# Life Insurance Cash Flows with Policyholder Behavior

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## Abstract

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## 1. Introduction

## 2. Life Insurance Setup

**Definition 1.**The prospective reserve at time t for state $i\in \mathrm{\mathcal{J}}$ is denoted ${V}_{i}\left(t\right)$ and given as:

**Proposition 1.**The prospective reserve at time t given $Z\left(t\right)=i$, $i\in \mathrm{\mathcal{J}}$, satisfies,

**Proposition 2.**The prospective reserve at time t given $Z\left(t\right)=i$, $i\in \mathrm{\mathcal{J}}$, satisfies Thiele’s differential equation,

**Proof.**See Formula (4.7) in [11]. ☐

**Remark 1.**If a time point $T\ge 0$ exists, such that ${b}_{i}\left(t\right)={b}_{ij}\left(t\right)=0$ for $t>T$ and all $i,j\in \mathrm{\mathcal{J}}$, then the boundary conditions ${V}_{i}\left(T\right)=0$ for $i\in \mathrm{\mathcal{J}}$ are used with Thiele’s differential equation.

**Definition 2.**The cash flow at time t associated with the payment process ${\left(B\left(t\right)\right)}_{t\ge 0}$, conditional on $Z\left(t\right)=i$, $i\in \mathrm{\mathcal{J}}$, is the function $s\mapsto {A}_{i}(t,s)$, given by:

**Proposition 3.**The cash flow ${A}_{i}(t,s)$ satisfies,

**Proof.**The first result is proven in Proposition 2.3 in [1] using integration by parts. The second result follows from the first result and from Proposition 1. ☐

**Proposition 4.**The transition probabilities ${p}_{ij}(t,s)$, for $i,j\in \mathrm{\mathcal{J}}$, are unique solutions to Kolmogorov’s backward differential equation,

**Proof.**See [10], Theorem 2.3.4. ☐

**Remark 2.**The payments are for notational simplicity assumed to be continuous throughout the paper. It is straightforward to include single payments at deterministic time points, which allows for, e.g., an endowment insurance. For example, if a single state-dependent payment at time T is included, $\Delta {B}_{i}$, Formula (1) would read:

#### 2.1. Technical Basis and Market Basis

^{^}are associated with the technical basis. Thus, $V\left(t\right)$ is the prospective reserve for the market basis, and $\widehat{V}\left(t\right)$ is the prospective reserve for the technical basis.

#### 2.2. The Policyholder Options

## 3. The Survival Model

#### 3.1. Survival Model with Surrender Modeling

#### 3.2. Survival Model with Surrender and Free Policy Modeling

- The first line is the value of the original cash flow Equation (2) without policyholder behavior, reduced by the probability of not surrendering and not becoming a free policy.
- The second line is the value of the surrender payments, when not a free policy.
- The third line is the benefit payments as a free policy, i.e., the positive payments reduced with the free policy factor $\rho \left(\tau \right)$ at the time τ of the free policy transition.
- The fourth line is the surrender payments if surrender occurs after the free policy transition.

- The original cash flows $\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{A}^{+}(t,s)$ and $\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{A}^{-}(t,s)$.
- The prospective reserve on the technical basis ${\widehat{V}}^{+}\left(s\right)$ and ${\widehat{V}}^{-}\left(s\right)$, for all future time points $s\ge t$, which allow us to determine the surrender payments and the free policy factor $\rho \left(s\right)$.
- The factor ${r}^{\rho}(t,s)$, which is a simple integral of the free policy transition rate.

#### 3.3. Free Policy Modeling When Surrender Is Already Modeled

#### 3.4. Approximate Method

## 4. A General Disability Markov Model

**Figure 4.**The eight-state Markov model, with disability, surrender and free policy. The transition rates between States 0, 1 and 2 are identical to the transition rates between States 4–6. The two surrender states can be considered one state, and then, this model is known as the so-called “seven-state model”.

