# Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Multivariate Gram–Charlier Model

**0**mean and ${I}_{n}$ (identity or order n) matrix,

#### 2.2. The Dynamic Conditional Correlation Model

#### 2.3. Risk Performance Model

## 3. Empirical Application

#### 3.1. Cryptocurrencies

#### 3.2. Data Description and Analysis

#### 3.2.1. Value at Risk and Median Shortfall

#### 3.2.2. Expected Shortfall

#### 3.2.3. Summary of Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Kraft, D.F.; Engle, R.F. Autoregressive Conditional Heteroskedaticity in Multiple Time Series; Department of Economics University of California: Berkeley, CA, USA, 1982. [Google Scholar]
- Engle, R.F.; Ng, V.K.; Rothschild, M. Asset Pricing with a factor ARCH covariance structure: Empirical estimates for treasure bills. J. Econom.
**1990**, 52, 245–266. [Google Scholar] - Bollerslev, T. Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH approach. Rev. Econ. Stat.
**1990**, 72, 498–505. [Google Scholar] [CrossRef] - Engle, R.F.; Kroner, K. Multivariate simultaneous GARCH. Econ. Theory
**1995**, 11, 122–150. [Google Scholar] [CrossRef] - Engle, R.F. Dynamic conditional correlation – A simple class of multivariate GARCH models. J. Bus. Econ. Stat.
**2002**, 20, 339–350. [Google Scholar] [CrossRef] - Engle, R.F.; Kelly, B. Dynamic equicorrelation. J. Bus. Econ. Stat.
**2012**, 30, 212–228. [Google Scholar] [CrossRef] - Bauwens, L.; Laurent, S.; Rombouts, J.V.K. Multivariate GARCH models: A survey. J. Appl. Econom.
**2006**, 21, 79–109. [Google Scholar] [CrossRef] [Green Version] - Engle, R.F.; González-Rivera, G. Semi-parametric ARCH models. J. Bus. Econ. Stat.
**1991**, 9, 345–359. [Google Scholar] - Fang, K.-T.; Kotz, S.; Ng, K. Symetric multivariate and related distributions. In Chapman and Hall/CRC, London.; Chapman & Hall: London, UK, 1990. [Google Scholar]
- Engle, R.F.; Sheppard, K. Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. NBER Working Paper No. 8554; National Bureau of Economic Research: Cmabridge, MA, USA, 2001. [Google Scholar]
- Sarabia, J.M.; Gómez-Déniz, E. Construction of multivariate distributions: A review of some recent results. Stat. Oper. Res. Trans.
**2008**, 32, 3–36. [Google Scholar] - Embrechts, P.; Lindskog, F.; McNeil, A. Modeling dependence with copulas and applications to risk management. In Handbook of Heavy Tailed Distributions in Finance; Rachev, T.S., Ed.; Elsevier: Amsterdam, The Netherlands, 2003; pp. 330–383. [Google Scholar]
- Jondeau, E.; Poon, S.-H.; Rockinger, M. Financial Modeling under Non-Gaussian Distributions. Springer Finance Series; Springer Science and Bussiness Media: Berlin, Germany, 2007. [Google Scholar]
- Kendall, M.; Stuart, A. The Advanced Theory of Statistics, 4th ed.; Griffin & Co.: London, UK, 1977; Volume I. [Google Scholar]
- Hald, A. The early history of the cumulants and the Gram-Charlier series. Int. Stat. Rev.
**2000**, 68, 137–153. [Google Scholar] [CrossRef] - Sauer, P.W.; Heydt, G.T. A conveniente Multivariate Gram-Charlier Type A Series. IEEE Trans. Commun.
**1979**, 27. [Google Scholar] - Mauleón, I. Modeling multivariate moments in European stock markets. Eur. J. Financ.
**2006**, 12, 241–263. [Google Scholar] [CrossRef] - Perote, J. The multivariate Edgeworth-Sargan density. Span. Econ. Rev.
**2004**, 6, 77–96. [Google Scholar] [CrossRef] - Del Brio, E.B.; Ñíguez, T.-M.; Perote, J. Gram–Charlier densities: A multivariate approach. Quant. Financ.
**2009**, 9, 855–868. [Google Scholar] [CrossRef] [Green Version] - Del Brio, E.B.; Ñíguez, T.M.; Perote, J. Multivariate semi-nonparametric distributions with dynamic conditional correlations. Int. J. Forecast.
**2011**, 27, 347–364. [Google Scholar] [CrossRef] - Weng, R.C. Expansions for multivariate densities. J. Stat. Plan. Inference
**2015**, 167, 174–181. [Google Scholar] [CrossRef] - Ñíguez, T.-M.; Perote, J. Multivariate moments expansion density: Application of the dynamic equicorrelation model. J. Bank. Financ.
