# Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. The Capital Asset Pricing Model

#### 2.2. Elliptical Distributions

**Definition**

**1.**

**Definition**

**2.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Property**

**1.**

**Property**

**2.**

**Example**

**5.**

- (a)
- If $X\sim {\mathcal{N}}_{p}(\mu ,\Omega )$, we have that ${R}^{2}\sim {\chi}^{2}\left(p\right)$ and therefore ${c}_{0}=1$. Then, $\mathrm{E}\left(X\right)=\mu $. $\mathrm{Cov}\left(X\right)=\Omega $, and the function that generates the CF is $\varphi \left(u\right)={\mathrm{e}}^{-\frac{u}{2}}$, for $u\ge 0$.
- (b)
- If $X\sim {t}_{p}(\mu ,\Omega ,\nu )$, then ${R}^{2}/p\sim \mathcal{F}(p,\nu )$. Thus, ${c}_{0}=\nu /\left(\nu -2\right)$, $\mathrm{E}\left(X\right)=\mu $, and $\mathrm{Cov}\left(X\right)=(\nu /(\nu -2\left)\right)\Omega $, for $\nu >2$. For the Student-t distribution, the CF is given in [51].
- (c)
- If $X\sim {\mathrm{NC}}_{p}(\mu ,\Omega ,w,\gamma )$, we have ${c}_{0}=1+w(\gamma -1)$, $\mathrm{Cov}\left(X\right)=(1+w(\gamma -1\left)\right)\Omega $, $\mathrm{E}\left(X\right)=\mu $, and $\varphi \left(u\right)=(1-w){\mathrm{e}}^{-\frac{u}{2}}+w{\mathrm{e}}^{-\frac{u\gamma}{2}}$, for $u\ge 0.$
- (d)
- If $X\sim {\mathrm{PE}}_{p}(\mu ,\Omega ,\nu )$, then $\mathrm{E}\left({R}^{s}\right)=({2}^{\frac{s(1+\nu )}{2}}\Gamma ((p+s)(\nu +1)/2))/\Gamma (p(\nu +1)/2).$ For $s>0$ (integer), we obtain $\mathrm{Cov}\left(X\right)=({2}^{(1+\nu )}\Gamma ((p+2)(\nu +1)/2))/(p\Gamma (p(\nu +1)/2)\Omega )$ and $\mathrm{E}\left(X\right)=\mu $. The CF of the PE distribution is given in [52].

**Property**

**3.**

**Example**

**6.**

- (a)
- Let $X\sim {t}_{p}(\mu ,\Omega ,\nu )$. Then, we have that $\mathrm{E}\left({R}^{2}\right)=p\nu /\left(\nu -2\right)=-2p{\varphi}^{\prime}\left(0\right)$, for $\nu >2$, and $\mathrm{E}\left({R}^{4}\right)={\nu}^{2}p(p+2)/\left((\nu -2)(\nu -4)\right)=4p(p+2){\varphi}^{\u2033}\left(0\right)$, for $\nu >4$. Therefore, we have ${\varphi}^{\prime}\left(0\right)=-(1/2)(\nu /\left(\nu -2\right))$, for $\nu >2$, and ${\varphi}^{\u2033}\left(0\right)={\nu}^{2}/\left(4(\nu -2)(\nu -4)\right)$, for $\nu >4$.
- (b)
- Let $X\sim {\mathrm{PE}}_{p}(\mu ,\Omega ,\nu )$. Then, we get $\mathrm{E}\left({R}^{2}\right)=\left({2}^{(1+\nu )}\Gamma ((p+2)(\nu +1)/2)\right)/\Gamma (p(\nu +1)/2)$$=-2p{\varphi}^{\prime}\left(0\right)$, $\mathrm{E}\left({R}^{4}\right)=\left({2}^{2(1+\nu )}\Gamma ((p+4)(\nu +1)/2)\right)/\Gamma (p(\nu +1)/2)=4p(p+2){\varphi}^{\u2033}\left(0\right).$ The above implies that ${\varphi}^{\prime}\left(0\right)=({2}^{\nu}/p)(\Gamma ((p+2)(\nu +1)/2)/\Gamma (p(\nu +1)/2))$ and ${\varphi}^{\u2033}\left(0\right)={2}^{2\nu}/\left(p(p+2)\right)(\Gamma \left((p+4)(\nu +1)/2\right)/\Gamma (p(\nu +1)/2)).$ Note that when $\nu =0$, the normal case is recovered, that is, ${\varphi}^{\prime}\left(0\right)=-1/2$ and ${\varphi}^{\u2033}\left(0\right)=1/4$.

