# Modeling System Risk in the South African Insurance Sector: A Dynamic Mixture Copula Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Marginal Expected Shortfall (MES)

#### Expected Shortfall (ES)

#### 2.2. Dynamic Mixture Copula-Marginal Expected Shortfall (DMC-MES)

#### 2.2.1. Copula

#### 2.2.2. Construction of the DMC-MES

- ${\epsilon}_{m,t}$ and ${\epsilon}_{i,t}$ are each independent and identically distributed (i.i.d) with unspecified, static distribution ${G}_{m}$ and ${G}_{i}$
- $E\left({\epsilon}_{m,t}\right)=E\left({\epsilon}_{i,t}\right)=0$, $Var\left({\epsilon}_{m,t}\right)=Var\left({\epsilon}_{i,t}\right)=1$ and
- $p\left({\epsilon}_{i,t}\le {\epsilon}_{i},{\epsilon}_{m,t}\le {\epsilon}_{m}\right)={C}_{t}\left(\underset{{v}_{i}}{\underset{\u23df}{{G}_{i}\left({\epsilon}_{i}\right)}},\underset{{v}_{m}}{\underset{\u23df}{{G}_{m}\left({\epsilon}_{m}\right)}},{\theta}_{t}\right)$ with dynamic copula parameter ${\theta}_{t}$.

#### 2.3. Estimation Technique of the DMC-MES

**Step 1**

**Step 2**

**Step 3**

#### 2.4. The Symmetrized Joe-Clayton (SJC) Copula

#### 2.5. Robustness Test

#### 2.5.1. Clayton Copula

#### 2.5.2. Gumbel Copula

#### 2.5.3. Vector Autoregressive Model and Impulse Responses

- ${Y}_{t}=\left({y}_{1t},{y}_{2t},\dots ,{y}_{nt}\right)\prime $: an $\left(n\times 1\right)$ vector of time series variables;
- $a:$ an $\left(n\times 1\right)$ vector of intercepts;
- ${A}_{i}\left(i=1,2,\dots ,p\right):$ an $\left(n\times n\right)$ coefficient matrices;
- ${\epsilon}_{t}:$ an $\left(n\times 1\right)$ vector of unobservable $i.i.d$ zero mean error term.

## 3. Empirical Analysis

#### 3.1. Data

#### 3.2. Marginal Distributions

#### 3.3. Estimation of the Copula Models

#### 3.4. Estimation of the DMC-MES

#### 3.5. Robustness Test

#### 3.5.1. Bivariate Copula

#### 3.5.2. Impulse Response Function

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scatter plots of the marginals. Note that the horizontal axis represents the marginal of the insurers, while the vertical axis represents the marginal of the market.

Company | Symbol | Sector |
---|---|---|

Discovery Limited | DSY | Life Insurance |

Liberty Holdings Limited | LBH | Life Insurance |

Momentum Metropolitan Holdings | MTM | Life Insurance |

Sanlam Limited | SLM | Life Insurance |

Santam Limited | SNT | Nonlife Insurance |

Discovery | Liberty | Momentum | Sanlam | Santam | |
---|---|---|---|---|---|

Mean | 0.041 | −0.005 | 0.005 | 0.03 | 0.026 |

Std.Dev. | 1.957 | 1.892 | 1.868 | 1.959 | 1.712 |

Skewness | −0.527 | 0.129 | −0132 | −0.429 | −0.474 |

Kurtosis | 12.695 | 17.853 | 7.183 | 7.703 | 14.13 |

JB test p-value | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |

Model | Ljung–Box Test | Arch LM Test | ||
---|---|---|---|---|

Discovery | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.4106 0.06036 | 0.3931 |

Liberty | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.4117 0.2674 | 0.8041 |

Momentum | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.1185 0.0823 | 0.6793 |

Sanlam | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.6952 0.7390 | 0.5538 |

Santam | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.9213 0.7559 | 0.5491 |

Market | AR(1)-GJRGARCH(1,1) | Standardized residuals Standardized squared residuals | 0.31172 0.4614 | 0.4527 |

AR (1) | GJR-GARCH (1,1) | Kurtosis | Skewness | |||||
---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{0}$ | ${\mathit{\beta}}_{0}$ | $\mathit{\omega}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | $\mathit{\gamma}$ | |||

Discovery | 0.0476 (0.0252) | −0.0058 (0.0183) | 0.0385 *** (0.0115) | 0.0517 *** (0.0126) | 0.9058 *** (0.0133) | 0.0669 *** (0.0182) | 4.9419 *** (1.571) | −0.2145 *** (0.0259) |

