# Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Contract Combining Reverse Annuity with Critical Health Insurance

#### 2.1. Multistate Markov Models

- the insured is alive and healthy;
- the insured is dead.

- the insured is alive and healthy;
- the insured became mildly ill during the last year;
- the insured has been mildly ill for at least one year;
- the insured became critically ill during the last year;
- the insured has been critically ill for at least one year;
- the insured is dead (D—dead).

#### 2.2. The Probability Structure of the Model

- ${q}_{x+k}$—the probability of death;
- ${\phi}_{x+k}$—the dread disease mortality rate;
- ${\chi}_{x+k}$—the dread disease incidence rate (the morbidity rate);
- ${\psi}_{x+k}$—the percentage of patients diagnosed in the critical stage;
- ${\xi}_{x+k}$—the probability of health deterioration to the critical state;
- ${d}_{x+k}^{\left(i,j\right)}$—the fatality rate in the population of the critically ill.

## 3. Two Version of Combining Contract

#### 3.1. Marriage Contract

**I**is an identity matrix with n+1 rows and columns. In this case, N = 2. The formula for the reverse annuity contract benefit in the classical notation is presented in (e.g., Marciniuk et al. 2020).

**D,**describing the probability structure, is calculated on the basis of the matrix of transition probabilities given by (2) for the husband and wife separately. Thus, the number of columns in matrix

**D**is reduced to N. The matrix of all premiums paid during the term of the contract is denoted as ${C}_{out}\in {R}^{\left(n+1\right)\times N}$. The matrix ${C}_{in}^{\left(1\right)}\in {R}^{(n+1)\times N}$ equals 1 in the second and the fourth columns.

- $\stackrel{..}{b}\text{\hspace{0.17em}}-{p}_{X}-{p}_{Y}$ (Both spouses are healthy);
- $\stackrel{..}{b}\text{\hspace{0.17em}}-{p}_{Y}$ (The husband is healthy; the wife is ill or dead);
- $\stackrel{..}{b}\text{\hspace{0.17em}}-{p}_{X}$ (The husband is ill or dead; the wife is healthy);
- $\stackrel{..}{b}\text{\hspace{0.17em}}-{p}_{X}+{c}_{Y}$ (The husband has become sick or his health has worsened; the wife is healthy);
- $\stackrel{..}{b}\text{\hspace{0.17em}}-{p}_{Y}-{p}_{X}$ (The husband is healthy; the wife has become sick or her health has worsened);
- $\stackrel{..}{b}\text{\hspace{0.17em}}+{c}_{Y}$ (The husband has become sick or his health has worsened; the wife is sick or dead);
- $\stackrel{..}{b}\text{\hspace{0.17em}}+{c}_{X}$ (The husband is sick or dead; the wife has become sick or her health has worsened);
- $\stackrel{..}{b}\text{\hspace{0.17em}}+{c}_{X}+{c}_{Y}$ (The spouses have both become sick or their health has worsened).

#### 3.2. Two Individual Contracts

**D**is obtained on the basis of the transition probability matrix given by (1) separately for the spouses.

- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}-{p}_{X}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}-{p}_{Y}^{II}$ (Both spouses are healthy);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}-{p}_{Y}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}$ (The husband is healthy; the wife is ill);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}-{p}_{Y}^{II}$ (The husband is healthy; the wife is dead);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}-{p}_{X}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}$ (The husband is ill; the wife is healthy);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}-{p}_{X}^{II}$ (The husband is dead; the wife is healthy);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}-{p}_{X}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}+{c}_{Y}^{II}$ (The husband has become sick or his health has worsened; the wife is healthy);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}+{c}_{X}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}-{p}_{Y}^{II}$ (The husband is healthy; the wife has become sick or her health has worsened);
- $\text{\hspace{0.17em}}\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}+{c}_{Y}^{II}+\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}$ (The husband has become sick or his health has worsened; the wife is sick);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}+{c}_{Y}^{II}$ (The husband has become sick or his health has worsened; the wife is dead);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}+{c}_{X}^{II}+\text{\hspace{0.17em}}\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}$ (The husband is sick; the wife has become sick or her health has worsened);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}+{c}_{X}^{II}$ (The husband is dead; the wife has become sick or her health has worsened);
- $\stackrel{..}{b}{\text{\hspace{0.17em}}}_{X}\text{\hspace{0.17em}}+{c}_{X}^{II}+\text{\hspace{0.17em}}\stackrel{..}{b}{\text{\hspace{0.17em}}}_{Y}\text{\hspace{0.17em}}+{c}_{Y}^{II}$ (The spouses have become sick or their health has worsened).

