# Design of an Insurance Policy Model Applied to Natural Stone Facade Claddings

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Materials and Methods

- Risk assessment, in which service life prediction models are used to estimate the risk of failure of stone claddings over time and according to their characteristics. These models allow identifying the claddings’ expected condition in each instant, as well as the instant after which they present any given probability of reaching the end of their service life;
- The definition of the claims; in this case, the claims are related with the need of performing different maintenance actions according to the cladding’s degradation during the period of insurance;
- The estimation of the annual premium; the insurer reserves a monetary amount (provision) for the eventuality of the occurrence of the worst-case scenario to its customer portfolio and the respective indemnity payment. This amount depends on three factors: (a) number of subscribers; (b) cost of replacing the cladding (€/m
^{2}); (c) area of the facade in inadequate conditions (m^{2}).

#### 3.1. Proposal of Deterministic and Stochastic Actuarial Models

#### 3.1.1. Overall Approach

_{w}represents the severity of degradation of the building component, expressed as a percentage, k

_{n}is the multiplying factor of anomaly “n”, as a function of their degradation level, within the range K = {0, 1, 2, 3, 4} (Table 1), k

_{a,n}is the weighting factor corresponding to the relative weight of the anomaly detected (Table 2), A

_{n}is the area of cladding affected by an anomaly “n”, in m

^{2}, A is the facade area, in m

^{2}, and k is the multiplying factor corresponding to the highest degradation level of a coating of area A.

_{w}thresholds considered in this study correspond to different levels of claims for the insurance company. The first level, corresponding to a S

_{w}of 10%, reveals a stone cladding with visual anomalies and slight cracking in the cladding, thus requiring a simple cleaning action. The second level, corresponding to a S

_{w}of 20%, reveals a stone cladding with more serious defects, requiring a major intervention and partial replacement of the cladding. The third level, corresponding to a S

_{w}of 40%, requires the replacement of the stone cladding.

_{w}values. In this model, the of estimated values are independent of the cladding’s characteristics, since the model only encompasses the age of the building as explanatory variable of the stone claddings’ degradation phenomenon.

_{w}represents the claddings’ severity of degradation, I their age, M distance from the sea, A area of the cladding, H exposure to damp, and TP type of stone. To apply the model shown in Equation (2), the variables M, A, H, and TP should be replaced by their numerical quantification, presented in Table 4 [15].

_{w}levels. For a S

_{w}of 40%, a discrete cumulative distribution function is obtained, due to the small number of case studies with this degradation level. With this information, it is possible to define, for different risk thresholds, the probability of a given cladding reaching the specific degradation values (Table 5).

#### 3.1.2. Models’ Assumptions Concerning the Coverage of the Insurance Policy

_{w}thresholds, including the following maintenance actions: (i) for S

_{w}of 10% (slight degradation), a cleaning action is performed; (ii) for S

_{w}of 20% (moderate degradation), a major intervention is performed; and iii) for S

_{w}of 40% (generalised degradation), a replacement of the cladding is required.

_{w}value. Based on the results obtained, the insurer decides to claim maintenance actions and the related budget, or not. The different levels of maintenance include the following actions:

- Cleaning: includes scaffolding installation, cleaning with water jet and brushing [18], repair of cracking with a width ≤ 1 mm and replacement of 20% of joints;
- Major intervention: includes all the previous actions but with the replacement of 30% of joints. Additionally, it includes the cleaning of the remainder 70% joints and the replacement of 20% of the stone plates;
- Replacement: includes scaffolding installation and the complete replacement of the cladding, with the application of the new cladding and the transportation to a landfill of the old cladding.

- During the duration of the policy, the claim of each maintenance action occurs only once, i.e., after a cleaning action, the insurer will only intervene when the next degradation level is reached, thus performing a major intervention;
- It is considered, as a simplification, that the maintenance actions have no effect on the cladding’s degradation condition (in terms of the S
_{w}value), i.e., the age in which the next maintenance action occurs is the same before and after the maintenance action; - It is assumed there are no periodic maintenance actions by the insured;
- In the case of a condominium, the insurance is equally shared by the individual unit owners that pay the same premium;
- An annual policy with renewal option is adopted. The premium is fixed during the period of subscription, independently of the rate’s volatility, which is beneficial for the insured, since he/she knows the total cost of the insurance policy, without surprises, hassles, or extra calculations.

