Evaluation of variable annuity guarantees with the effect of jumps in the asset price process

Abstract

Financial crisis in 2007–2008 have caused losses to life insurance companies issuing variable annuities with guarantees. This is partly due to failure of variable annuity (VA) issuers to anticipate the large variations in asset prices during the financial crisis times in their pricing framework and also setting a higher guaranteed rate. This study aims to investigate the pricing of the guaranteed minimum death and accumulation benefits embedded in flexible premium VA. We compare the prices from calibrated Black–Scholes model to that of calibrated jump-diffusion model. Although both models assume constant volatility, the fact that Black–Scholes model ignores abnormal asset price changes due to jumps is likely to under-price the VA. We also conduct a case study to analyse the impact on guarantee fees for different stock market performance and regional mortality rates.

Keywords:

• jump-diffusion model
• variable annuity

Public Interest Statement

This study investigates the pricing of the guaranteed minimum benefits embedded in variable annuity (VA). Financial crisis in 2007–2008 have caused losses to life insurance companies issuing variable annuities with guarantees. The study assesses the effects of jumps in the asset price process on the VA guarantee fees. We also analyse the impact on the guarantee fees from stock market performance and regional mortality rates. The study confirms that by ignoring jumps in the asset price, Black–Scholes model leads to under-pricing of the VA guarantees. This implies that Life Insurance Company has a high probability of suffering losses due to abnormal asset price changes if it uses Black–Scholes model in pricing. Also the study shows that high-performing stock markets reduces the cost of the VA guarantees, while higher mortality rates of annuitants increase the cost of the VA guarantees.

Acknowledgements

We would like to thank in advance the editor and anonymous reviewers for their supportive comments and suggestions.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Mussa Juma

Mussa Juma received his BA (Hons) in Statistics & Economics and MSc in Insurance and Risk Management. He is currently studying PhD in science at Universiti Tunku Abdul Rahman, Malaysia in the area of insurances and econometrics.

Min Cherng Lee

Min Cherng Lee received his BS (Hons) in Mathematics with Actuarial Science and PhD in Statistics. Currently, he is an assistant professor of the Department of Mathematical and Actuarial Sciences at Universiti Tunku Abdul Rahman of Malaysia. His research interests include modelling and data analytics.

Seong Tah Chin

Seong Tah Chin received his BS (Hons) in Mathematics, Master and PhD in Mathematics. Currently, he is an adjunct associate professor of the Department of Mathematical and Actuarial Sciences at Universiti Tunku Abdul Rahman of Malaysia, specializing in stochastic processes.

Kian Wah Liew

Kian Wah Liew received his BS (Hons) in Mathematics, Master and PhD in Statistics. Currently, he is an assistant professor of the Department of Applied Mathematics at University of Nottingham, Malaysia campus in the area of derivatives pricing.