**2016**, 72, S216–S232. [Google Scholar] [CrossRef] [Green Version] - Mora-Valencia, A.; Ñíguez, T.M.; Perote, J. Multivariate approximations to portfolio return distributions. Comput. Math. Organ. Theory
**2017**, 23, 347–361. [Google Scholar] [CrossRef] - Del Brio, E.B.; Mora-Valencia, A. The kidnapping of Europe: High-order moments’ transmission between developed and emerging markets. Emerg. Mark. Rev.
**2017**, 31, 96–115. [Google Scholar] [CrossRef] - Dharmani, B.C. Multivariate generalized Gram-Charlier series in vector notations. J. Math. Chem.
**2018**, 56, 1631–1655. [Google Scholar] [CrossRef] [Green Version] - Del Brio, E.B.; Mora-Valencia, A.; Perote, J. Expected shortfall assessment in commodity (L)ETF portfolios with semi-nonparametric specifications. Eur. J. Financ.
**2019**, 25, 1746–1764. [Google Scholar] [CrossRef] - Ñíguez, T.M.; Perote, J. Forecasting heavy-tailed densities with Positive Edgeworth and Gram-Charlier expansions. Oxf. Bull. Econ. Stat.
**2012**, 74, 600–627. [Google Scholar] [CrossRef] - Glosten, L.R.; Jagannathean, R.; Runkle, D.E. On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. J. Financ.
**1993**, 48, 1779–1801. [Google Scholar] [CrossRef] - Jiménez, I.; Mora-Valencia, A.; Perote, J. Risk quantification and validation for Bitcoin. Oper. Res. Lett.
**2020**, 48, 534–541. [Google Scholar] [CrossRef] - Christoffersen, P.F. Evaluating Interval Forecasts. Int. Econ. Rev.
**1998**, 39, 841. [Google Scholar] [CrossRef] - Engle, R.F.E.; Manganelli, S. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles. J. Bus. Econ. Stat.
**2004**, 22, 367–381. [Google Scholar] [CrossRef] - Kratz, M.; Lok, Y.H.; McNeil, A.J. Multinomial VaR backtests: A simple implicit approach to backtesting expected shortfall. J. Bank. Financ.
**2018**, 88, 393–407. [Google Scholar] [CrossRef] [Green Version] - Dimitriadis, T.; Bayer, S. Regression-Based Expected Shortfall Backtesting. J. Financ. Econ.
**2020**. [Google Scholar] [CrossRef] - Donley, M.G.; Spanos, P. Dynamic Analysis of Non-Linear Structures by the Method of Statistical Quadratization; (Lectures N); Springer: Berlin, Germany, 1990. [Google Scholar]
- Zoia, M.G.; Biffi, P.; Nicolussi, F. Value at risk and expected shortfall based on Gram-Charlier like expansions. J. Bank. Financ.
**2018**, 93, 92–104. [Google Scholar] [CrossRef] - Acereda, B.; Leon, A.; Mora, J. Estimating the expected shortfall of cryptocurrencies: An evaluation based on backtesting. Financ. Res. Lett.
**2019**, 1–6. [Google Scholar] [CrossRef] - León, Á.; Ñíguez, T.M. Modeling asset returns under time-varying semi-nonparametric distributions. J. Bank. Financ.
**2020**, 118. [Google Scholar] [CrossRef] - Cappiello, L.; Engle, R.F.; Sheppard, K. Asymmetric dynamics in the correlations of global equity and bond returns. J. Financ. Econom.
**2006**, 4, 537–572. [Google Scholar] [CrossRef] - Jorion, P. Risk management lessons from long-term capital management. Eur. Financ. Manag.
**2000**, 6, 277–300. [Google Scholar] [CrossRef] - Barnard, R.W.; Pearce, K.; Trindade, A.A. When is tail mean estimation more efficient than tail median? Answers and implications for quantitative risk management. Ann. Oper. Res.
**2018**, 262, 47–65. [Google Scholar] [CrossRef] - So, M.K.P.; Wong, C.M. Estimation of multiple period expected shortfall and median shortfall for risk management. Quant. Financ.
**2012**, 12, 739–754. [Google Scholar] [CrossRef] [Green Version] - Kou, S.; Peng, X. Expected shortfall or median shortfall. J. Financ. Eng.
**2014**, 1. [Google Scholar] [CrossRef] - Emmer, S.; Kratz, M.; Tasche, D. What is the best risk measure in practice? A comparison of standard measures. J. Risk
**2015**, 18, 31–60. [Google Scholar] [CrossRef] [Green Version] - Nakamoto, S. Bitcoin: Un Sistema de Dinero en Efectivo Electrónico Peer-to-Peer. 2008, pp. 1–9. Available online: www.bitcoin.org (accessed on 30 October 2020).
- Dyhrberg, A.H. Bitcoin, gold and the dollar - A GARCH volatility analysis. Financ. Res. Lett.
