# On the Realized Risk of Foreign Exchange Rates: A Fractal Perspective

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## Abstract

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“There are strong pragmatic reasons to begin the study of economic distributions and time series by those that satisfy the law of Pareto.”(Benoit Mandelbrot, mathematician known for the Theory of Roughness and Fractals)

## 1. Introduction

## 2. Data

## 3. Methodology

#### 3.1. Estimating Power Law Exponents Using Maximum Likelihood Estimation

_{PL}denotes the percentage of sample observations governed by a power law process. The last column reports the 95% confidence interval (CI) intervals for the estimated power law exponents.

#### 3.2. Testing the Power Law Model

#### 3.3. An Empirical Comparison of Distributions

#### 3.4. Robustness Checks

## 4. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Panel A | ||||||

Distribution | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | ${\mathit{x}}_{\mathit{M}\mathit{I}\mathit{N}}$ | $\mathit{D}$ | N_{PL} | 95% CI |

AUD/USD | 2.4139 | 0.1666 | 6.2002 | 0.0545 | 15.65% | [2.0873, 2.7405] |

EUR/USD | 2.4325 | 0.0934 | 1.3364 | 0.0691 | 51.09% | [2.2493, 2.6156] |

GBP/USD | 2.5637 | 0.1196 | 1.4831 | 0.0416 | 37.17% | [2.3293, 2.7981] |

NOK/USD | 3.2158 | 0.2432 | 6.0219 | 0.0553 | 18.04% | [2.7391, 3.6926] |

NZD/USD | 2.4391 | 0.1085 | 3.4412 | 0.0582 | 38.26% | [2.2265, 2.6518] |

USD/CAD | 3.0643 | 0.2522 | 3.8276 | 0.0632 | 14.57% | [2.5700, 3.5586] |

USD/CHF | 2.7203 | 0.1640 | 2.9686 | 0.0462 | 23.91% | [2.3989, 3.0418] |

USD/JPY | 2.4901 | 0.1268 | 2.3796 | 0.0452 | 30.00% | [2.2415, 2.7387] |

USD/SEK | 2.3217 | 0.1058 | 3.2904 | 0.0736 | 33.91% | [2.1143, 2.5291] |

Panel B | ||||||

Distribution | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | ${\mathit{x}}_{\mathit{M}\mathit{I}\mathit{N}}$ | $\mathit{D}$ | N_{PL} | 95% CI |

AUD/USD | 3.3773 | 0.2389 | 3.0557 | 0.0453 | 21.52% | [2.9090, 3.8456] |

EUR/USD | 2.7133 | 0.1342 | 1.1669 | 0.0481 | 35.43% | [2.4503, 2.9764] |

GBP/USD | 2.7723 | 0.1682 | 2.0690 | 0.0476 | 24.13% | [2.4426, 3.1020] |

NOK/USD | 2.8486 | 0.1591 | 3.1635 | 0.0389 | 29.34% | [2.5367, 3.1604] |

NZD/USD | 3.1287 | 0.2139 | 2.9721 | 0.0732 | 21.52% | [2.7094, 3.5481] |

USD/CAD | 3.0508 | 0.1772 | 1.2762 | 0.0650 | 29.13% | [2.7036, 3.3981] |

USD/CHF | 3.0085 | 0.1804 | 1.3947 | 0.0474 | 26.95% | [2.6550, 3.3620] |

USD/JPY | 2.8305 | 0.2538 | 3.2205 | 0.0528 | 11.30% | [2.3330, 3.3281] |

USD/SEK | 3.4147 | 0.2491 | 3.0119 | 0.0689 | 20.43% | [2.9266, 3.9029] |

_{PL}denotes the percentage of sample observations governed by a power law process. The last column reports the 95% confidence interval (CI) intervals for the estimated power law exponents.

