Damage actuarial

The non-life insurance mathematics is a branch of actuarial mathematics . While in life insurance only the time of death is random, in non- life insurance , in addition to the time of damage, the amount of damage is also random and difficult to predict. The mathematical theory behind non-life insurance mathematics is called risk theory , often also ruin theory . It makes extensive use of the theory of stochastic processes .

The risk process

Assume an insurance company starts with an initial capital , here called initial reserve . In non-claim periods, this reserve increases due to the (assumed constant) inflow of insurance premiums . At random points in time , damage occurs with a random amount of damage that must be paid for by the insurance company. The capital reserve available at the time is called the risk process and is described by ${\displaystyle T_{0}=0}$ ${\displaystyle U}$ ${\displaystyle P}$ ${\displaystyle T_{i};\ i=1,2,\dotsc \ }$ ${\displaystyle X_{i};\ i=1,2,\dotsc \ }$ ${\displaystyle t\in [0,\infty )}$ ${\displaystyle R_{t}}$

Sketch of a risk process

${\displaystyle R_{t}=U+T_{1}P-X_{1}+(T_{2}-T_{1})P-X_{2}+\cdots +(t-T_{N_{t}})P=U+Pt-\sum _{i=1}^{N_{t}}X_{i}}$.

Here is the random number of claims in ( claim number process ). The consequence is called the process of the damage or claim times ( claim arrival process ). With the amount of the total claims is described in ( accumulated claim process ). Is z. If , for example, it has become negative after a lot of major damage , one speaks of ruin . Naturally, the insurance company wants to keep the probability of ruin very low . ${\displaystyle N_{t}}$ ${\displaystyle [0,t]}$ ${\displaystyle T_{1},T_{2},\cdots \ }$ ${\displaystyle Z_{t}=\sum _{i=1}^{N_{t}}X_{i}}$ ${\displaystyle [0,t]}$ ${\displaystyle R_{t}}$ ${\displaystyle P(\inf _{t\in [0,\infty )}R_{t}<0)}$

Model assumptions and distribution of the total damage

See e.g. B. It is interested in the distribution of the total damage , i. H. the probability . If one assumes that a Markov chain and the individual requirements are stochastically independent of each other with distribution functions , then we get for ${\displaystyle Z_{t}}$ ${\displaystyle P(Z_{t}\leq z)}$ ${\displaystyle N_{t}}$ ${\displaystyle X_{i}}$ ${\displaystyle F_{i};\ i=1,2,\dotsc }$ ${\displaystyle z\geq 0}$

${\displaystyle P(Z_{t}\leq z)=P(N_{t}=0)+\sum _{k=1}^{\infty }P(N_{t}=k)\circledast _{j=1}^{k}F_{j}(z)}$.

Here is the -fold convolution of the distribution functions . Specifically, when a homogeneous Poisson process with intensity is then obtained for a process Poisson compound (Compound Poisson process) with the distribution ${\displaystyle \circledast _{j=1}^{k}}$ ${\displaystyle k}$ ${\displaystyle F_{j}}$ ${\displaystyle N_{t}}$ ${\displaystyle \lambda }$ ${\displaystyle Z_{t}}$

${\displaystyle P(Z_{t}\leq z)=\mathrm {e} ^{-\lambda t}+\sum _{k=1}^{\infty }\mathrm {e} ^{-\lambda t}{\frac {(\lambda t)^{k}}{k!}}\circledast _{j=1}^{k}F_{j}(z)}$.

If the individual claims are independent and identically exponentially are the parameters , we obtain the well in the queuing theory known Erlangmodell ${\displaystyle 1/\mu }$

${\displaystyle P(Z_{t}\leq z)=\mathrm {e} ^{-\lambda t}+\sum _{k=1}^{\infty }\mathrm {e} ^{-\lambda t}{\frac {(\lambda t)^{k}}{k!}}\Gamma (k,{\frac {1}{\mu }};z)}$,

where is the distribution function of the gamma distribution with the parameters and . ${\displaystyle \Gamma (k,{\frac {1}{\mu }};z)}$ ${\displaystyle k}$ ${\displaystyle 1/\mu }$

Wald's equations

They provide formulas for the expected value and variance of the total damage. If the individual claims are distributed independently and identically, d. H. all are distributed like a prototype , then Wald's formula and the Blackwell-Girshick equation apply : ${\displaystyle X_{i}}$ ${\displaystyle X}$

${\displaystyle \operatorname {E} Z_{t}=\operatorname {E} N_{t}\cdot \operatorname {E} X;\quad \operatorname {Var} Z_{t}=(\operatorname {E} X)^{2}\cdot \operatorname {Var} N_{t}+\operatorname {E} N_{t}\cdot \operatorname {Var} X}$.

This results especially for the Erlang model

${\displaystyle \operatorname {E} Z_{t}=\lambda \mu t;\quad \operatorname {Var} Z_{t}=2\lambda \mu ^{2}t}$.

Ruin problem

There are essentially three methods for calculating the probability of ruin : ${\displaystyle P(\inf _{t\in [0,\infty )}R_{t}<0)}$

• Integral equations , see Cramér
• Renewal theory , see Feller
• Martingales , see e.g. B.

reinsurance

One speaks of reinsurance when the primary insurer does not want to bear the risk alone. Then he transfers some of the risk to a reinsurance company. There are various types of reinsurance, see Proportional reinsurance (eg, quota share reinsurance ) and non-proportional reinsurance (eg, Stop Loss ).

literature

• Bühlmann, H. (1970): Mathematical Methods in Risk Theory , Springer
• Embrechts, P .; Klüppelberg, C. and Mikosch, T. (1997): Modeling Extremal Events for Insurance and Finance , Springer
• Rolski, T .; Schmidli, H .; Schmidt, V. and Teugels, J. (1999): Stochastic Processes for Insurance and Finance , Wiley

Individual evidence

1. ^ Straub, E. (1988): Non-Life Insurance Mathematics , Springer
2. ↑ Cramer, H. (1955): Collective Risk Theory: A Survey of the Theory from Point of view of the Theory of Stochstic Processes , Esselte Reklam, Stockholm
3. ^ Feller, W. (1966): An Introduction to Probability Theory and Its Application , Vol. II, Wiley
4. ^ Gerber, HU (1979): An Introduction to Mathematical Risk Theory , Homewood, Irwin