**2016**, 16, 85–92. [Google Scholar] [CrossRef] [Green Version] - Corbet, S.; Meegan, A.; Larkin, C.; Lucey, B.; Yarovaya, L. Exploring the dynamic relationships between cryptocurrencies and other financial assets. Econ. Lett.
**2018**, 165, 28–34. [Google Scholar] [CrossRef] - Dyhrberg, A.H. Hedging capabilities of bitcoin. Is it the virtual gold? Financ. Res. Lett.
**2016**, 16, 139–144. [Google Scholar] [CrossRef] [Green Version] - Yermack, D. Is Bitcoin a Real Currency? An Economic Appraisal. In Handbook of Digital Currency: Bitcoin, Innovation, Financial Instruments, and Big Data; Elsevier Inc.: Amsterdam, The Netherlands, 2015; pp. 31–43. ISBN 9780128023518. [Google Scholar]
- Baek, C.; Elbeck, M. Bitcoins as an investment or speculative vehicle? A first look. Appl. Econ. Lett.
**2015**, 22, 30–34. [Google Scholar] [CrossRef] - Dwyer, G.P. The economics of Bitcoin and similar private digital currencies. J. Financ. Stab.
**2015**, 17, 81–91. [Google Scholar] [CrossRef] [Green Version] - Gkillas, K.; Katsiampa, P. An application of extreme value theory to cryptocurrencies. Econ. Lett.
**2018**, 164, 109–111. [Google Scholar] [CrossRef] [Green Version] - Katsiampa, P. Volatility estimation for Bitcoin: A comparison of GARCH models. Econ. Lett.
**2017**, 158, 3–6. [Google Scholar] [CrossRef] [Green Version] - Lahmiri, S.; Bekiros, S.; Salvi, A. Long-range memory, distributional variation and randomness of bitcoin volatility. Chaossolitons Fractals
**2018**, 107, 43–48. [Google Scholar] [CrossRef] - Stavroyiannis, S. Volatility Modeling and Risk Assessment of the Major Digital Currencies. Ssrn Electron. J.
**2018**. [Google Scholar] [CrossRef] - Balcilar, M.; Bouri, E.; Gupta, R.; Roubaud, D. Can volume predict Bitcoin returns and volatility? A quantiles-based approach. Econ. Model.
**2017**, 64, 74–81. [Google Scholar] [CrossRef] [Green Version] - Blau, B.M. Price dynamics and speculative trading in Bitcoin. Res. Int. Bus. Financ.
**2018**, 43, 15–21. [Google Scholar] [CrossRef] - Guesmi, K.; Saadi, S.; Abid, I.; Ftiti, Z. Portfolio diversification with virtual currency: Evidence from bitcoin. Int. Rev. Financ. Anal.
**2019**, 63, 431–437. [Google Scholar] [CrossRef] - Canh, N.P.; Wongchoti, U.; Thanh, S.D.; Thong, N.T. Systematic risk in cryptocurrency market: Evidence from DCC-MGARCH model. Financ. Res. Lett.
**2019**, 29, 90–100. [Google Scholar] [CrossRef] - Aslanidis, N.; Bariviera, A.F.; Martínez-Ibañez, O. An analysis of cryptocurrencies conditional cross correlations. Financ. Res. Lett.
**2019**, 31, 130–137. [Google Scholar] [CrossRef] [Green Version] - Qureshi, S.; Aftab, M.; Bouri, E.; Saeed, T. Dynamic interdependence of cryptocurrency markets: An analysis across time and frequency. Phys. A: Stat. Mech. Its Appl.
**2020**, 559, 125077. [Google Scholar] [CrossRef] - Novales, A.; Garcia-Jorcano, L. Backtesting extreme value theory models of expected shortfall. Quant. Financ.
**2019**, 19, 799–825. [Google Scholar] [CrossRef] [Green Version] - Guegan, D.; Hassani, B. Distortion risk measure or the transformation of unimodal distributions into multimodal functions. Int. Ser. Oper. Res. Manag. Sci.
**2015**, 211, 71–88. [Google Scholar] [CrossRef] - Guegan, D.; Hassani, B.K. More accurate measurement for enhanced controls: VaR vs. ES? J. Int. Financ. Mark. Inst. Money
**2018**, 54, 152–165. [Google Scholar] [CrossRef] [Green Version] - Gourieroux, C.; Laurent, J.P.; Scaillet, O. Sensitivity analysis of Values at Risk. J. Empir. Financ.
**2000**, 7, 225–245. [Google Scholar] [CrossRef] [Green Version] - Hallerbach, W. Decomposing portfolio value-at-risk: A general analysis. J. Risk
**2003**, 5, 1–18. [Google Scholar] [CrossRef] [Green Version] - Tasche, D.; Tibiletti, L. A shortcut to sign incremental value at risk for risk allocation. J. Risk Financ.
**2003**, 4, 43–46. [Google Scholar] [CrossRef] - Scaillet, O. Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall. Math. Financ.
**2004**, 14, 115–129. [Google Scholar] [CrossRef] - Zhang, Y.; Rachev, S. Risk Attribution and Portfolio Performance Measurement-An Overview. J. Appl. Funct. Anal.
**2006**, 4, 373–402. [Google Scholar]

