# Actuarial Applications and Estimation of Extended CreditRisk+

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## Abstract

**:**

## 1. Introduction

## 2. An Alternative Stochastic Mortality Model

#### 2.1. Basic Definitions and Notation

#### 2.2. Some Classical Stochastic Mortality Models

#### 2.3. An Additive Stochastic Mortality Model

**Remark**

**1.**

**Definition 1**(Additive stochastic mortality model)

**.**

**Remark**

**2.**

**Remark**

**3.**

**Definition 2**(Trend families for central death rates and weights)

**.**

**Remark 4**(Long-term projections)

**.**

## 3. Parameter Estimation

**Definition 3**(Time independence and risk factors)

**.**

#### 3.1. Estimation via Maximum Likelihood

**Lemma 1**(Likelihood function)

**.**

**Proof.**

**Definition 4**(Maximum likelihood estimates)

**.**

#### 3.2. Estimation via a Maximum a Posteriori Approach

**Lemma 2**(Posterior density)

**.**

**Proof.**

**Definition 5**(Maximum a posteriori estimates)

**.**

**Lemma 3**(Conditions for maximum a posteriori estimates)

**.**

**Proof.**

#### 3.3. Estimation via MCMC

#### 3.4. Estimation via Matching of Moments

**Assumption 1**(i.i.d. setting)

**.**

**Lemma**

**4.**

**Proof.**

**Definition 6**(Matching of moments estimates for risk factor variances)

**.**

## 4. Applications

#### 4.1. Prediction of Underlying Death Causes

#### 4.2. Forecasting Death Probabilities

**a**) of Figure 2.

**b**) in Figure 2, or from a parabola with $k=3$, see subfigure (

**c**) in Figure 2. The usage of higher order differences for graduation of statistical estimates goes back to the Whittaker–Henderson method. Taking $k=2,3$ unfortunately yields negative prior correlations amongst certain parameters which is why we do not recommend their use. Of course, there exist many further possible choices for prior distributions. However, in our example, we set ${\epsilon}_{\alpha}={\epsilon}_{\beta}={\epsilon}_{\zeta}={\epsilon}_{\eta}={\epsilon}_{\gamma}=0$ as this yields accurate results whilst still being reasonably smooth.

## 5. A Link to the Extended CreditRisk${}^{+}$ Model and Applications

#### 5.1. The ECRP Model

**Definition 7**(Policyholders and number of deaths)

**.**

**Definition 8**(Portfolio quantities)

**.**

**Remark**

**5.**

- (a)
- For applications in the context of internal models we may set ${Y}_{i}$ as the best estimate liability, i.e., discounted future cash flows, of policyholder i at the end of the period. Thus, when using stochastic discount factors or contracts with optionality, for example, portfolio quantities may be stochastic.
- (b)
- In the context of portfolio payment analysis we may set ${Y}_{i}$ as the payments (such as annuities) to i over the next period. We may include premiums in a second dimension in order to get joint distributions of premiums and payments.
- (c)
- For applications in the context of mortality estimation and projection we set ${Y}_{i}=1$.
- (d)
- Using discretisation which preserves expectations (termed as stochastic rounding in (Schmock 2017, section 6.2.2), we may assume ${Y}_{i}$ to be ${[0,\infty )}^{d}$-valued .

