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Derivation of biometrically dependent cash flows

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Abstract

The article deals with cash flows of future payments based on a promised payment in the area of life or pension insurance. Such a cash flow therefore consists of biometrically dependent values in respective future periods. The article presents a new approach and an iterative procedure for deriving formulae to determine these values. The application of the approach is exemplified for a central unit present value in the Heubeck mortality table model. The use of cash flows for the duration of future payments, the forecast of present values as well as premium reserves, and for the full yield curve approach is illustrated.

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Notes

  1. See, for example, [1,2,3] and [8].

  2. Payments of premiums into an insurance policy are also generally dependent on biometric factors.

  3. This approach is not followed in [1]. Instead, it is proposed there to take into account compound interest during the year. However, consistency with the Heubeck mortality table model cannot be achieved with this approach.

  4. Therefore, the IFRS standard IAS 19 (rev. 2018) in IAS 19.147 letter c) requires, at least among other disclosures, the disclosure of the weighted average duration of the obligations of a pension plan.

  5. The classic approach uses for the service cost and the interest cost the aggegated interest rate derived from a yield curve for the PBO. For more details, see [3].

  6. This denotes the present value of an active person’s entitlement to a retirement pension from the final age of annual amount 1 if disability occurs before the final age is reached. For the definition, see also [12], p. 321.

  7. The iteration procedure of Theorem 2 is also applicable for the so-called fluctuation model (s. [6]) according to Heubeck. Here, the status of internal beneficiaries and external beneficiaries must be distinguished. The respective transition or retention probabilities can also be found in [2].

  8. Compare person populations in [6].

  9. See also the Remark below.

  10. An adjustment factor of this kind is often not part of the agreed payment obligation itself, but part of the actuarial assumptions.

  11. An inflation curve provides for each \(n>0\) the average expected annual inflation over n annual periods. From this, the expectation for inflation in a given period can be derived.

  12. See for the meaning of k(t) also [5], page 493 ff.

  13. This is known as Fisher–Weil Duration in the case of non-biometrically dependent cash flows. (see [4], p. 415).

  14. In practice, as a rule, only the intra-year payments of old-age pensions in vesting period or current pensions are considered (see Sect. 6).

  15. If only the present value of a cash flow is known, but not its coefficients, the Macaulay duration can be approximately determined using present values at interest rates close to \(i_0\) (see [11]). See also [1], Sect. 9 for the determination of the Macaulay duration with known coefficients.

  16. A detailed exposition can be found in [11].

  17. The German Teilwert pursuant to Sect. 6 of the German Income Tax Act (EStG) and the actuarial premium reserve in an insurance policy is generally a prospective expression using constant premiums. The DBO can be understood as a retrospectively expressed premium reserve using the respective service costs as premiums.

  18. In general, \(v_{n-m} = \frac{v_n}{v_m}\) does not hold, so the assumption of a uniform rate of return is necessary here.

  19. In IFRS accounting standards, the abbreviation CSC is used for current service cost. In US-GAAP, SC stands for Service Cost. For practical reasons we use the abbreviation SC.

  20. For the derivation of the yield curve and the interest rates \(i_k\), various methods are commonly used, based on index values of corporate bonds at the valuation date \(t_0\).

  21. For more details, please refer to [3].

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  1. Freelance Author, member of DAV, aba and Fachvereinigung Mathematische Sachverständige (aba), Duesseldorf, Germany

    M. Matthias Schmitt

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Schmitt, M.M. Derivation of biometrically dependent cash flows. Eur. Actuar. J. 12, 779–812 (2022). https://doi.org/10.1007/s13385-021-00303-z

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