**Figure 1.**Cryptocurrency prices and returns for Bitcoin, Litecoin, and Ripple series. Daily prices from 4 August 2013 to 6 March 2020 (2407 observations).

**Figure 2.**Portfolio returns (black line) compared to 99% Value at Risk (VaR) (red line) and 97.5%-VaR (green line) for the four portfolios with both dynamic conditional correlation (DCC) and dynamic equicorrelation (DECO) models. Portfolio I corresponds to equally weighted portfolio; Portfolio II is 25% Bitcoin (BTC), 25% Litecoin (LTC), and 50% Ripple (XRP); Portfolio III is a combination of 25% BTC, 50% LTC, and 25% XRP; and Portfolio IV is 50% BTC, 25% LTC, and 25% XRP. Panel

**A**(

**B**) displays the DCC (DECO) model for every portfolio.

**Figure 3.**Portfolio returns (black line) compared to 99% median shortfall (MS) (red line) and 97.5%-MS (green line) for four portfolios with both dynamic conditional correlation (DCC) and dynamic equicorrelation (DECO). Portfolio I corresponds to equally weighted portfolio; Portfolio II is 25% Bitcoin (BTC), 25% Litecoin (LTC), and 50% Ripple (XRP); Portfolio III is a combination of 25% BTC, 50% LTC, and 25% XRP; and Portfolio IV is 50% BTC, 25% LTC, and 25% XRP. Panel

**A**(

**B**) displays the SNP (DECO) model for every portfolio.

**Figure 4.**Portfolio returns (black line) compared to 97.5% expected shortfall (ES) for four portfolios with both DCC (red line) and DECO (green line). Portfolio

**I**corresponds to equally weighted portfolio; Portfolio

**II**is 25% Bitcoin (BTC), 25% Litecoin (LTC), and 50% Ripple (XRP); Portfolio

**III**is 25% BTC, 50% LTC, and 25% XRP; and Portfolio

**IV**is 50% BTC, 25% LTC, and 25% XRP.

Bitcoin | Litecoin | Ripple | P-I | P-II | P-III | P-IV | |
---|---|---|---|---|---|---|---|

