 to 0.29. The same is true for the other KUJ preference suggested in this study. Employing Equation (14), we find that with the KUJ domestic peer group e ffect, for α = 2 and β = 0.5, the optimal investment weight in the US decreases from 0.49 with no peer group e ffect to 0.39 with the peer group e ffect (this figure is not reported in a table). Thus, with the KUJ preference with a positive cross derivative, adding the domestic peer group e ffect decreases rather than increases the optimal US investment weight. Hence, with these specific KUJ preferences, we cannot rationalize the home bias empirically, which confirms the assertion that a positive cross derivative and CL are generally not equivalent. With a positive cross derivative, the investor wishes to increase the correlation by increasing the domestic investment weight but may not do it because other parameters (e.g., mean return) may induce a decrease in the bivariate expected utility by such action. Thus, if all investors have CRRA preference with various risk aversion parameters, the IHB is even intensified with the peer e ffect, despite the fact that a positive cross derivative implies that, other things being held unchanged, the American investor wants her portfolio to be as close as possible to the S&P stock index.

We turn now to examine whether incorporating the peer e ffect with a negative cross derivative

may increase the optimal domestic investment weight. We employ the function [ *w*1−<sup>α</sup> 1−<sup>α</sup> ]/[ *w*1−<sup>β</sup> *p* 1−β ] (with α < 1 and β < 1). It is easy to verify that the derivative with respect to w is positive (monotonicity) and that the cross derivative is negative. Once again, counter intuitively, we find that with this bivariate utility function with a negative cross derivative, the optimal domestic investment weight of the American investor in the US increases rather than decreases, due to the peer e ffect. For example, for α = 2 and γ = 0.5, we obtain with this function that the domestic investment weight increases from 0.49 to 0.54. Therefore, we conclude that, with historical data, a positive cross derivative is neither su fficient (see Proposition 1) nor necessary (as demonstrated empirically with a preference with a negative cross derivative) for IHB rationalization.