**Property**

**4.**

**Definition 3**(SMN distribution).

## 3. Estimation and Diagnostics in the Elliptical CAPM

#### 3.1. EM Algorithm

- First step (E): Calculate $Q=\left(\theta \right|\widehat{\theta})$ previously defined for $\widehat{\theta}={\theta}^{(r-1)}$, that is, $Q\left(\theta \right|{\theta}^{(r-1)})$$=\mathrm{E}\left(\mathcal{L}\left(\theta \right|{Y}_{\mathrm{c}})\right|{Y}_{\mathrm{o}},{\theta}^{(r-1)})$.
- Second step (M): Choose ${\theta}^{\left(r\right)}$ that maximizes $Q\left(\theta \right|\widehat{\theta})$ such that $Q\left({\theta}^{\left(r\right)}\right|\widehat{\theta})\ge Q\left(\theta \right|\phantom{\rule{4pt}{0ex}}\widehat{\theta}).$

#### 3.2. Local Influence Analysis

#### 3.3. Generalized Leverage

## 4. Empirical Application

#### 4.1. Descriptive Analysis

#### 4.2. Descriptive Measures of Monthly Returns

#### 4.3. Descriptive Graphs for Monthly Returns of CMPC

#### 4.4. CAPM under Normality Assumption

#### 4.5. CAPM under Elliptical Distributions

#### 4.6. Diagnostic Analysis of Selected Models

## 5. Economic Rationale for the Results of the Diagnostic Analysis

#### 5.1. Context

#### 5.2. January 2009

#### 5.3. May 2009

#### 5.4. September 2009

#### 5.5. December 2009

#### 5.6. October 2011

#### 5.7. May 2013

#### 5.8. October 2017

#### 5.9. November 2017

#### 5.10. December 2017

#### 5.11. May 2019

#### 5.12. Lessons for Dummies

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Acronyms

Acronyms | Meaning |
---|---|

AIC | Akaike information criterion |

BCI | Credit and Investment Bank of Chile |

CAPM | Capital asset pricing model |

CCU | The United Breweries Company of Chile |

CMPC | Chilean Paper Manufacturing Company |

ECL | Engie Energia Chile SA |

ED | Elliptical distribution |

EM | Expectation-maximization |

ENELAM | ENEL Americas SA Company |

ENTEL | Chilean telecommunications company |

IAM | Investments Metropolitan Waters |

IBOVESPA | Brazilian stock market index |

IMCE | Monthly Indicator of Business Confidence |

IPSA | Selective stock price index, Chilean market |

PE | Power-exponential |

SMN | Scale mixture of normals |

SMU | Supermarket retail company in Chile |

SQMB | Chemical and Mining Society of Chile |

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**Figure 1.**Plot of the expected return treasury bills and the market portfolio. Source: [15].

**Figure 7.**Scatter plot of monthly returns for the IPSA and CMPC with fitted CAPM lines under the Student-t and normal distributions.

**Figure 8.**Scatter plot of monthly returns for the IPSA and CMPC with fitted CAPM line under the PE distribution.

**Figure 9.**Scatter plot of monthly returns for the IPSA and CMPC with fitted CAPM lines under the normal, Student-t, and PE distributions.

**Figure 11.**Plots of total local influence ${C}_{h}\left(\theta \right)$ under case-weight perturbations in the indicated model with data of monthly returns.

**Figure 12.**Plots of total local influence ${C}_{h}\left(\theta \right)$ under the scale perturbations in the indicated model with data of monthly returns.