**Definition 9**(Aggregated portfolio quantities)

**.**

**Remark**

**6.**

**Definition 10**(The ECRP model)

**.**

- (a)
- Consider independent random common risk factors ${\Lambda}_{1},\cdots ,{\Lambda}_{K}:\Omega \to [0,\infty )$ which have a gamma distribution with mean ${e}_{k}=1$ and variance ${\sigma}_{k}^{2}>0$, i.e., with shape and inverse scale parameter ${\sigma}_{k}^{-2}$. Also the degenerate case with ${\sigma}_{k}^{2}=0$ for $k\in \{1,\cdots ,K\}$ is allowed. Corresponding weights ${w}_{i,0},\cdots ,{w}_{i,K}\in [0,1]$ for every policyholder $i\in \{1,\cdots ,E\}$. Risk index zero represents idiosyncratic risk and we require ${w}_{i,0}+\cdots +{w}_{i,K}=1$.
- (b)
- Deaths ${N}_{1,0},\cdots ,{N}_{E,0}:\Omega \to {\mathbb{N}}_{0}$ are independent from one another, as well as all other random variables and, for all $i\in \{1,\cdots ,E\}$, they are Poisson distributed with intensity ${m}_{i}{w}_{i,0}$, i.e.,$$\mathbb{P}\left(\bigcap _{i=1}^{E}\{{N}_{i,0}={\widehat{N}}_{i,0}\}\right)=\prod _{i=1}^{E}{e}^{-{m}_{i}{w}_{i,0}}\frac{{\left({m}_{i}{w}_{i,0}\right)}^{{\widehat{N}}_{i,0}}}{{\widehat{N}}_{i,0}!}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\widehat{N}}_{1,0},\cdots ,{\widehat{N}}_{E,0}\in {\mathbb{N}}_{0}\phantom{\rule{0.166667em}{0ex}}.$$
- (c)
- Given risk factors, deaths ${\left({N}_{i,k}\right)}_{i\in \{1,\cdots ,E\},k\in \{1,\cdots ,K\}}:\Omega \to {\mathbb{N}}_{0}^{E\times K}$ are independent and, for every policyholder $i\in \{1,\cdots ,E\}$ and $k\in \{1,\cdots ,K\}$, they are Poisson distributed with random intensity ${m}_{i}{w}_{i,k}{\Lambda}_{k}$, i.e.,$$\mathbb{P}\left(\bigcap _{i=1}^{E}\bigcap _{k=1}^{K}\{{N}_{i,k}={\widehat{N}}_{i,k}\}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\Lambda}_{1},\cdots ,{\Lambda}_{K}\right)=\prod _{i=1}^{E}\prod _{k=1}^{K}{e}^{-{m}_{i}{w}_{i,k}{\Lambda}_{k}}\frac{{\left({m}_{i}{w}_{i,k}{\Lambda}_{k}\right)}^{{\widehat{N}}_{i,k}}}{{\widehat{N}}_{i,k}!}\phantom{\rule{1.em}{0ex}}\mathrm{a}.\mathrm{s}.,$$
- (d)
- For every policyholder $i\in \{1,\cdots ,E\}$, the total number of deaths ${N}_{i}$ is split up additively according to risk factors as ${N}_{i}={N}_{i,0}+\cdots +{N}_{i,K}$. Thus, by model construction, $\mathbb{E}\left[{N}_{i}\right]={m}_{i}({w}_{i,0}+\cdots +{w}_{i,K})={m}_{i}$.

**Remark**

**7.**

#### 5.2. Application I: Mortality Risk, Longevity Risk and Solvency II Application

#### 5.3. Application II: Impact of Certain Health Scenarios in Portfolios

#### 5.4. Application III: Forecasting Central Death Rates and Comparison With the Lee–Carter Model

#### 5.5. Considered Risks

#### 5.6. Generalised and Alternative Models

## 6. Model Validation and Model Selection

#### 6.1. Validation via Cross-Covariance

#### 6.2. Validation via Independence

#### 6.3. Validation via Serial Correlation

#### 6.4. Validation via Risk Factor Realisations

#### 6.5. Model Selection

## 7. Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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2. | http://www.abs.gov.au/AUSSTATS/[email protected]/DetailsPage/3101.0Jun%202013?OpenDocument, accessed on May 10, 2016. |

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4. | |

5. |

**Figure 1.**MCMC chains and corresponding density histograms for the variance of risk factor for deaths due to external causes of injury and poisoning ${\sigma}_{9}^{2}$ in subfigure (

