# Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- -
- What should the initial premium level be for new policyholders?
- -
- Which premium adjustment criterion is better in the proposed risk models: premiums adjusted by aggregate claims amount or premiums adjusted by claim frequency only?
- -
- What is the likely impact of the initial external environment condition on the risk of ruin when the proposed premium adjustment rules are implemented?

- -
- The choice of initial premium level for new policyholders is not an easy task. A low initial premium level tends to be very risky when the company’s initial capital amount is small. However, when the initial capital amount is sufficiently large, the insurer has more flexibility to lower the initial premium level that can attract more new policyholders and help with boosting the insurance business without significantly increasing the risk of insolvency.
- -
- The initial external environment condition does have a significant impact on the risk of ruin under the proposed premium adjustment rules, but the impact could be different from our first guess.
- -
- Adjusting premiums according to claim frequency can be riskier than the case of adjusting premiums by aggregate claims.

## 2. Models and Assumptions

**c**denote a premium level set where $\mathbf{c}={\left\{{c}_{i}\right\}}_{i\in \mathcal{\U0001d4db}}$, $\mathcal{\U0001d4db}=\{1,2,\dots ,l\}$, $l\in {\mathbb{N}}^{+}$, ${c}_{i}\in {\mathbb{R}}^{+}$. Here ${c}_{i}$, $i=1,\dots ,l$, are premium levels per unit volume of risk. Let the external economic environmental status in time period $[t-1,t)$, $t\in {\mathbb{N}}^{+}$ be represented by a homogeneous and irreducible discrete-time Markov chain ${\left\{{J}_{t}\right\}}_{t\in \mathbb{N}}$ with a finite state space $\mathbf{R}=\{1,2,\dots ,r\}$ and a transition probability matrix ${P}_{J}={\left[{p}_{J}(g,h)\right]}_{g,h\in \mathbf{R}}$, where ${p}_{J}(g,h)=\mathbb{P}({J}_{t}=h|{J}_{t-1}=g)$. The stationary probability distribution of the Markov process is denoted by $\mathit{\lambda}=[{\lambda}_{1},\dots ,{\lambda}_{r}]$ where $0\le {\lambda}_{i}\le 1,i=1,2,\dots ,r$, and ${\sum}_{i=1}^{r}{\lambda}_{i}=1$. Let ${\left\{{L}_{t}\right\}}_{t\in {\mathbb{N}}^{+}}$ be a stochastic process monitoring the premium levels that the insurance company charges over time. Here ${L}_{t}\in \mathbf{c}$ for any $t\in {\mathbb{N}}^{+}$ and this premium level applies in the time period $[t-1,t)$.

#### 2.1. Premiums Adjusted by Aggregate Claims

**Remark**

**1.**

#### 2.2. Premiums Adjusted by Claim Frequency

## 3. Finite-Time Ruin Probabilities

#### 3.1. Premiums Adjusted by Aggregate Claims

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.2. Premiums Adjusted by Claim Frequency

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 4. Lundberg Inequalities for Ruin Probabilities

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Remark**

**2.**

## 5. The Joint Distribution of Premium Level and Environment State at Ruin

#### 5.1. Premiums Adjusted by Aggregate Claims

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

#### 5.2. Premiums Adjusted by Claim Frequency

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**5.**

## 6. Some Numerical Results

#### 6.1. An Example for Premiums Adjusted by Aggregate Claims

- Economic state 1 (normal): mean $=10$, variance $=101.743$;
- Economic state 2 (deflation): mean $=5$, variance $=54.664$;
- Economic state 3 (inflation): mean $=15$, variance $=268.187$.

- If the recorded aggregate claims in the current period is no more than the 30th percentile of the aggregate claim distribution, then the premium level for the next period will move to the lower premium level or stay in the lowest one;
- If the recorded aggregate claims in the current period is more than the 30th percentile but no more than the 70th percentile of the aggregate claim distribution, then the premium level for the next period will remain in the current premium level;
- If the recorded aggregate claims in the current period is more than the 70th percentile of the aggregate claim distribution, then the premium level for the next period will move to the higher premium level or stay in the highest one.

