# Proposal to Extend Access to Loans for Serious Illnesses Using Open Data

^{*}

^{†}

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*data scientist*.

## Abstract

**:**

## 1. Introduction

## 2. Overall Picture Based on French Mortality Benchmarks

## 3. A Generic Method to Estimate Loan Insurance Premiums for Patients

#### 3.1. Mortality Risk of Patients in the General Population According to Various Risk Factors

#### 3.2. Transposition to Borrowers

#### 3.2.1. Theoretical Considerations and Definition of a Relative Risk Multiplier

#### 3.2.2. Mathematical Definition of the Multiplier and How to Obtain it

- -
- ${q}_{x}{}^{B}$ (annual mortality rate of borrowers) and ${q}_{x}{}^{D}$ (annual mortality rate in the general population) are functions of age and possibly gender. ${q}_{x}{}^{B}$ is an average of placeholder borrower tables calculated by the laboratory; an insurer can apply its own experience table;
- -
- ${q}_{x}{}^{C}$ (annual mortality rate of persons with the pathology) depends on age and possibly gender but also on disease-related variables, such as how long ago the diagnosis was made and elements of disease severity;
- -
- ${q}_{x}{}^{A}$ (annual mortality rate of borrowers with the pathology) has all these variables;
- -
- $\rho $ is theoretically a function of age, wealth, history of disease, and any other characteristic associated with the loan applicant.

#### 3.2.3. Temporary Disability

- For incidence, we considered a logistic regression giving the annual probability of a work stoppage greater than three days. We chose a logistic regression because this annual incidence is a probability is between 0 and 1, but other models could have been chosen. The three-day threshold was chosen by expert judgement to differentiate temporary disability from pure work stoppage.
- For duration, we considered a gamma regression, giving the duration of a work stoppage greater than 3 days. We chose a gamma regression because it leads to durations that are positive, but other models could have been chosen.

#### 3.2.4. Permanent Disability

#### 3.2.5. Premium Calculation

## 4. Application to Two Diseases: Breast Cancer and Diabetes

#### 4.1. Breast Cancer

#### 4.1.1. Mortality by Risk Factor

- -
- Age at diagnosis;
- -
- Oestrogen receptor function;
- -
- TNM stage;
- -
- SBR grade.

- -
- Age between 45 and 59;
- -
- Stage T of 1;
- -
- Stage N of 0;
- -
- Stage M of 0;
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- Oestrogen receptor dysfunction;
- -
- SBR grade of 1.

- -
- 42.5% were in the higher socio-professional status (SPS+) group defined as “executives, middle professional group, and clerical employees”. Their survival 5 and 7 years post-diagnosis was ${(1-{{q}_{x}}^{{A}^{\prime}})}^{5}=88.2\%$ and ${(1-{{q}_{x}}^{{A}^{\prime}})}^{7}=83.0\%$, thus on average ${{q}_{x}}^{{A}^{\prime}}=1-88.2{\%}^{\frac{1}{5}}$ and ${q}_{x}{}^{{A}^{\prime}}=1-83.0{\%}^{\frac{1}{7}}$;
- -
- 16.6% did not have a specified SPS;
- -
- 40.9% were in the lower socio-professional status (SPS−) group defined as “famers, artisans, manual workers, unemployed”; Their survival 5 and 7 years post diagnosis was $77.4\%$ and 69.4%.

#### 4.1.2. Short-Term Disability

- -
- A logistic regression (glm function in R, using “family=binomial(logit)”) yields the following average incidence rate:$${i}_{x}=\frac{1}{1+{\mathrm{e}}^{-\left(\alpha +\beta x+\gamma {1}_{breastcancer}\right)}}$$
- -
- A gamma regression (glm function in R, using “family = Gamman(link = ‘log’)”) yields the following average duration:$${d}_{x}={\mathrm{e}}^{-\left(\alpha +\beta x+\gamma {1}_{breastcancer}\right)}$$

#### 4.1.3. Long-Term Disability

- -
- ${p}_{x}{}^{C}=1-{(1-0.084)}^{1/3}=2.88\%$ for $t$ between $0$ and $3$ years.
- -
- ${p}_{x}{}^{C}=1-{\left(\frac{1-0.108}{1-0.084}\right)}^{1/7}=0.38\%$ for $t$ beyond $3$ years.