#### 3.1.3. Determination of the Annual Premium for the Insurance Policy

_{t, nom}

_{(C)}, C

_{t, nom}

_{(MI)}, and C

_{t,nom}

_{(R)}correspond to the nominal costs of the maintenance actions of the different degradation thresholds, t

_{C}, t

_{MI}, and t

_{R}, indicating the year in which those costs occur and r

_{nom}represents the nominal discount rate, which includes the global inflation risk, opportunity costs, and other costs. The C

_{t,nom}terms include the effect of inflation, which affects the nominal discount rate, as explained by Fisher’s theory [20], as shown in Equation (4).

_{s}) of 3% and a global inflation rate (i

_{g}) of 2% are adopted, in accordance with the Portuguese context. To simplify the presentation of this study results, real values are used. Equation (3) can be used to determine the expected costs and premiums, as shown in Equations (5) and (7), respectively.

_{t,real}

_{(10)}, C

_{t,real}

_{(20)}, and C

_{t,real}

_{(40)}are the real costs of the maintenance actions of the different degradation thresholds, which are the same for all the proposed models; t

_{10}, t

_{20}and t

_{40}are the years in which those costs occur; r

_{real,cost}is the real discount rate, equal for all the models and obtained through Equation (4). Equation (6) presents the discount rate applied to the cost, only considering rate i

_{s}.

_{t,premium}is the annual premium in €/m

^{2}, whose definition is the main objective of applying the proposed models.

_{real,premium}is the real discount rate, equal for all the models, considering not only rate i

_{s}but also rate i

_{g}, as shown in Equation (8).

_{t,prémio}) is obtained by equalling Equations (5) and (7). The value obtained is converted to Euros paid per household by multiplying the premium by the cladding’s area and then dividing by the number of individual apartments. The final definition of the premium value should also consider the insurance company’s profit margin. The different models proposed allow transforming the probability of the claddings reaching the end of their service life into profit margins.

## 4. Application of the Methodology to a Real Case Study

^{2}), is located less than 5 km from the sea, and has a high exposure to damp.

- The cladding’s characteristics do not produce significant deviations between the remaining time before action obtained by the deterministic models. The comparison between the remaining time before action obtained by the single-parameter stochastic model and multiparameter stochastic model, for the threshold S
_{w}= 10%, for any risk margin is also inconclusive; - In the multiparameter stochastic model, the risk’s reduction is equivalent to an anticipation of the maintenance action’s schedule. This effect is less noticeable in the second and third degradation levels;
- For the thresholds S
_{w}= 20% and S_{w}= 40%, in both stochastic models, when a lower level of risk is assumed, naturally a lower remaining time before action is obtained. This conclusion is more evident for the single-parameter stochastic model, for replacement actions (S_{w}= 40%); - For a fixed risk margin, in both the second and third degradation thresholds, the expected remaining time before action is higher in the multiparameter stochastic model than in the single-parameter one. The difference is substantial for lower risk margins (5%) and for more profound maintenance actions, such as the replacement.

_{w}= 40%. The procedure will be exemplified with the single-parameter deterministic model, as shown in Equation (9).

^{2}, as shown in Equation (11).

## 5. Results

- An insurance product based on a deterministic model will not be able to compete against an insurance based on a stochastic model with a 50% risk margin, because it provides the same coverage and risk charging, with a higher premium;
- In the deterministic approach, the commercial premium’s difference between single-parameter and multiparameter is below 2€, which reveals the lack of preponderance of the discrimination of the cladding’s characteristics in these models;
- In the stochastic models, the lesser the assumed risk margin, the greater the premium. This aspect is more evident in the single-parameter model;
- For a fixed risk margin, the premium is lower in the multiparameter stochastic model than in the single-parameter one. The difference is more expressive for smaller risk margins (5%);
- These results reveal that the design of an insurance product knowing the characteristics of the insured object allows reducing the costs for the policyholder, which also increases the possibility of acquiring this insurance policy.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Degradation’s evolution using the severity of degradation level for the 142 stone claddings inspected (data sourced from Silva et al. [12]).

