# Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Models

#### 2.1. Financial Market Model

#### 2.2. Mortality Model

#### 2.3. Surrender Model

## 3. Products and Approximations

#### 3.1. Product Definitions and Characteristics

#### 3.2. Product Pricing and Required Approximations

## 4. Model Calibration

#### 4.1. Financial Market Model

#### 4.2. Mortality Model

#### 4.3. Surrender Model

## 5. Numerical Example

## 6. Emergency Fund Extension

## 7. Conclusion and Future Research

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ATM | at-the-money |

bn | billion |

CDF | cumulative distribution function |

DAX | Deutscher Aktienindex |

EURIBOR | Euro Interbank Offered Rate |

GMAB | guaranteed minimum accumulation benefit |

GMDB | guaranteed minimum death benefit |

GMLB | guaranteed minimum living benefit |

GMWB | guaranteed minimum withdrawal benefit |

GMXB | guaranteed minimum benefit |

MC | Monte Carlo Simulation |

SB | surrender benefit |

U.K. | United Kingdom |

U.S. | United States |

VA | Variable annuity |

## Appendix A. Required Distributions

#### Appendix A.1. Financial Market Model

#### Appendix A.2. Insurance Market Model

## Appendix B. Required Theorems

#### Appendix B.1. Exponential of Truncated Univariate Gaussian

- if $b>0$:$$\begin{array}{cc}\hfill {\mathbb{E}}_{\mathbb{P}}\left[exp\left(a+bX\right){\mathbb{1}}_{\left\{a+bX\ge 0\right\}}\right]& =exp\left(a+b\mu +\frac{1}{2}{b}^{2}{\sigma}^{2}\right)\mathsf{\Phi}\left(\frac{a/b+b{\sigma}^{2}+\mu}{\sigma}\right),\hfill \\ \hfill {\mathbb{E}}_{\mathbb{P}}\left[exp\left(-a-bX\right){\mathbb{1}}_{\left\{a+bX\ge 0\right\}}\right]& =exp\left(-a-b\mu +\frac{1}{2}{b}^{2}{\sigma}^{2}\right)\mathsf{\Phi}\left(\frac{a/b-b{\sigma}^{2}+\mu}{\sigma}\right),\hfill \end{array}$$
- if $b<0$:$$\begin{array}{cc}\hfill {\mathbb{E}}_{\mathbb{P}}\left[exp\left(a+bX\right){\mathbb{1}}_{\left\{a+bX\ge 0\right\}}\right]& =exp\left(a+b\mu +\frac{1}{2}{b}^{2}{\sigma}^{2}\right)\mathsf{\Phi}\left(-\frac{a/b+b{\sigma}^{2}+\mu}{\sigma}\right),\hfill \\ \hfill {\mathbb{E}}_{\mathbb{P}}\left[exp\left(-a-bX\right){\mathbb{1}}_{\left\{a+bX\ge 0\right\}}\right]& =exp\left(-a-b\mu +\frac{1}{2}{b}^{2}{\sigma}^{2}\right)\mathsf{\Phi}\left(-\frac{a/b-b{\sigma}^{2}+\mu}{\sigma}\right).\hfill \end{array}$$