Liberty | 0.0159 (0.0251) | −0.0664 *** (0.0165) | 0.0366 *** (0.0054) | 0.0001 (0.0036) | 0.9573 *** (0.0024) | 0.0603 *** (0.0024) | 12.32 *** (0.330) | −0.0801 *** (0.0246) |

Momentum | 0.0321 (0.0263) | −0.0683 *** (0.0181) | 0.0567 *** (0.0214) | 0.0564 *** (0.0173) | 0.9138 *** (0.0191) | −0.0318 * (0.0171) | 8.3628 *** (0.9165) | 0.0101 *** (0.0027) |

Sanlam | 0.0221 (0.0259) | −0.0690 *** (0.0180) | 0.0526 *** (0.0154) | 0.0353 *** (0.0123) | 0.9098 *** (0.0138) | 0.0849 *** (0.0182) | 4.7317 *** (1.3719) | −0.1677 *** (0.0153) |

Santam | 0.0299 (0.0242) | −0.1172 *** (0.0182) | 0.4707 *** (0.1510) | 0.2065 *** (0.0048) | 0.6849 *** (0.0737) | 0.0292*** (0.0046) | 16.038 *** (2.195) | −0.6265 ** (0.249) |

**Table 5.**Estimation Results of the Dynamic Mixture Copula of the Rotated Clayton and Clayton (RC&C): Between Market and Insurers.

Weight | GAS (1,1) | ||||||
---|---|---|---|---|---|---|---|

$\tilde{\mathit{w}}$ | ${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{A}}_{11}$ | ${\mathit{A}}_{22}$ | ${\mathit{B}}_{11}$ | ${\mathit{B}}_{22}$ | |

Discovery | 0.392 *** (0.031) | 0.109 *** (0.078) | 0.002 *** (0.0001) | 0.233 ** (0.118) | 0.012 *** (0.004) | 0.849 *** (0.107) | 0.998 *** (0.002) |

Liberty | 0.485 *** (0.039) | 0.0005 *** (0.00001) | 0.0206 (0.021) | 0.036 *** (0.009) | 0.136 (0.097) | 0.998 *** (0.004) | 0.959 *** (0.045) |

Momentum | 0.463 *** (0.065) | −0.003 *** (0.0001) | 0.007 *** (0.0001) | 0.067 (0.104) | 0.023 *** (0.0012) | 1.003 *** (0.0001) | 0.989 *** (0.0004) |

Sanlam | 0.347 *** (0.029) | −0.002 *** (0.0001) | −0.001 *** (0.0005) | 0.011 (0.012) | 0.044 *** (0.008) | 1.0014 *** (0.0001) | 1.0013 *** (0.0003) |

Santam | 0.423 *** (0.057) | −0.002( 0.004) | 0.0001 (0.0004) | 0.094 (0.09) | 0.034 (0.041) | 0.995 *** (0.011) | 0.998 *** (0.003) |

**Table 6.**Estimation Results of Dynamic Mixture Copula of the Rotated Gumbel and Gumbel (RG&G): Between Markets and Insurers.

Weight | GAS (1,1) | ||||||
---|---|---|---|---|---|---|---|

$\tilde{\mathit{w}}$ | ${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{A}}_{11}$ | ${\mathit{A}}_{22}$ | ${\mathit{B}}_{11}$ | ${\mathit{B}}_{22}$ | |

Discovery | 0.639 *** (0.044) | −0.006 (0.026) | 0.082 (0.113) | 0.144 *** (0.093) | 0.175 (0.235) | 0.659 *** (0.221) | 0.539 (0.606) |

Liberty | 0.555 *** (0.049) | −0.132 (0.105) | −0.327 (0.241) | 0.229 (0.159) | −0.175 (0.227) | 0.497 (0.329) | 0.394 (0.641) |

Momentum | 0.567 *** (0.0009) | −0.0006 *** (0.00001) | −0.002 ** (0.0001) | −0.017 *** (0.0001) | 0.048 ** (0.003) | 1.0004 *** (0.0001) | 0.991 *** (0.0004) |

Sanlam | 0.696 *** (0.045) | 0.019 *** (0.009) | 0.039 (0.029) | 0.032 ** (0.012) | −0.154 (0.114) | 0.982 *** (0.009) | 0.961 *** (0.029) |

Santam | 0.643 *** (0.086) | −0.003 (0.006) | −1.212 ** (0.524) | 0.042 (0.047) | 2.159 (1.528) | 0.996 *** (0.007) | 0.948 *** (0.316) |

