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## 金融代写|风险理论代写Risk theory代考|The Bühlmann–Straub Model

The Bühlmann-Straub model extends the Bühlmann model by allowing different risk volumes $w_{i m}$. Examples of risk volumes are allowed maximal mileage in car insurance, area in fire insurance, total amount of persons insured (or total wages) in collective health or accident insurance, and annual premium written by the cedent in reinsurance. In all these examples, it seems reasonable to assume $\mathbb{E}\left[X_{i m} \mid Z_i\right]=$ $\mu\left(Z_i\right) w_{i m}$. Further, one can often think of $X_{i m}$ as being decomposable into a number of independent parts proportional to the risk volume, which leads to $\operatorname{Var}\left[X_{i m} \mid Z_i\right]=$ $\sigma^2\left(Z_i\right) w_{i m}$. For technical reasons, it is tradition to replace $X_{i m}$ by $X_{i m} / w_{i m}$, and the basic assumptions of the Bühlmann-Straub model then become
$$\mathbb{E}\left[X_{i m} \mid Z_i\right]=\mu\left(Z_i\right), \quad \operatorname{Var}\left[X_{i m} \mid Z_i\right]=\sigma^2\left(Z_i\right) / w_{i m},$$
to which one adds similar independence and conditional independence assumptions as for the Bühlmann model.

To compute the credibility premium in the Bühlmann-Straub model, we may assume $M=1$ and consider thus $X_0=\mu(Z), X_1, \ldots, X_n$ with
$$\mathbb{E}\left[X_i \mid Z=\zeta\right]=\mu(\zeta), \quad \mathbb{a r}\left[X_i \mid Z=\zeta\right]=\sigma^2(\zeta) / w_i, \quad i \neq 0$$
Similar calculations as in the proof of Theorem $3.1$ show that the second-order structure is given by
$$\mathbb{E} X_i=\mu_0, \quad \operatorname{Var} X_i=\sigma^2 / w_i+\tau^2, \operatorname{Cov}\left(X_i, X_j\right)=\tau^2$$
for $i=1, \ldots, n$, where $\mu_0=\mathbb{E} \mu(Z), \sigma^2=\mathbb{E} \sigma^2(Z), \tau^2=\mathbb{V}$ ar $\mu(Z)$, whereas
$$\mathbb{E} X_0=\mu_0, \quad \operatorname{Var} X_0=\tau^2, \operatorname{Cov}\left(X_0, X_j\right)=\tau^2, j \neq 0$$

## 金融代写|风险理论代写Risk theory代考|Bonus-Malus Systems

The main example of bonus-malus systems is car insurance, and we will stick to that setting. The key feature is that the portfolio is divided into $K$ classes, such that the premium $H_k$ in class $k=1, \ldots, K$ depends on the class, and that the class of a driver is dynamically adjusted according to his claim experience (a driver entering the portfolio is assigned class $k_0$ for some fixed $k_0$ ).

We will think of the driver’s reliability (as estimated by the company) as being a decreasing function of $k$ and thereby $H_k$ as being a increasing function, such that class 1 contains the ‘best’ drivers and class $K$ the ‘worst’. Of course, this may not always reflect reality since a good driver can by randomness have many claims and thereby be allocated a high bonus class.

For a particular driver, we denote by $X_n$ his bonus class in year $n$ and $M_n$ his number of claims. Many systems then determine the bonus class $X_{n+1}$ in year $n$ on the basis of $X_n$ and $M_n$ alone. That is, $X_{n+1}=\kappa(k, m)$ when $X_n=k, M_n=m$. For the whole of the section, one may safely think of $X_n$ as being Poisson $(\lambda)$, where $\lambda$ is the risk parameter of the driver under consideration, and of the expected size of a claim as being 1 (this is just a normalization).

Example $4.1$ For systems with a small $K$, a very common system is $-1 /+2$ where one moves down one class after a claim-free year and up two classes per claim. In formulas, $\kappa(k, m)=\max (1, k-1)$ if $m=0, \kappa(k, m)=\min (K, k+2 m)$ if $m>0$. For $K=5$, this gives the $\kappa(k, n)$ in Table 2 (here $n=2+$ means two or more claims).

Many (but not all!) systems in practical use have a higher number of classes, typically 12-25. The bonus rules are then most often based on similar ideas as in the $-1 /+2$ system but with some modifications for $k$ close to 1 or $K$. For example, a system which has been used in Norway (see Sundt [166] p. 73) has $K=13$, $k_0=8$ and the rule that after a claim-free year, one moves down one class (except from class 12 or 13 where one moves to 10 , and from class 1 where one cannot move down), and that one moves up two classes for each claim when possible. This gives the rules in Table 3. Another system, introduced by five of the largest Dutch insurance companies in 1982 (see Kaas et al. [97] p. 127), has $K=14, k_0=2$ and the rules in Table 4. Here $H_k$ are the premiums in percentages of the premium in the initial class $k_0$ (such normalized $H_k$ are often denoted relativities).

# 风险理论代考

## 金融代写|风险理论代写Risk theory代考|The Bühlmann–Straub Model

Bühlmann-Straub 模型通过允许不同的风险量扩展了 Bühlmann 模型 $w_{i m}$. 风险量的示例包括汽车保险中允许的最大里程、火灾保险中的区域、集体健康或意外保 险中的被保险人总数（或总工资) 以及分出人在再保险中承担的年度保费。在所有这些例子中，假设 $\mathbb{E}\left[X_{i m} \mid Z_i\right]=\mu\left(Z_i\right) w_{i m}$. 此外，人们经常可以想到 $X_{i m}$ 可 分解为与风险量成正比的多个独立部分，从而导致 $\operatorname{Var}\left[X_{i m} \mid Z_i\right]=\sigma^2\left(Z_i\right) w_{i m}$. 出于技术原因，更换是传统 $X_{i m}$ 经过 $X_{i m} / w_{i m}$ ，然后 Bühlmann-Straub 模型的 基本假设变为
$$\mathbb{E}\left[X_{i m} \mid Z_i\right]=\mu\left(Z_i\right), \quad \operatorname{Var}\left[X_{i m} \mid Z_i\right]=\sigma^2\left(Z_i\right) / w_{i m}$$

$$\mathbb{E}\left[X_i \mid Z=\zeta\right]=\mu(\zeta), \quad \operatorname{ar}\left[X_i \mid Z=\zeta\right]=\sigma^2(\zeta) / w_i, \quad i \neq 0$$

$$\mathbb{E} X_i=\mu_0, \quad \operatorname{Var} X_i=\sigma^2 / w_i+\tau^2, \operatorname{Cov}\left(X_i, X_j\right)=\tau^2$$

$$\mathbb{E} X_0=\mu_0, \quad \operatorname{Var} X_0=\tau^2, \operatorname{Cov}\left(X_0, X_j\right)=\tau^2, j \neq 0$$

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