Zillmer method

The Zillmer method (also Zillmerungs method ) is a mathematical formula for calculating the economic value of the obligation that an insurer has from a life insurance contract . It is a simplified calculation method that can therefore only be used for certain simply structured life insurances. The Zillmer method is approved in some countries, especially Central Europe , for determining the actuarial reserve in the balance sheet of the annual financial statements of an insurer for such life insurance policies. It was originally derived through a refinement from the first developed net contribution method. This refinement is also known as the Zillmerization of this procedure, the associated verb is Zillmer . The procedure is called Zillmerei out of date .

The term “Zillmerung” is sometimes used to denote the fact that the economic value of a life insurance contract is initially well below the sum of the premiums already paid. The term is used in this sense regardless of whether the contract value was actually determined using the Zillmer method or another formula for determining the contract value. The term here refers to a property of economic value, not the mathematical approach to calculating it. In literature, there is often no differentiation between the two meanings.

Purpose of the Zillmer process

The Zillmer method is mainly used to calculate the actuarial reserve of certain simple life insurance contracts for the balance sheet of the insurer's financial statements . The aim is to determine the value of such a contract with very little technical effort using the correspondingly simple procedure. This enabled the necessary calculations to be carried out on paper in the time before computers were available with significantly reduced effort.

scope of application

The Zillmer method can only be used for very simple life insurance contracts (conventional term life insurance , mixed life insurance and annuity insurance with a fixed amount of benefits). The premium must be paid annually for the entire duration of the contract. Furthermore, costs may only arise once at the beginning and evenly distributed over the term of the contract. There are corresponding formulas for health insurance .

Historical development of the Zillmer process

The formula is named after the actuary August Zillmer (1831–1893), who introduced the method in 1863. The background to this was the initial one-off commission in life insurance. Previously, the commissions were usually paid over the term. The procedures customary up to then assessed the insurer's obligation for the balance sheet in the case of one-off commissions either too high or too low. This repeatedly brought insurers into financial difficulties. Too high actuarial reserves prevented newly established insurers from growing sufficiently, while too low actuarial reserves could lead to over-indebtedness even with established insurers if too many terminations or insufficient investment income were made . Zillmer improved the existing procedures for the Zillmer procedure in such a way that an appropriate approximation for the value of the contract always resulted for the contracts within the scope of the procedure.

Since the Zillmer procedure was particularly in favor of newly founded insurers, the established insurers did not support the Zillmer procedure. In addition, it delivered amounts that were between the results of the procedures customary up to that point. Therefore it was too cautious for some and too careless for others. A conflict that lasted about 40 years developed until even more sophisticated methods showed that the Zillmer method delivers correct results in its area of ​​application. Ultimately, the Zillmer procedure was recognized by the VAG in 1901 and subsequently became a matter of course.

Nevertheless, during this dispute, the public had gained the impression that the Zillmer trial would reduce the value of the contract. Contributions that actually have to be deferred would be used to cover acquisition costs. Although mathematicians disagreed and stated that the procedure directly calculates the value of the contract correctly, the Zillmer procedure still has a bad reputation today. It is not perceived that the initially low contract value is economically justified and does not result from the valuation method.

The background to this was that Zillmer himself imposed certain restrictions on his method, which in some cases gave too high values. Although he recognized that the value of the contract is often negative for a while in the beginning, he set as a secondary condition in the calculation that regardless of this, the actuarial reserve should be positive by the end of the first year at the latest. He limited the lower Zillmer contribution he used in the calculation instead of the contractual contribution . This gave the impression that Zillmerisation needed a limitation, that it was therefore altogether doubtful, although this limitation has no economic justification. On the other hand, the Zillmer procedure did not ensure that future ongoing costs would be sufficiently taken into account in the contract value. Zillmer did not set the necessary secondary condition.

The German Insurance Supervision Act , passed in 1901, allowed the Zillmer procedure to be used when determining the actuarial reserve with the restrictions proposed by Zillmer. Other countries soon followed. Due to the First World War , however, the extensive use of the Zillmer process was limited to Central Europe. In the rest of the world, actuarial methods have long developed differently, in which the value of the contract is represented by two or more separately calculated balance sheet items combined. In recent times, the insurance project of the International Accounting Standards Board (IASB) has brought the procedure closer to the basic idea of ​​the Zillmer method.