Year | Author(s) | Title | Topic |
---|---|---|---|

1964 | Sharpe [2] | Capital asset prices: a theory of market equilibrium under conditions of risk | CAPM |

1965 | Fama [18] | The behavior of stock market prices | CAPM |

1970 | Kelker [26] | Distribution theory of spherical distributions | ED |

and a location-scale parameter generalization | |||

1974 | Blattberg and Gonedes [19] | Comparison of the stable and Student-t distributions as | CAPM |

statistical models for stock prices | & ED | ||

1977 | Kariya and Eaton [27] | Robust tests for spherical symmetry | ED |

1980 | Muirhead [28] | The effects of elliptical distributions on some standard | ED |

procedures involving correlation coefficients | |||

1985 | Hayakana and Puri [29] | Asymptotic distributions of likelihood ratio criteria for testing latent roots | ED |

and latent vectors of a covariance matrix under an elliptical population | |||

1990 | Fang and Zhang [30] | Generalized multivariate analysis | ED |

1993 | Zhou [20] | Asset pricing tests under alternative distributions | CAPM |

2002 | Hodgson et al. [37] | Semiparametric elliptical CAPM | CAPM & ED |

2003 | Cademartori et al. [17] | Robust estimation of systematic risk using the t distribution | CAPM |

in the Chilean stock markets | & ED | ||

2003 | Díaz-García et al. [22,23,24] | Multivariate elliptical models | ED |

2003 | Galea et al. [31] | On influence diagnostic in univariate elliptical linear regression model | ED |

2004 | Hung et al. [12] | CAPM, higher co-moment and factor models of UK stock returns | CAPM |

2008 | Galea et al. [38] | Influence diagnostics in the elliptical CAPM | CAPM & ED |

2008 | Xu and Hou [39] | CAPM with generalized elliptical distribution | CAPM & ED |

2008 | Hamada and Valdez [40] | CAPM and option pricing with elliptically contoured distributions | CAPM & ED |

2009 | Castro and Silveira [41] | CAPM with IBOVESPA and S&P500 data | CAPM |

2009 | Paula and Cysneiros [42] | Systematic risk estimation in symmetric models | CAPM & ED |

2011 | Riquelme et al. [32] | Influence diagnostics on the coefficient of variation | ED |

of elliptically contoured distributions | |||

2015 | Ang [13] | Analyzing financial data and implementing financial models using R | CAPM |

2015 | Riquelme et al. [33] | Robust linear functional mixed models | ED |

2015 | Ruppert and Matteson [14] | Statistics and data analysis for financial engineering | CAPM |

2016 | Tzang et al. [43] | Systematic risk and volatility skew | CAPM |

2017 | Arashi and Nadarajah [34] | Generalized elliptical distributions | ED |

2017 | Brealey et al. [15] | Principles of corporate finance | CAPM |

2017 | Zhang [16] | The investment CAPM | CAPM |

2019 | Galea and Giménez [44] | Local influence diagnostics in the CAPM | CAPM |

2019 | Ventura et al. [35] | Log-symmetric regression models: information criteria and application | ED |

to the movie business and industry data with economic implications | |||

2020 | Galea et al. [25] | Robust inference in the multivariate Student-t CAPM | CAPM & ED |

2020 | Lesniewska-Choquet et al. [36] | On elliptical possibility distributions | ED |

2021 | Chen et al. [11] | Research and analysis of asset pricing model based | CAPM |

on the empirical test of stock price |

Monthly Return | Monthly Return | |
---|---|---|

CMPC | IPSA | |

Minimum | −0.16740 | −0.10463 |

First quartile | −0.03590 | −0.02360 |

Median | 0.00125 | 0.00196 |

Mean | 0.01012 | 0.00606 |

Third quartile | 0.04990 | 0.03640 |

Maximum | 0.20780 | 0.16087 |

Range | 0.37520 | 0.26550 |

Variance | 0.00502 | 0.00190 |

Standard deviation | 0.07086 | 0.04361 |

Coefficient of variation | 6.99936 | 7.19695 |

Coefficient of skewness | 1.01904 | 1.05948 |

Coefficient of kurtosis | 0.22305 | 0.70998 |

Measure | Value |
---|---|

D-statistic | 0.08 |

p-value | 0.05 |

Coefficient | Estimate | Estimated Standard Error | p-Value |
---|---|---|---|

Intercept ($\alpha $) | $0.00384$ | $0.00452$ | $0.400$ |

Slope $\beta $ | $1.11175$ | $0.10409$ | <0.001 |

Estimated scale parameter: 0.0519 |

Distribution | Intercept ($\widehat{\mathit{\alpha}}$) | Slope ($\widehat{\mathit{\beta}}$) | ${\widehat{\mathit{\sigma}}}^{2}$ | AIC |
---|---|---|---|---|

