# Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The DR Portfolio Optimization Strategy

#### 2.1. Model for Extreme Risks

#### 2.2. DR Strategy

#### 2.3. Estimation of DR Strategy

**Theorem**

**1.**

## 3. Preliminary Data Analysis of Stock Losses

#### 3.1. Data

#### 3.2. Independence

^{−1}; see Leadbetter et al. (2012). Formally, the extremal index is defined as follows. Let ${\left\{{X}_{n}\right\}}_{n\in \mathbb{Z}}$ be a strictly stationary time series and ${M}_{n}=max\{{X}_{1},{X}_{2},\dots ,{X}_{n}\}$. Let ${\left\{{\tilde{X}}_{n}\right\}}_{n\in \mathbb{Z}}$ be an i.i.d. sequence with same marginal distribution as ${\left\{{X}_{n}\right\}}_{n\in \mathbb{Z}}$ and ${\tilde{M}}_{n}=max\{{\tilde{X}}_{1},{\tilde{X}}_{2},\dots ,{\tilde{X}}_{n}\}$. For some norming sequences ${c}_{n}>0$ and ${d}_{n}$, if

#### 3.3. Heavy Tailedness

## 4. Empirical Study

#### 4.1. Analysis for All Stocks

#### 4.2. Analysis of Grouped Stocks

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof

**Proof**

**of Theorem 1.**

#### Appendix A.2. Tables and Figures

**Figure A1.**Portfolio values for grouped stocks with tail indices $\alpha \le 2.5$. (

**a**) Portfolio with $\alpha \le 2.5$ in the entire backtest period; (

**b**) portfolio with $\alpha \le 2.5$ in the crisis period.

**Table A1.**Performance comparisons for DR, EW, MV, ERI, MDP, and S&P 500 index for stocks with $\alpha \le 2.5$.

Metric | DR | EW | MV | ERI | MDP | S&P 500 |
---|---|---|---|---|---|---|

Cumulative return | 305.59% | 324.20% | 219.87% | 154.42% | 406.11% | 154.00% |

Annualized return | 9.12% | 9.43% | 7.52% | 5.99% | 10.64% | 5.98% |

Annualized Sharpe ratio | 0.5941 | 0.6070 | 0.6279 | 0.4928 | 0.7259 | 0.2849 |

Annualized ${\mathrm{STARR}}_{0.95}$ | 0.2512 | 0.2469 | 0.2601 | 0.2048 | 0.3028 | 0.1139 |

Maximum drawdown | 47.69% | 46.62% | 38.06% | 42.21% | 41.22% | 56.78% |

Concentration coefficient | 8.1558 | 84.0000 | 13.1616 | 7.1420 | 18.2519 | N/A |

Average turnover | 0.0590 | 0.0000 | 0.0474 | 0.0450 | 0.0279 | N/A |

CR with $\theta =0.1\%$ | 252.50% | 324.20% | 187.16% | 129.16% | 374.40% | N/A |

CR with $\theta =0.5\%$ | 100.97% | 324.20% | 86.49% | 50.77% | 266.19% | N/A |

CR with $\theta =1\%$ | −0.62% | 324.20% | 8.70% | −10.76% | 164.92% | N/A |

PCA | 11.27% | 38.34% | 28.32% | 17.65% | 14.74% | N/A |

Skewness | −0.2476 | −0.1166 | −0.8514 | −0.8160 | −0.4977 | −0.2556 |

Kurtosis | 8.4234 | 8.7792 | 11.6180 | 10.5831 | 7.0308 | 13.9341 |

**Figure A2.**Portfolio values with transaction cost for grouped stocks with tail indices $\alpha \le 2.5$ when $\theta =0.1\%,0.5\%,1\%$, ${\theta}_{fc}=1.5\theta $.