**a**) and for the death probability intercept parameter of females aged 55 to 59 years ${\alpha}_{2,\mathrm{f}}$ in subfigure (

**b**).

**Figure 2.**Correlation structure of Gaussian priors with penalisation for deviation from ordinate with $\epsilon =1/100$ in subfigure (

**a**), straight line with $\epsilon =1/2000$ in subfigure (

**b**), and parabola $\epsilon =1/50000$ in subfigure (

**c**).

**Figure 3.**Logarithm of death central death rates (

**a**) for 2013 and forecasts for 2063 in Australia as well as parameter values for $\alpha ,\beta ,\zeta ,\eta $ and $\gamma $ in subfigures (

**b**), (

**c**), (

**d**), (

**e**) and (

**f**), respectively.

**Figure 5.**Forecasted and true death rates using the ECRP model (AM) and the Lee–Carter model (LC) for females aged 50 to 54 years.

Death Cause | Factor | Death Cause | Factor | Death Cause | Factor | Death Cause | |
---|---|---|---|---|---|---|---|

infectious | 1.25 | neoplasms | 1.00 | endocrine | 1.01 | mental | 0.78 |

nervous | 1.20 | circulatory | 1.00 | respiratory | 0.91 | digestive | 1.05 |

genitourinary | 1.14 | external | 1.06 | not elsewhere | 1.00 |

**Table 2.**Estimates for selected risk factor standard deviations $\sigma $ using matching of moments (MM), Approximation (11) (appr.) and MCMC mean estimates (mean), as well as corresponding standard deviations (stdev.) and 5% and 95% quantiles (5% and 95%).

MM | Appr. | Mean | 5% | 95% | Stdev. | |
---|---|---|---|---|---|---|

infectious | 0.1932 | 0.0787 | 0.0812 | 0.0583 | 0.1063 | 0.0147 |

neoplasms | 0.0198 | 0.0148 | 0.0173 | 0.0100 | 0.0200 | 0.0029 |

mental | 0.1502 | 0.1357 | 0.1591 | 0.1200 | 0.2052 | 0.0265 |

circulatory | 0.0377 | 0.0243 | 0.0300 | 0.0224 | 0.0387 | 0.0053 |

respiratory | 0.0712 | 0.0612 | 0.0670 | 0.0510 | 0.0866 | 0.0110 |

external | 0.1044 | 0.0912 | 0.1049 | 0.0787 | 0.1353 | 0.0176 |

**Table 3.**Selected estimated weights in years 2011 and 2031 for ages 80 to 84 years. 5 and 95% MCMC quantiles are given in brackets.

2011, Male | 2031, Male | 2011, Female | 2031, Female | |
---|---|---|---|---|

neoplasms | 0.327 $\left(\begin{array}{c}0.328\\ 0.319\end{array}\right)$ | 0.385 $\left(\begin{array}{c}0.392\\ 0.363\end{array}\right)$ | 0.263 $\left(\begin{array}{c}0.267\\ 0.258\end{array}\right)$ | 0.295 $\left(\begin{array}{c}0.319\\ 0.287\end{array}\right)$ |

circulatory | 0.324 $\left(\begin{array}{c}0.330\\ 0.320\end{array}\right)$ | 0.169 $\left(\begin{array}{c}0.181\\ 0.164\end{array}\right)$ | 0.340 $\left(\begin{array}{c}0.348\\ 0.337\end{array}\right)$ | 0.145 $\left(\begin{array}{c}0.158\\ 0.140\end{array}\right)$ |

respiratory | 0.106 $\left(\begin{array}{c}0.111\\ 0.102\end{array}\right)$ | 0.090 $\left(\begin{array}{c}0.101\\ 0.083\end{array}\right)$ | 0.101 $\left(\begin{array}{c}0.104\\ 0.096\end{array}\right)$ | 0.129 $\left(\begin{array}{c}0.139\\ 0.113\end{array}\right)$ |

