# Modeling and Forecasting Realized Portfolio Diversification Benefits

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Realized Measure of Diversification Benefits

#### 2.1. Quantifying Diversification Benefits

#### 2.2. Realized Measures for Diversification

**Proposition**

**1.**

## 3. Time Series Model for Diversification Measures

## 4. Empirical Application

#### 4.1. Data and Construction of Portfolios

#### 4.2. Time Series Modeling

#### 4.3. Economic Evaluation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 1

#### Appendix A.2. Further Asymptotic Results

## References

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**Figure 1.**In-sample: autocorrelation functions of ${\widehat{L}}_{t}$ (

**left**) and ${\widehat{w}}_{t}$ (

**right**).

**Figure 2.**Out-of-sample: autocorrelation functions of ${\widehat{L}}_{t}$ (

**left**) and ${\widehat{w}}_{t}$ (

**right**).

**Figure 3.**${\widehat{L}}_{t}$, in-sample autocorrelations of HAR residuals (

**left**) and their squares (

**right**).

**Figure 4.**${\widehat{L}}_{t}$, in-sample autocorrelations of AR(5) residuals (

**left**) and their squares (

**right**).

**Figure 5.**${\widehat{L}}_{t}$, in-sample autocorrelations of AR(1) residuals (

**left**) and their squares (

**right**).

**Figure 6.**${\widehat{w}}_{t}$, in-sample autocorrelations of HAR residuals (

**left**) and their squares (

**right**).

**Figure 7.**${\widehat{w}}_{t}$, in-sample autocorrelations of AR(5) residuals (

**left**) and their squares (

**right**).

**Figure 8.**${\widehat{w}}_{t}$, in-sample autocorrelations of AR(1) residuals (

**left**) and their squares (

**right**).

**Figure 9.**Ratios ${\sigma}_{t}^{2}({\tilde{w}}_{t},{\tilde{L}}_{t})/{\sigma}_{\mathrm{low},t}^{2}$ for the HAR, AR(5), and AR(1) models, from above to below. Note: the red lines correspond to a ratio of one, indicating equal variances.

**Figure 10.**Ratios ${\sigma}_{t}^{2}({\tilde{w}}_{t},{\tilde{L}}_{t})/{\sigma}_{ew,t}^{2}$ for the HAR, AR(5), and AR(1) models, from above to below. Note: the red lines correspond to a ratio of one, indicating equal variances.

Block A: Monte Carlo Simulation Results for ${\mathit{T}}_{\mathit{D}}$ | |||||||

m | ${\mathit{\sigma}}_{\mathit{pa},\mathit{t}}$ | Mean | Variance | Skewness | Kurtosis | ${\mathit{p}}_{\mathit{KS}}$ | ${\mathit{p}}_{\mathit{JB}}$ |

78 | 0 | −0.064 | $1.188$ | −0.979 | $5.374$ | $0.051$ | 0 |

0.3 | −0.072 | $1.133$ | −0.855 | $4.421$ | $0.183$ | 0 | |

0.5 | −0.098 | $1.203$ | −0.999 | $5.067$ | $0.098$ | 0 | |

390 | 0 | −0.056 | 1.054 | −0.198 | $3.113$ | $0.576$ | $0.029$ |

0.3 | −0.033 | $1.001$ | −0.383 | $3.290$ | $0.458$ | 0 | |

0.5 | −0.039 | $1.004$ | −0.419 | $3.359$ | $0.341$ | 0 | |

Block B: Monte Carlo Simulation Results for ${\mathit{T}}_{\mathit{L}}$ | |||||||

$\mathit{m}$ | ${\mathit{\sigma}}_{\mathit{pa},\mathit{t}}$ | Mean | Variance | Skewness | Kurtosis | ${\mathit{p}}_{\mathit{KS}}$ | ${\mathit{p}}_{\mathit{JB}}$ |

78 | 0 | $0.105$ | $1.049$ | $0.210$ | $3.234$ | $0.048$ | $0.008$ |

0.3 | $0.122$ | $0.998$ | $0.403$ | $3.288$ | $0.070$ | 0 | |

0.5 | $0.138$ | $0.997$ | $0.557$ | $3.462$ | $0.019$ | 0 | |

390 | 0 | $0.018$ | $1.046$ | $0.239$ | $3.088$ | $0.732$ | $0.007$ |

0.3 | $0.051$ | $0.971$ | $0.141$ | $3.066$ | $0.357$ | $0.175$ | |

0.5 | $0.060$ | $0.963$ | $0.195$ | $3.018$ | $0.378$ | $0.042$ |

Company | In-Sample | Out-of-Sample | % Change from In- to Out-of-Sample | Portfolio |
---|---|---|---|---|

Bank of America | 1.68 | 8.63 | 414.64 | $\left.\phantom{\begin{array}{c}\hfill c\end{array}}\right\}{P}_{\mathrm{high}}$ |