**Figure 2.**Cumulative distribution functions concerning the probability of 142 stone claddings reaching the end of their service life, considering three risk thresholds.

Degradation Level | Anomaly | % Area of Cladding Affected | |
---|---|---|---|

Level 0 (S_{w} ≤ 1%) | No visible degradation | - | |

Level 1 Good (1% < S _{w} ≤ 8%) | Aesthetic degradation anomalies | Surface dirt | >10% |

Moisture stains | ≤15% | ||

Localized stains | |||

Colour change | |||

Flatness deficiencies | ≤10% | ||

Loss-of-integrity anomalies | Material degradation (*) ≤ 10% plate thickness | ≤20% | |

Cracking width ≤ 0.2 mm | |||

Level 2 Slight degradation (8% < S _{w} ≤ 20%) | Aesthetic degradation anomalies | Moisture stains | >15% |

Localized stains | |||

Colour change | |||

Biological growth | ≤30% | ||

Parasitic vegetation | |||

Efflorescence | |||

Flatness deficiencies | > 10% e ≤ 50% | ||

Joints anomalies | Joints material degradation | ≤30% | |

Loss of material—open joint | ≤10% | ||

Fastening to the substrate anomalies | Scaling of stone near the edges Partial loss of stone material | ≤20% | |

Loss-of-integrity anomalies | Material degradation (*) ≤ 10% plate thickness | >20% | |

Material degradation (*) > 10% and ≤ 30% plate thickness | ≤20% | ||

Cracking width ≤ 0.2 mm | >20% | ||

Cracking width > 0.2 mm and ≤ 3 mm | ≤20% | ||

Fracture | ≤5% | ||

Level 3 Moderate degradation (20% < S _{w} ≤ 45%) | Aesthetic degradation anomalies | Biological growth | >30% |

Parasitic vegetation | |||

Efflorescence | |||

Flatness deficiencies | >50% | ||

Joints anomalies | Joints material degradation | >30% | |

Loss of material—open joint | >10% | ||

Fastening to the substrate anomalies | Scaling of stone near the edges Partial loss of stone material | >20% | |

Detachment | ≤10% | ||

Loss-of-integrity anomalies | Material degradation (*) > 10% and ≤ 30% plate thickness | >20% | |

Material degradation (*) > 30% plate thickness | ≤20% | ||

Cracking width > 0.2 mm and ≤ 3 mm | >20% | ||

Cracking width ≥ 3 mm | ≤20% | ||

Fracture | > 5% e ≤ 10% | ||

Level 4 Generalized degradation (S _{w} ≥ 45%) | Fastening to the substrate anomalies | Detachment | >10% |

Loss-of-integrity anomalies | Material degradation (*) > 30% plate thickness | >20% | |

Cracking width > 3 mm | |||

Fracture | >10% |

**Table 2.**Weighting coefficients corresponding to the relative weight of the anomaly detected in natural stone claddings (data sourced from Silva et al. [12]).

Anomaly | Repair Operation (Cost in €/m^{2}) | Ratio between Repair Cost and Replacement Cost ^{(*)} | Weighting Coefficient k_{a,n} | |
---|---|---|---|---|

Visual or surface degradation | Cleaning (11.7 €/m^{2}) | 13% | 0.13 | |

Joints | Degradation of filling material | Joint repair (23.4 €/m^{2}) | 25% | 0.25 |

Loss of filling material | Replacement of the joint material in cladding directly adhered to the substrate involves some risks, and may damage the natural stone | 100% | 1.0 | |

Bond to substrate | Replacement of stone plates always costs at least as much as executing a new cladding, and may cost more because of having to remove the degraded original cladding | 120% | 1.2 | |

Loss of integrity | Repairing loss-of-integrity anomalies may involve only a surface repair (epoxy resins or equivalent) or the replacement of the stone plate | 100% | 1.0 |

^{2}.

**Table 3.**Estimated age of natural stone claddings for three different S

_{w}values (single-parameter deterministic model).