endocrine | 0.047 $\left(\begin{array}{c}0.049\\ 0.045\end{array}\right)$ | 0.073 $\left(\begin{array}{c}0.085\\ 0.070\end{array}\right)$ | 0.053 $\left(\begin{array}{c}0.053\\ 0.050\end{array}\right)$ | 0.071 $\left(\begin{array}{c}0.074\\ 0.061\end{array}\right)$ |

nervous | 0.044 $\left(\begin{array}{c}0.047\\ 0.043\end{array}\right)$ | 0.058 $\left(\begin{array}{c}0.068\\ 0.055\end{array}\right)$ | 0.054 $\left(\begin{array}{c}0.057\\ 0.052\end{array}\right)$ | 0.080 $\left(\begin{array}{c}0.089\\ 0.071\end{array}\right)$ |

infectious | 0.015 $\left(\begin{array}{c}0.016\\ 0.014\end{array}\right)$ | 0.020 $\left(\begin{array}{c}0.027\\ 0.019\end{array}\right)$ | 0.015 $\left(\begin{array}{c}0.018\\ 0.015\end{array}\right)$ | 0.019 $\left(\begin{array}{c}0.028\\ 0.020\end{array}\right)$ |

mental | 0.042 $\left(\begin{array}{c}0.046\\ 0.037\end{array}\right)$ | 0.115 $\left(\begin{array}{c}0.130\\ 0.078\end{array}\right)$ | 0.063 $\left(\begin{array}{c}0.068\\ 0.055\end{array}\right)$ | 0.168 $\left(\begin{array}{c}0.188\\ 0.118\end{array}\right)$ |

Age in 2013 | 0 (Newborn) | 20 | 40 | 60 | 80 |
---|---|---|---|---|---|

male | $83.07$ | $63.33$ | $43.62$ | $24.44$ | $8.26$ |

female | $89.45$ | $69.05$ | $48.20$ | $27.76$ | $9.88$ |

**Table 5.**Quantiles, execution times (speed) and total variation distance (accuracy) of Monte Carlo with Bernoulli deaths and $\mathrm{50,000}$ simulations, as well as the extended CreditRisk${}^{+}$ (ECRP) model with Poisson deaths, given a simple portfolio.

Quantiles | Speed | Accuracy | |||||
---|---|---|---|---|---|---|---|

1% | 10% | 50% | 90% | 99% | |||

Bernoulli (MC), ${w}_{i,0}=1$ | 450 | 472 | 500 | 528 | 552 | $22.99$ s | $0.0187$ |

Poisson (ECRP), ${w}_{i,0}=1$ | 449 | 471 | 500 | 529 | 553 | $0.01$ s | $0.0125$ |

Bernoulli (MC), ${w}_{i,1}=1$ | 202 | 310 | 483 | 711 | 936 | $23.07$ s | $0.0489$ |

Poisson (ECRP), ${w}_{i,1}=1$ | 204 | 309 | 483 | 712 | 944 | $0.02$ s | ≤0.0500 |

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## Share and Cite

**MDPI and ACS Style**

Hirz, J.; Schmock, U.; Shevchenko, P.V.
Actuarial Applications and Estimation of Extended CreditRisk+. *Risks* **2017**, *5*, 23.
https://doi.org/10.3390/risks5020023

**AMA Style**

Hirz J, Schmock U, Shevchenko PV.
Actuarial Applications and Estimation of Extended CreditRisk+. *Risks*. 2017; 5(2):23.
https://doi.org/10.3390/risks5020023

**Chicago/Turabian Style**

Hirz, Jonas, Uwe Schmock, and Pavel V. Shevchenko.
2017. "Actuarial Applications and Estimation of Extended CreditRisk+" *Risks* 5, no. 2: 23.
https://doi.org/10.3390/risks5020023