Alcoa | 3.32 | 6.30 | 89.76 | |

American Express | 3.16 | 5.47 | 73.23 | |

J.P. Morgan | 3.93 | 6.00 | 52.69 | |

Exxon | 1.73 | 2.36 | 36.15 | |

General Electric | 2.69 | 3.62 | 34.20 | $\left.\phantom{\begin{array}{c}\hfill c\end{array}}\right\}{P}_{\mathrm{low}\phantom{\rule{4.pt}{0ex}}}$ |

DuPont | 2.25 | 2.76 | 22.93 | |

IBM | 2.05 | 1.84 | −10.49 | |

Microsoft | 2.91 | 2.07 | −28.82 | |

Coca Cola | 1.68 | 1.19 | −29.18 | |

${P}_{ew\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}$ | 1.10 | 1.92 | 75.13 | |

${P}_{\mathrm{low}\phantom{\rule{4.pt}{0ex}}}$ | 1.21 | 1.27 | 5.01 | |

${P}_{\mathrm{high}}$ | 1.31 | 3.28 | 150.29 |

Parameter | In-Sample | Full Sample | ||||
---|---|---|---|---|---|---|

HAR | AR(5) | AR(1) | HAR | AR(5) | AR(1) | |

c | $-\underset{(0.1443)}{0.4029}$ *** | $-(0.1067)0.8895$ *** | $-\underset{(0.0797)}{1.5080}$ *** | $-\underset{(0.1215)}{0.4756}$ *** | $-\underset{(0.0893)}{1.0520}$ *** | $-\underset{(0.0674)}{1.8687}$ *** |

${\varphi}_{1}$ or ${\beta}_{d}$ | $\underset{(0.0347)}{0.1039}$ *** | $\underset{(0.0314)}{0.1877}$ *** | $\underset{(0.0299)}{0.2941}$ *** | $\underset{(0.0234)}{0.1174}$ *** | $\underset{(0.0211)}{0.1958}$ *** | $\underset{(0.0201)}{0.3112}$ *** |

${\varphi}_{2}$ | $\underset{(0.0316)}{0.1356}$ *** | $\underset{(0.0214)}{0.1376}$ *** | ||||

${\varphi}_{3}$ | $\underset{(0.0318)}{0.0708}$ ** | $\underset{(0.0215)}{0.0845}$ *** | ||||

${\varphi}_{4}$ | $\underset{(0.0317)}{0.1461}$ *** | $\underset{(0.0214)}{0.1005}$ *** | ||||

${\varphi}_{5}$ | ${\underset{(0.0314)}{0.0466}}^{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}}$ | $\underset{(0.0211)}{0.0957}$ *** | ||||

${\beta}_{w}$ | $\underset{(0.0981)}{0.3743}$ *** | $\underset{(0.0675)}{0.4651}$ *** | ||||

${\beta}_{m}$ | $\underset{(0.1234)}{0.5586}$ *** | $\underset{(0.0852)}{0.4962}$ *** | ||||

AIC | $3598.5$ | $3677.5$ | $3762.0$ | $8896.7$ | $9002.7$ | $9189.6$ |

BIC | $3623.0$ | $3711.9$ | $3776.7$ | $8925.2$ | $9042.7$ | $9206.8$ |

adjusted ${\overline{R}}^{2}$ | $0.160$ | $0.147$ | $0.086$ | $0.176$ | $0.163$ | $0.096$ |

**Table 4.**${\widehat{L}}_{t}$, in-sample residual diagnostic test statistics. LB, Ljung–Box; SW, Shapiro–Wilk.

HAR | AR(5) | AR(1) | |
---|---|---|---|

LB(5) | 7.299 (0.199) | 3.255 (0.661) | 70.890 (6.7 $\times {10}^{-14}$) |

ARCH-LM | 1.686 (0.891) | 3.866 (0.569) | 1.408 (0.923) |

SW | 0.843 (<2.2 $\times {10}^{-16}$) | 0.844 (<2.2 $\times {10}^{-16}$) | 0.850 (<2.2 $\times {10}^{-16}$) |

Parameter | In-Sample | Full Sample | ||||
---|---|---|---|---|---|---|

HAR | AR(5) | AR(1) | HAR | AR(5) | AR(1) | |

c | $\underset{(0.0304)}{0.1283}$ *** | $\underset{(0.0227)}{0.2012}$ *** | $\underset{(0.0162)}{0.3278}$ *** | $\underset{(0.0062)}{0.0104}$ * | $\underset{(0.0061)}{0.0256}$ *** | $\underset{(0.0066)}{0.0759}$ *** |

${\varphi}_{1}$ or ${\beta}_{d}$ | $\underset{(0.0374)}{0.1377}$ *** | $\underset{(0.0315)}{0.2202}$ *** | $\underset{(0.0297)}{0.3179}$ *** | $\underset{(0.0253)}{0.1468}$ *** | $\underset{(0.0210)}{0.2886}$ *** | $\underset{(0.0151)}{0.7014}$ *** |

${\varphi}_{2}$ | $\underset{(0.0320)}{0.1492}$ *** | $\underset{(0.0216)}{0.1961}$ *** | ||||

${\varphi}_{3}$ | $\underset{(0.0323)}{0.0598}$ * | $\underset{(0.0219)}{0.1306}$ *** | ||||