**Proposition 5.**The cash flow in State 0, $\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{A}^{\mathrm{f}}(t,s)$, for payments at time s valued at time t, is given by:

**Proof.**See Appendix A.3. ☐

**Proposition 6.**The quantities ${p}_{ij}^{\rho}(t,s)$ satisfy the forward differential equation, for $i\in \{0,1,2\}$ and $j\in \{4,5,6\}$,

**Proof.**See Appendix A.4. ☐

**Remark 3.**In Remark 2, single payments at time T were allowed. This can be included in the above results: if we assume single payments at time T, $\Delta {B}_{0}$ in state alive and $\Delta {B}_{1}$ in state disabled, the following terms must be added to the cash flow Equation (15),

## 5. Numerical Example

- A disability annuity consisting of an annual payment of 100.000 while disabled, until age 65.
- A life annuity consisting of an annual payment of 100.000 while alive, from age 65 until death.

- The three-state disability Markov-model, as shown in Figure 5;
- Interest rate of 1%;
- Transition rates, where x is the age,$$\begin{array}{cc}\hfill {\widehat{\mu}}_{01}\left(x\right)& =\left(0.0004+{10}^{4.54+0.06x-10}\right){1}_{\{x\le 65\}}\hfill \\ \hfill {\widehat{\mu}}_{10}\left(x\right)& =\left(2.0058\xb7{e}^{-0.117x}\right){1}_{\{x\le 65\}}\hfill \\ \hfill {\widehat{\mu}}_{02}\left(x\right)& =0.0005+{10}^{5.88+0.038x-10}\hfill \\ \hfill {\widehat{\mu}}_{12}\left(x\right)& ={\widehat{\mu}}_{02}\left(x\right)\left(1+{1}_{\{x\le 65\}}\right)\hfill \end{array}$$

- active to dead, ${\mu}_{02}\left(x\right)$: the mortality benchmark from 2012 from The Danish FSA,
- active to disabled: ${\mu}_{01}\left(x\right)={1}_{\{x\le 65\}}\left({10}^{5.662015+0.033462x-10}\right)$,
- disabled to active: ${\mu}_{10}\left(x\right)=4.0116{e}^{-0.117x}$,
- disabled to dead: ${\mu}_{12}\left(x\right)=0.010339+{10}^{5.070927+0.05049x-10}$.

**Figure 6.**Transition rates (left) and transition probabilities (right). In the left figure, the transition rates between the states in the disability model, (0, active, 1, disabled, and 2, dead) are shown, as well as the surrender (active to surrender (as)) and free policy (active to free policy (af)) transition rates. In the right figure, the full lines are the non-free policy states, and the dashed lines are corresponding free policy states. The active, free policy and surrender states are dominant.

**Figure 7.**Cash flows (left) and market value for different parallel shifts in the interest rate structure (right), measured in basis points. In the figures, we see that the modeling of policyholder behavior has a major impact. Furthermore, it seems difficult to distinguish the approximate and the correct method using the eye-ball norm only. The approximation yields a slightly larger cash flow and market value than the correct modeling does.

**Table 1.**Market value of the cash flow and dollar durations (DV01), without policyholder behavior (PHB), with the approximative method and the correct method. The duration is greatly reduced when policyholder behavior is included, and the approximation is close to the result from the correct method.

Results | Without PHB | Approximation | Correct |
---|---|---|---|

Market value | −183.798 | −76.599 | −72.641 |

DV01 | 130.792 | 42.462 | 44.239 |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## A. Proofs

#### A.1. Cash Flow for Section 3.2

#### A.2. Proof of Equation (14)

#### A.3. Proof of Proposition 5

**Proof.**The cash flow is given as:

#### A.4. Proof of Proposition 6

**Proof.**We differentiate ${p}_{ij}^{\rho}(t,s)$ for $i\in \{0,1,2\}$ and $j\in \{4,5,6\}$,

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^{1}The free policy option is sometimes referred to as a “paid-up policy” in the literature.

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**MDPI and ACS Style**

Buchardt, K.; Møller, T. Life Insurance Cash Flows with Policyholder Behavior. *Risks* **2015**, *3*, 290-317.
https://doi.org/10.3390/risks3030290

**AMA Style**

Buchardt K, Møller T. Life Insurance Cash Flows with Policyholder Behavior. *Risks*. 2015; 3(3):290-317.
https://doi.org/10.3390/risks3030290

**Chicago/Turabian Style**

Buchardt, Kristian, and Thomas Møller. 2015. "Life Insurance Cash Flows with Policyholder Behavior" *Risks* 3, no. 3: 290-317.
https://doi.org/10.3390/risks3030290