# Impending Doom: The Loss of Diversification before a Crisis

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## Abstract

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## 1. Introduction

- The first principal component (PC1) with the largest eigenvalue represents a market wide effect that influences all stocks. In the financial literature, this is often called the systematic risk.
- A variable number of principal components (PCs) following the market component, which represent synchronized fluctuations associated with specific groups of stocks.
- The remaining PCs indicate randomness in the price fluctuations (noise). There were believed to contain no useful financial information and hence were eliminated from further investigation.

- Literature focused on contagion, spillover effects and joint crashes in financial markets (Adrian 2007; Billio et al. 2012; Kritzman et al. 2011; Wang et al. 2011). Those studies were based on the analysis of interconnectedness among market security returns and our analysis below follows in their steps.
- Empirical studies on systemic risk that include research on the auto-correlation in the number of bank defaults, bank returns, and fund withdrawals (Brandt and Hartmann 2000; Kenett et al. 2012; Lehar 2005).
- Research on bank capital ratios and bank liabilities (Aguirre and Saidi 2004; Bahansali et al. 2008; Brana and Lahet 2009).

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Kaiser–Meyer–Olkin Test

`cor`and

`cov`functions, respectively, in the stat package in base

`R`(R Core Team 2014) on a rolling window of 504 trading days, which is equivalent to two calendar years. The correlation matrices were for use with the Kaiser–Meyer–Olkin (KMO) test described in this section and the two tests involving PCA. The covariance matrices were for use with the diversification ratio described in Section 2.5 below.

#### 2.3. Principal Component Analysis

`eigen`in base

`R`on a rolling window of 504 trading days. For a correlation matrix, the total variation is equal to the number of variables in the matrix. Thus, for our matrices, this was 156. To obtain the percentage of variance explained by PC1 if ${E}_{1}$ is the eigenvalue of PC1, then

#### 2.4. PCA Stock Selection

- Apply PCA to the correlation matrix of a stock market.
- Associate one stock with the highest coefficient in absolute value with each of the last m principal components that have eigenvalues less than a certain numerical value, called the deletion criteria, and then delete those m stocks.
- A second or subsequent PCA is performed on the retained stocks. The same procedure described in step 2 is applied to the output of the PCA and, if necessary, further stocks are deleted.
- The procedure is repeated until no further deletions are considered necessary based on a stopping criteria that is a pre-determined minimum eigenvalue of the last principal component.

#### 2.5. Diversification Ratio

## 3. Results

#### 3.1. KMO Test

#### 3.2. Market Wide Effects: Principal Component One

#### 3.3. Number of Stocks Required for a Diversified Portfolio

#### 3.4. Diversification Ratio

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ASX200 | Australian Stock Exchange 200 Index |

KMO | Kaiser–Meyer–Olkin test |

PCA | Principal Component Analysis |

PC | Principal Component |

## Appendix A

- We created a new variable associated with each stock called the Dividend Factor. We started with a factor of 1 and, every time a dividend was paid, we multiplied the Dividend Factor,$$\begin{array}{cc}\hfill \mathrm{Daily}\phantom{\rule{4.pt}{0ex}}\mathrm{Dividend}\phantom{\rule{4.pt}{0ex}}{\mathrm{Factor}}_{i}\left(t\right)& =\left\{\begin{array}{cc}1,\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\mathrm{no}\mathrm{dividend},\hfill \\ 1+\frac{{D}_{i}\left(t\right)}{{P}_{i}\left(t\right)},\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{dividend},\hfill \end{array}\right\},\hfill \end{array}$$$$\begin{array}{cc}\hfill \mathrm{Cumulative}\phantom{\rule{4.pt}{0ex}}\mathrm{Dividend}\phantom{\rule{4.pt}{0ex}}{\mathrm{Factor}}_{i}\left(t\right)& =\prod _{j=1}^{t}\left(\mathrm{Daily}\phantom{\rule{4.pt}{0ex}}\mathrm{Dividend}\phantom{\rule{4.pt}{0ex}}{\mathrm{Factor}}_{i}\left(t\right)\right),\hfill \end{array}$$
- We adjusted the price series with the dividend factor, and the adjusted price was calculated by$${\mathrm{PNEW}}_{i}\left(t\right)={P}_{i}\left(t\right)\times \mathrm{Cumulative}\phantom{\rule{4.pt}{0ex}}\mathrm{Dividend}\phantom{\rule{4.pt}{0ex}}{\mathrm{Factor}}_{i}\left(t\right).$$
- The return series for a given stock i was calculated as$${\mathrm{R}}_{i}\left(t\right)=\frac{{\mathrm{PNEW}}_{i}(t+1)-{\mathrm{PNEW}}_{i}\left(t\right)}{{\mathrm{PNEW}}_{i}\left(t\right)}.$$

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1. |

**Figure 1.**The KMO measure of sampling adequacy for 156 stocks (left hand axis, black line) and the variance explained by PC1 (right hand axis, red line). Both measures were calculated weekly using a rolling window approach with a window size of two years (504 trading days).

**Figure 2.**The variance explained by PC1 (right hand aixs, red line) with the ASX200 index value (left hand axis, black line). The plot of index values includes the first two-year period used for the estimation of the PCA.

**Figure 3.**The variance explained by PC1 ( right hand axis, red line) and the ASX200 index returns (left hand axis, blue line). Both the variance explained by PC1 and the ASX200 index returns were calculated weekly using a rolling window size of two years (equivalent to 504 trading days).

**Figure 4.**The number of stocks selected by PCA over time. A stocks selection procedure of 0.7 deletion criteria and a 0.5 stop criteria was used. The selection procedure was applied on a rolling window basis with window size of two years (504 trading days).

**Figure 5.**The variance explained by PC1 (right hand axis, red line) and the diversification ratio (left hand axis, black line). The variance explained by PC1 was calculated weekly using a rolling window size of two years (equivalent to 504 trading days). The diversification ratio was calculated using Equation (3).

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**MDPI and ACS Style**

Yang, L.; Rea, W.; Rea, A.
Impending Doom: The Loss of Diversification before a Crisis. *Int. J. Financial Stud.* **2017**, *5*, 29.
https://doi.org/10.3390/ijfs5040029

**AMA Style**

Yang L, Rea W, Rea A.
Impending Doom: The Loss of Diversification before a Crisis. *International Journal of Financial Studies*. 2017; 5(4):29.
https://doi.org/10.3390/ijfs5040029

**Chicago/Turabian Style**

Yang, Libin, William Rea, and Alethea Rea.
2017. "Impending Doom: The Loss of Diversification before a Crisis" *International Journal of Financial Studies* 5, no. 4: 29.
https://doi.org/10.3390/ijfs5040029