## 4. Results

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof.**

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The Number of State | Estimators of Transition Probabilities |
---|---|

1 | ${q}_{11}=1-\left({q}_{x+k}-{\phi}_{x+k}\right)-{\chi}_{x+k}$$,\text{}{q}_{12}={\chi}_{x+k}\left(1-{\psi}_{x+k}\right)$ ${q}_{13}={\chi}_{x+k}\cdot {\psi}_{x+k}$$,\text{}{q}_{16}={q}_{x+k}-{\phi}_{x+k}$ |

2 and 3 | ${q}_{ij}=1-{q}_{x+k}-{\xi}_{x+k}$, for i = 2, 3 and j = 3 ${q}_{ij}={\xi}_{x+k}$, for i = 2, 3 and j = 4 ${q}_{i6}={q}_{x+k}$, for i = 2, 3 |

4 | ${q}_{{45}^{\left(1\right)}}=1-{d}_{x+k}^{\left(4,{5}^{\left(1\right)}\right)}$$,\text{}{q}_{46}={d}_{x+k}^{\left(4,{5}^{\left(1\right)}\right)}$ |

${5}^{\left(1\right)},\dots ,{5}^{\left(h\right)}$ | ${q}_{{5}^{\left(i\right)}{5}^{\left(j\right)}}=1-{d}_{x+k}^{\left({5}^{\left(i\right)},{5}^{\left(j\right)}\right)}$, for i = 1,2,… h − 1 and j = i + 1 ${q}_{{5}^{\left(l\right)}6}={d}_{x+k}^{\left({5}^{\left(l\right)}6\right)}$, for l = 1, 2, … h. |

For a Wife | For a Husband |
---|---|

${H}^{X}\left(k\right)=\left(\begin{array}{cccc}{p}_{x+k}& {q}_{x+k}& {p}_{x+k}& {q}_{x+k}\\ 0& 1& 0& 1\\ 0& 0& {p}_{x+k}& {q}_{x+k}\\ 0& 0& 0& 1\end{array}\right)$ | ${H}^{Y}\left(k\right)=\left(\begin{array}{cccc}{p}_{y+k}& {p}_{y+k}& {q}_{y+k}& {q}_{y+k}\\ 0& {p}_{y+k}& 0& {q}_{y+k}\\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right)$ |

First Variant—Marriage Contract | |||||||

x = y | Annuity$\stackrel{\mathbf{..}}{\mathit{b}}$ | Benefit for Woman${\mathit{c}}_{\mathit{x}}$ | Benefit for Man${\mathit{c}}_{\mathit{y}}$ | Premium for Woman${\mathit{p}}_{\mathit{x}}$ | Premium for Man${\mathit{p}}_{\mathit{y}}$ | ||

65 | 4085.4 | 8417.8 | 4945.8 | 12.6 | 28.2 | ||

70 | 4628.7 | 9558.4 | 5413.1 | 14.3 | 32.0 | ||

75 | 5458.0 | 11,903.0 | 6803.5 | 16.9 | 37.7 | ||

80 | 6717.5 | 15,593.0 | 10,410.0 | 20.8 | 46.4 | ||

85 | 8612.4 | 20,667.0 | 16,076.0 | 26.6 | 59.5 | ||

Second Variant—Individual Contracts | |||||||

x = y | Annuity for Woman${\stackrel{\mathbf{..}}{\mathit{b}}}_{\mathit{x}}$ | Annuity for Man${\stackrel{\mathbf{..}}{\mathit{b}}}_{\mathit{y}}$ | Total Annuity$\stackrel{\mathbf{..}}{\mathit{b}}$ | Benefit for Woman${\mathit{c}}_{\mathit{x}}$ | Benefit for Man${\mathit{c}}_{\mathit{y}}$ | Premium for Woman${\mathit{p}}_{\mathit{x}}$ | Premium for Man${\mathit{p}}_{\mathit{y}}$ |

