# A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk

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## Abstract

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## 1. Introduction

## 2. Projection in General

## 3. Valuation Bases and Insurance Model

## 4. Participating Life Insurance

#### 4.1. Non-Guaranteed Payments (Bonus)

#### 4.2. Product Specification

#### 4.3. Two-Account Model

#### 4.4. Bonus Mechanisms

#### 4.5. Technical Reserves

**Example 1**(Survival model). We consider a simple example that provides the basis for numerical illustrations later on. The state of the policy is described by the classical survival model with two states, 0 (alive) and 1 (dead). The payments of the policy consist of a constant continuous premium payment π while alive, a term insurance sum ${b}^{ad}$ upon death before expiration T and a pure endowment sum ${b}^{a}$ upon survival until expiration T. Under the bonus scheme “additional benefits”, bonus is used to increase the endowment sum. There are no payments in the death state. For simplicity, we write $I={I}_{1}$, $N={N}_{01}$, ${\mu}^{\u25cb}={\mu}_{01}^{\u25cb}$ and ${p}^{\u25cb}={p}_{00}^{\u25cb}$ for $\u25cb=*,m$, and we have

#### 4.6. Cash Flows

**Example 2**(Survival model continued). For the simple policy in Example 1, the time 0 market premium and benefit cash flows read

#### 4.7. Two-Account Projection

**Example 3**(Survival model continued). For the simple policy in Example 1–2, the adjustment term α reads

#### 4.8. Procedure for Determining the Technical Interest Rate and the Upscaling Factor

**Example 4**(Survival model continued). For the simple policy in Example 1–3, the expected technical reserve ${V}^{*,\u25cb}\left(\right)open="("\; close=")">\xb7,\rho $ reads

**Remark 1.**In Section 4.2, we mentioned that our setup does not allow for policyholder behavior options, such as surrender or free policy. However, it is not particularly complicated to include surrender, since it is an absorbing state. For the sake of clarity, we will not go into details on how. We just mention that, under the bonus scheme “additional benefits”, the surrender cash flow needs to be split into an upscaled and non-upscaled part. Furthermore, under the bonus scheme “consolidation”, the bonus suddenly raises the guarantee through a higher surrender value (typically equal to the technical reserve), and the market cash flows need to be recalculated every time the policy is consolidated to account for the higher surrender value.

#### 4.9. Bonus Allocation and Guarantee Fee

#### 4.10. Application of Projections

#### 4.11. Numerical Examples

#### 4.11.1. One-Policy Portfolio

- the expected evolution of the upscaling factor $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">{k}^{\left(\right)}$,
- the expected evolution of the assets $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">X\left(t\right)$, market reserve $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">V\left(t\right)$, technical reserve $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">Y\left(t\right)$ and collective bonus potential $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">K\left(t\right)$,
- the expected level for the guarantee injections ${\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">g\left(t\right)$ and guarantee fees ${\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">{\pi}_{g}\left(t\right)$,$t=1,...,T$.

**Figure 3.**Approximated expected guarantee injection g and guarantee fee π

_{g}as a function of time.

**Figure 4.**Approximated expected assets X, technical reserve Y, market reserve V and collective bonus potential K as a function of time.

#### 4.11.2. Two-Policy Portfolio

#### 4.11.3. Constant Guarantee Fee Fraction

#### 4.11.4. Period-Dependent Guarantee Fee Fraction

**Figure 6.**Approximated expected guarantee injection g and guarantee fee π

_{g}as a function of time.

**Figure 8.**Approximated expected assets X, technical reserve Y, market reserve V and collective bonus potential K as a function of time.

## 5. Unit-Linked Insurance

#### 5.1. Product Specification and Two-Account Model

**Example 5**(Survival model). We consider a unit-linked version of the simple participating life insurance policy in Examples 1–4. This example provides the basis for numerical illustrations later on. The state of the policy is described by the classical survival model with two states, 0 (alive) and 1 (dead). The policy expires at the retirement date, i.e., $T=R$. The payments of the policy consist of a constant continuous premium payment π while alive, a term insurance sum ${b}^{ad}$ upon death before expiration T and a pure endowment sum upon survival until expiration T. The size of the endowment sum is equal to the value of the assets at expiration divided by the (market) probability of surviving to expiration. There are no payments in the death state. For simplicity, we write $I={I}_{1}$, $N={N}_{01}$, ${\mu}^{m}={\mu}_{01}^{m}$ and ${p}^{m}={p}_{00}^{m}$, and we have

#### 5.2. Cash Flows

**Example 6**(Survival model continued). For the simple policy in Example 5, the time 0 market premium and benefit cash flows read

#### 5.3. Two-Account Projection

#### 5.4. Applications of Projections

#### 5.5. Numerical Example

- the expected evolution of the assets $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">X\left(t\right)$ and the guarantee account $t\mapsto {\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">Y\left(t\right)$,
- the expected level for the guarantee upgrade ${\text{\mathbb{E}}}^{Q}\left(\right)open="["\; close="]">u\left(t\right)$, $t=1,...,40$.

#### 5.5.1. Unit-Linked versus Participating Life

**Figure 11.**Approximated expected cash flows for the participating life insurance policy and the unit-linked insurance policy as a function of time.

**Figure 12.**Empirical distribution of final payments for the participating life insurance policy and the unit-linked insurance policy.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Jensen, N.R.; Schomacker, K.J.
A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk. *Risks* **2015**, *3*, 183-218.
https://doi.org/10.3390/risks3020183

**AMA Style**

Jensen NR, Schomacker KJ.
A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk. *Risks*. 2015; 3(2):183-218.
https://doi.org/10.3390/risks3020183

**Chicago/Turabian Style**

Jensen, Ninna Reitzel, and Kristian Juul Schomacker.
2015. "A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk" *Risks* 3, no. 2: 183-218.
https://doi.org/10.3390/risks3020183