$t\left(4\right)$ | $0.00061$ | $1.04640$ | $0.00173$ | $-400.08$ |

$t\left(5\right)$ | $0.00101$ | $1.05472$ | $0.00185$ | $-401.59$ |

$t\left(6\right)$ | $0.00131$ | $1.06128$ | $0.00194$ | $-402.40$ |

$t\left(7\right)$ | $0.00155$ | $1.06657$ | $0.00201$ | $-402.86$ |

$t\left(8\right)$ | $0.00175$ | $1.07092$ | $0.00207$ | $-403.12$ |

$t\left(9\right)$ | $0.00192$ | $1.07454$ | $0.00212$ | $-403.28$ |

$t\left(10\right)$ | $0.00206$ | $1.07759$ | $0.00216$ | $-403.38$ |

$t\left(11\right)$ | $0.00218$ | $1.08020$ | $0.00219$ | $-403.43$ |

$t\left(12\right)$ | $0.00228$ | $1.08246$ | $0.00222$ | $-403.46$ |

$t\left(13\right)$ | $0.00238$ | $1.08442$ | $0.00225$ | $-403.47$ |

$t\left(14\right)$ | $0.00246$ | $1.08615$ | $0.00228$ | $-403.47$ |

$t\left(15\right)$ | $0.00253$ | $1.08767$ | $0.00230$ | $-403.46$ |

$t\left(16\right)$ | $0.00260$ | $1.08903$ | $0.00231$ | $-403.45$ |

$t\left(17\right)$ | $0.00266$ | $1.09025$ | $0.00233$ | $-403.43$ |

$t\left(18\right)$ | $0.00271$ | $1.09134$ | $0.00235$ | $-403.41$ |

$t\left(19\right)$ | $0.00276$ | $1.09234$ | $0.00236$ | $-403.39$ |

$t\left(20\right)$ | $0.00280$ | $1.09324$ | $0.00237$ | $-403.37$ |

$t\left(21\right)$ | $0.00285$ | $1.09406$ | $0.00238$ | $-403.35$ |

$t\left(22\right)$ | $0.00288$ | $1.09482$ | $0.00240$ | $-403.33$ |

$t\left(23\right)$ | $0.00292$ | $1.09551$ | $0.00241$ | $-403.31$ |

$t\left(24\right)$ | $0.00295$ | $1.09615$ | $0.00241$ | $-403.30$ |

$t\left(25\right)$ | $0.00298$ | $1.09674$ | $0.00242$ | $-403.28$ |

$t\left(26\right)$ | $0.00301$ | $1,09729$ | $0.00243$ | $-403.26$ |

$t\left(27\right)$ | $0.00304$ | $1,09780$ | $0.00244$ | $-403.24$ |

$t\left(28\right)$ | $0.00306$ | $1,09828$ | $0.00244$ | $-403.22$ |

$t\left(29\right)$ | $0.00309$ | $1,09872$ | $0.00245$ | $-403.21$ |

$t\left(30\right)$ | $0.00311$ | $1.09914$ | $0.00246$ | $-403.19$ |

$t\left(50\right)$ | $0.00338$ | $1.10408$ | $0.00253$ | $-402.97$ |

$t\left(100\right)$ | $0.00360$ | $1.10788$ | $0.00259$ | $-402.75$ |

Normal | $0.00384$ | $1.11175$ | $0.00265$ | $-402.48$ |

Distribution | Intercept ($\widehat{\mathit{\alpha}}$) | Slope ($\widehat{\mathit{\beta}}$) | ${\widehat{\mathit{\sigma}}}^{2}$ | AIC |
---|---|---|---|---|

