# On Smoothing and Habit Formation of Variable Life Annuity Benefits

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. The General Framework

#### 2.2. Introducing Lifetime Uncertainty

## 3. Three Strategies for Consumption and Investment

#### 3.1. The Classical Strategy

#### 3.2. The Habit Strategy

#### 3.3. The Hybrid Strategy

**Remark 1.**

## 4. The Development of Consumption over Time

#### 4.1. Consumption Dynamics in the Classical Strategy

**Theorem 1.**

**Proof of Theorem 1.**

#### 4.2. Consumption Dynamics in the Habit Strategy

**Theorem 2.**

**Proof of Theorem 2.**

**Theorem 3.**

**Proof of Theorem 3.**

#### 4.3. Consumption Dynamics in the Hybrid Strategy

**Theorem 4.**

**Proof of Theorem 4.**

#### 4.4. Comparing Consumption Dynamics in the Habit and the Hybrid Strategy

**Theorem 5.**

**Proof of Theorem 5.**

## 5. Designing a Smooth Pension Product

#### 5.1. First Approach

#### 5.2. Second Approach

#### 5.3. Valuation

## 6. Numerical Examples

#### 6.1. Numerical Setup

#### 6.2. The Classical Strategy

#### 6.3. The First Approach of a Smooth Pension Product

#### 6.4. The Second Approach of a Smooth Pension Product

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Studying Optimal Consumption

#### Appendix A.1. Proof of Theorem 1—Consumption Dynamics in the Classical Strategy

#### Appendix A.2. Proof of Theorem 2—Consumption Dynamics in the Habit Strategy

## References

- Bruhn, Kenneth, and Mogens Steffensen. 2011. Household consumption, investment and life insurance. Insurance: Mathematics and Economics 48: 315–25. [Google Scholar] [CrossRef]
- Bruhn, Kenneth, and Mogens Steffensen. 2013. Optimal smooth consumption and annuity design. Journal of Banking & Finance 37: 2693–701. [Google Scholar]
- Cejnek, Georg, Richard Franz, Otto Randle, and Neal M. Stoughton. 2014. A survey of university endowment management research. Journal of Investment Management. Third Quarter. [CrossRef]
- DFSA. 2022a. Finanstilsynets Aktuelle Risikobillede. Available online: https://www.finanstilsynet.dk/Nyheder-og-Presse/Pressemeddelelser/2023/Risikobillede_260123 (accessed on 25 May 2023).
- DFSA. 2022b. Levetidsmodel. Available online: https://www.finanstilsynet.dk/tal-og-fakta/oplysninger-for-virksomheder/oplysningstal-om-forsikring-og-pension/levetidsmodel (accessed on 15 May 2023).
- Kashif, Muhammad, Francesco Menoncin, and Iqbal Owadally. 2020. Optimal portfolio and spending rules for endowment funds. Review of Quantitative Finance and Accounting 55: 671–93. [Google Scholar] [CrossRef]
- Konicz, Agnieszka Karolina, David Pisinger, Kourosh Marjani Rasmussen, and Mogens Steffensen. 2015. A combined stochastic programming and optimal control approach to personal finance and pensions. OR Spectrum 37: 583–616. [Google Scholar] [CrossRef]
- Kraft, Holger, and Mogens Steffensen. 2008. Optimal consumption and insurance: A continuous-time Markov chain approach. ASTIN Bulletin: The Journal of the IAA 38: 231–57. [Google Scholar]
- Merton, Robert C. 1969. Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics 51: 247–57. [Google Scholar] [CrossRef]
- Merton, Robert C. 1971. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3: 373–413. [Google Scholar] [CrossRef]
- Munk, Claus. 2008. Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences. Journal of Economic Dynamics and Control 32: 3560–89. [Google Scholar] [CrossRef]
- Richard, Scott F. 1975. Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics 2: 187–203. [Google Scholar] [CrossRef]
- Steffensen, Mogens, and Julie Bjørner Søe. 2023. What is the value of the annuity market? In Decisions in Economics and Finance. Berlin: Springer. [Google Scholar] [CrossRef]

**Figure 1.**Wealth and consumption under the classical strategy simulated with parameter values from Table 1. The solid lines show the mean of 100,000 simulations, and the dashed lines show the $5\%$ and $95\%$ quantile.

**Figure 2.**Wealth and consumption under the first approach of a smooth pension product simulated with different values of the interest rate ${r}_{w}$, $\alpha ={k}_{\eta}=0.2$ and parameter values from Table 1. The solid lines show the mean of 100,000 simulations, and the dashed lines show the $5\%$ and $95\%$ quantile.

**Figure 3.**Investment portfolio under the first approach of a smooth pension product simulated with different values of the interest rate ${r}_{w}$, $\alpha ={k}_{\eta}=0.2$ and parameter values from Table 1. The solid lines show the mean of 100,000 simulations, and the dashed lines show the $5\%$ and $95\%$ quantile.

**Figure 4.**The function $\eta $ and the weights, w and y, under the first approach of a smooth pension product with different values of the interest rate ${r}_{w}$, $\alpha ={k}_{\eta}=0.2$ and parameter values from Table 1.

**Figure 5.**Wealth and consumption under the second approach of a smooth pension product simulated with different values of the interest rate ${r}_{w}$, $\alpha =0.2$ and parameter values from Table 1. The solid lines show the mean of 100,000 simulations, and the dashed lines show the $5\%$ and $95\%$ quantile.

**Figure 6.**Investment portfolio under the second approach of a smooth pension product simulated with different values of the interest rate ${r}_{w}$, $\alpha =0.2$ and parameter values from Table 1. The solid lines show the mean of 100,000 simulations, and the dashed lines show the $5\%$ and $95\%$ quantile.

**Figure 7.**The annuity ${a}_{w}$ and the weights, w and y, under the second approach of a smooth pension product with different values of the interest rate ${r}_{w}$, $\alpha =0.2$ and parameter values from Table 1.

Parameter | Describtion | Value |
---|---|---|

${x}_{0}$ | Initial wealth | 100 |

T | Fixed time horizon | 45 |

$\rho $ | Impatience factor | 0.04 |

$\gamma $ | Risk aversion parameter | 2 |

r | Drift of the risk-free asset | 0.02 |

$\lambda $ | Drift of the risky asset | 0.05 |

$\sigma $ | Volatility of the risky asset | 0.2 |

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## Share and Cite

**MDPI and ACS Style**

Steffensen, M.; Vikkelsøe, S.H.
On Smoothing and Habit Formation of Variable Life Annuity Benefits. *J. Risk Financial Manag.* **2024**, *17*, 75.
https://doi.org/10.3390/jrfm17020075

**AMA Style**

Steffensen M, Vikkelsøe SH.
On Smoothing and Habit Formation of Variable Life Annuity Benefits. *Journal of Risk and Financial Management*. 2024; 17(2):75.
https://doi.org/10.3390/jrfm17020075

**Chicago/Turabian Style**

Steffensen, Mogens, and Savannah Halling Vikkelsøe.
2024. "On Smoothing and Habit Formation of Variable Life Annuity Benefits" *Journal of Risk and Financial Management* 17, no. 2: 75.
https://doi.org/10.3390/jrfm17020075