65 | 2287.5 | 2786.9 | 5074.4 | 15,253.0 | 4882.6 | 22.9 | 27.9 |

70 | 2657.4 | 3257.5 | 5914.9 | 17,760.0 | 5513.1 | 26.6 | 32.6 |

75 | 3252.0 | 3952.1 | 7204.1 | 22,956.0 | 7129.0 | 32.5 | 39.5 |

80 | 4195.0 | 4984.0 | 9179.0 | 31,513.0 | 11,177.0 | 42.0 | 49.9 |

85 | 5673.3 | 6487.8 | 12,161.1 | 44,058.0 | 17,526.0 | 56.7 | 64.9 |

First Variant—Marriage Contract | Second Variant—Individual Contracts | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

x | X | ||||||||||

y | 65 | 70 | 75 | 80 | 85 | y | 65 | 70 | 75 | 80 | 85 |

65 | 4085 | 4394 | 4717 | 5006 | 5226 | 65 | 5074 | 5444 | 6039 | 6982 | 8460 |

70 | 4224 | 4629 | 5084 | 5526 | 5889 | 70 | 5545 | 5915 | 6510 | 7453 | 8931 |

75 | 4340 | 4839 | 5458 | 6115 | 6712 | 75 | 6240 | 6610 | 7205 | 8147 | 9625 |

80 | 4428 | 5008 | 5788 | 6718 | 7647 | 80 | 7272 | 7641 | 8237 | 9179 | 10,657 |

85 | 4493 | 5133 | 6050 | 7241 | 8612 | 85 | 8775 | 9145 | 9740 | 10,683 | 12,161 |

**Table 5.**Critical illness insurance payment depending on the age of spouses in two variants of contract.

First Variant—Marriage Contract | Second Variant—Individual Contract | |||||
---|---|---|---|---|---|---|

Critical illness insurance payment for man | ||||||

x | different x | |||||

y | 65 | 70 | 75 | 80 | 85 | |

65 | 4946 | 5320 | 5711 | 6060 | 6326 | 4882.6 |

70 | 4940 | 5413 | 5946 | 6463 | 6887 | 5513.1 |

75 | 5410 | 6032 | 6804 | 7622 | 8367 | 7129.3 |

80 | 6862 | 7761 | 8970 | 10,410 | 11,850 | 11,171 |

85 | 8387 | 9581 | 11,293 | 13,516 | 16,076 | 17,526 |

Critical illness insurance payment for woman | ||||||

x | different y | |||||

y | 65 | 70 | 75 | 80 | 85 | |

65 | 8418 | 9075 | 10,288 | 11,619 | 12,539 | 15,253 |

70 | 8704 | 9558 | 11,088 | 12,828 | 14,132 | 17,760 |

75 | 8943 | 9993 | 11,903 | 14,194 | 16,107 | 22,596 |

80 | 9124 | 10,342 | 12,624 | 15,593 | 18,350 | 31,513 |

85 | 9258 | 10,600 | 13,194 | 16,807 | 20,667 | 44,058 |

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**MDPI and ACS Style**

Marciniuk, A.; Zmyślona, B.
Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget. *Risks* **2022**, *10*, 140.
https://doi.org/10.3390/risks10070140

**AMA Style**

Marciniuk A, Zmyślona B.
Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget. *Risks*. 2022; 10(7):140.
https://doi.org/10.3390/risks10070140

**Chicago/Turabian Style**

Marciniuk, Agnieszka, and Beata Zmyślona.
2022. "Marriage and Individual Equity Release Contracts with Dread Disease Insurance as a Tool for Managing the Pensioners’ Budget" *Risks* 10, no. 7: 140.
https://doi.org/10.3390/risks10070140