#### 4.1.4. Results

#### 4.2. Diabetes

#### 4.2.1. Mortality by Type of Diabetes

#### 4.2.2. Short-Term Disability by Age and Age of Diagnosis

- -
- A logistic regression yields the following average incidence rate:$${i}_{x}=\frac{1}{1+{\mathrm{e}}^{-\left(\alpha +\beta x+\gamma {1}_{diabetestype1}+\delta \tau \right)}}$$
- -
- A gamma regression yields the following average duration:$${d}_{x}={\mathrm{e}}^{-\left(\alpha +\beta x+\gamma {1}_{diabetestype1}+\delta \tau \right)}$$

#### 4.2.3. Long-Term Disability

- -
- ${p}_{x}{}^{C}=1-{(1-0.033)}^{1/3}=1.11\%$ for $t$ between $0$ and $3$ years.
- -
- ${p}_{x}{}^{C}=1-{\left(\frac{1-0.076}{1-0.033}\right)}^{1/7}=0.65\%$ for $t$ beyond $3$ years.

#### 4.2.4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | For insured risks this gender breakdown is based on expert opinion, as the data at our disposal does not distinguish between the sexes. |

2 | Average, maximum and minimum of best estimate actuarial tables among a group of French insurer for the mortality and temporary disability incidence of borrowers. Female risks were obtained by dividing by 1.5 (expert judgement) and male risks where deducted by considering that the tables contained 55% males (expert judgement). |

3 | Since the conditional transition probabilities depend on the time spent in the state, semi-Markovian models are better suited, but given the context of this study, this simplification is acceptable. |

4 | Surveillance Research Program, National Cancer Institute SEER* Stat software (seer.cancer.gov/seerstat) version 8.3.9. |

5 | In blue and red for men and women respectively. The solid lines represent those without kidney disease. The numbers associated with the curves (15, 30 and 45) are the ages at diagnosis. |

## References

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**Figure 1.**Annual mortality rate in France between the ages of 25 and 65 for different populations (55% men and 45% women)1.

**Figure 6.**SMR by age at diagnosis and current age5.

**Figure 7.**Relative risks according to socio-professional category with executives as reference (source: Piffaretti et al. 2016).

**Figure 9.**Additional pure premium $p$ for mortality risk as a function of applicant age and whether diagnosis is recent or not (three cases).

**Figure 10.**Additional pure premium $p$ for disability risk as a function of applicant age, if diagnostic is recent or not (three cases).

Population with a Pathology | Healthy Population (*) | Relative Mortality Risk Due to Disease | |
---|---|---|---|

Higher socio-professional status | ${q}_{x}{}^{{A}^{\prime}}$ | ${q}_{x}{}^{{B}^{\prime}}$ | ${q}_{x}{}^{{A}^{\prime}}/{q}_{x}{}^{{B}^{\prime}}$ |

General population | ${q}_{x}{}^{{C}^{\prime}}$ | ${q}_{x}{}^{{D}^{\prime}}$ | ${q}_{x}{}^{{C}^{\prime}}/{q}_{x}{}^{{D}^{\prime}}$ |

**Table 2.**Percentage of patients in permanent disability 3 and 10 years after the start of long-term sickness in France.

Rate (%) | ||
---|---|---|

Long-Term Sickness | 3 Years Later | 10 Years Later |

Multiple sclerosis | 14.0 | 23.4 |

Incapacitating stroke | 19.8 | 21.9 |

Severe active rheumatoid arthritis | 10.1 | 7.1 |

Chronic arteriopathies with ischemic manifestations | 10.6 | 17.0 |

Coronary artery disease | 9.9 | 15.1 |

Heart failure, severe heart disease | 9.6 | 14.3 |

Severe chronic kidney disease and nephrotic syndrome | 7.6 | 13.2 |

Severe forms of neurological conditions, severe epilepsy | 8.8 | 13.0 |

Long-term psychiatric conditions | 9.9 | 13.0 |

Severe chronic respiratory failure | 9.0 | 12.6 |

Severe ankylosing spondylitis | 8.0 | 12.5 |

Malignant tumors | 8.4 | 10.8 |

Severe high blood pressure | 5.1 | 9.8 |

Chronic active liver disease and cirrhosis | 5.9 | 8.6 |

Type 1 and 2 diabetes | 3.3 | 7.6 |

Crohn’s disease and active ulcerative colitis | 2.2 | 4.7 |

Severe primary immunodeficiency, AIDS | 1.9 | 3.6 |

$1-{\mathit{q}}_{\mathit{x}}{}^{\mathit{A}}-{\mathit{p}}_{\mathit{x}}{}^{\mathit{A}}$ | ${\mathit{p}}_{\mathit{x}}{}^{\mathit{A}}$ | ${\mathit{q}}_{\mathit{x}}{}^{\mathit{A}}$ |
---|---|---|

