## 1 Introduction

^{1}The problem has been studied in the mean-variance literature with several solutions proposed, including a robust optimization counterpart of the classical mean-variance model (Goldfarb and Iyengar 2003). Robust portfolio selection models have also been developed with higher order moments.

^{2}These works develop efficient frontiers that are robust to the data ambiguity. Tangency portfolios are used to identify a unique point on the frontier that maximizes a performance ratio such as Sharpe in mean-variance analysis (Sharpe 1994). In this paper, we advance the robust portfolio optimization literature (Mulvey et al. 1995) with a robust mean-to-CVaR (MtC) model for stable distributions (Farinelli et al. 2008; Rachev et al. 2008) with interval ambiguity in means and covariance following Ben-Tal et al. (2009).

^{3}

## 2 Robust MtC portfolio selection model

### 2.1 Preliminaries

^{4}:

### 2.2 Robust model formulation

## 3 Controlled experiments

^{7}We study the impact of ambiguity in means or standard deviations on asset allocations. When we consider ambiguity in the means, we solve model (7) where \(\Sigma _{-}=\Sigma _{+}\) and equal to the unambiguous covariance. For ambiguity in standard deviations, we solve the model with \(\bar{r}_{-}=\bar{r}_{+}\) and equal to the unambiguous mean returns.

### 3.1 Varying correlation

^{8}We solve robust MtC for varying shrinkage factors and display the allocation to asset A in Fig. 1. Panel A (column figures) shows results when the mean returns are ambiguous, with the standard deviations fixed at 20%. Panel B shows results with ambiguous standard deviations, with mean returns at 7%.

### 3.2 Varying ambiguity interval

^{9}

## 4 An application to the equity home bias puzzle

### 4.1 Data

^{10}The monthly data span from January 1999 to December 2019. We also construct two indices for a second example, using equally-weighted returns of an MSCI subset of emerging markets. We construct equally-weighted portfolios EME\(_1\) of Asian markets, and EME\(_2\) of European and Latin American markets.

^{11}

Country | Mean | Std | Skew | Kurt | SR | CVaR | MtC | Corr |
---|---|---|---|---|---|---|---|---|

Home | 0.06 | 0.15 | − 0.64 | 1.02 | 0.40 | 0.10 | 0.050 | – |

RoW | 0.09 | 0.19 | − 0.64 | 2.69 | 0.47 | 0.12 | 0.063 | 0.82 |

EME\(_1\) | 0.12 | 0.21 | − 0.23 | 2.01 | 0.57 | 0.13 | 0.077 | 0.75 |

EME\(_2\) | 0.12 | 0.22 | − 0.53 | 2.42 | 0.55 | 0.13 | 0.077 | 0.70 |

^{12}Then, we calculate the mean returns and covariance matrix for each sampled time series of returns and obtain the maximum ambiguity interval between the minimum and maximum values. In Table 2, we report the minimum, maximum, and size of the ambiguity intervals of the means (Panel A) and covariance (Panel B) for the US and RoW.

(A) Mean | (B) Covariance | ||||||||
---|---|---|---|---|---|---|---|---|---|

Min | Max | Size | |||||||

RoW | Home | RoW | Home | RoW | Home | RoW | Home | ||

Min | − 0.09 | − 0.08 | RoW | 0.02 | 0.01 | 0.07 | 0.05 | 0.05 | 0.04 |

Max | 0.27 | 0.19 | Home | 0.01 | 0.01 | 0.05 | 0.04 | 0.04 | 0.03 |

Size | 0.36 | 0.27 |

### 4.2 The US equity home bias puzzle

^{13}We use the ambiguity interval data from Table 2 and solve the robust model for varying values of the shrinkage factors of the US and RoW countries. We test whether the optimal asset allocations for ambiguity in means or covariance match the observed US home bias.