Commutation values

Excerpt from the DAV - mortality table 1994 T Men from 0 to 10 years. The corresponding commutation values ​​are shown based on the then valid interest rate of 3.25%.

As Kommutationswerte even Kommutationszahlen be in the life and health insurance mathematics appropriate auxiliary variables referred by which fast present value calculations are made.

The calculation of benefit and contribution cash values ​​is usually very time-consuming, as each summand contains a product with further factors . In principle, this simplification does not play a role in computer-aided calculation . Nevertheless, commutation values ​​enable the complex calculations to be broken down into clear partial expressions, since they occur at different points in calculations and are used several times.

Commutation values ​​are used in operational systems , life tables and in formulas . They immediately take into account the equivalence principle of the pension calculation , according to which the equivalence of two payment flows is included by comparing the present values ​​or the final values. (→ basic formulas of pension calculation )

history

Introduction to the calculation of annuities , 1785 ( digitized version )

The introduction of the commutation numbers is based on the two-volume, almost thousand-page textbook

Introduction to the calculation of annuities and entitlements that depend on the life and death of one or more perſons

by the German-Danish philosopher, mathematician and natural scientist Johannes Nikolaus Tetens , which appeared in Leipzig in 1785 and 1786. Its recursive structure made manual tariff calculation easier . The use of tables should also minimize the sources of calculation errors. His work is based, among other things, on the work of Johann Peter Süßmilch and Leonhard Euler ; The latter carried out the calculation of widows' pensions in 1760 .

The starting point for this publication was the examination of the Calenbergische Witwencasse, which was established in 1767 and had run into difficulties in what is now Lower Saxony , and was transferred to Tetens . In an extensive foreword, he explains the calculation bases he created for life insurance mathematics in a generally understandable manner. Tetens justified the concept of risk, which is defined as the product of the size of the profit and the probability of occurrence.

Tetens does not use the term commutation values ​​literally. However, he uses the auxiliary variable several times in his textbook. He also uses discounted figures for the first time to determine the present value. In addition to derivations and strict proofs, the book consists of a large number of (auxiliary) tables and practical calculation examples to explain how to use the tables.

Tetens life table with the tabulated commutation values

For example, Tetes provided the following calculation scheme to easily calculate the defense of the liaison pensions :

“This table is only made according to the sweet milk mortality table , but it can easily be set up for any other order of mortality. [...] From the latter, the one can be made in less than an hour. It's all in a simple addition of numbers. All the work, which can be called a little laborious, consists in calculating the discontinued numbers of the living (the numbers in column E) for individual persons in the first table . So everything else is well established. [...] "

1. ↑ The Suessmilch-Baumann life table from 1775 was used in Germany for over a century.
2. ↑ The tables used by Tetes are similar to today's mortality tables. For the annuity calculation he used a five-column table with the following columns:
   Column A: years , by which he meant age; Column B: Decremente , by which he meant the deceased; Position C: differences in decrements ; Column D: sums of the sums of the living ; Column E: Sums of the sums of the discontinued numbers of the living .

construction

To construct the commutation values, one considers the lifetime and the associated family of random variables , which are distributed independently and identically for all . It is used as the stock size of the newborn and as the lifespan of persons . ${\displaystyle T:\Omega \to \mathbb {R} _{+}}$ ${\displaystyle \{Z_{i}\}_{i\in \{1,\cdots ,\ell _{0}\}}}$ ${\displaystyle P_{Z_{i}}=P_{T}}$ ${\displaystyle i\in 1,\cdots ,\ell _{0}}$ ${\displaystyle \ell _{0}}$ ${\displaystyle Z_{i}}$ ${\displaystyle i}$

For be: ${\displaystyle x\in \mathbb {N} _{0}}$

${\displaystyle \ell _{x}:=E{\Big \lbrack }\sum _{i=1}^{\ell _{0}}\chi _{\{x<Z_{i}\}}{\Big \rbrack }}$

${\displaystyle d_{x}:=E{\Big \lbrack }\sum _{i=1}^{\ell _{0}}\chi _{\{x<Z_{i}\leq x+1\}}{\Big \rbrack }}$

${\displaystyle \ell _{x}}$is the expected number of survivors with age , in short: number of living and the expected number of dying in the year of life , in short: number of dead. In life tables, normalization is usually used with 100,000 living. ${\displaystyle x}$ ${\displaystyle d_{x}}$ ${\displaystyle x+1}$ ${\displaystyle \ell _{0}}$

application

Life actuarial

In addition to the one-year mortality probabilities, commutation values ​​are shown in mortality tables . It has prevailed in the calculation to discount the living and the dead instead of the sum insured and the contributions . Age is chosen as the reference date without restriction of the general public . ${\displaystyle 0}$