PE($-0.3$) | $0.00738$ | $1.171$ | $0.0045$ | $-396.311$ |

PE($-0.2$) | $0.00595$ | $1.148$ | $0.0038$ | $-399.367$ |

PE($-0.1$) | $0.00479$ | $1.128$ | $0.00317$ | $-401.340$ |

PE(0) | $0.00384$ | $1.111$ | $0.00265$ | $-402.489$ |

PE($0.1$) | $0.00306$ | $1.099$ | $0.00222$ | $-402.991$ |

PE($0.2$) | $0.00241$ | $1.090$ | $0.00185$ | $-402.980$ |

PE($0.3$) | $0.00184$ | $1.083$ | $0.00155$ | $-402.561$ |

PE($0.4$) | $0.00131$ | $1.077$ | $0.00129$ | $-401.819$ |

PE($0.5$) | $0.00078$ | $1.071$ | $0.00107$ | $-400.822$ |

PE($0.6$) | $0.00027$ | $1.064$ | $0.00088$ | $-399.627$ |

PE($0.7$) | $-0.00010$ | $1.050$ | $0.00088$ | $-398.278$ |

PE($0.8$) | $-0.00065$ | $1.025$ | $0.00060$ | $-396.821$ |

PE($0.9$) | $-0.00178$ | $0.973$ | $0.00049$ | $-395.345$ |

PE(1) | $-0.00189$ | $0.967$ | $0.00040$ | $-393.906$ |

Normal | $0.00384$ | $1.111$ | $0.00265$ | $-402.489$ |

Coefficient | Estimate | Estimated Standard Error | p-Value |
---|---|---|---|

$t\left(13\right)$ distribution | |||

Intercept ($\alpha $) | 0.00237 | 0.00441 | 0.59099 |

Slope $\beta $ | 1.08442 | 0.10175 | <0.001 |

Estimated scale parameter (standard error): 0.00225 (0.00030) | |||

PE(0.1) distribution | |||

Intercept ($\alpha $) | 0.00306 | 0.00446 | 0.49321 |

Slope $\beta $ | 1.09978 | 0.10273 | <0.001 |

Estimated scale parameter (standard error): 0.00221 (0.00028) |

**Table 8.**Generalized leverage influencing values in the indicated model with data of monthly returns.

Month | Model | ||
---|---|---|---|

ID | Normal | t(13) | PE(0.1) |

5 | $0.103469824$ | $0.0937170021$ | $0.089626797$ |

107 | $0.056541808$ | $0.0671930662$ | $0.060098533$ |

34 | $0.053399058$ | $0.0662188435$ | $0.064053963$ |

108 | $0.053121313$ | $0.065466301$ | $0.06090873$ |

33 | $0.047313871$ | $0.0365877282$ | $0.039377753$ |

12 | $0.042417174$ | $0.0514758602$ | $0.045672545$ |

99 | $0.041406406$ | $0.0518224811$ | $0.056815278$ |

**Table 9.**Total local influence ${C}_{h}\left(\theta \right)$ under case-weight perturbations influencing values in the indicated model with data of monthly returns.

Month | Model | ||
---|---|---|---|

ID | Normal | t(13) | PE(0.1) |

53 | $0.860949117$ | $0.40570635$ | $0.681177036$ |

106 | $0.716484295$ | $0.405246402$ | $0.591312213$ |

9 | $0.426412238$ | $0.33028374$ | $0.37581537$ |

125 | $0.378345502$ | $0.265678507$ | $0.318202773$ |

1 | $0.320817151$ | $0.295408047$ | $0.296590614$ |

5 | $0.286899237$ | $0.304962844$ | $0.290597895$ |

7 | $0.183248408$ | $0.182029441$ | $0.172338372$ |

**Table 10.**Values of total local influence ${C}_{h}\left(\theta \right)$ under scale perturbations in the indicated model with data of monthly returns.

Month | Model | ||
---|---|---|---|

ID | Normal | t(13) | PE(0.1) |

53 | $1.0072186178960$ | $0.1406609575389$ | $0.6778691678711$ |

106 | $0.8428877736200$ | $0.15463982224536$ | $0.5903240985097$ |

9 | $0.5092675356114$ | $0.1704991756999$ | $0.3800676606066$ |

125 | $0.4612890324318$ | $0.14683653584494$ | $0.3306427541719$ |

1 | $0.3817210438169$ | $0.17909510321507$ | $0.2978430086623$ |

5 | $0.3002488531891$ | $0.26009152074061$ | $0.2514919330143$ |

7 | $0.2383909956002$ | $0.12823410225654$ | $0.1903168970739$ |

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## Share and Cite

**MDPI and ACS Style**

Leal, D.; Jiménez, R.; Riquelme, M.; Leiva, V.
Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale. *Mathematics* **2023**, *11*, 1394.
https://doi.org/10.3390/math11061394

**AMA Style**

Leal D, Jiménez R, Riquelme M, Leiva V.
Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale. *Mathematics*. 2023; 11(6):1394.
https://doi.org/10.3390/math11061394

**Chicago/Turabian Style**

Leal, Danilo, Rodrigo Jiménez, Marco Riquelme, and Víctor Leiva.
2023. "Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale" *Mathematics* 11, no. 6: 1394.
https://doi.org/10.3390/math11061394