# Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis

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## Abstract

**:**

^{®}Europe 600. A comparison between the slope coefficients evaluated using the ordinary Least Squares (LS) approach and the new Least Quartic (LQ) technique shows that the perception of market risk exposure is best captured by the proposed estimator during market turmoil, and it seems to anticipate the market risk increase typical of these periods. Moreover, by analyzing the out-of-sample risk-adjusted returns we show that the proposed method outperforms the ordinary LS estimator in terms of the most common performance indices. Finally, a bootstrap analysis suggests that significantly different Sharpe ratios between LS and LQ yields and Value at Risk estimates can be considered more accurate in the LQ framework. This study adds insights into market analysis and helps in identifying more precisely potentially risky assets whose extreme behavior is strongly dependent on market behavior.

## 1. Introduction

^{®}Europe 600. Specifically, an empirical comparison between the linear and quartic estimators is pointed out. Finally, Section 5 concludes.

## 2. The Role of Higher Moments in Financial Risk Measurement

## 3. Methodology

#### 3.1. Co-Moments

_{2,1}∈ ℜ. The corresponding standardized moment is defined as ${\lambda}_{1,2}=\frac{{\mu}_{1,2}}{{\sigma}_{x}{\sigma}_{y}^{2}}$ and similarly for ${\lambda}_{2,1}$. Figure 1 illustrates graphically the behavior of positive and negative co-skewness showing the two density functions and the associated scatter diagrams.

#### 3.2. The Least Quartic Criterion

## 4. Least Quartic vs. Least Squares Estimators: An Empirical Comparison

^{®}Europe 600 index stocks, covering the period from January 2001 to December 2016. Based on daily quotes, log-returns are computed. The unconditional distribution of all the considered daily returns series, and the benchmark itself, fall outside the normal distribution schema, thus confirming the conclusions of most empirical studies. Both Kolmogorov–Smirnov and Shapiro–Wilks tests reject the hypothesis of normality with more than 1% significance level for all returns series. The observed deviations from the Gaussian provide the main justification for using the proposed LQ method in beta estimation; this is reinforced by the marked prevalence of negative skew (65% of the assets) and of positive excess kurtosis (95% of the assets). Almost all values of both skewness and kurtosis are significantly different from the Gaussian distribution reference values at the usual significance levels.

^{®}Europe 600 index over the whole sample period. All estimates are significantly different from zero with less than 5% significance level. Looking at the two patterns showed in Figure 5, it is clear that the perception of market risk exposure is best captured by the LQ estimator during market turmoil, like the recent financial crisis in 2007 and the sovereign debt crisis in 2011.

- estimate the optimal vector of weights ${\mathit{w}}_{k,{l}_{j}}^{\prime}$ using the estimation window composed of the previous $M$ daily returns of each component, where $M$ is the rolling window length (three different equal-sized sections of 100–200 and 500 data points are tested);
- compute the returns for the following out-of-sample window, which ends at ${l}_{j+1}$, keeping fixed the optimal set of weights.

^{®}Europe 600 Index:

^{®}Europe 600 index. The outcomes from the rolling strategies are coherent with those of the whole period, experiencing better indicators when the window length $M$ increases and significantly improving the strategies’ overall out-of-sample annualized return and risk-adjusted performance measured by the Sharpe ratio. In terms of turnover, not surprisingly, we find that the value of $TR$ decreases with $M$ for all the employed strategies; however, in this case, the LQ approach does not differ significantly from the LS solution.

^{®}Europe 600 index stocks. Secondly, we fit an LQ regression model and calculate a regression coefficient from each bootstrap sample (in other words we obtain 500 bootstrap replicates of the regression coefficients obtained with the proposed procedure). Thirdly, we obtain the bootstrap estimates of standard errors of the regression parameters and compute the empirical p-values. If we consider statistical significance at the 5% level, in around 93% of the cases LQ regression coefficient estimates are significant. Finally, the null hypothesis ${H}_{0}:{\widehat{SR}}_{LQ}={\widehat{SR}}_{LS}$ is tested. Around 80% of the simulations yielded significantly different Sharpe ratios at the 5% significance level. Specifically, since the Sharpe ratio is simply the return per unit of risk (represented by the standard deviation), the higher it is, the better the combined performance of risk and return. Therefore, this ratio minimizes the probability that a future portfolio return falls below the risk-free rate. Moreover, since leptokurtic and asymmetric distributions often occur in financial time series, we can assume that if, in the regression coefficient estimates, we also take into account third and fourth moments of returns, we may obtain a better measurement of risk. The fact that the bootstrap hypothesis testing leads to the rejection of the null hypothesis that ${\widehat{SR}}_{LQ}={\widehat{SR}}_{LS}$ in most of the cases examined, suggests that the risk assessment obtained using our proposed method outperforms the classic one. In particular, the risk estimation made using the Sharpe ratio based on the quartic estimator is better than the one obtained using the ordinary LS procedure in 85% of the analyzed time series. Moreover, if we consider a 95% VaR over a one-day holding horizon and we backtest the loss forecasted by Value-at-Risk compared to the actual ones, we observe that in 93% of the time series considered the frequency of exceedances is consistent with the specified confidence level when the LQ estimator is used. The accuracy reduces to 72% when the LS one is taken into account. Consequently, VaR estimates are more accurate in the LQ framework.

