# Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Literature

#### 2.1. Measures of Risk

#### 2.2. Risk Transmission between Financial Markets

#### 2.3. The Expectile Approach

## 3. Methodology and Data

#### 3.1. Traditional Downside Risk Measures

#### 3.2. Expectile Based VaR

#### 3.3. Expectile Regression

- Let ${\beta}^{\left[0\right]}$ be an appropriate initial guess of the parameter $\beta \left(\theta \right)$.
- Given ${\beta}^{\left[k\right]}$, the kth guess, ${\beta}^{[k+1]}$ is calculated as$${\beta}^{[k+1]}={\left({w}^{\left[k\right]}{\mathit{x}\mathit{x}}^{\prime}\right)}^{-1}{w}^{\left[k\right]}\mathit{x}y,$$$${w}^{\left[k\right]}=\left(\begin{array}{c}|\theta -{\mathbf{1}}_{\{{y}_{1}-{x}_{1}^{\prime}{\beta}^{\left[k\right]}\le 0\}}|\\ \vdots \\ |\theta -{\mathbf{1}}_{\{{y}_{T}-{x}_{T}^{\prime}{\beta}^{\left[k\right]}\le 0\}}|\end{array}\right),\phantom{\rule{4pt}{0ex}}\mathit{x}=\left(\begin{array}{c}{x}_{1},\dots ,{x}_{T}\end{array}\right),\phantom{\rule{4pt}{0ex}}y=\left(\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{T}\end{array}\right).$$
- Repeat Step 2, for instance, until $|{\beta}^{[k+1]}-{\beta}^{\left[k\right]}|$ is small enough.

#### 3.4. Variables

- $y=$ ASX 200 index return;
- ${x}^{\left(2\right)}=$ Nikkei 225 index return;
- ${x}^{\left(3\right)}=$ S&P 500 index return; and
- ${x}^{\left(4\right)}=$ CSI 300 index return.

- $y=$ Nikkei 225 index return;
- ${x}^{\left(2\right)}=$ ASX 200 index return;
- ${x}^{\left(3\right)}=$ S&P 500 index return; and
- ${x}^{\left(4\right)}=$ CSI 300 index return.

#### 3.5. Data and Sample Statistics

## 4. Empirical Results

#### 4.1. The Effect of Risk Factors

#### 4.1.1. Results for ASX 200 Index

#### 4.1.2. Results for Nikkei 225 Index

#### 4.1.3. The Comparison

#### 4.2. EVaR Estimates

#### 4.3. Out-of-Sample Estimation

#### 4.4. ES Estimates

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | https://www.asx.com.au/about, accessed on 10 September 2023. |

2 | https://www.bis.org/bcbs/basel3.htm, accessed on 10 September 2023. |

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**Figure 2.**In-sample $\nu (\theta |{\mathcal{F}}_{t})$ of ASX 200 index for $\theta =0.0067\phantom{\rule{3.33333pt}{0ex}}(\alpha =0.01)$ and $0.0254\phantom{\rule{3.33333pt}{0ex}}(\alpha =0.05)$. In-sample period: 1 October 2012 to 30 September 2023.

**Figure 3.**In-sample $\nu (\theta |{\mathcal{F}}_{t})$ of Nikkei 225 index for $\theta =0.0030\phantom{\rule{3.33333pt}{0ex}}(\alpha =0.01)$ and $0.0213\phantom{\rule{3.33333pt}{0ex}}(\alpha =0.05)$. In-sample period: 1 October 2012 to 30 September 2023.

**Figure 4.**In-sample period: 1 October 2013 to 30 September 2020, the (

**left**) side of the bar, and out-of-sample period: 1 October 2020 to 30 September 2023, the (

**right**) side of the bar.

**Figure 5.**In-sample period: 1 October 2013 to 30 September 2020, the (

**left**) side of the bar, and out-of-sample period: 1 October 2020 to 30 September 2023, the (

**right**) side of the bar.

Index | ASX 200 | Nikkei 225 | S&P 500 | CSI 300 |
---|---|---|---|---|

Mean $(\times {10}^{-4})$ | 1.18 | 3.24 | 3.73 | 1.79 |

Median $(\times {10}^{-4})$ | 6.5 | 7.81 | 5.74 | 2.23 |

Max $(\times {10}^{-2})$ | 6.77 | 7.73 | 8.97 | 6.5 |

Min $(\times {10}^{-2})$ | −10.2 | −8.25 | −12.77 | −9.15 |

S.dev $(\times {10}^{-2})$ | 0.97 | 1.26 | 1.11 | 1.4 |

Skew. | −1.05 | −0.15 | −0.81 | −0.8 |

Kurt. | 12.42 | 4.28 | 16.19 | 6.03 |

ADF | −1.92 | −1.14 | −0.81 | −2.31 |

(p-value) | $\left(0.32\right)$ | $\left(0.70\right)$ | $\left(0.82\right)$ | $\left(0.17\right)$ |

**Table 2.**The expectile regression results for ASX 200 and Nikkei 225 indices. “s.e.” denotes the standard error, ***, **, * label significance at 1%, 5%, and 10% levels, respectively.

