# Credibility Methods for Individual Life Insurance

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

- Denote losses by ${X}_{j}$ and assume that we have observed independent losses $\mathbf{X}=({X}_{1},{X}_{2},\dots ,{X}_{n})$. Note that ${X}_{j}$ might be the annual loss amount from policyholder j, or the loss in the ${j}^{th}$ period, depending on the context.
- Let $\xi =E[{X}_{j}]$ and ${\sigma}^{2}=\mathrm{Var}({X}_{j})$.
- Let $S={\sum}_{j=1}^{n}{X}_{j}$ and let $\overline{X}=\frac{S}{n}$ be the sample mean.
- -
- Observe that $E[S]=n\xi ,E[\overline{X}]=\xi ,\mathrm{Var}(S)=n{\sigma}^{2}$, and $\mathrm{Var}(\overline{X})=\frac{{\sigma}^{2}}{n}$.

- Let M be some other estimate of the mean for this group. M might be based on industry data or large experience studies on groups similar to the risk class in question.

#### 1.2. Significance

#### 1.3. Overview of Paper

## 2. Brief Overview of Credibility Methods

#### 2.1. Limited Fluctuation Credibility

#### 2.1.1. Full Credibility

- $\frac{1+p}{2}=0.95,$
- ${y}_{p}=1.645,$
- ${\left(\frac{{y}_{p}}{r}\right)}^{2}\approx 1082$,

#### 2.1.2. Application to Life Insurance

**Remark**

**1.**

#### 2.1.3. Partial Credibility

**Remark**

**2.**

- 1.
- If full credibility is not justified (i.e., if $n<{n}_{f}$), the partial credibility factor Z is the square root of the ratio of the number of observations n to the number of observations ${n}_{f}$ required for full credibility.
- 2.
- Observe that as σ increases, Z decreases. Thus, lower credibility is awarded when the observations are more variable. Again, this is consistent with our intuition.
- 3.
- In Equation (6), the term:$$\sqrt{n}\left(\frac{\xi}{\sigma}\right)=\frac{\xi}{(\sigma /\sqrt{n})}=\frac{E[\overline{X}]}{\sqrt{Var(\overline{X})}}$$
- 4.
- We can write the formula Equation (6) succinctly to include both the full and partial credibility cases by writing:$$\begin{array}{ccc}Z\hfill & =\hfill & min\left\{1,\sqrt{n}\left(\frac{\xi}{\sigma}\right)\left(\frac{r}{{y}_{p}}\right)\right\}=min\left\{1,\frac{E[estimator]}{\sqrt{Var(estimator)}}\left(\frac{r}{{y}_{p}}\right)\right\}\hfill \\ & =\hfill & min\left\{1,\sqrt{\frac{n}{{n}_{f}}}\right\}\hfill \end{array}.$$

#### 2.1.4. Strengths and Weaknesses of the Limited Fluctuation Approach

- There is no justification for choosing an estimate of the form Equation (1).
- There is no guarantee of the reliability of the estimate M, and the method does not account for the relative soundness of M versus $\overline{X}$.
- The choices of p and r are completely arbitrary. Note that as $r\to 0$ or $p\to 1$, ${n}_{f}\to \infty $. Thus, given any credibility standard ${n}_{f}$, one can select a value of r and p to justify it!

#### 2.2. Greatest Accuracy (Bühlmann) Credibility

- $Z\to 1$ as $n\to \infty $.
- For more homogeneous risk classes (i.e., those whose value of a is small relative to $\nu $), Z will be closer to 0. In other words, the value of $\mu $ is a more valuable predictor for a more homogenous population. However, for a more heterogeneous group (i.e., those whose value of a is large relative to $\nu $), Z will be closer to 1. This result is appealing. If risk classes are very similar to each other (a is small relative to $\nu $), the population mean $\mu $ should be weighted more heavily. If the risk classes are very different from each other (a is large relative to $\nu $), the experience data should get more weight.

#### Strengths and Weaknesses of the Greatest Accuracy Method

#### 2.3. Other Credibility Methods

## 3. Previous Literature: Application of Credibility to Company Mortality Experience Data

#### 3.1. Notation

- ${f}_{i}$ is the fraction of the year for which the ith life was observed.
- ${d}_{i}=1$ if life i died during the year; otherwise, ${d}_{i}=0$.
- ${q}_{i}$ is the observed mortality rate.
- ${q}_{i}^{s}$ is the standard table mortality rate.
- We assume that ${q}_{i}={m}_{c}{q}_{i}^{s}$ (i.e., that the actual mortality is a constant multiple of the table).
- ${A}_{c}={\sum}_{i=1}^{n}{d}_{i}=$ the actual number of deaths.
- ${E}_{c}={\sum}_{i=1}^{n}{f}_{i}{{q}_{i}}^{s}=$ the expected number of deaths.
- ${\widehat{m}}_{c}=\frac{{A}_{c}}{{E}_{c}}=$ the estimated actual-to-expected (A/E) mortality ratio based on claim counts.

- ${A}_{d}={\sum}_{i=1}^{n}{b}_{i}{d}_{i}=$ the actual amount paid.
- ${E}_{d}={\sum}_{i=1}^{n}{b}_{i}{f}_{i}{{q}_{i}}^{s}=$ the expected amount paid.
- ${\widehat{m}}_{d}=\frac{{A}_{d}}{{E}_{d}}=$ the estimated actual-to-expected (A/E) mortality ratio based on claim amounts.

