# The Investment Home Bias with Peer Effect

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Bivariate First-Degree Stochastic Dominance (BFSD) and the IHB

#### 3.1. The Sufficient Conditions for BFSD Implying the IHB Rationalization

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

**Example**

**1.**

#### 3.2. Discussion

## 4. The IHB with Some Specific KUJ Preferences

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

#### 4.2. The KUJ Preferences (for $\alpha >1$)

#### 4.3. Data and Results

## 5. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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 |

## Appendix B

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 |

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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. |

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. |

5 | Note 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), Galí (1994), Campbell and Cochrane (1999), Bolton and Ockenfels (2000), Dupor and Liu (2003), Zizzo (2003), and Demarzo et al. (2008). |

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 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. |

9 | 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). |

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 ${\Delta}_{i}=0$. 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 ${U}_{12}<0,$ in some range, one can always find a bivariate preference, such that outside this range the cross derivative is close to zero; hence, ${\Delta}_{i}$ is negative. Therefore, to guarantee that ${\Delta}_{i}$ is non-negative, the cross derivative cannot be negative. |

12 | Generally, if $F\left(x\right)\le G\left(x\right)$ for all values x and there is at least one strict inequality, we say that F dominates G by first degree stochastic dominance (FSD). |

13 | 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. |

**Table 1.**Joint Probability and Joint Cumulative Probability Functions Corresponding to the two Examples Given in the Text.

Correlation +1 | Correlation −1 | ||||
---|---|---|---|---|---|

Part (a): The First Example Given in the Text | |||||

The Probability Functions | |||||

${\mathit{F}}_{\mathit{A}}$ | ${\mathit{G}}_{\mathit{M}}$ | ||||

w$\backslash {w}_{p}$ | 3 | 4 | w$\backslash {w}_{p}$ | 3 | 4 |

2 | 0.5 | 0 | 2 | 0 | 0.5 |

5 | 0 | 0.5 | 5 | 0.5 | 0 |

The Cumulative Bivariate Probability Functions | |||||

w/${w}_{p}$ | 3 | 4 | w/${w}_{p}$ | 3 | 4 |

2 | 0.5 | 0.5 | 2 | 0 | 0.5 |

5 | 0.5 | 1 | 5 | 0.5 | 1 |

Part (b): The Second Example Given in the Text | |||||

The probability Functions | |||||

w$/{w}_{p}$ | 3 | 4 | w$/{w}_{p}$ | 3 | 4 |

2 | 0.5 | 0 | 2 | 0 | 0.5 |

5 | 0 | 0.5 | 10 | 0.5 | 0 |

The Cumulative Bivariate Probability Functions | |||||

w$/{w}_{p}$ | 3 | 4 | w$/{w}_{p}$ | 3 | 4 |

2 | 0.5 | 0.5 | 2 | 0 | 0.5 |

5 | 0.5 | 1 | 10 | 0.5 | 1 |

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

α | (CRRA Univariate Preference γ = 0) | 0.5 | 1 | 2 | |
---|---|---|---|---|---|

γ | |||||

1 * | 0.00 | 0.00 | 0.00 | 0.00 | |

2 | 0.49 | 0.39 | 0.29 | 0.13 | |

5 | 0.95 | 0.82 | 0.72 | 0.57 |

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**MDPI and ACS Style**

Levy, H.
The Investment Home Bias with Peer Effect. *J. Risk Financial Manag.* **2020**, *13*, 94.
https://doi.org/10.3390/jrfm13050094

**AMA Style**

Levy H.
The Investment Home Bias with Peer Effect. *Journal of Risk and Financial Management*. 2020; 13(5):94.
https://doi.org/10.3390/jrfm13050094

**Chicago/Turabian Style**

Levy, Haim.
2020. "The Investment Home Bias with Peer Effect" *Journal of Risk and Financial Management* 13, no. 5: 94.
https://doi.org/10.3390/jrfm13050094