**Proof.**

#### Appendix B.2. Exponential of Truncated Multivariate Gaussian

**Proof.**

#### Appendix B.3. First Order Moments of Truncated Bivariate Gaussian

**Proof.**

**Proof.**

#### Appendix B.4. Second Order Moments of Truncated Bivariate Gaussian

**Proof.**

**Proof.**

#### Appendix C. Approximation Proofs

#### Appendix C.1. Proof of ${S}_{2}^{{\mathbb{Q}}^{T}}$ (D, α, β, **t**)

#### Appendix C.2. Proof of ${S}_{2}^{{\mathbb{Q}}^{T}}$ (D, α, β, **t**)

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^{1}Source: White Mountain Insurance Group Report 2010. In this case the expected number of policyholders entitled to the final payoff increases and therefore, the present value of liabilities rises as well.^{2}This is more or less a general assessment. Cancellation could be rational and utility-maximizing for specific policyholders, however, these personal reasons for cancellation are not included in the model.^{3}For the underlying data set see http://www.ons.gov.uk/ons/rel/lifetables/historic-and-projected-mortality-data-from-the-uk-life-tables/2010-based/rft-qx-principal.xls^{4}In the suggested modeling approach the mortality intensity can become negative with positive probability. This probability can be calculated analytically, see Appendix A.2. However, in practical applications, like for the parameters used in our example (see Section 4), this probability is negligible (less than ${10}^{-5}$).^{5}This will be the range for the corresponding expressions in our numerical case studies.^{6}The data can be downloaded from http://www.bankofengland.co.uk/statistics/Pages/yieldcurve/archive.aspx.^{7}For the parameters in Table 5 the actual boundaries are 1%–4.88% and 4.88%–9.82%.^{8}Delta (abs.) represents the difference between the approximated and the corresponding simulated prices.^{9}Delta (rel.) is equal to the ratio of Delta (abs.) and the simulated price. Hence, it refers to the relative price deviations.

**Figure 2.**Approximation Error for Equation (13).

Maturity | 2Y | 3Y | 4Y | 5Y | 6Y | 7Y | 8Y | 9Y | 10Y |
---|---|---|---|---|---|---|---|---|---|

Volatility | 99.7 | 66.1 | 61.2 | 55.4 | 50.0 | 45.3 | 41.6 | 38.5 | 36.1 |

${\mathit{a}}_{\mathit{r}}$ | ${\mathit{\sigma}}_{\mathit{r}}$ | ${\mathit{\rho}}_{\mathbf{Sr}}$ |
---|---|---|

0.0799 | 0.0079 | −0.0403 |

${\mathit{T}}_{\mathit{i}}$ | 01/18/2013 | 02/15/2013 | 03/15/2013 | 06/21/2013 | 09/20/2013 | 12/20/2013 |

${\sigma}_{S}\left({T}_{i}\right)$ | 0.1368 | 0.1232 | 0.1557 | 0.1712 | 0.1898 | 0.1993 |

${\mathit{T}}_{\mathit{i}}$ | 06/20/2014 | 12/19/2014 | 06/19/2015 | 12/18/2015 | 12/16/2016 | 12/15/2017 |

${\sigma}_{S}\left({T}_{i}\right)$ | 0.2179 | 0.2146 | 0.2367 | 0.2624 | 0.2432 | 0.2237 |

Parameter | b | z | κ | γ | ${\mathit{\sigma}}_{\mathit{\xi}}$ | ${\mathit{\gamma}}_{\mathit{m}}$ |
---|---|---|---|---|---|---|