## Notes

1 | See https://www.statista.com/statistics/247328/activity-per-trading-day-on-the-global-currency-market/ (accessed on 1 January 2023). |

2 | Other related studies are Alexander (1995) and Bauwens et al. (2005), who used (G)ARCH-type models to explore common volatility in the foreign exchange rate market as well as the impact of scheduled and unscheduled news announcements on foreign exchange rate return volatility. |

3 | Note that for the sake of readability, we skip in our notation here the index j denoting the week. |

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AUD/USD | EUR/USD | GBP/USD | NOK/USD | NZD/USD | USD/CAD | USD/CHF | USD/JPY | USD/SEK | |
---|---|---|---|---|---|---|---|---|---|

Mean | 3.3170 | 1.6138 | 1.8063 | 3.2683 | 3.2357 | 1.5839 | 2.1195 | 1.9851 | 2.9710 |

Median | 1.6427 | 1.0225 | 1.1017 | 1.9579 | 1.8758 | 0.9303 | 1.0874 | 1.0876 | 1.7309 |

Std.Dev. | 8.7209 | 1.9674 | 3.3995 | 4.8487 | 4.8140 | 2.3482 | 9.2697 | 3.1163 | 4.0658 |

Kurtosis | 187.8401 | 27.0404 | 289.4048 | 90.3521 | 91.2411 | 64.0870 | 618.1327 | 57.6324 | 25.8182 |

Skewness | 12.3553 | 4.1499 | 13.9099 | 7.5595 | 7.2678 | 6.5452 | 23.5305 | 5.7512 | 4.3111 |

Minimum | 0.0676 | 0.0333 | 0.0427 | 0.0398 | 0.0641 | 0.0230 | 0.0081 | 0.0155 | 0.0870 |

Maximum | 161.8294 | 21.9666 | 78.6839 | 77.0604 | 82.8622 | 31.4039 | 255.0862 | 47.5975 | 39.5884 |

Obs | 920 | 920 | 920 | 920 | 920 | 920 | 920 | 920 | 920 |

Distribution | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | ${\mathit{x}}_{\mathit{M}\mathit{I}\mathit{N}}$ | $\mathit{D}$ | N_{PL} | 95% CI |
---|---|---|---|---|---|---|

AUD/USD | 2.4834 | 0.0911 | 2.9452 | 0.0242 | 28.80% | [2.3048, 2.6620] |

EUR/USD | 3.5968 | 0.3126 | 4.3242 | 0.0497 | 7.50% | [2.9841, 4.2095] |

GBP/USD | 2.6704 | 0.1092 | 1.9530 | 0.0319 | 25.43% | [2.4563, 2.8844] |

NOK/USD | 3.1845 | 0.1978 | 6.1729 | 0.0389 | 13.26% | [2.7968, 3.5721] |

NZD/USD | 2.8648 | 0.1617 | 5.4697 | 0.0509 | 14.46% | [2.5479, 3.1818] |

USD/CAD | 2.5048 | 0.0869 | 1.4408 | 0.0535 | 32.61% | [2.3345, 2.6751] |

USD/CHF | 2.7312 | 0.1256 | 2.4059 | 0.0275 | 20.65% | [2.4851, 2.9774] |

USD/JPY | 2.5888 | 0.1276 | 3.0011 | 0.0485 | 16.85% | [2.3387, 2.8390] |

USD/SEK | 2.5553 | 0.1019 | 3.2868 | 0.0462 | 25.33% | [2.3556, 2.7550] |

Power Law | Lognormal | Exponential | |
---|---|---|---|

EUR/USD | 0.96 | 0.05 | 0.00 |

AUD/USD | 0.76 | 0.40 | 0.08 |

GBP/USD | 0.64 | 0.38 | 0.14 |

NOK/USD | 0.86 | 0.60 | 0.00 |

NZD/USD | 0.30 | 0.90 | 0.10 |

USD/CAD | 0.00 | 0.46 | 0.00 |

USD/CHF | 0.98 | 0.26 | 0.44 |

USD/JPY | 0.16 | 0.64 | 0.32 |

USD/SEK | 0.04 | 0.86 | 0.40 |

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**MDPI and ACS Style**

Fathi, M.; Grobys, K.; Kolari, J.W.
On the Realized Risk of Foreign Exchange Rates: A Fractal Perspective. *J. Risk Financial Manag.* **2024**, *17*, 79.
https://doi.org/10.3390/jrfm17020079

**AMA Style**

Fathi M, Grobys K, Kolari JW.
On the Realized Risk of Foreign Exchange Rates: A Fractal Perspective. *Journal of Risk and Financial Management*. 2024; 17(2):79.
https://doi.org/10.3390/jrfm17020079

**Chicago/Turabian Style**

Fathi, Masoumeh, Klaus Grobys, and James W. Kolari.
2024. "On the Realized Risk of Foreign Exchange Rates: A Fractal Perspective" *Journal of Risk and Financial Management* 17, no. 2: 79.
https://doi.org/10.3390/jrfm17020079