## 5. Conclusions

^{®}Europe 600. The empirical analysis, based on the least quartic estimation of the slope coefficient, adds insights into market analysis and helps in identifying more precisely potentially risky assets whose extreme behavior is strongly dependent on the market behavior. By comparing the slope coefficients calculated using the traditional CAPM expression based on the ordinary LS method and the alternative LQ technique, it emerges that the perception of market risk exposure is best captured by the LQ estimator during market turmoil (e.g., the financial crisis in 2007 and the sovereign debt crisis in 2011). The out-of-sample analysis, based on 1000 simulated portfolios, also shows how the LQ criterion in most cases outperforms the traditional LS optimization strategy with consistently higher mean returns, lower variability and higher values both of the Sharpe ratio and of the correlation with the STOXX

^{®}Europe 600 index. Finally, through the bootstrapping methodology, we show that the VaR and risk estimation obtained using the Sharpe ratio based on the quartic estimator are better than those obtained using the ordinary LS procedure in most of the time series analyzed at the 5% significance level.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Various types of co-skewness: (

**a**) positive left-tail co-skewness; (

**b**) positive right-tail co-skewness; (

**c**) negative co-skewness. Scatter diagrams (upper pane); probability density functions (lower panel).

**Figure 2.**Various types of co-kurtosis: (

**a**) positive lepto co-kurtosis; (

**b**) positive plati co-kurtosis; (

**c**) negative co-kurtosis. Scatter diagrams (upper panel); probability density functions (lower panel).

**Figure 3.**The plot of a quartic polynomial. For the sake of illustration we set ${\mu}_{4,0}=1$; ${\mu}_{3,1}=-4$; ${\mu}_{2,2}=6$; ${\mu}_{1,3}=2$ and ${\mu}_{0,4}=1$.

**Figure 4.**Co-skewness and co-kurtosis pattern for a sample-asset: (

**a**) ${\lambda}_{1,2}$; (

**b**) ${\lambda}_{2,1}$; (

**c**) ${\kappa}_{1,3}$; (

**d**) ${\kappa}_{2,2}$; (

**e**) ${\kappa}_{3,1}$.

Global | Rolling | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{M}=100$ | $\mathit{M}=200$ | $\mathit{M}=500$ | ||||||

LS | LQ | LS | LQ | LS | LQ | LS | LQ | |

$\widehat{r}$ | 4.963 | 5.884 | 1.946 | 1.995 | 2.283 | 2.333 | 6.748 | 7.219 |

${\widehat{\sigma}}_{r}$ | 23.821 | 22.955 | 29.013 | 28.993 | 27.075 | 26.622 | 24.943 | 24.469 |

${\widehat{\sigma}}_{r}^{-}$ | 0.969 | 0.933 | 1.073 | 1.065 | 1.056 | 1.036 | 0.974 | 0.953 |

$\widehat{SR}$ | 0.242 | 0.284 | 0.066 | 0.069 | 0.089 | 0.093 | 0.974 | 0.953 |

${\rho}_{\widehat{r},M}$ | 0.899 | 0.908 | 0.797 | 0.810 | 0.823 | 0.826 | 0.859 | 0.861 |

$Tr$ | - | - | 0.144 | 0.143 | 0.115 | 0.115 | 0.058 | 0.058 |

**Table 2.**Percentages of cases where the LQ estimates are better than the corresponding LS alternatives.

Global | Rolling | |||
---|---|---|---|---|

$\mathit{M}=100$ | $\mathit{M}=200$ | $\mathit{M}=500$ | ||

$\widehat{r}$ | 58 | 53 | 55 | 52 |

${\widehat{\sigma}}_{r}$ | 74 | 62 | 61 | 69 |

${\widehat{\sigma}}_{r}^{-}$ | 70 | 60 | 63 | 68 |

$\widehat{SR}$ | 57 | 53 | 57 | 61 |

${\rho}_{\widehat{r},M}$ | 56 | 55 | 57 | 61 |

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**MDPI and ACS Style**

Arbia, G.; Bramante, R.; Facchinetti, S.
Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis. *Risks* **2020**, *8*, 95.
https://doi.org/10.3390/risks8030095

**AMA Style**

Arbia G, Bramante R, Facchinetti S.
Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis. *Risks*. 2020; 8(3):95.
https://doi.org/10.3390/risks8030095

**Chicago/Turabian Style**

Arbia, Giuseppe, Riccardo Bramante, and Silvia Facchinetti.
2020. "Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis" *Risks* 8, no. 3: 95.
https://doi.org/10.3390/risks8030095