Time-Varying SJC Copula | |||||
---|---|---|---|---|---|

Discovery | Liberty | Momentum | Sanlam | Santam | |

${\omega}^{U}$ | 1.958 (1.58) | 2.032 ** (0.96) | 0.094 * (0.05) | 8.707 (23.14) | −0.040 (0.37) |

${\alpha}^{U}$ | −9.999 (8.21) | −9.969 ** (4.99) | −0.493 * (0.28) | 9.761 *** (1.24) | −0.733 (0.84) |

${\beta}^{U}$ | 0.431 (0.62) | −0.424 (0.64) | 0.976 *** (0.02) | 4.477 (39.03) | 0.832 *** (0.25) |

${\omega}^{L}$ | 2.849 ** (1.41) | 0.111 *** (0.03) | 2.944 *** (0.48) | 9.984 *** (0.99) | −0.094 (0.39) |

${\alpha}^{L}$ | −9.999 (9.79) | −0.526 *** (0.17) | −9.997 *** (2.45) | 9.889 (92.22) | −1.546 (1.95) |

${\beta}^{L}$ | −0.040 (0.33) | 0.974 *** (0.01) | −0.647 *** (0.13) | −9.796 *** (2.15) | 0.143 (0.34) |

Copulas | tvSJC | RC&C | RG&G |
---|---|---|---|

Copula Likelihood | $\mathit{\mathcal{L}}\mathit{\mathcal{L}}$ | $\mathit{\mathcal{L}}\mathit{\mathcal{L}}$ | $\mathit{\mathcal{L}}\mathit{\mathcal{L}}$ |

Discovery | −1115.9 | −1093.4 | −1117.8 |

Liberty | −768.0 | −762.9 | −778.4 |

Momentum | −910.7 | −903.3 | −925.6 |

Sanlam | −2344.0 | −2693.7 | −2774.4 |

Santam | −304.1 | −305.9 | −307.5 |

Mean | Std.Dev | Min | Max | Ranking | |
---|---|---|---|---|---|

Discovery | 0.983 | 7.709 | 0.0551 | 334.3986 | 2 |

Liberty | 0.583 | 5.748 | 0.0375 | 216.4342 | 4 |

Momentum | 0.805 | 5.580 | 0.0618 | 223.8966 | 3 |

Sanlam | 1.942 | 18.132 | 0.1128 | 894.1638 | 1 |

Santam | −0.007 | 0.100 | −2.3490 | 4.3039 | 5 |

**Table 10.**Parameters for the Copulas and Their 95% Confidence Intervals between Sanlam and the Market, and Sanlam and the Other Insurers.

Copula | Parameters | 95% CI | AIC | BIC |
---|---|---|---|---|

Gumbel(MKT) | 3.431 | [3.343, 3.519] | 5160.0 | 5170.0 |

Clayton(MKT) | 3.568 | [3.462, 3.673] | 4510.0 | 4520.0 |

Gumbel(DSY) | 1.450 | [1.412, 1.487] | 780.4 | 786.4 |

Clayton(DSY) | 0.781 | [0.726, 0.837] | 802.1 | 808.1 |

Gumbel(LBH) | 1.384 | [1.349, 1.419] | 617.6 | 623.7 |

Clayton(LBH) | 0.636 | [0.582, 0.689] | 594.0 | 600.1 |

Gumbel(MTM) | 1.455 | [1.418, 1.492] | 791.2 | 797.2 |

Clayton(MTM) | 0.753 | [0.697, 0.808] | 765.3 | 771.3 |

Gumbel(STN) | 1.775 | [1.149, 1.206] | 175.5 | 181.6 |

Clayton(STN) | 0.354 | [0.305, 0.403] | 233.9 | 240.0 |

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**MDPI and ACS Style**

Muteba Mwamba, J.W.; Angaman, E.S.E.F.
Modeling System Risk in the South African Insurance Sector: A Dynamic Mixture Copula Approach. *Int. J. Financial Stud.* **2021**, *9*, 29.
https://doi.org/10.3390/ijfs9020029

**AMA Style**

Muteba Mwamba JW, Angaman ESEF.
Modeling System Risk in the South African Insurance Sector: A Dynamic Mixture Copula Approach. *International Journal of Financial Studies*. 2021; 9(2):29.
https://doi.org/10.3390/ijfs9020029

**Chicago/Turabian Style**

Muteba Mwamba, John Weirstrass, and Ehounou Serge Eloge Florentin Angaman.
2021. "Modeling System Risk in the South African Insurance Sector: A Dynamic Mixture Copula Approach" *International Journal of Financial Studies* 9, no. 2: 29.
https://doi.org/10.3390/ijfs9020029