The Zillmer method, like the earlier methods, is designed in such a way that it requires relatively few calculation steps. This was essential for determining the value of a large number of life insurance contracts before computers became available. However, modern computers have practically eliminated the need to severely limit the number of calculation steps. Therefore, much more precise and flexible procedures can be used today, with which any desired contract design can be mapped. The Zillmer process is therefore technically obsolete. However, it is still used extensively today for simple contracts, which have been common since the time of Zillmer, since there has been no reason to introduce the much more cost-intensive modern systems. The further development of the regulations on solvency and accounting suggest that the Zillmer procedure will disappear in practice in the future. In 2008, the German VVG made it impossible to use the Zillmer method to determine the surrender value, since according to this, costs have to be spread over 5 years. These specifications no longer correspond to the scope of the Zillmer process.

The value of the obligation of a contract to be determined by means of the Zillmer method

The Zillmer method is intended to determine the economic value of an insurer's obligation from a life insurance contract. The economic value of an obligation from a contract is generally determined by the following formula (§ 341f HGB ):

Economic value = current value of future expenses based on the contract, i.e. here the future expected insurance benefits and costs , minus the current value of future income based on the contract, i.e. here the contractual contributions

The current value of the obligation therefore corresponds to the current value of the future expenses based on the contract, insofar as they are not covered by future income from the contract. The current value of future payments is determined as the present value of these payments. In the case of uncertain future payments, these are to be weighted actuarially with their probability . The payments based on insurance contracts are characterized by a particularly high level of uncertainty, especially the benefits.

This economic view corresponds to the mathematical approach, which is referred to in mathematics as the prospective method . The prospective method takes into account all future payments (incoming and outgoing payments) under the contract and this with the expected value . The future payments are discounted to the calculation date , for accounting purposes the balance sheet date .

The determination of the value of the obligation from a life insurance contract therefore requires the consideration of a great number of payments, often in the future and at the same time very uncertain. The most precise mathematical implementation possible requires extensive assumptions and estimates and involves a very large number of calculation steps . Even today, these can only be carried out with powerful computers. In particular, stochastic models are often required. The Zillmer method is a formula that simplifies the very time-consuming prospective method and reduces it to a relatively small number of calculation steps that can be carried out in a reasonable time without a computer. But it also reduces the complexity and thus the computational effort and the susceptibility to errors when using computers .

The case of a negative value of the obligation in the initial period of the contract

An insurer is legally obliged to agree the premiums with the customer at least so high that they can cover all expenses that are expected to arise from the contract (e.g. in Germany determined by Section 11 (1) VAG ). The current value of the expected expenses when the contract is concluded is therefore no higher than the current value of the future contributions. The premiums must also be determined so carefully that the insurer expects to be able to cover all expenses even in the event of very unfavorable developments. Normally, and especially in all favorable cases, the contributions are too high. In most contracts worldwide, the excessive premiums are for the most part reimbursed to the policyholders later as a premium refund. When the contract is concluded, the insurer's economic value of the obligation, i.e. the difference between the two numbers, is therefore usually negative.

The first payments are made immediately upon conclusion of the contract. This is because the first premium must be paid at the start of the contract (e.g. determined in Germany by Section 33 VVG ), so that the first of the expected income is made immediately. However, the conclusion of the contract and the collection of the first contribution are also connected with the first expenses. In particular, if the contract was brokered through an insurance agent , the commission agreed with him must be paid to him (e.g. determined in Germany by Section 92 (4) HGB). Both of these change the value of the remaining future income and expenses in the following calculation, as some of them have already taken place.

Most of the time, the expenses that occur immediately at the beginning are higher than the first receipt. As a result, the first payments at the start of the contract further reduce the already negative value of the obligation from the insurer's point of view. It is now mostly clearly negative and this only changes over the course of a few contributions. Because with every contribution payment, the deductible element, the current value of future income, becomes smaller and thus the value, i.e. the difference between the current value of future expenses minus the current value of future income, increases. The first link, the value of future expenses, however, hardly decreases, since almost all life insurance contracts incur only a few expenses in the beginning.

