Einladung zur Vortragsreihe aus Finanz- und Versicherungsmathematik
Prof. Pavel V. Shevchenko
Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit and capital protection options via stochastic control optimization
In this talk we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. A variable annuity with GMWB promises to return the entire initial investment through cash withdrawals during the policy life plus the remaining account balance at maturity, regardless of the portfolio performance. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss-Hermite quadrature method with cubic spline. As expected, we found that the fair fee in the case of optimal policyholder withdrawals is significantly higher than in the case of static (pre-determined) withdrawals commonly assumed in the industry practice for valuation of these products. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value – there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic installment of fixed amount, and it is cheaper than buying a separate life insurance.
We also present numerical algorithm and results for pricing a protected capital option offered by many asset managers for investment portfolios to take advantage of market growth and protect savings. Pricing of protected capital option is similar to pricing GMWB. However, few differences such as anniversary reset of the protected capital to the portfolio value if the later is higher and different penalty for withdrawals above a certain threshold make numerical evaluation more challenging. Under optimal withdrawal policyholder behavior the pricing of such a product is an optimal stochastic control problem that cannot be solved using Monte Carlo method. To evaluate products with capital protection option, it is common industry practice to assume static withdrawals and use Monte Carlo method. As a result, the fair fee is underpriced if policyholder behaves optimally. We found that extra fee that has to be charged to counter the optimal policyholder behavior is most significant at smaller interest rate and higher volatility levels, and it is sensitive to the penalty threshold. At low interest rate and a moderate penalty threshold level (15% of the portfolio value per annum) typically set in practice, the extra fee due to optimal withdrawal can be as high as 40% and more on top of the base case of no withdrawal or the case of fixed withdrawals at the penalty threshold.
The talk is based on recent papers
Zur Person: Prof Pavel Shevchenko is a Senior Principal Research Scientist in the Division of Computational Informatics of CSIRO (The Commonwealth Scientific and Industrial Research Organisation of Australia). Prof Shevchenko joined CSIRO in 1999 to work in the area of financial risk. He leads research and commercial projects on modelling of operational and credit risks; retirement products; option pricing; insurance; modelling commodities and foreign exchange; and the development of relevant numerical methods and software. He received a MSc from Moscow Institute of Physics and Technology and Kapitza Institute for Physical Problems in 1994; a PhD from The University of New South Wales in 1999 in theoretical physics. He is currently an adjunct Professor at School of Mathematics and Statistics University of NSW, adjunct Professor at School of Mathematical Sciences University of Technology Sydney, Honorary Senior Research Associate in University College London, and associate editor of the Journal of Operational Risk. Prof Shevchenko has published extensively in academic journals, consults for major financial institutions and is a frequent presenter at industry and academic conferences. His publication records include one book monograph, two co-authored monographs, two book chapters, over 50 journal papers and over 70 technical reports.
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Dipl.-Ing. Manfred Rapf
o.Univ.-Prof. Dr. Walter Schachermayer
Univ.-Prof. Dr. Thorsten Rheinländer
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