# Continuous and Jump Betas: Implications for Portfolio Diversification

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Modeling Framework

#### 3.1. The Estimators in Discrete Time

#### 3.2. Testing for Jumps

#### 3.3. Choices of Parameter Values

## 4. Estimated Betas

#### 4.1. Summary Statistics of the Estimated Betas

#### 4.2. Robustness Analysis

## 5. Portfolio Diversification with Betas

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

^{1.}Existing studies use standard deviation [5,6,7], mean absolute deviation [8,9], semi-variance [10], terminal wealth standard deviation [11,12], residual variance from the CAPM [13,14], mean realised dispersion [15], and realised volatility [16]. For a comprehensive list of different risk measures used in portfolio diversification, see ([17] p. 44).^{2.}There are 900 stocks in the original dataset of stocks that have been represented on the S&P500 at any time during the sample period. We additionally delete stocks that have altered currency of trade, are over-the-counter or listed on alternative exchanges. See Dungey et al. [44] for further database details.^{4.}The parameter τ in (7) can be more loosely specified such that $2\tau $ is above the Blumenthal-Getoor index, which covers the set of $\{\tau :\tau \ge 1\}$. However, in practice, higher values of τ are usually chosen to avoid friction caused by discrete sampling. For further details, see Todorov and Bollerslev [24].^{5.}The small sample properties of the two discrete estimators have been investigated in Alexeev et al. [42]. The authors show that the estimation bias becomes a concern only when the difference between ${\beta}^{c}$ and ${\beta}^{d}$ exceeds unity. This is not the case in the current application.^{6.}Alternatively, if microstructure noise, nonsynchronicity and intraday volatility patterns are persistent, one could use the robust two-time scale estimator of Boudt and Zhang [54] to take these into account. The estimator is implemented using the modified Lee and Mykland [51] jump statistic proposed in Boudt et al. [55].^{7.}Given the difference in variability in betas with firm characteristics, future research could explore a stratified sampling scheme to further the diversification benefits.^{8.}A portfolio size resulting in a 10-fold reduction in the normalised IQR can be inferred from Figure 3b through the intersection of the blue and red curves with the horizontal line at 0.1. Figures similar to Figure 3b are constructed for each time period, and the corresponding portfolio sizes are determined from each of these figures and plotted in Figure 4a.

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**Figure 1.**Distributions of monthly ${\widehat{\beta}}^{c}$ and ${\widehat{\beta}}^{d}$ for all investible S&P500 stocks. Red dots indicate the maxima and minima, solid black lines denote the 2.5%–97.5% inter-percentile range, blue blocks represent the interquantile ranges and the hollow black circles are the medians. Gaps in certain months of the distributions of estimated ${\widehat{\beta}}^{d}$s are indicative of “no-jump” months.

**Figure 2.**Signature plots of jump contribution for the pre-averaged versus equi-distant data handling approaches. However, to demonstrate the effects of the pre-averaging versus equi-distant approaches on the data in terms of jump contributions, we selected four exemplars (AKL, CLF, GME and NFLX). In each case, the higher signature plot pertains to the equi-distant data and the lower to the pre-averaged data. At the 5-min sampling frequency (indicated on each panel by the dashed vertical line), the difference in the contribution of jumps to total volatility for both methods is trivially different.

**Figure 4.**Portfolio sizes, n, required to reduce beta spreads. Plots above depict required portfolio sizes to reduce IQR (left panels) and $IP{R}_{95\%-5\%}$ (right panels) by a factor of 10 (top panels) and by a factor of five (bottom panels). For example, in panel (c), we show that investors seeking to reduce exposure of their portfolios to systematic discontinuous risk by a factor of five should hold smaller portfolios than when the same reduction is required in the continuous systematic risk component.

© 2016 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/).

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Alexeev, V.; Dungey, M.; Yao, W. Continuous and Jump Betas: Implications for Portfolio Diversification. *Econometrics* **2016**, *4*, 27.
https://doi.org/10.3390/econometrics4020027

**AMA Style**

Alexeev V, Dungey M, Yao W. Continuous and Jump Betas: Implications for Portfolio Diversification. *Econometrics*. 2016; 4(2):27.
https://doi.org/10.3390/econometrics4020027

**Chicago/Turabian Style**

Alexeev, Vitali, Mardi Dungey, and Wenying Yao. 2016. "Continuous and Jump Betas: Implications for Portfolio Diversification" *Econometrics* 4, no. 2: 27.
https://doi.org/10.3390/econometrics4020027