The value of the obligation of life insurance contracts therefore remains mathematically negative for a long time after the conclusion of the contract. This reflects the economic situation from the insurer's point of view. The insurer has already made more expenses than it earned from the contract. However, he expects that the future contributions will not only cover the expected future expenditure, but also this additional initial expenditure that has already taken place. Even if this is booked economically as an obligation and thus negatively, the insurer actually expects future contribution margins, i.e. income.

Accounting for negative values ​​of the obligation

If the value of an obligation is already negative when the contract is signed, the provision for the obligation is to be set at zero, as negative provisions are not permitted in the balance sheet. In principle, no claim may be set in this amount. Because the expected surplus from the contract represented by the negative value may only be recognized in accordance with the fulfillment of the contract, not in advance when the contract is concluded ( realization principle ).

This is different if the value of the obligation falls directly after the conclusion of the contract due to expenses. In this case, the insurer has the expectation that expenses already made will be covered in the future by income (income) from the contract. This economic background is also taken into account in the accounting. Because the expectation of future income covering these initial expenses is accounted for as a receivable, although there is no obligation under the law of obligations on the future contributions. By accounting for the expected contributions, the income statement in the annual financial statements is balanced despite the payment of the initial expenses. Without this approach, an insurer would first have to report a new contract that is economically profitable as a loss. This would not be information about this transaction that corresponds to the actual economic circumstances for readers of the annual financial statements. In this exceptional case, it is therefore permissible to show a claim that does not exist under the law of obligations in the balance sheet.

The actuarial reserve is for a long time lower than the sum of the contributions already paid, if it is not even zero and a claim is reported. This always arises when the initial expenses exceed the initial contributions. The (permissible) method used to determine the actuarial reserve is irrelevant. Since original methods could not determine this value correctly, but this was only achieved by Zillmerization of this method, i.e. the application of the Zillmer method, this fact has historically been associated with the term "Zillmerization". It is not a particular characteristic of the Zillmer process itself, but rather the economic value of the contract brought about by the particular circumstances.

The Zillmer method

Need for an approximation method

The prospective procedure is very time-consuming, especially since a large number of possible expenses and income have to be weighted with their probability. The calculations must be carried out individually for all (often hundreds of thousands) contracts of an insurer (in Germany determined by Section 252 (1) No. 3 HGB). Therefore, before computers became available, mathematicians tried to simplify the calculation using approximation methods (permitted in Germany by Section 341e, Paragraph 3 of the German Commercial Code) in order to reduce the effort involved in the calculation and the susceptibility to errors.

The first step was to summarize similar contracts and only calculate the value of the sum of these contracts. This already restricted the possible variety of contracts, as otherwise only a few contracts would have been similar. But even this simplification led to such a large number of different values ​​that had to be calculated that a simplification of the calculation itself was inevitable.

Therefore an attempt was made to reduce the number of computation steps required. This was done by appropriately summarizing intermediate results, which are calculated only once, then recorded in tables and used again and again. These intermediate results are called commutation values ​​in traditional actuarial mathematics . In traditional actuarial mathematics, all calculations consist of a combination of such tabulated interim results, including the Zillmer method. The use of such commutation values ​​is not an approximation, but leads to the exact result. Commutation values ​​can only be used if the contracts are very simple. The design of the contracts, in particular all calculation bases used (referred to as calculation bases), has been optimized with as few calculation steps as possible with regard to predictability. Originally, all sizes were only allowed to be proportional to the sum insured or to the contributions, which in turn are proportional to the sum insured.

However, even with extensive use of tables, even this significant reduction in the number of calculation steps was not sufficient to enable manual calculation in mass business. In a further step, the formulas were simplified by approximations. While the very simple services could be represented well by commutation values ​​and nothing could be simplified here, the consideration of the future expenses for the insurer's own costs (for contract conclusion and administration) definitely meant a doubling or even tripling of the number of calculation steps and the Commutation values ​​to be linked, an inappropriately high technical effort for this. Therefore, in the early days of actuarial mathematics, for the sake of simplicity, all expenses for future costs were simply ignored in the calculation.

