Abstract
In this paper, we analyze the latest guarantee feature in the variable annuities market: guaranteed minimum withdrawal benefits for life (GMWB for life) which are also called guaranteed lifelong withdrawal benefits (GLWB). This option gives the client the right to deduct a certain amount annually from the policy’s account value until death—even if a unit-linked account value drops to zero. We show how such products can be analyzed within a general framework presented in Bauer et al. (ASTIN Bull. 38(2):621–651, 2008). We price the embedded guarantee for different product designs and parameters under both, deterministic and optimal client behavior.
Zusammenfassung
Das vorliegende Paper analysiert spezielle Garantien in „Variable Annuity“-Produkten, so genannte Guaranteed Minimum Withdrawal Benefits for Life (GMWB for life) oder Guaranteed Lifelong Withdrawal Benefits (GLWB). Diese Produkte geben dem Versicherungsnehmer das Recht auf jährliche Entnahmen einer gewissen Mindesthöhe, selbst wenn das Fondsguthaben zwischenzeitlich aufgebraucht ist. Wir zeigen, wie derartige Produkte in einem allgemeinen Modellrahmen aus Bauer et al. (ASTIN Bull. 38(2):621–651, 2008) analysiert werden können und bewerten die enthaltenen Garantien für unterschiedliche Produktdesigns und Parameter unter der Annahme eines deterministischen bzw. optimalen Kundenverhaltens.
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Notes
Since G W is not needed for GLWB options, two processes G E and one process G W are sufficient.
Contracts where the step-up is applied based on the account value before withdrawals can be considered by letting \(G_{t}^{E +} = \max\{ G_{t}^{E-} ;x_{W} \cdot A_{t}^{ -} \}\).
Note that surrender charges only apply to the exceeding part. Therefore surrender charges were not considered in case (c) above.
Whenever a non-guaranteed withdrawal occurs, future guarantees are also reduced. We here describe a so-called pro-rata reduction which is the predominant form in the market.
Here, IR + denotes the non negative real numbers (including zero); furthermore we let \(\mathit{IR}_{ +}^{\infty} = \mathit{IR}_{ +} \cup\{ \infty\}\).
We understand rational in the sense of maximizing the economic value of a contract. Thus, rationality is defined independent of any utility functions.
For an introduction to Monte Carlo methods see, e.g., Glasserman (2003).
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Holz, D., Kling, A. & Ruß, J. GMWB for life an analysis of lifelong withdrawal guarantees. ZVersWiss 101, 305–325 (2012). https://doi.org/10.1007/s12297-012-0193-3
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DOI: https://doi.org/10.1007/s12297-012-0193-3