Min. | −26.620 | −51.393 | −61.627 | −30.186 | −29.132 | −35.021 | −29.297 |

1st Qu. | −1.249 | −2.062 | −2.257 | −1.559 | −1.704 | −1.698 | −1.441 |

Median | 0.177 | −0.040 | −0.266 | 0.098 | 0.028 | 0.048 | 0.149 |

Mean | 0.186 | 0.126 | 0.155 | 0.156 | 0.155 | 0.148 | 0.163 |

3rd. Qu | 1.810 | 1.920 | 1.964 | 1.871 | 1.885 | 1.881 | 1.833 |

Max. | 35.745 | 82.897 | 102.736 | 41.781 | 54.809 | 52.063 | 40.275 |

Std. Dev. | 4.207 | 6.447 | 7.199 | 4.762 | 5.102 | 4.994 | 4.445 |

Variance | 17.696 | 41.567 | 51.819 | 22.673 | 26.031 | 24.939 | 19.755 |

Exc. Kurtosis | 8.133 | 25.865 | 30.380 | 11.716 | 14.161 | 14.106 | 10.493 |

Skewness | −0.108 | 1.723 | 2.081 | 0.518 | 0.885 | 0.732 | 0.193 |

Return | Bitcoin | Litecoin | Ripple |
---|---|---|---|

Bitcoin | 1.000 | 0.665 | 0.377 |

Litecoin | 0.665 | 1.000 | 0.368 |

Ripple | 0.377 | 0.368 | 1.000 |

Portfolio I | ||||

Exc. | CC | DQ | AE | |

Panel A: SNP-DCC | ||||

99%-VaR | 11 | (0.033) | (0.004) | 2.200 |

97.5%-VaR | 15 | (0.158) | (0.098) | 1.200 |

Panel B: SNP-DECO | ||||

99%-VaR | 11 | (0.033) | (0.003) | 2.200 |

97.5%-VaR | 13 | (0.119) | (0.065) | 1.040 |

Portfolio II | ||||

Panel A: SNP-DCC | ||||

99%-VaR | 9 | (0.230) | (0.406) | 1.800 |

97.5%-VaR | 15 | (0.158) | (0.101) | 1.200 |

Panel B: SNP-DECO | ||||

99%-VaR | 9 | (0.230) | (0.328) | 1.800 |

97.5%-VaR | 11 | (0.445) | (0.681) | 0.880 |

Portfolio III | ||||

Panel A: SNP-DCC | ||||

99%-VaR | 10 | (0.059) | (0.013) | 2.000 |

97.5%-VaR | 15 | (0.158) | (0.075) | 1.200 |

Panel B: SNP-DECO | ||||

99%-VaR | 10 | (0.059) | (0.011) | 2.000 |

97.5%-VaR | 14 | (0.144) | (0.076) | 1.120 |

Portfolio IV | ||||

Panel A: SNP-DCC | ||||

99%-VaR | 11 | (0.033) | (0.002) | 2.200 |

97.5%-VaR | 14 | (0.641) | (0.571) | 1.120 |

Panel B: SNP-DECO | ||||

99%-VaR | 11 | (0.033) | (0.002) | 2.200 |

97.5%-VaR | 13 | (0.627) | (0.559) | 1.040 |

Portfolio I | ||||

Exc. | CC | DQ | AE | |

Panel A: SNP-DCC | ||||

99%-MS | 8 | (0.019) | (0.003) | 3.200 |

97.5%-MS | 11 | (0.110) | (0.046) | 1.760 |

Panel B: SNP-DECO | ||||

99%-MS | 7 | (0.059) | (0.009) | 2.800 |

97.5%-MS | 11 | (0.110) | (0.036) | 1.760 |

Portfolio II | ||||

Panel A: SNP-DCC | ||||

99%-MS | 7 | (0.059) | (0.028) | 2.800 |

97.5%-MS | 10 | (0.311) | (0.436) | 1.600 |

Panel B: SNP-DECO | ||||

99%-MS | 6 | (0.159) | (0.106) | 2.400 |

97.5%-MS | 10 | (0.311) | (0.347) | 1.600 |

Portfolio III | ||||

Panel A: SNP-DCC | ||||

99%-MS | 7 | (0.059) | (0.024) | 2.800 |

97.5%-MS | 12 | (0.068) | (0.034) | 1.920 |

Panel B: SNP-DECO | ||||

99%-MS | 4 | (0.661) | (0.566) | 1.600 |

97.5%-MS | 11 | (0.110) | (0.030) | 1.760 |

Portfolio IV | ||||

Panel A: SNP-DCC | ||||

99%-MS | 7 | (0.059) | (0.035) | 2.800 |

97.5%-MS | 11 | (0.110) | (0.027) | 1.760 |