${\varphi}_{4}$ | $\underset{(0.0320)}{0.1297}$ *** | $\underset{(0.0216)}{0.1467}$ *** | ||||

${\varphi}_{5}$ | ${\underset{(0.0315)}{0.0212}}^{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}}$ | $\underset{(0.0210)}{0.1353}$ *** | ||||

${\beta}_{w}$ | $\underset{(0.0732)}{0.2794}$ *** | $\underset{(0.0481)}{0.3692}$ *** | ||||

${\beta}_{m}$ | $\underset{(0.0854)}{0.3131}$ *** | $\underset{(0.0432)}{0.4376}$ *** | ||||

AIC | $-23.614$ | $-24.731$ | $33.051$ | $-441.542$ | $-352.927$ | $207.223$ |

BIC | $0.924$ | $9.741$ | $47.837$ | $-413.018$ | $-312.940$ | $224.366$ |

adjusted ${\overline{R}}^{2}$ | $0.157$ | $0.155$ | $0.100$ | $0.620$ | $0.605$ | $0.492$ |

HAR | AR(5) | AR(1) | |
---|---|---|---|

LB(5) | 11.929 (0.036) | 2.886 (0.718) | 64.117 (1.7 $\times {10}^{-12}$) |

ARCH-LM | 8.353 (0.138) | 7.347 (0.196) | 10.760 (0.056) |

SW | 0.997 (0.119) | 0.997 (0.088) | 0.998 (0.163) |

HAR | ${\widehat{\mathit{L}}}_{\mathit{t}}<\ell $ | ${\widehat{\mathit{L}}}_{\mathit{t}}\ge \ell $ |

${\tilde{L}}_{t}<\ell $ | ||

${\tilde{L}}_{t}\ge \ell $ | ||

66.31% correct predictions | ||

AR(5) | ${\widehat{L}}_{t}<\ell $ | ${\widehat{L}}_{t}\ge \ell $ |

${\tilde{L}}_{t}<\ell $ | ||

${\tilde{L}}_{t}\ge \ell $ | ||

63.69% correct predictions | ||

AR(1) | ${\widehat{L}}_{t}<\ell $ | ${\widehat{L}}_{t}\ge \ell $ |

${\tilde{L}}_{t}<\ell $ | ||

${\tilde{L}}_{t}\ge \ell $ | ||

59.43% correct predictions |

**Table 8.**Ratios of ${\sigma}_{t}^{2}({\tilde{w}}_{t},{\tilde{L}}_{t})$ to different benchmark variances.

Benchmark | Mean Ratio | Std. Deviation of Ratio | % with >1 | % with <1 | % with =1 |
---|---|---|---|---|---|

HAR to get ${\tilde{L}}_{t}$ and ${\tilde{w}}_{t}$ | |||||

${\sigma}_{t}^{2}({\widehat{w}}_{t})$ | 1.068 | 0.1029 | 1 | 0 | - |

${\sigma}_{\mathrm{low},t}^{2}$ | 0.9576 | 0.0958 | 0.1230 | 0.4377 | 0.4393 |

${\sigma}_{ew,t}^{2}$ | 0.817 | 0.2102 | 0.1779 | 0.8221 | - |

AR(5) to get ${\tilde{L}}_{t}$ and ${\tilde{w}}_{t}$ | |||||

${\sigma}_{t}^{2}({\widehat{w}}_{t})$ | 1.0884 | 0.1303 | 1 | 0 | - |

${\sigma}_{\mathrm{low},t}^{2}$ | 0.9751 | 0.1083 | 0.2992 | 0.3385 | 0.3623 |

${\sigma}_{ew,t}^{2}$ | 0.8258 | 0.1963 | 0.1705 | 0.8295 | - |

AR(1) to get ${\tilde{L}}_{t}$ and ${\tilde{w}}_{t}$ | |||||

${\sigma}_{t}^{2}({\widehat{w}}_{t})$ | 1.1833 | 0.264 | 1 | 0 | - |

${\sigma}_{\mathrm{low},t}^{2}$ | 1.0588 | 0.2094 | 0.5025 | 0.3270 | 0.1705 |

${\sigma}_{ew,t}^{2}$ | 0.8714 | 0.1454 | 0.1434 | 0.8566 | - |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Golosnoy, V.; Hildebrandt, B.; Köhler, S. Modeling and Forecasting Realized Portfolio Diversification Benefits. *J. Risk Financial Manag.* **2019**, *12*, 116.
https://doi.org/10.3390/jrfm12030116

**AMA Style**

Golosnoy V, Hildebrandt B, Köhler S. Modeling and Forecasting Realized Portfolio Diversification Benefits. *Journal of Risk and Financial Management*. 2019; 12(3):116.
https://doi.org/10.3390/jrfm12030116

**Chicago/Turabian Style**

Golosnoy, Vasyl, Benno Hildebrandt, and Steffen Köhler. 2019. "Modeling and Forecasting Realized Portfolio Diversification Benefits" *Journal of Risk and Financial Management* 12, no. 3: 116.
https://doi.org/10.3390/jrfm12030116