Cover capital
In actuarial mathematics, the actuarial reserve is the designation for an actuarially calculated value that is assigned to an insurance contract at a certain point in the insurance process. However, this assignment of a value is not objectively and clearly determined. Rather, depending on the point of view, the choice of assumptions for the valuation and the purpose of the valuation for a given insurance contract, very different values can arise at one point in time, each of which would be referred to as coverage capital. The term cover capital therefore only describes what the value should mean, but does not contain a specific definition of a value.
The mathematical symbol in traditional actuarial mathematics for the coverage capital at the end of the m-th year of an insured person who is x-year-old at the start of the contract is .
Background of the concept
Originally, insurance contracts were valued during the course of the insurance in order to be able to determine the extent to which the insurer had to make investments in order to adequately cover the obligation under the contract, i.e. to ensure compliance with it. This value of the required (not actually available) capital investments was therefore referred to as cover capital . This term was then generally used for each value assigned to an insurance contract at a certain point in the insurance history, since the calculation of these values is largely the same, even if the resulting values are very different from the principle of the mathematical procedure, as with regard to the evaluation process can.
In the legal language, cover capital is only used as a general term for a contract value to be specified in more detail (Section 169 of the Insurance Contract Act (Germany) and Section 81k of the Insurance Supervision Act (Austria) ). The value of the capital investments that an insurer has to make to cover its obligations is legally referred to in Germany and Switzerland as the target amount of the tied assets and in Austria as the cover requirement .
Evaluation process
In actuarial mathematics there are traditionally two types of cover capital, prospective cover capital and retrospective cover capital .
Prospective reserve capital
The prospective reserve capital is an evaluation of the contract with a view to future contract execution. It is therefore described as an evaluation of the difference between the future outgoing payments (services and possibly costs) and future incoming payments ( contributions ). Past payment flows from the contract, e.g. B. the contributions paid so far for the contract, play no role here. In almost all practical applications, the prospective reserve capital is the only appropriate calculation method. Thus, the cash surrender value , as the repurchase of the rights under the contract to be determined value of the future by those rights payments to be made to calculate the prospective actuarial reserves, as well as the provision for future policy and other economic assessments of the contract such as the fair value . In addition to the theoretically pure form of prospective reserve capital, where all conceivable future payment flows of the contract are taken into account under all imponderables, simplifications are used in practice, but these are always chosen in such a way that this leads to an increase in the resulting reserve capital. In practice, not all imponderables are taken into account (calculation of cover capital using a stochastic model ), in particular not all interactions (cover capital calculation using a deterministic model ). In addition, the future costs for the administration of the contract, in particular, are not explicitly estimated, but are taken into account with the flat rate cautiously applied in the contribution ( implicit procedure or zillmerized net contribution procedure ).
Retrospective reserve capital
The retrospective reserve capital is an evaluation of the contract with a view to the cash flows exchanged in the past. It describes how much the policyholder (less often the insurer) has paid more than he has received. In the insurance sector, this procedure can only be used sensibly at the collective level, since the insurer's main benefit, the insurance benefit, is only paid to individuals in the group. In addition, it does not describe the economic value of the contract, but only provides a view of the past. For this reason, the retrospective reserve capital is only used for contracts in which the entitlements to future payments are derived directly from previous payments, e.g. B. in unit-linked life insurance . Otherwise the retrospective reserve capital may not be used for legal purposes.
use cases
In order to determine the actuarial reserve of a contract in the annual financial statements of an insurer, a prospective reserve capital, which may also be determined using the implicit method, must be calculated in accordance with commercial law principles. The assumptions on future cash flows must meet the requirements of commercial law, particularly with regard to the principle of prudence , on the balance sheet date . In order to determine the surrender value for contracts concluded in Germany from January 1, 2008, however, a prospective reserve capital must be calculated using the assumptions about future cash flows that were used as a basis for the premium calculation when the contract was concluded in accordance with the contractual agreements. The same applies in Austria and Switzerland. In the case of contracts concluded earlier, the surrender value in Germany was always to be paid as at least the current value. The fair value of the claims from an insurance contract is also a prospective reserve capital, but with the assumptions about future cash flows that a market participant would use. To the extent that the fair value is to be used for balance sheet purposes , assumptions of the market participants are used - here, however, selected in accordance with the accounting requirements .