Panel B: SNP-DECO | ||||

99%-MS | 6 | (0.159) | (0.119) | 2.400 |

97.5%-MS | 11 | (0.110) | (0.022) | 1.760 |

Model | ESR-Intercept | Multinomial (N = 8) | ||
---|---|---|---|---|

One | Two | Pearson | Nass | |

Portfolio I | ||||

SNP-DCC | (0.077) | (0.153) | 17.70 | 13.53 |

SNP-DECO | (0.245) | (0.489) | 17.86 | 13.66 |

Portfolio II | ||||

SNP-DCC | (0.241) | (0.481) | 8.74 | 6.68 |

SNP-DECO | (0.445) | (0.891) | 10.34 | 7.91 |

Portfolio III | ||||

SNP-DCC | (0.163) | (0.327) | 11.30 | 8.64 |

SNP-DECO | (0.315) | (0.631) | 10.10 | 7.72 |

Portfolio IV | ||||

SNP-DCC | (0.079) | (0.159) | 21.62 | 16.53 |

SNP-DECO | (0.203) | (0.405) | 16.58 | 12.67 |

Portfolio | I | II | III | IV |
---|---|---|---|---|

99%-VaR | −$30,307.43 | −$40,480.70 | −$43,347.10 | −$37,401.91 |

97.5%-VaR | −$23,184.31 | −$30,876.36 | −$33,254.79 | −$28,606.09 |

Portfolio I | Portfolio II | Portfolio III | Portfolio IV | ||||
---|---|---|---|---|---|---|---|

Panel A: SNP-DCC | |||||||

99%-VaR | −$23,029.54 | −$30,795.56 | −($7766.02) | −$34,248.99 | −($11,219.46) | −$28,648.27 | −($5618.74) |

97.5%-VaR | −$18,650.38 | −$24,849.60 | −($6199.21) | −$27,736.32 | −($9085.93) | −$23,110.72 | −($4460.33) |

Panel B: SNP-DECO | |||||||

99%-VaR | −$24,015.39 | −$32,148.16 | −($8132.77) | −$35,414.07 | −($11,398.68) | −$29,807.34 | −($5791.95) |

97.5%-VaR | −$19,450.48 | −$25,942.31 | −($6491.83) | −$28,681.50 | −($9231.03) | −$24,049.10 | −($4598.63) |

Portfolio I | Portfolio II | Portfolio III | Portfolio IV | ||||
---|---|---|---|---|---|---|---|

Panel A: SNP-DCC | |||||||

97.5%-direct ES | −$24,082.91 | −$32,142.66 | −($8059.75) | −$35,983.31 | −($11,900.40) | −$29,983.53 | −($5900.62) |

97.5%-ES aprox. | −$21,958.25 | −$29,911.32 | −($7953.07) | −$32,726.86 | −($10,768.61) | −$27,427.08 | −($5468.83) |

Panel B: SNP-DECO | |||||||

97.5%-direct ES | −$25,113.45 | −$33,554.14 | −($8440.69) | −$37,206.94 | −($12,093.49) | −$31,195.81 | −($6082.36) |

97.5%-ES aprox. | −$22,898.66 | −$31,225.27 | −($8326.60) | −$33,840.54 | −($10,941.88) | −$28,537.48 | −($5638.81) |

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**MDPI and ACS Style**

Jiménez, I.; Mora-Valencia, A.; Ñíguez, T.-M.; Perote, J.
Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies. *Mathematics* **2020**, *8*, 2110.
https://doi.org/10.3390/math8122110

**AMA Style**

Jiménez I, Mora-Valencia A, Ñíguez T-M, Perote J.
Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies. *Mathematics*. 2020; 8(12):2110.
https://doi.org/10.3390/math8122110

**Chicago/Turabian Style**

Jiménez, Inés, Andrés Mora-Valencia, Trino-Manuel Ñíguez, and Javier Perote.
2020. "Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies" *Mathematics* 8, no. 12: 2110.
https://doi.org/10.3390/math8122110