Life actuarial

The Life Insurance Mathematics is a branch of actuarial mathematics . Put simply, there are two types of life insurance , insurance against death (to protect the bereaved) and insurance against survival (as a provision for old age). There are then further differences in detail, e.g. B. between insurance for one or more lives (e.g. marriage, groups, ...) and life insurance policies with one or more risks(e.g. disability, death, ...). The following is an example of life insurance for a life at risk. Life actuarial mathematics is a coupling of deterministic financial mathematics (in particular interest calculation due to long term) and probability theory (due to the random time of death).

Benefits and rewards

Premium is the actuarial term for insurance premium. Let the interest rate assumed as fixed , the discount factor and the time of death. In the following, we consider the sum of the discounted insurance benefits at the start of the contract and the corresponding discounted premium payments. For the sake of simplicity, it is assumed that benefits and premiums are granted or demanded at the end of the year or the beginning of the year (the so-called sub - annual view is then only a refinement). In this context, the interval is referred to as " the year " after the start of the contract. ${\displaystyle i}$ ${\displaystyle v=1/(i+1)}$ ${\displaystyle t\in (0,\infty )}$ ${\displaystyle (k-1,k];\quad k=1,2,\cdots }$ ${\displaystyle k\ }$

Insurance against death

If the person dies in the year after the contract was signed, d. H. , then at the end of the year , d. H. in arrears , the performance is due. The discounted insurance benefit at the start of the contract is therefore ${\displaystyle k}$ ${\displaystyle t\in (k-1,k]}$ ${\displaystyle k}$ ${\displaystyle \delta _{k}}$

${\displaystyle \Delta (t)=\sum _{k=1}^{\infty }v^{k}\delta _{k}\chi _{(k-1,k]}(t)={\begin{cases}v\delta _{1}&{\text{falls Tod im Jahr}}\ 1\\v^{2}\delta _{2}&{\text{falls Tod im Jahr}}\ 2\\\vdots \\v^{k}\delta _{k}&{\text{falls Tod im Jahr}}\ k\\\vdots \end{cases}}}$.

The indicator function denotes . The consequence is called a (subsequent) benefit plan in the event of death. ${\displaystyle \chi }$ ${\displaystyle \delta _{1},\delta _{2},\cdots \quad }$

Survival insurance

If the person is still alive in the year , at the beginning of the year , i. H. in advance , the benefit is due. The service discounted at the start of the contract is then ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \lambda _{k}}$

${\displaystyle \Lambda (t)=\sum _{k=1}^{\infty }v^{k-1}\lambda _{k}\chi _{(k-1,\infty )}(t)={\begin{cases}\lambda _{1}&{\text{falls Tod im Jahr}}\ 1\\\lambda _{1}+v\lambda _{2}&{\text{falls Tod im Jahr}}\ 2\\\vdots \\\sum _{k=1}^{m}v^{k-1}\lambda _{k}&{\text{falls Tod im Jahr}}\ m\\\vdots \end{cases}}}$.

The result is called a (advance) benefit plan in the event of survival. ${\displaystyle \lambda _{1},\lambda _{2},\cdots \quad }$

Mixed insurance

For example, we consider a -year life insurance combined with a life insurance deferred for years. The discounted benefit is then ${\displaystyle m}$ ${\displaystyle m}$

${\displaystyle \Delta (t)+\Lambda (t)=\sum _{k=1}^{m}v^{k}\delta _{k}\chi _{(k-1,k]}(t)+\sum _{k=m}^{\infty }v^{k-1}\lambda _{k}\chi _{(k-1,\infty )}(t)}$.

Bonuses

At the beginning of the year , i.e. in advance, the premium is due if the person is still alive during the year . The total premium payment is discounted to the start of the contract ${\displaystyle k}$ ${\displaystyle \pi _{k}}$ ${\displaystyle k}$

${\displaystyle \Pi (t)=\sum _{k=1}^{\infty }v^{k-1}\pi _{k}\chi _{(k-1,\infty )}(t)}$.

The result is called the (advance) premium plan. ${\displaystyle \pi _{1},\pi _{2},\cdots \quad }$

Equivalence principle

The policyholder perceives it as fair if the expected premium payments match the expected benefits. This agreement is called the equivalence principle . The sizes depend on the time of death . Since this point in time is random, probability theory now comes into play. It denotes the age of the person at the start of the contract and the random remaining life of this -year-old. The equivalence principle demands ${\displaystyle \Delta (t),\ \Lambda (t),\ \Pi (t)}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle T_{x}}$ ${\displaystyle x}$

${\displaystyle \operatorname {E} \Delta (T_{x})+\operatorname {E} \Lambda (T_{x})=\operatorname {E} \Pi (T_{x})}$.

Denotes the expected value operator . The above expected values ​​can easily be calculated from the fact that is known in probability theory that a random variable always applies: ${\displaystyle \operatorname {E} }$ ${\displaystyle X}$ ${\displaystyle \operatorname {E} \chi _{A}(X)=P(X\in A)}$

${\displaystyle \operatorname {E} \Delta (T_{x})=\sum _{k=1}^{\infty }v^{k}\delta _{k}P(k-1<T_{x}\leq k);\quad \operatorname {E} \Lambda (T_{x})=\sum _{k=1}^{\infty }v^{k-1}\lambda _{k}P(k-1<T_{x});\quad \operatorname {E} \Pi (T_{x})=\sum _{k=1}^{\infty }v^{k-1}\pi _{k}P(k-1<T_{x})}$.

The probabilities in these formulas need to be estimated. This is usually done with the help of life tables .

Premium calculation

For the benefits of the insurance contract you can use the equivalence principle to calculate the necessary premium plan. These premiums are called net premiums . For policyholders and insurance companies, however, it is not sufficient to only consider the expected values ​​of premiums and benefits in the premium calculation. There is long-term security only if the volatility of these variables (e.g. their variance ) is taken into account with a security margin . The net premium plus safety surcharge results in the so-called risk premium . The premium that the policyholder ultimately has to pay, the so-called gross premium , is even higher because, in addition to the risk premium, B. operating costs and the calculated corporate profit are added.

Bibliography

• Gerber, HU (1986): Life Insurance Mathematics , Springer Berlin-Heidelberg-New York
• Milbrodt, H. and Helbig, M. (1999): Mathematical methods of personal insurance, DeGruyter, Berlin
• Schmidt, KD (2009, third edition): Versicherungsmathematik , Springer Dordrecht-Heidelberg-London-New York