#### 3.2. Limited Fluctuation Formulae

#### 3.3. Greatest Accuracy Formulae

## 4. LF Analysis Applied to M Financial’s Data: Qualitative Results

## 5. Qualitative Comparison of Credibility Methods Using a Simulated Data Set

#### 5.1. Overview

#### 5.2. Generating the “Universe”

- The ratio ${m}_{h}$ of actual deaths to expected deaths over the 20-year time period, $h=1,\dots ,20$. In other words, we computed the hypothetical mean for each of the 20 risk classes. These values ranged from 0.71 to 1.28.
- The overall A/E ratio $\mu =0.95$ for the universe of 1 million individuals.

#### 5.3. Generating the Experience Data and Computing the LF and GA Credibility Factors

#### 5.4. An Intuitively Appealing Benchmark “Credibility Factor”

- Let h refer to the risk class or population; $h=1,\dots ,20$.
- Let $n=$ number of policyholders in the company’s experience data; $n=500,1500,\dots $
- Let $i=$ trial; $i=1,\dots ,2000$.
- Let $\mu =$ the A/E ratio for the universe of 1 million individuals. In the simulation, we had $\mu \approx 0.95$.
- Let ${\widehat{m}}_{hni}=$ the observed A/E ratio for company h, trial i, when there are n policyholders in the group.
- Let ${m}_{h}$ be the true A/E ratio for population h. Recall that this was prescribed a priori when we generated the 20 populations.
- Let ${Z}_{hn}$ be the credibility factor for company h when there are n policyholders in the group.

- Let ${d}_{hni}=$ actual number of deaths for company h, trial i, when there are n policyholders.
- Let ${d}_{hn}=\frac{1}{2000}{\sum}_{i=1}^{2000}{d}_{hni}$. That is, ${d}_{hn}$ is the average actual number of deaths over the 2000 trials for company h when there are n policyholders.
- Denote ${d}_{hn}$ by ${d}_{h}$. We will suppress the index n. Observe that ${d}_{h}$ is analogous to ${A}_{h}$, the actual number of claims for company h, from (Klugman et al. 2009).
- Let ${\tilde{Z}}_{hd}^{B}$ be the benchmark credibility factor for company h when there are d claims. Thus we express the credibility factor $\tilde{Z}$ as a function of ${d}_{h}={d}_{hn}$. This will make the comparison with the LF and GA results meaningful.
- Let ${\tilde{Z}}_{hd}^{LF}$ and ${\tilde{Z}}_{hd}^{GA}$ be the credibility factors for company h when there are d claims computed via the LF and GA methods, respectively. We computed these factors in Section 5.3.

#### 5.5. Qualitative Comparison of the Credibility Methods

- The exception to Observation 1 occurred when the hypothetical mean ${m}_{h}$ was close to the overall population mean $\mu \approx 0.95$—see Figure 4. This is not surprising, as the benchmark factor is the relative frequency in 2000 trials that the observed A/E ratio ${\widehat{m}}_{h}$ is closer to the true hypothetical mean ${m}_{h}$ than the overall population mean $\mu $. When ${m}_{h}$ is very close to $\mu $, it is unlikely that ${\widehat{m}}_{h}$ will land closer—that is, the event $|{\widehat{m}}_{h}-{m}_{h}|<|\mu -{m}_{h}|$ is unlikely, resulting in a smaller benchmark Z.
- For our simulated data set, and for the real data in (Klugman et al. 2009), the GA method produced significantly higher credibility factors at low numbers of claims than the LF method with $r=0.05$. Of course, the difference was even more pronounced when we chose the Canadian standard $r=0.03$.

## 6. Conclusions and Recommendations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A/E | actual-to-expected mortality ratio |

ASOP | Actuarial Standard of Practice |

GA | greatest accuracy (GA) or Bühlmann credibility |

LF | limited fluctuation credibility |

NAIC | National Association of Insurance Commissioners |

PBR | principle-based reserving |

SVL | Standard Valuation Law |

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**Figure 1.**We show the limited fluctuation (LF) and greatest accuracy (GA) credibility factors for three “companies” (risk classes) from our simulated data set and contrast our results with the results from (Klugman et al. 2009), which are based on real data. The results are remarkably consistent.

**Figure 2.**We contrast the Benchmark, GA, and LF credibility factors for “company” (risk class) 1, whose hypothetical mean is ${m}_{1}\approx 0.707$.

**Figure 3.**We contrast the Benchmark, GA, and LF credibility factors for “company” (risk class) 20, whose hypothetical mean is ${m}_{20}\approx 1.278$.

**Figure 4.**We contrast the Benchmark, GA, and LF credibility factors for “company” (risk class) 9, whose hypothetical mean is ${m}_{9}\approx 0.953$.

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## Share and Cite

**MDPI and ACS Style**

Gong, Y.; Li, Z.; Milazzo, M.; Moore, K.; Provencher, M.
Credibility Methods for Individual Life Insurance. *Risks* **2018**, *6*, 144.
https://doi.org/10.3390/risks6040144

**AMA Style**

Gong Y, Li Z, Milazzo M, Moore K, Provencher M.
Credibility Methods for Individual Life Insurance. *Risks*. 2018; 6(4):144.
https://doi.org/10.3390/risks6040144

**Chicago/Turabian Style**

Gong, Yikai (Maxwell), Zhuangdi Li, Maria Milazzo, Kristen Moore, and Matthew Provencher.
2018. "Credibility Methods for Individual Life Insurance" *Risks* 6, no. 4: 144.
https://doi.org/10.3390/risks6040144