Estimated Value | 12.1104 | 76.1390 | 0.4806 | 0.0195 | 0.0254 | 10.6482 |

Case 1 | Case 2 | Case 3 | |
---|---|---|---|

α | 0.00 | 1.00 | 0.25 |

β | 0.00 | 0.04 | 0.20 |

C | 0.00 | 0.01 | 0.05 |

Insured | man, aged 50 years |
---|---|

Premium | 100 |

Maturity | 15 years |

δ | 1.00 % |

Termination dates | 1, …, 14 (once per year) |

Repayment Dates | 1, …, 15 (once per year) |

Surrender Fees | 7%, …, 1% (years 1–7, linear) |

0% (years 8–14, no fee) |

GMAB | |||

Case 1 | Case 2 | Case 3 | |

Analytic Sol. | 110.6804 | x | x |

Approximation | 110.6804 | 79.2336 | 36.2581 |

Sim. (mean, 500K) | 110.9925 | 81.2484 | 39.0351 |

Sim. (std, 500K) | 0.1200 | 0.0662 | 0.0303 |

Delta (abs.)8 | −0.3121 | −2.0148 | −2.7771 |

Delta (rel.)9 | −0.0028 | −0.0248 | −0.0711 |

SB | |||

Case 1 | Case 2 | Case 3 | |

Analytic Sol. | 0 | x | x |

Approximation | 0 | 26.3468 | 62.9797 |

Sim. (mean, 500K) | 0 | 27.7328 | 63.2259 |

Sim. (std, 500K) | 0 | 0.0397 | 0.0519 |

Delta (abs.) | 0 | −1.3860 | −0.2462 |

Delta (rel.) | x | −0.0500 | −0.0039 |

GMDB | |||

Case 1 | Case 2 | Case 3 | |

Analytic Sol. | 12.2560 | x | x |

Approximation | 12.2560 | 10.3374 | 6.9443 |

Sim. (mean, 500K) | 12.2589 | 10.4157 | 7.0785 |

Sim. (std, 500K) | 0.0078 | 0.0049 | 0.0027 |

Delta (abs.) | −0.0029 | −0.0783 | −0.1343 |

Delta (rel.) | −0.0002 | −0.0075 | −0.0190 |

VA = GMAB + SB + GMDB | |||

Case 1 | Case 2 | Case 3 | |

Analytic Sol. | 122.9363 | x | x |

Approximation | 122.9363 | 115.9177 | 106.1820 |

Sim. (mean, 500K) | 123.2514 | 119.3968 | 109.3395 |

Sim. (std, 500K) | 0.1270 | 0.1007 | 0.0731 |

Delta (abs.) | −0.3150 | −3.4791 | −3.1575 |

Delta (rel.) | −0.0026 | −0.0291 | −0.0289 |

Case 4 | Case 5 | |
---|---|---|

α | 1.00 | 1.00 |

β | 0.04 | 0.04 |

$\tilde{\alpha}$ | 0.10 | 0.10 |

$\tilde{\beta}$ | 0.20 | 0.20 |

$\tilde{C}$ | 0.03 | 0.03 |

l | −0.25 | −0.50 |

GMAB | |||

Case 2 | Case 4 | Case 5 | |

Approximation | 79.2336 | 75.4547 | 78.1821 |

Sim. (mean, 500 K) | 81.2484 | 74.6267 | 77.2791 |

Sim. (std, 500 K) | 0.0662 | 0.0680 | 0.0678 |

Delta (abs.) | −2.0148 | 0.8280 | 0.9031 |

Delta (rel.) | −0.0248 | 0.0111 | 0.0117 |

SB | |||

Case 2 | Case 4 | Case 5 | |

Approximation | 26.3468 | 28.5614 | 26.8172 |

Sim. (mean, 500 K) | 27.7328 | 31.5358 | 29.6096 |

Sim. (std, 500 K) | 0.0397 | 0.0367 | 0.0381 |

Delta (abs.) | −1.3860 | −2.9744 | −2.7924 |

Delta (rel.) | −0.0500 | −0.0943 | −0.0943 |

GMDB | |||

Case 2 | Case 4 | Case 5 | |

Approximation | 10.3374 | 9.9925 | 10.2123 |

Sim. (mean, 500 K) | 10.4157 | 9.9483 | 10.1630 |

Sim. (std, 500 K) | 0.0049 | 0.0051 | 0.0050 |

Delta (abs.) | −0.0783 | 0.0442 | 0.0493 |

Delta (rel.) | −0.0075 | 0.0044 | 0.0048 |

VA = GMAB + SB + GMDB | |||

Case 2 | Case 4 | Case 5 | |

Approximation | 115.9177 | 114.0086 | 115.2116 |

Sim. (mean, 500 K) | 119.3968 | 116.1108 | 117.0517 |

Sim. (std, 500 K) | 0.1007 | 0.1017 | 0.1021 |

Delta (abs.) | −3.4791 | −2.1022 | −1.8401 |

Delta (rel.) | −0.0291 | −0.0181 | −0.0157 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Escobar, M.; Krayzler, M.; Ramsauer, F.; Saunders, D.; Zagst, R. Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs. *Risks* **2016**, *4*, 41.
https://doi.org/10.3390/risks4040041

**AMA Style**

Escobar M, Krayzler M, Ramsauer F, Saunders D, Zagst R. Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs. *Risks*. 2016; 4(4):41.
https://doi.org/10.3390/risks4040041

**Chicago/Turabian Style**

Escobar, Marcos, Mikhail Krayzler, Franz Ramsauer, David Saunders, and Rudi Zagst. 2016. "Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs" *Risks* 4, no. 4: 41.
https://doi.org/10.3390/risks4040041