**Remark**

**3.**

- The transition matrix of economic state given above is a hypothetical one. Different matrices will generate different sequences of premiums in the future. How to obtain a reliable estimate of such a transition matrix in real-life is beyond the scope of this study. Econometric studies could possibly provide answers to this question.
- The above set of premium rules is again a hypothetical one and is a much simplified version of the real-life bonus-malus rules. This helps to simplify the computational process and can sufficiently showcase our key results obtained in the main text before.
- The premium loadings given in c do not indicate the bonus or malus cases directly. Only when the initial premium level (or base level) is chosen, then we can tell whether a given premium level is a bonus (lower than base level) or a malus case (higher than base level).

**Scenario**

**1.**

**Scenario**

**2.**

#### 6.2. An Example for Premiums Adjusted by Claim Frequency

- The claim frequency is modelled by Poisson distribution with mean 1.57, 0.785 and 2.355 for weather state 1 (normal condition), 2 (less severe condition) and 3 (severe weather condition) respectively.
- The one-step transition probability matrix of weather states (environment state) with the corresponding stationary probability distribution are the same as the ones for the external economic states in the previous example. Similar to the previous example, the weather state is assumed to be fixed over a year from the beginning and the premiums depend on the current weather state.
- The individual claim size distribution under each weather state is assumed to be geometric with P.M.F. ${\mathbb{P}}_{W}\left(w\right)=\left(\frac{1.57}{10}\right){(1-\frac{1.57}{10})}^{(w-1)}$ for $w\ge 1$ and mean $\frac{10}{1.57}$. Then the expected aggregate claim amount under weather state 1, 2, and 3 is 10, 5 and 15, respectively. The expected long-term aggregate claim amount is 10, which is the same as the one in the previous example as well.

- If the number of claims in the current period is 0, then the premium level for the next period will move to the lower premium level or stay in the lowest one;
- If the number of claims in the current period is greater than 0 but no more than 2, then the premium level for the next period will remain in the current premium level;
- If the number of claims in the current period is more 2, then the premium level for the next period will move to the higher premium level or stay in the highest one.

## 7. Conclusions and Discussions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Theorem**

**2.**

## Appendix B

**Proof of Theorem**

**5.**

## Appendix C

## Appendix D

## References

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u | ${\mathit{\psi}}_{1,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,1}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.581516 | 0.485600 | 0.370290 | 0.278787 | 0.220787 | 0.982500 |

10 | 0.346148 | 0.268051 | 0.189482 | 0.135426 | 0.106381 | 0.823486 |

20 | 0.202262 | 0.147489 | 0.097952 | 0.067067 | 0.052281 | 0.690207 |

30 | 0.117224 | 0.081516 | 0.051458 | 0.034011 | 0.026317 | 0.578500 |

40 | 0.067836 | 0.045466 | 0.027558 | 0.017698 | 0.013597 | 0.484872 |

50 | 0.039369 | 0.025658 | 0.015062 | 0.009450 | 0.007212 | 0.406397 |

70 | 0.013508 | 0.008491 | 0.004769 | 0.002893 | 0.002181 | 0.285494 |

90 | 0.004775 | 0.002943 | 0.001609 | 0.000954 | 0.000713 | 0.200560 |

120 | 0.001052 | 0.000638 | 0.000340 | 0.000197 | 0.000146 | 0.118091 |

150 | 0.000240 | 0.000144 | 0.000075 | 0.000043 | 0.000031 | 0.069532 |

200 | 0.000021 | 0.000012 | 0.000006 | 0.000004 | 0.000003 | 0.028761 |

u | ${\mathit{\psi}}_{1,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,2}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.602651 | 0.530232 | 0.432010 | 0.346695 | 0.290467 | 0.982500 |