**Table A2.**Performance comparisons for DR, EW, MV, ERI, MDP, and S&P 500 index for stocks with $2.5<\alpha \le 3.5$.

Metric | DR | EW | MV | ERI | MDP | S&P 500 |
---|---|---|---|---|---|---|

Cumulative return | 742.46% | 388.84% | 189.04% | 284.35% | 545.97% | 154.00% |

Annualized return | 14.21% | 10.40% | 6.84% | 8.76% | 12.33% | 5.98% |

Annualized Sharpe ratio | 0.8510 | 0.6035 | 0.5980 | 0.7096 | 0.8258 | 0.2849 |

Annualized ${\mathrm{STARR}}_{0.95}$ | 0.3611 | 0.2477 | 0.2480 | 0.3046 | 0.3401 | 0.1139 |

Maximum drawdown | 46.62% | 52.52% | 38.50% | 42.16% | 51.96% | 56.78% |

Concentration coefficient | 7.8232 | 195.0000 | 16.4092 | 6.5321 | 18.6969 | N/A |

Average turnover | 0.0697 | 0.0000 | 0.0527 | 0.0473 | 0.0321 | N/A |

CR with $\theta =0.1\%$ | 618.14% | 388.84% | 157.13% | 244.67% | 500.34% | N/A |

CR with $\theta =0.5\%$ | 278.86% | 388.84% | 61.01% | 122.79% | 347.81% | N/A |

CR with $\theta =1\%$ | 70.01% | 388.84% | −10.33% | 28.98% | 210.39% | N/A |

PCA | 8.94% | 39.19% | 25.99% | 19.34% | 15.75% | N/A |

Skewness | −0.2755 | −0.0556 | −1.1798 | −0.5133 | −0.6143 | −0.2556 |

Kurtosis | 8.1260 | 7.7376 | 11.0068 | 4.5851 | 7.6877 | 13.9341 |

**Figure A4.**Portfolio values with transaction cost for grouped stocks with tail indices $2.5<\alpha \le 3.5$ when $\theta =0.1\%,0.5\%,1\%$, ${\theta}_{fc}=1.5\theta $.

**Table A3.**Performance comparisons for DR, EW, MV, ERI, MDP, and S&P 500 index for stocks with $\alpha >3.5$.

Metric | DR | EW | MV | ERI | MDP | S&P 500 |
---|---|---|---|---|---|---|

Cumulative return | 414.42% | 375.12% | 314.14% | 356.74% | 714.50% | 154.00% |

Annualized return | 10.75% | 10.20% | 9.26% | 9.93% | 13.97% | 5.98% |

Annualized Sharpe ratio | 0.5735 | 0.5575 | 0.6558 | 0.6698 | 0.7861 | 0.2849 |

Annualized ${\mathrm{STARR}}_{0.95}$ | 0.2505 | 0.2327 | 0.2683 | 0.2870 | 0.3326 | 0.1139 |

Maximum drawdown | 48.03% | 54.70% | 54.34% | 45.40% | 49.61% | 56.78% |

Concentration coefficient | 6.6431 | 82.0000 | 9.7144 | 5.7968 | 13.1654 | N/A |

Average turnover | 0.0510 | 0.0000 | 0.0376 | 0.0362 | 0.0221 | N/A |

CR with $\theta =0.1\%$ | 353.40% | 375.12% | 280.57% | 319.34% | 673.63% | N/A |

CR with $\theta =0.5\%$ | 173.39% | 375.12% | 171.35% | 197.86% | 529.65% | N/A |

CR with $\theta =1\%$ | 45.01% | 375.12% | 77.77% | 94.07% | 386.70% | N/A |

PCA | 22.79% | 42.73% | 29.79% | 23.27% | 35.33% | N/A |

Skewness | 0.3535 | 0.0349 | −0.8981 | −0.3410 | −0.3370 | −0.2556 |

Kurtosis | 10.2420 | 7.6200 | 10.6307 | 4.9313 | 6.1345 | 13.9341 |

**Figure A6.**Portfolio values with transaction cost for grouped stocks with tail indices $\alpha >3.5$ when $\theta =0.1\%,0.5\%,1\%$, ${\theta}_{fc}=1.5\theta $.

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**Figure 1.**Estimates of extremal index for stock losses with different frequencies. (

**a**) Daily losses; (

**b**) alternate-day losses; (

**c**) weekly losses.

**Figure 3.**The values of portfolios of 361 stocks under the strategies of DR, EW, MV, ERI, and MDP, and the values of S&P 500 index. The initial value for each portfolio is set as $100. (

**a**) The entire backtest period 2005–2020; (

**b**) the financial crisis of 2007–2009 period.

**Figure 6.**Portfolio value of 361 stocks with transaction cost $\theta =0.1\%,0.5\%,1\%$ and ${\theta}_{fc}=1.5\theta $.