However, this results in a value that is much too low, because the contributions to be deducted from the expenses also cover the costs. While the correct value is benefits + costs - contributions , benefits - contributions is far too low and, above all, negative for far too long. The negative values ​​of young contracts were even simply offset against the positive values ​​of older contracts, so that some insurers hardly made any provisions at the time. This approach was widespread internationally, but put insurers in economic danger if too many new contracts were terminated. This procedure has therefore been banned in Germany and some other countries.

As an alternative, the contributions were artificially reduced to the value that was sufficient to cover only the benefits, the so-called net contribution, for the sole purpose of calculation . This was a reasonable solution until the 19th century when there were initial commission fees instead of running costs only. The resulting value was much too high for such cases. Only established insurers could afford to set up such actuarial reserves and this meant that it was hardly possible to set up new insurers. So the search began for a simple procedure that would always approximately correctly determine the value of the obligation, even with initial costs.

The approximation by the Zillmer method

Zillmer looked for the right value between the two traditional methods, one of which gives too high and the other too low results. He did not reduce the contractual contribution quite as much or he increased the net contribution somewhat, to the so-called Zillmer contribution or gezillmer net contribution . In such a way that this resulted in roughly the correct value of the obligation, even if initial acquisition costs were incurred. Because services + costs - contributions roughly correspond to services - Zillmer contribution . The difference between the Zillmer contribution and the contractual contribution is the cost surcharge in the contribution, which roughly corresponds to the expected future (ongoing) costs. If you leave out both, the result hardly changes. This omission of the surcharge and future costs is the actual mathematical view and is known as implicit consideration of costs . On the other hand, from the German point of view at the time, something was added to the net contribution, the so-called Zillmer surcharge , and this addition was actually called zillmern after Zillmer, and the process was even called Zillmerei . This means that contribution - cost surcharge = Zillmer contribution = net contribution + Zillmer surcharge .

The formula looks very different depending on the type and expected timing of the insurance benefits.

The Zillmer procedure and closing costs

With the Zillmer procedure, the costs incurred evenly over the duration of the contract are omitted from the expenses and the contribution portion intended to cover these costs from the income. These are normally the administrative costs (in the parlance of accounting, administrative expenses , described in Germany in Section 43 (3 ) RechVersV ), and the so-called administrative surcharge for contributions. Thus, in the formula of the Zillmer procedure, only the so-called closing cost surcharge remains as a cost surcharge in the context of the future income taken into account, which serves to cover the closing costs (in the parlance of accounting, closing expenses , described in Germany in Section 43 (2) RechVersV). This highlighting of the acquisition fee surcharge created the impression that the Zillmer procedure is actually about the acquisition costs. In fact, only for the sake of simplicity, the administrative costs and the surcharges covering them are omitted without significantly changing the result. Therefore, in parlance, zillmerization is used as a synonym for the fact that the value of the insurer's obligation is lower than the sum of the premiums already paid for a long time due to the prospectively taken into account contract cost surcharges. In fact, all methods used to calculate the economic value of a contract anticipate future revenue to cover expenses already incurred. This is a defining characteristic of a prospective procedure and reflects the generally applicable economic characteristic of a contract value. This specifically has nothing to do with the Zillmer process.

Solving the problem of initially negative contract values

Zillmer solved the problem of initially negative contract values ​​by proposing to limit the acquisition surcharge so that the value of the contract was positive by the end of the first insurance year at the latest. For an average contract, he calculated a rate of 1.25% of the sum insured as the limit. Such a limit ( Höchstzillmätze ) was later provided for by law in the various states, in Germany in 1901. However, even at the beginning of the 20th century, negative contract values ​​were no longer a problem. Because the actually problematic offsetting of positive actuarial reserves with negative values ​​is not allowed at all in modern accounting because of the offsetting ban . The negative values ​​can at most be shown separately as a claim. Since at the same time, in Germany as well from 1901, the (positive) actuarial reserves had to be covered with qualified capital investments, it was no longer possible to offset or balance the balance sheet between the negative contract values ​​of new contracts and the actuarial reserves of older contracts. Therefore, the limitation proposed by Zillmer in 1863 was actually already obsolete when the Zillmer process was generally approved. However, the limit was introduced in the VAG in 1901 because these connections had not yet been recognized. The limit is still in place today in order to economically prevent insurers from paying excessively high acquisition commissions and to avoid reductions in profit sharing. So this has nothing to do with Zillmer's original idea.