10 | 0.340618 | 0.280003 | 0.210953 | 0.159843 | 0.132489 | 0.823486 |

20 | 0.194130 | 0.151662 | 0.107550 | 0.077895 | 0.063776 | 0.690207 |

30 | 0.110690 | 0.083187 | 0.056257 | 0.039292 | 0.031786 | 0.578500 |

40 | 0.063296 | 0.046186 | 0.030090 | 0.020401 | 0.016316 | 0.484872 |

50 | 0.036402 | 0.025979 | 0.016437 | 0.010875 | 0.008605 | 0.406397 |

70 | 0.012333 | 0.008554 | 0.005196 | 0.003313 | 0.002573 | 0.285494 |

90 | 0.004325 | 0.002954 | 0.001750 | 0.001087 | 0.000832 | 0.200560 |

120 | 0.000946 | 0.000638 | 0.000369 | 0.000223 | 0.000168 | 0.118091 |

150 | 0.000215 | 0.000143 | 0.000082 | 0.000049 | 0.000036 | 0.069532 |

200 | 0.000019 | 0.000012 | 0.000007 | 0.000004 | 0.000003 | 0.028761 |

u | ${\mathit{\psi}}_{1,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,3}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.536216 | 0.441881 | 0.338071 | 0.259681 | 0.209647 | 0.982500 |

10 | 0.362565 | 0.284586 | 0.209476 | 0.157582 | 0.127362 | 0.823486 |

20 | 0.240562 | 0.181306 | 0.129259 | 0.095593 | 0.077312 | 0.690207 |

30 | 0.157427 | 0.114621 | 0.079529 | 0.057972 | 0.046900 | 0.578500 |

40 | 0.101979 | 0.072065 | 0.048833 | 0.035150 | 0.028439 | 0.484872 |

50 | 0.065557 | 0.045126 | 0.029942 | 0.021312 | 0.017240 | 0.406397 |

70 | 0.026650 | 0.017546 | 0.011225 | 0.007835 | 0.006334 | 0.285494 |

90 | 0.010669 | 0.006769 | 0.004198 | 0.002881 | 0.002327 | 0.200560 |

120 | 0.002651 | 0.001606 | 0.000957 | 0.000643 | 0.000519 | 0.118091 |

150 | 0.000647 | 0.000377 | 0.000217 | 0.000144 | 0.000116 | 0.069532 |

200 | 0.000060 | 0.000033 | 0.000018 | 0.000012 | 0.000009 | 0.028761 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.758260 | 0.066721 | 0.017378 | 0.004019 | 0.000892 |

$h=2$ | 0.031033 | 0.015421 | 0.005128 | 0.001372 | 0.000344 |

$h=3$ | 0.062983 | 0.026394 | 0.007770 | 0.001865 | 0.000421 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.000113 | 0.000726 | 0.004665 | 0.047550 | 0.800367 |

$h=2$ | 0.000037 | 0.000231 | 0.001394 | 0.008976 | 0.038254 |

$h=3$ | 0.000408 | 0.001594 | 0.005976 | 0.024634 | 0.065076 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.016169 | 0.053910 | 0.071874 | 0.054714 | 0.043763 |

$h=2$ | 0.002011 | 0.008437 | 0.013755 | 0.012849 | 0.012611 |

$h=3$ | 0.098078 | 0.218514 | 0.204083 | 0.118145 | 0.071086 |

u | ${\mathit{\psi}}_{1,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,1}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,1}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.605971 | 0.509785 | 0.394719 | 0.299570 | 0.235311 | 0.971992 |

10 | 0.388786 | 0.299805 | 0.209603 | 0.146053 | 0.110407 | 0.731630 |

20 | 0.236054 | 0.167432 | 0.106238 | 0.068367 | 0.050195 | 0.550706 |

30 | 0.137875 | 0.090424 | 0.052377 | 0.031307 | 0.022445 | 0.414523 |

40 | 0.078166 | 0.047692 | 0.025389 | 0.014180 | 0.009959 | 0.312016 |

50 | 0.043249 | 0.024708 | 0.012176 | 0.006393 | 0.004407 | 0.234858 |

70 | 0.012487 | 0.006372 | 0.002750 | 0.001299 | 0.000865 | 0.133065 |

90 | 0.003391 | 0.001581 | 0.000614 | 0.000266 | 0.000172 | 0.075391 |

120 | 0.000441 | 0.000186 | 0.000064 | 0.000025 | 0.000015 | 0.032152 |

150 | 0.000053 | 0.000021 | 0.000007 | 0.000002 | 0.000001 | 0.013712 |

200 | 0.000001 | 0.000001 | 0.000000 | 0.000000 | 0.000000 | 0.003313 |

u | ${\mathit{\psi}}_{1,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,2}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,2}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.647608 | 0.600217 | 0.511647 | 0.414121 | 0.332302 | 0.971992 |

