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On the Bayesian Risk Evaluation of Minimum Guarantees in Variable Annuities

  • Byoung Hark Yoo , Bangwon Ko EMAIL logo and Hyuk-Sung Kwon
Published/Copyright: December 10, 2015
Asia-Pacific Journal of Risk and Insurance
From the journal Asia-Pacific Journal of Risk and Insurance Volume 10 Issue 1

Abstract

Long-term equity return models have a substantial impact on calculating reserves and capital requirements for minimum guarantees in variable annuities (VAs). Under a Bayesian statistical framework, we reinvestigate the standard long-term equity return models (independent lognormal, regime-switching lognormal and stochastic log volatility models) with the Korean equity market data. Our empirical analysis shows that the long-term behaviors of the Korean market are best described by the regime-switching lognormal models, and there can be significant difference in the amount of the reserves and the capital requirements depending on the model selection. The long-term nature of the VA contracts requires that more caution be paid for modeling the mean part of the stochastic log volatility model.

Keywords: Bayesian approach; regime-switching model; stochastic log volatility model; lognormal model; GMAB

Acknowledgement

This work was supported by the Soongsil University Research Fund of 2013.

Appendix A. Estimation Algorithm

A1. RSLN Model

Yt+1|ρt=μρt+σρtZt∼Nμρt,σρt2

  1. Generate μ from pμ|Y,ρ,σ2~ Normal

  2. Generate σ2 from p(σ2|Y,ρ,μ)~ Inverse Gamma

  3. Generate p from p(p|ρ)~ Beta

  4. Generate ρ from p(ρ|Y,μ,σ2,p)~ multi-move

  5. Repeat (i)-(iv)

A2. SLV Model

Yt=γ+eht/2ZY,t,whereZY,t∼N(0,1)
ht=μ+ϕht−1+σhZh,t, where ZY,t∼N(0,1)

  1. Generate γ from p(γ|Y,h)~ Normal

  2. Generate μ and ϕ from p(μ,ϕ|h,σ2)~ Normal

  3. Generate σ2 from p(σ2|h,μ,ϕ)~ Inverse Gamma

  4. Generate h from p(h|Y,γ,μ,ϕ,σ2)~ multi-move

  5. Repeat (i)-(iv)

Appendix B. Marginal Likelihood Calculation

B1. RSLN Model

lnp(Y)=lnp(Θ∗)+lnp(Y|Θ∗)−lnp(Θ∗|Y)
lnp(Θ∗|Y)=lnp(μ∗|Y)+lnp(σ2∗|Y,μ∗)+lnp(p∗|Y,μ∗,σ2∗)
lnp(p∗|Y,μ∗,σ2∗)=1G∑g=1Gp(p∗|Y,μ∗,σ2∗,ρ(g))

B2. SLV Model

lnp(Y)=lnp(Θ∗,h∗)+lnp(Y|Θ∗,h∗)−lnp(Θ∗,h∗|Y)
lnp(Θ∗,h∗)=lnp(Θ∗)+lnp(h∗|Θ∗)
lnp(Θ∗,h∗|Y)=lnp(γ∗|Y)+lnp(μ∗,ϕ∗|Y,γ∗)+lnp(σ2∗|Y,γ∗,μ∗,ϕ∗)
+lnp(h∗|Y,μ∗,ϕ∗,σ2∗)
lnp(h∗|Y,μ∗,ϕ∗,σ2∗)=1G∑g=1Gp(h∗|Y,μ∗,ϕ∗,σ2∗,z(g))

Here Z(g) is the index variable for mixed normal distributions. See Kim, Shephard, and Chib (1998)) for more detail.

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Published Online: 2015-12-10
Published in Print: 2016-1-1

©2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Featured Articles
  3. Forecasting Mortality using Imputed Data: The Case of Taiwan
  4. On the Bayesian Risk Evaluation of Minimum Guarantees in Variable Annuities
  5. On the Use of Long-Term Risk Measures as an Approach to Communicating Risks
  6. Effective Structure of Reinsurance Function for Disaster Risk in the Asia-Oceania Region
  7. Forecasting Thai Mortality by Using the Lee-Carter Model
  8. Long-Term Care: Is There Crowding Out of Informal Care, Private Insurance as Well as Saving?
https://doi.org/10.1515/apjri-2015-0007
Keywords for this article
Bayesian approach; regime-switching model; stochastic log volatility model; lognormal model; GMAB

Articles in the same Issue

  1. Frontmatter
  2. Featured Articles
  3. Forecasting Mortality using Imputed Data: The Case of Taiwan
  4. On the Bayesian Risk Evaluation of Minimum Guarantees in Variable Annuities
  5. On the Use of Long-Term Risk Measures as an Approach to Communicating Risks
  6. Effective Structure of Reinsurance Function for Disaster Risk in the Asia-Oceania Region
  7. Forecasting Thai Mortality by Using the Lee-Carter Model
  8. Long-Term Care: Is There Crowding Out of Informal Care, Private Insurance as Well as Saving?
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