The coverage capital in traditional actuarial mathematics
Traditional actuarial mathematics determines the coverage capital as prospective coverage capital with a deterministic model and mostly with the implicit method.
The traditional actuarial approach
Traditional actuarial mathematics assigns a value for insurance benefits to each insurance year, separated according to the type of insured event (death, experience, occupational disability, etc.). This value is determined by multiplying the probability of the occurrence of the insured event by the amount of the service to be provided in accordance with the contract. The probability is taken from a table according to the current characteristics, mostly just age and gender, the “elimination order” (referred to as a life table for insurances on death or survival ). The probability that the insurance year in question will be experienced at all is also taken into account.
Due to the interest effect, cash flows have different values if they are due at different times. Since the payments in the individual insurance years are made at different times, namely in a different insurance year, they must be standardized to the same point in time by discounting or compounding interest. In traditional actuarial mathematics, a fixed interest rate is used for all payment dates, known as the "calculation interest", taking into account the compound interest .
These values are used in traditional actuarial mathematics to determine the required net single premium as the present value of all values for each insurance year at the beginning of the contract (net single premium according to the principle of equivalence), which is a mathematical symbol . The "n" stands for the term of the contract. “Single premium” means that the policyholder pays a single, single premium for the entire contractual insurance coverage at the start of the contract. “Net” means that all expenses incurred in fulfilling the contract for insurance operations have been ignored. The equivalence principle states that the contribution of the contract was determined exactly according to the imputed need. However, the imputed need is regularly determined extremely carefully, so that these contributions almost certainly lead to surpluses.
The gross one-time premium is obtained by taking into account the expenses for insurance operations in addition to the services. These are in particular the initial closing expenses and the ongoing expenses for collection and administration. Flat rates, the so-called “cost surcharges”, are used for this.
Elimination regulations, calculation interest and cost surcharges form the calculation bases of traditional actuarial mathematics.
Most contracts do not provide for single contributions, but regular contributions, ie contributions in "installments", as these single contributions are too high to be paid all at once. Traditional actuarial mathematics takes these “installment” contributions (called “ongoing contributions”) into account as annual contributions. Shorter contribution periods, e.g. B. monthly contributions are not taken into account in traditional actuarial mathematics, but determined by simply dividing the annual contribution by 12 and applying a flat rate surcharge. The current contributions carry the designation . The calculation is based on the "annuity" of the single premium. The calculation is the same as if the insurer were to provide the policyholder with a single premium loan, and the ongoing premiums are the repayments of that loan. However, since mortality is also taken into account, d. H. in the event of death, the loan is canceled, resulting in a pension formula. This means that the annual premium is equal to the single premium divided by the present value of the pension.
In general, traditional actuarial mathematics does an intensive “recycling” of formulas. So-called commutation values are determined based on the calculation bases . These are arithmetic operations “stored” in tables. Traditional actuarial science was developed long before calculating machines were invented (in the 17th and 18th centuries). The number of individual arithmetic operations described above that had to be carried out by hand in a single contract with a term of 20 years is several hundred. Since, apart from the sum insured, there are very few parameters (age, gender and term), you can tabulate many calculations, mostly values for lifelong contracts. For example, the value for a contract with a 20-year term for an insured person who is 30 years old at the start is obtained by subtracting the tabular value for a 50-year-old with a lifelong term from the table value for a 30-year-old with a lifelong term, with only two arithmetic operations remaining demanding correction for interest rate and survival rate is required. The difference is exactly the 20-year contract period between the age of 30 and the age of 50. However, these table calculations are hardly necessary today because of the high computing speed of computers.
The price for this calculation simplification is, however, that the insurance products have to be extremely simple and no differentiation is possible for the interest rate according to duration. However, these simple products are extremely cheap to manage, so that they are prevalent in Germany to this day.
The determination of the reserve capital
The calculation method of traditional actuarial mathematics makes it possible to determine a cover capital at that point in time according to the calculation bases used. Traditional actuarial mathematics generally assumes that the contributions stated in the formulas were determined according to the equivalence principle, i.e. the same calculation bases are used for all calculations. Then the prospective reserve capital corresponds to the retrospective.
The actuarial reserve can simply be described as the one-time contribution that would be required in addition to the future current contributions in order to be able to make future payments. The formulas for the reserve capital are designed accordingly. Since the calculation of the reserve capital must be carried out annually for each contract, the formulas have been made particularly efficient; H. with as few arithmetic operations as possible through intensive use of already tabulated values. Traditional actuarial mathematics was developed in particular for the efficient annual calculation of the actuarial reserves for the purpose of determining the actuarial reserve .