10 | 0.410970 | 0.362287 | 0.281517 | 0.204813 | 0.150624 | 0.731630 |

20 | 0.251122 | 0.211238 | 0.150889 | 0.099501 | 0.067739 | 0.550706 |

30 | 0.148774 | 0.119900 | 0.079446 | 0.047947 | 0.030482 | 0.414523 |

40 | 0.085828 | 0.066553 | 0.041270 | 0.023024 | 0.013769 | 0.312016 |

50 | 0.048375 | 0.036238 | 0.021204 | 0.011040 | 0.006249 | 0.234858 |

70 | 0.014483 | 0.010237 | 0.005448 | 0.002533 | 0.001305 | 0.133065 |

90 | 0.004064 | 0.002744 | 0.001359 | 0.000580 | 0.000276 | 0.075391 |

120 | 0.000550 | 0.000353 | 0.000162 | 0.000063 | 0.000027 | 0.032152 |

150 | 0.000069 | 0.000042 | 0.000018 | 0.000007 | 0.000003 | 0.013712 |

200 | 0.000002 | 0.000001 | 0.000000 | 0.000000 | 0.000000 | 0.003313 |

u | ${\mathit{\psi}}_{1,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{2,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{3,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{4,3}(\mathit{u},40)$ | ${\mathit{\psi}}_{5,3}(\mathit{u},40)$ | UB |
---|---|---|---|---|---|---|

0 | 0.555437 | 0.430304 | 0.315517 | 0.231635 | 0.179284 | 0.971992 |

10 | 0.354335 | 0.249205 | 0.167191 | 0.115193 | 0.087273 | 0.731630 |

20 | 0.212928 | 0.136511 | 0.084156 | 0.054739 | 0.040872 | 0.550706 |

30 | 0.122699 | 0.072029 | 0.040953 | 0.025268 | 0.018672 | 0.414523 |

40 | 0.068509 | 0.037011 | 0.019486 | 0.011455 | 0.008399 | 0.312016 |

50 | 0.037306 | 0.018650 | 0.009134 | 0.005138 | 0.003744 | 0.234858 |

70 | 0.010434 | 0.004543 | 0.001955 | 0.001019 | 0.000736 | 0.133065 |

90 | 0.002749 | 0.001066 | 0.000411 | 0.000202 | 0.000144 | 0.075391 |

120 | 0.000343 | 0.000116 | 0.000039 | 0.000018 | 0.000013 | 0.032152 |

150 | 0.000040 | 0.000012 | 0.000004 | 0.000002 | 0.000001 | 0.013712 |

200 | 0.000001 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.003313 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.788065 | 0.069503 | 0.012257 | 0.002256 | 0.000437 |

$h=2$ | 0.041473 | 0.010975 | 0.002007 | 0.000384 | 0.000077 |

$h=3$ | 0.045738 | 0.020304 | 0.005216 | 0.001088 | 0.000220 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.000410 | 0.001394 | 0.005025 | 0.039455 | 0.863448 |

$h=2$ | 0.000204 | 0.000681 | 0.002484 | 0.010320 | 0.034591 |

$h=3$ | 0.000116 | 0.000389 | 0.001216 | 0.006851 | 0.033414 |

$\mathit{j}=1$ | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

$h=1$ | 0.066714 | 0.193377 | 0.205458 | 0.115465 | 0.057055 |

$h=2$ | 0.017125 | 0.033472 | 0.034556 | 0.020260 | 0.010142 |

$h=3$ | 0.020572 | 0.071252 | 0.082757 | 0.047871 | 0.023924 |

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## Share and Cite

**MDPI and ACS Style**

Osatakul, D.; Wu, X.
Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment. *Risks* **2021**, *9*, 26.
https://doi.org/10.3390/risks9010026

**AMA Style**

Osatakul D, Wu X.
Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment. *Risks*. 2021; 9(1):26.
https://doi.org/10.3390/risks9010026

**Chicago/Turabian Style**

Osatakul, Dhiti, and Xueyuan Wu.
2021. "Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment" *Risks* 9, no. 1: 26.
https://doi.org/10.3390/risks9010026