Indices | ASX 200 | Nikkei 225 | ||
---|---|---|---|---|

$\mathit{\alpha}$ | 1% | 5% | 1% | 5% |

$\mathit{\theta}$ | 0.67% | 2.54% | 0.3% | 2.13% |

Variables | Estimate | Estimate | Estimate | Estimate |

(s.e.) | (s.e.) | (s.e.) | (s.e.) | |

Const. | −0.0081 | −0.0059 | −0.0206 | −0.0098 |

(0.0014) *** | (0.0011) *** | (0.0022) *** | (0.0014) *** | |

${y}_{t-1}^{+}$ | −0.0781 | −0.0275 | −0.1201 | −0.0691 |

(0.2363) | (0.1210) | (0.1240) | (0.0914) | |

${y}_{t-1}^{-}$ | 0.3097 | 0.1114 | −0.0314 | 0.0498 |

(0.2185) | (0.2031) | (0.1513) | (0.0820) | |

${y}_{t-2}^{+}$ | −0.7581 | −0.5408 | −0.2603 | −0.2693 |

(0.1854) *** | (0.1566) *** | (0.1339) * | (0.1044) *** | |

${y}_{t-2}^{-}$ | 0.6885 | 0.434 | 0.4659 | 0.5004 |

(0.2962) ** | (0.1884) ** | (0.0887) *** | (0.0849) *** | |

${x}_{t-1}^{\left(2\right)+}$ | −0.0758 | −0.0903 | 0.0615 | −0.1439 |

(0.0884) | (0.0841) | (0.3145) | (0.1737) | |

${x}_{t-1}^{\left(2\right)-}$ | 0.0035 | −0.0166 | 0.5495 | 0.4419 |

(0.0884) | (0.0781) | (0.2405) ** | (0.1701) *** | |

S&P ${500}_{t-1}^{+}$ | −0.2258 | −0.1205 | −0.1587 | 0.3057 |

(0.1226) | (0.1708) | (0.9359) | (0.1504) ** | |

>S&P ${500}_{t-1}^{-}$ | 0.7614 | 0.6062 | 0.6437 | 0.5785 |

(0.1776) *** | (0.1124) *** | (0.2245) *** | (0.1169) *** | |

$\mathrm{CSI}\phantom{\rule{4.pt}{0ex}}{200}_{t-1}^{+}$ | −0.1036 | −0.0871 | 0.2205 | 0.0381 |

(0.1127) | (0.0593) | (0.1588) | (0.0759) | |

$\mathrm{CSI}\phantom{\rule{4.pt}{0ex}}{200}_{t-1}^{-}$ | 0.0648 | 0.0671 | −0.0404 | 0.0297 |

(0.0563) | (0.0437) | (0.0499) | (0.0498) |

$\mathit{\alpha}$ | 1% | 5% | ||||
---|---|---|---|---|---|---|

Mean | Max | Tail Prob. | Mean | Max | Tail Prob. | |

ASX 200 | 0.0187 | 0.1829 | 2.729% | 0.0131 | 0.1316 | 6.606% |

Nikkei 225 | 0.0276 | 0.1544 | 1.432% | 0.0163 | 0.1376 | 5.524% |

**Table 4.**Comparison of the number of exceedances over the estimated expectiles between in-sample and out-of -sample. In-sample period: 1 October 2013 to 30 September 2020. Out-of-sample period: 1 October 2020 to 30 September 2023.

Indices | ASX 200 | Nikkei 225 | ||
---|---|---|---|---|

$\alpha $ | 1% | 5% | 1% | 5% |

$\theta $ | 0.67% | 3.0% | 0.28% | 2.3% |

In-sample | 49/1770 | 132/1770 | 26/1709 | 95/1709 |

(2.8%) | (7.5%) | (1.5%) | (5.6%) | |

Out-of-sample | 16/758 | 45/758 | 3/735 | 29/735 |

(2.1%) | (5.9%) | (0.4%) | (3.9%) |

$\mathit{\alpha}$ | 1% | 5% | ||||
---|---|---|---|---|---|---|

Mean | Max | Tail Prob. | Mean | Max | Tail Prob. | |

ASX 200 | 0.0312 | 0.3061 | 0.237% | 0.0201 | 0.2021 | 1.899% |

Nikkei 225 | 0.0358 | 0.2005 | 0.245% | 0.0234 | 0.1987 | 1.795% |

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## Share and Cite

**MDPI and ACS Style**

Marumo, K.; Li, S.
Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression. *J. Risk Financial Manag.* **2024**, *17*, 189.
https://doi.org/10.3390/jrfm17050189

**AMA Style**

Marumo K, Li S.
Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression. *Journal of Risk and Financial Management*. 2024; 17(5):189.
https://doi.org/10.3390/jrfm17050189

**Chicago/Turabian Style**

Marumo, Kohei, and Steven Li.
2024. "Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression" *Journal of Risk and Financial Management* 17, no. 5: 189.
https://doi.org/10.3390/jrfm17050189