Legal basis of the Zillmer procedure in the EU and using Germany as an example

Since the approximation through the Zillmer procedure increases the actuarial reserve in principle, it is simply not permitted. According to the principles of commercial law, provisions may not be overvalued beyond the necessary caution. Therefore, a legal permit is required to unnecessarily increase the actuarial reserve by omitting the administrative costs for the expenses and the excess administrative cost surcharge for the income in the legally prescribed prospective procedure. The basis for the annual financial statements of a German insurer is EU law. From a commercial law perspective, Article 18 (2) of EU Directive 91/674 / EEC allows the Zillmer procedure. Since the annual financial statements of a German insurer, in addition to the normal commercial reporting purpose, also serve as the basis for the prudent solvency provisions that are harmonized in the EU , the relevant EU regulations also apply here. According to Article 20 (1) E of EU Directive 2002/83 / EC, such an increase by omitting the administrative costs, the so-called implicit procedure, is permissible. This permission for the implicit procedure is implemented in German law in Section 25 (1) RechVersV. Instead of the term implicit process , the synonymous German term Zillmerungsverfahren is used. This reference to European law defines this term, which is otherwise not defined as binding in German law.

Consideration of negative values ​​as a receivable when calculating the actuarial reserve

The approach mentioned in § 4 Paragraph 1 DeckRV of a claim for reimbursement of the one-time acquisition costs paid, which are shown in the balance sheet as not yet due claims according to § 15 RechVersV , is not specific for the use of the Zillmer method when calculating the Actuarial reserve, but also results from the prospective procedure and every other permissible procedure. The approach to this requirement is based on general commercial law principles.

According to general commercial law principles, expected income based on contractual agreements in the case of a partially fulfilled pending transaction may in future be recognized as a receivable to cover expenses already incurred , but only up to the amount of the expenses actually incurred. The prerequisite for the existence of a partially fulfilled pending transaction is that the business activity, in the context of which the expenses were incurred, is a contractual part of the contract and that the expected contractual income is contractually provided as payment for this part of the contract. In this case, the contract must stipulate that the insurance company's conclusion activities take place within the framework of the contract and that the contributions are also remuneration for these activities. A negative value resulting from the calculation of the actuarial reserve can then be recognized as a claim.

Maximum zillmer rate

With the prospective method and all other permissible methods, future contractual contributions are taken into account in the balance sheet in advance in order to be able to offset the acquisition costs already incurred in the income statement. However, the policyholder is not obliged to pay these premiums in the future (Section 165 (1) VVG). As a result, the claim stated in the amount is canceled immediately upon termination of the premium payment by the policyholder. The resulting loss cannot jeopardize the insurer's existence, but it does reduce the surpluses for the owners and for the community of policyholders. In addition, taking such amounts into account in the balance sheet makes it economically easier for CC to pay acquisition commissions. In order to limit both, in the context of state supervision, i.e. social control, not accounting, the consideration of contributions is limited in advance by the so-called maximum zillmer rate . It is determined by § 4 DeckRV and amounts to 2.5% of the sum of the contractual contributions from January 1, 2015. This means that future contributions may only be taken into account in advance in the actuarial reserve or in the claim to this extent. This applies regardless of whether the Zillmer method is used or not. The maximum zillmer rate is also to be taken into account when determining the surrender value in accordance with Section 169 (3) VVG. In any case, the Zillmer method cannot be used there because of the design. Since the German supervisory authority cannot determine foreign insurers, this provision does not apply to insurers based abroad. Foreign insurers may therefore agree surrender values ​​for contracts concluded in Germany, for which future premiums are taken into account in advance without restriction.

The DeckRV only determines the limitation of the consideration of future contributions. On the other hand, it does not define or regulate the Zillmer procedure itself, but only describes it to explain the problem that led to the legal determination of a maximum Zillmer rate.

"Gezillmerte surrender values"

The term "Zillmerung" is sometimes associated with the phenomenon that the economic value of a life insurance contract is initially well below the sum of the premiums already paid, even if this is not justified as an economic reality by the process with which the economic one Value is determined. So this is not a specific consequence of using the Zillmer method described above, but results from any method that is suitable for determining economic value.