Let the periodic discount factor be at the point in time , then the commutation values ​​formed with the survivors are usually designated as follows: ${\displaystyle v^{x}}$ ${\displaystyle x}$

1. Order: discounted living of old age ${\displaystyle D_{x}=\ell _{x}v^{x}}$ ${\displaystyle x}$
2. Order: Sum of the discounted living aged from to ${\displaystyle N_{x}=\sum _{k=0}^{\omega -x}D_{x+k}}$ ${\displaystyle x}$ ${\displaystyle \omega }$
3. Order: double the sum of the discounted living ${\displaystyle S_{x}=\sum _{k=0}^{\omega -x}N_{x+k}}$

The following commutation values ​​are formed with the dead:

1. Order: number of discounted dead in old age ${\displaystyle C_{x}=d_{x}v^{x+1}}$ ${\displaystyle x}$
2. Order: Sum of the discounted dead aged up to ${\displaystyle M_{x}=\sum _{k=0}^{\omega -x}C_{x+k}}$ ${\displaystyle x}$ ${\displaystyle \omega }$
3. Order: double the total of the discounted dead ${\displaystyle R_{x}=\sum _{k=0}^{\omega -x}M_{x+k}}$

The commutation values ​​discounted by one period more result from the stipulation that insurance benefits for deaths take place at the end of the year. Contribution and survival benefits are due at the beginning of the year. For true because of the relationship . Analogous relationships apply to the sums and double sums. ${\displaystyle 0\leq x\leq \omega }$ ${\displaystyle d_{x}=\ell _{x}-\ell _{x+1}}$ ${\displaystyle C_{x}=vD_{x}-D_{x+1}}$

The commutation values ​​of the 1st order are also referred to as discounted living or discounted dead . As shown above, the commutation values ​​for the living can be used to express those of the dead. The opposite way is not possible. For this reason, only the commutation values ​​for the living are used in business plans and notifications from life insurers to BaFin in accordance with Section 143 VAG .

The commutation values and relate to the end of the year of death and therefore only come into effect for payments at this point in time. Corrections are used to smooth the process so that payments can be made immediately after the time of death . ${\displaystyle C,M}$ ${\displaystyle R}$

Health insurance math

In addition to the commutation values ​​from life insurance mathematics, health insurance usually uses two additional and only there specific commutation values ​​that do not have a fixed way of speaking.

They are defined as:

${\displaystyle O_{x+m}:=D_{x+m}k_{x+m}}$

and

${\displaystyle U_{x+m}:=\sum _{v=m}^{\omega -x}O_{x+v}}$.

literature

• Karl Michael Ortmann: Practical Life Insurance Mathematics . Springer Spectrum, Wiesbaden 2016, ISBN 978-3-658-10199-2 , pp. 124-125.
• Hartmut Milbrodt: Mathematical methods of personal insurance . de Gruyter, 1999, ISBN 3-11-014226-0 , p. 321 ff.
• Jens Kahlenberg: Life Insurance Mathematics. Basic knowledge of the technology of German life insurance . Springer Gabler, 2018, ISBN 978-3-658-14657-3 , pp. 129-131.
• Klaus D. Schmidt: Versicherungsmathematik , Springer, 2002, ISBN 978-3-540-42731-5 , pp. 123 ff.
• Torsten Becker: Mathematics of private health insurance . Springer Spectrum, Wiesbaden 2017, ISBN 978-3-658-16665-6 , pp. 101-104.

Web links

Wikibooks: Formula collection financial mathematics: commutation values

• Volkert Paulsen: Actuarial Mathematics (PDF, pp. 14–15).

Individual evidence

1. ^ Peter Koch: History of Insurance Science in Germany. Verlag Versicherungswirtschaft, Karlsruhe 1998, ISBN 3-88487-745-3 , pp. 40–41.
2. ↑ Tetens: Introduction to the calculation of annuities and entitlements that depend on the life and death of one or more perſons. S. X ( digitized version ).
3. ↑ Tetens: Introduction to the calculation of annuities and entitlements that depend on the life and death of one or more perſons , p. 220 ( digitized version ).
4. ↑ Jens Kahlenberg: Life Insurance Mathematics. Basic knowledge of the technology of German life insurance . P. 130.
5. ^ Wolfgang Grundmann: Finance and Insurance Mathematics . Vieweg + Teubner Verlag, Wiesbaden 1996, ISBN 978-3-8154-2087-4 , p. 88.
6. ↑ Hartmut Milbrodt, Volker Röhrs: Actuarial Methods of German Private Health Insurance , Verlag Versicherungswirtschaft, Karlsruhe 2016, ISBN 978-3-89952-610-3 , p. 54.