Criticism of traditional actuarial mathematics
Traditional actuarial science was designed for a world without computers. Its main goal is to minimize the number of arithmetic operations. For this, significant limitations in the accuracy of the invoice and the flexibility of the products were accepted.
With today's demands on accuracy, it can hardly be justified that payments due in one month are discounted with the same interest rate as payments due in 10 years, instead of using a yield curve .
Outside of Germany, where the traditional actuarial method is referred to as the “deterministic method”, much more flexible methods that make use of the possibilities of modern computer technology are already in use. B. the "stochastic" or "analytical" methods. The products developed in this way can no longer be mapped using the methods of traditional actuarial mathematics. For such products, neither calculation bases nor cover capital in the traditional sense can be determined.
On the other hand, the products determined using traditional actuarial mathematics are still the simplest, and therefore also the most transparent and efficient, products available today. However, as soon as the purpose of the contract is no longer solely the risk protection and the provision of a certain capital (or pension) when experiencing, but financial speculation on market values in capital markets is desired, these methods are overwhelmed.
Cover capital in German law
The use of the cover capital concept in commercial law
The definition of prospective reserve capital corresponds to the normal valuation procedure in commercial law for claims and debts. According to European law, which Germany has implemented in Section 341 of the Commercial Code (HGB), the actuarial reserve is to be determined as the difference between the actuarially determined value of the future outgoing payments and the corresponding value of the future incoming payments. The actuarial reserve is therefore to be determined as actuarial capital. Commercial law does not specify the exact calculation method. Depending on the type of contract, both classic actuarial and more modern methods, such as B. stochastic modeling or the like can be used. It is essential that the assessment is carried out carefully. In order to meet commercial law requirements, all assumptions must include a certain safety margin. Assumptions other than those used in the premium calculation may be used without further ado. Possibly. the assumptions of the premium calculation may not be used if they are not careful enough from a commercial law perspective. The use of less cautious assumptions than those of the premium calculation is permissible if the assumptions are still sufficiently cautious from a commercial law point of view and if the contributions taken into account are capped by the appropriate amount ( realization principle ). Also insofar as insurance contracts are to be valued at fair value, e.g. For example, in the case of group mergers in accordance with IFRS 3 , cover capital methods are used. Ultimately, the fair value principle also represents a reserve capital. Only here the assumptions are to be selected as they would be established by a market participant when setting a price.
There is, however, a difference in language usage between actuarial law and commercial law. In actuarial mathematics, the actuarial reserve of the individual contract is often referred to as the actuarial reserve and only the entire balance sheet item as the actuarial reserve. From a commercial law perspective, this is not correct, as an actuarial reserve must be created individually for each contract and the balance sheet item only represents the total of all actuarial reserves. Actuarial mathematics also often refers to the actuarial reserve calculated using the calculation bases of the premium calculation as "the actuarial capital", as if there was no other.
The concept of cover capital in Section 169 VVG
The actuarial reserve on which the statutory minimum surrender value is based is determined in accordance with Section 169 (3) VVG using the calculation bases for the premium calculation and - according to the legal justification - is based on the methods to be used to calculate the actuarial reserve in accordance with Section 341 f of the German Commercial Code (HGB), i.e. prospective Coverage capital to calculate. Since commercial law does not require the methods of traditional actuarial mathematics, but only some kind of prospective method, the method that was used to calculate the contributions must be applied. Section 169 (3) VVG further stipulates that if contracts from abroad are offered in which a cover capital is not customary, the reference value customary there is to be used. What is overlooked here is that "the reserve capital" is a universal mathematical method, namely simply the evaluation of all future payment flows, which must be the basis of a meaningful contribution determination with mathematical methods everywhere in the world. In particular, the European Union stipulates the use of this method for calculating the actuarial reserve; it is therefore used in every country in the European Union for every life insurance contract. With Section 341 f of the German Commercial Code, Germany has simply adopted this provision.
For life insurances taken out before 2008 and always for unit-linked life insurances, the surrender value must be determined on the basis of the current value. According to the legal justification for the introduction of the regulation from 1994, the fair value is equated with the common value in tax law , which corresponds to a prospective reserve capital with market-conform assumptions.