Surrender values , which are initially significantly below the premiums already paid, are also referred to as zillmerized surrender values , regardless of the mathematical method used to calculate the values. The amount of these surrender values ​​only reflects the actual economic value of the contract. The legal basis for the surrender value in Germany from 1994 to 2007 was expressly the market value (Section 176 (3) VVG in the version from 1994 to 2007), i.e. the economic value. This must always be assessed using the prospective method and is therefore even lower than the value of the actuarial reserve, which is to be applied carefully. The discussion about the surrender value is not about the question of whether the Zillmer process is used or not. Rather, it is about the consequence of the phenomenon of economic value described above that the value of the contract is initially significantly lower than the contributions already paid. This is opposed to the view that the surrender value is based on the sum of the contributions already paid, i.e. should be higher than the economic value.

The surrender value of a life insurance contract had in Germany until 1994, according to the law analogous to be agreed in the actuarial provision. Until 2008 it could be contractually agreed that the surrender value, in favor of the policyholder, was determined differently from the statutory minimum value, the (economic) fair value, analogous to the value determined under commercial law, i.e. the actuarial reserve. However, since these values ​​were usually determined using the Zillmer method, the Zillmer method was also indirectly linked to the surrender value of contracts concluded before 2008.

Since 2008, the surrender value in Germany can no longer be easily agreed on the basis of the actuarial reserve, i.e. the economic value of the insurer's obligation determined according to commercial law principles, or the economic market value. According to the law, the surrender value must include the closing costs in the amount of the imputed closing cost surcharges for the contributions, spread over at least 5 years at the current value of the expenditure. (Section 169 (3) VVG) This calculation of the surrender value cannot be achieved with the Zillmer method. The very simple approximation method can only be used if the closing costs are incurred in full at the start of the contract. Nevertheless, the surrender value is initially still lower (albeit not quite as much) as the sum of the contributions already paid and therefore colloquially in the sense described above is still spoken of zillmerization when reference is made to the phenomenon that the surrender values are lower are than the contributions paid so far. In countries in which the surrender value is based on the value of the contract, even in those where the Zillmer method was never used, the surrender values ​​are significantly lower than in Germany.

literature

• Accounting and auditing of insurance companies. 5th edition. IDW Verlag, 2011, p. 96 ff.
• Engelbrecht: The cover capital in life insurance. In: Journal for the entire insurance science. 1907, p. 611 ff.

Individual evidence

1. ^ Zillmer: Contributions to the theory of the premium reserve in life insurance establishments. Publishing house von Th. Von der Rahmen, Stettin 1863.
2. ^ Zillmer: Contributions to the theory of the premium reserve in life insurance establishments. Verlag von Th. Von der Rahmen, Stettin 1863, p. 23 f.
3. ^ Heym: Yearbooks for Economics and Statistics. Verlag Fischer, Volume V 1882, p. 210 f.
4. ↑ Höckner under the pseudonym Logophilus: The dispute over the Zillmer method in life insurance. 1902, p. 78 ff.
5. ↑ Höckner under the pseudonym Logophilus: The dispute over the Zillmer method in life insurance. 1902, p. 79.
6. ↑ Engeländer: The economic theory of the actuarial reserve according to US-GAAP. In: Insurance Industry. 1997, p. 45.
7. ↑ ifrs.org
8. ↑ Faigle / Engeländer, Versicherungswirtschaft 2001, p. 1570.
9. ^ Zillmer: The mathematical calculations in life and pension insurance. 2nd Edition. Nicolaische Verlags-Buchhandlung, Berlin 1887, p. 112 ff.
10. ↑ Federal Council printed matter 823/94. P. 123 f.
11. ↑ Bundesrats-Drucksache 114/96, p. 10.
12. ↑ BT-Drs. 12/6959, p. 103.
13. ↑ Engeländer, NVersZ 2002, p. 436.
14. ↑ BVerfG 1 BVR 1317/96 para. 65.
15. ↑ BVerfG February 15, 2006 - 1 BvR 1317/96 In: NJW. 2006, p. 1783.

Please note the information on legal issues!