**Figure 7.**Optimal weights under the strategies DR, MV, ERI, and MDP for the 361-stock portfolio over the backtest period. (

**a**) DR; (

**b**) MV; (

**c**) ERI; (

**d**) MDP.

**Figure 8.**Percentage of variance explained by first principal component for 361-Stock portfolio over the backtest period.

p-Value Interval | LB-Q(5) | Co-Skewness | Co-Kurtosis | PE |
---|---|---|---|---|

$(0,0.001]$ | 144 | 316 | 293 | 115 |

$(0.001,0.01]$ | 62 | 7 | 18 | 64 |

$(0.01,0.05]$ | 66 | 8 | 8 | 61 |

$(0.05,0.1]$ | 26 | 7 | 8 | 29 |

$(0.1,0.3]$ | 28 | 10 | 13 | 50 |

$(0.3,1.0]$ | 35 | 12 | 20 | 42 |

**Table 2.**Performance comparisons for DR, EW, MV, ERI, MDP, and S&P 500 index for the 361-stock portfolio.

Metric | DR | EW | MV | ERI | MDP | S&P 500 |
---|---|---|---|---|---|---|

Cumulative return (CR) | 747.92% | 373.02% | 190.77% | 268.63% | 564.54% | 154% |

Annualized return | 14.26% | 10.17% | 6.88% | 8.47% | 12.53% | 5.98% |

Annualized Sharpe ratio | 0.8203 | 0.5974 | 0.6057 | 0.6986 | 0.8423 | 0.2849 |

Annualized ${\mathrm{STARR}}_{0.95}$ | 0.3617 | 0.2452 | 0.2498 | 0.2918 | 0.3578 | 0.1139 |

Maximum drawdown | 43.79% | 51.64% | 39.15% | 35.51% | 49.08% | 56.78% |

Concentration coefficient | 8.6449 | 361.0000 | 21.8908 | 8.1329 | 21.7522 | N/A |

Average turnover | 0.0679 | 0.0000 | 0.05176 | 0.0543 | 0.0343 | N/A |

CR with $\theta =0.1\%$ | 622.21% | 373.02% | 158.30% | 224.51% | 514.36% | N/A |

CR with $\theta =0.5\%$ | 279.76% | 373.02% | 60.83% | 94.77% | 348.74% | N/A |

CR with $\theta =1\%$ | 69.71% | 373.02% | −11.06% | 2.76% | 202.96% | N/A |

PCA | 28.96% | 39.39% | 20.13% | 14.19% | 22.44% | N/A |

Skewness | 0.6555 | −0.0494 | −1.1730 | −0.7705 | −0.6376 | −0.2556 |

Kurtosis | 14.3580 | 7.9947 | 11.4814 | 6.2075 | 7.8981 | 13.9341 |

Metric | EW | MV | ERI | MDP |
---|---|---|---|---|

Cosine Similarity [−1, 1] | 0.1538347 | 0.1862669 | 0.2448010 | 0.4276556 |

Correlation Coefficient [−1, 1] | 0.8041764 | 0.7697190 | 0.7314304 | 0.8724533 |

Tail Index Groupings | Number of Stocks |
---|---|

$\alpha \le 2.5$ | 84 |

$2.5<\alpha \le 3.5$ | 195 |

$3.5<\alpha $ | 82 |

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**MDPI and ACS Style**

Mehta, N.J.; Yang, F.
Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis. *Risks* **2022**, *10*, 101.
https://doi.org/10.3390/risks10050101

**AMA Style**

Mehta NJ, Yang F.
Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis. *Risks*. 2022; 10(5):101.
https://doi.org/10.3390/risks10050101

**Chicago/Turabian Style**

Mehta, Navya Jayesh, and Fan Yang.
2022. "Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis" *Risks* 10, no. 5: 101.
https://doi.org/10.3390/risks10050101