0 | 1$-{q}_{x}{}^{A}$ | ${q}_{x}{}^{A}$ |

0 | 0 | 1 |

T Stage | N Stage | M Stage | SBR Grade | Oestrogen Receptor Function | m |
---|---|---|---|---|---|

1 | 0 | 0 | 1 | Positive | 2.2 |

2 | 0 | 0 | 1 | Positive | 6.4 |

3 | 0 | 0 | 1 | Positive | 8.9 |

1 | 1 | 0 | 1 | Positive | 5.9 |

2 | 1 | 0 | 1 | Positive | 14.6 |

3 | 1 | 0 | 1 | Positive | 19.7 |

1 | 0 | 0 | 2 | Positive | 5.8 |

2 | 0 | 0 | 2 | Positive | 14.4 |

3 | 0 | 0 | 2 | Positive | 19.4 |

1 | 1 | 0 | 2 | Positive | 13.2 |

2 | 1 | 0 | 2 | Positive | 31.0 |

3 | 1 | 0 | 2 | Positive | 41.3 |

1 | 0 | 0 | 3 | Positive | 8.3 |

2 | 0 | 0 | 3 | Positive | 20.0 |

3 | 0 | 0 | 3 | Positive | 26.8 |

1 | 1 | 0 | 3 | Positive | 18.4 |

2 | 1 | 0 | 3 | Positive | 42.4 |

3 | 1 | 0 | 3 | Positive | 56.2 |

1 | 0 | 0 | 1 | Negative | 3.2 |

2 | 0 | 0 | 1 | Negative | 8.5 |

3 | 0 | 0 | 1 | Negative | 11.7 |

1 | 1 | 0 | 1 | Negative | 7.8 |

2 | 1 | 0 | 1 | Negative | 18.9 |

3 | 1 | 0 | 1 | Negative | 25.4 |

1 | 0 | 0 | 2 | Negative | 7.7 |

2 | 0 | 0 | 2 | Negative | 18.7 |

3 | 0 | 0 | 2 | Negative | 25.1 |

1 | 1 | 0 | 2 | Negative | 17.2 |

2 | 1 | 0 | 2 | Negative | 39.7 |

3 | 1 | 0 | 2 | Negative | 52.7 |

1 | 0 | 0 | 3 | Negative | 10.9 |

2 | 0 | 0 | 3 | Negative | 25.8 |

3 | 0 | 0 | 3 | Negative | 34.4 |

1 | 1 | 0 | 3 | Negative | 23.8 |

2 | 1 | 0 | 3 | Negative | 54.1 |

3 | 1 | 0 | 3 | Negative | 71.3 |

T Stage | N Stage | M Stage | SBR Grade | Oestrogen Receptor Function | p |
---|---|---|---|---|---|

- | - | 0 | - | - | 13.5 |

1 | - | 0 | - | - | 5.8 |

2 | - | 0 | - | - | 21.1 |

3 | - | 0 | - | - | 31.7 |

4 | - | 0 | - | - | 36.7 |

- | 1 | 0 | - | - | 7.2 |

- | 0 | 0 | - | - | 24.9 |

- | - | 0 | 1 | - | 3.5 |

- | - | 0 | 2 | - | 11.5 |

- | - | 0 | 3 | - | 21.1 |

- | - | 0 | - | Negative | 21.4 |

- | - | 0 | - | Positive | 11.3 |

South Korea | 108% |

United States | 111% |

Scotland | 123% |

France | 115% |

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**MDPI and ACS Style**

Planchet, F.; Debonneuil, É.; Péju, M.
Proposal to Extend Access to Loans for Serious Illnesses Using Open Data. *Risks* **2022**, *10*, 51.
https://doi.org/10.3390/risks10030051

**AMA Style**

Planchet F, Debonneuil É, Péju M.
Proposal to Extend Access to Loans for Serious Illnesses Using Open Data. *Risks*. 2022; 10(3):51.
https://doi.org/10.3390/risks10030051

**Chicago/Turabian Style**

Planchet, Frédéric, Édouard Debonneuil, and Marie Péju.
2022. "Proposal to Extend Access to Loans for Serious Illnesses Using Open Data" *Risks* 10, no. 3: 51.
https://doi.org/10.3390/risks10030051