Degradation Level S_{w} (%) | Estimated Age (years) |
---|---|

10 | 52 |

20 | 68 |

40 | 88 |

Independent Variables | Variables’ Quantification | |||
---|---|---|---|---|

Distance from the sea | ≤5 km: 0.96 | >5 km: 1.03 | ||

Dimensions of stone plates | Medium dimensions: 1.04 | Large dimensions: 0.94 | ||

Exposure to damp | Low: 1.03 | High: 0.91 | ||

Type of stone | Limestone: 1.04 | Marble: 0.96 | Granite: 1.39 |

**Table 5.**Estimated claddings age obtained by the single-parameter stochastic model for different S

_{w}values and risk margins.

Risk Margin | |||||
---|---|---|---|---|---|

5% | 10% | 20% | 50% | ||

S_{w} (%) | 10 | 34 | 39 | 43 | 51 |

20 | 53 | 57 | 61 | 68 | |

40 | 74 | 80 | 85 | 94 |

S_{w} Index | Details of the Maintenance Task | Costs (Year 0) (€/m^{2}) |
---|---|---|

10% | Scaffolding | 3.69 |

Cracking | 8.28 | |

Cleaning | 19.03 | |

Joint’s replacement (20%) | 2.05 | |

Total | 33.05 | |

20% | Scaffolding | 3.69 |

Cracking | 8.28 | |

Cleaning | 19.03 | |

Joint’s replacement (30%) + Joint’s repair (70%) | 6.67 | |

Cladding’s replacement (20%) | 11.23 | |

Total | 48.90 | |

40% | Scaffolding | 3.69 |

Cladding’s replacement (100%) | 56.14 | |

Total | 59.83 |

**Table 7.**Case study’s remaining time (in years) before action for the defined degradation levels, by model.

Deterministic | Single-Parameter Stochastic | Multiparameter Stochastic | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Single-Parameter | Multiparameter | Risk Margins | ||||||||||

5% | 10% | 20% | 50% | 5% | 10% | 20% | 50% | |||||

S_{w} | 10% | 38 | 34 | 20 | 25 | 29 | 37 | 19 | 24 | 28 | 36 | |

20% | 54 | 54 | 39 | 43 | 47 | 54 | 50 | 52 | 55 | 59 | ||

40% | 74 | 73 | 60 | 66 | 71 | 80 | 70 | 72 | 76 | 81 |

**Table 8.**Summary table with estimated times before action, costs, risk premium rates, risk premiums, risk coefficients, and commercial premiums per household for the different models.

Deterministic | Single-Parameter Stochastic | Multiparameter Stochastic | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Single-parameter | Multiparameter | 5% | 10% | 20% | 50% | 5% | 10% | 20% | 50% | ||

S_{w} | 10% | 38 | 34 | 20 | 25 | 29 | 37 | 19 | 24 | 28 | 36 |

20% | 54 | 54 | 39 | 43 | 47 | 54 | 50 | 52 | 55 | 59 | |

40% | 74 | 73 | 60 | 66 | 71 | 80 | 70 | 72 | 76 | 81 | |

VP,cost (€/m^{2}) | 28.63 | 30.19 | 45.26 | 39.35 | 34.85 | 27.82 | 38.81 | 35.15 | 31.63 | 26.59 | |

C_{t,premium} (€/m^{2}) | 0.53 | 0.56 | 0.98 | 0.79 | 0.67 | 0.49 | 0.75 | 0.66 | 0.58 | 0.46 | |

Risk premium by household (€) | 21.45 | 22.84 | 39.55 | 32.02 | 26.90 | 19.75 | 30.26 | 26.86 | 23.26 | 18.72 | |

Risk coefficients | 1.3*1.2 | 1.3*1.1 | 1.3*1.2 | 1.3*1.1 | |||||||

Commercial premium per household (€) | 33.47 | 32.66 | 61.69 | 49.96 | 41.96 | 30.80 | 43.28 | 38.41 | 33.26 | 26.77 | |

Final premium per household (€) | - | - | 33.89 | 32.72 | 31.92 | 30.80 | 28.42 | 27.93 | 27.42 | 26.77 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Macedo, M.; de Brito, J.; Silva, A.; Oliveira Cruz, C.
Design of an Insurance Policy Model Applied to Natural Stone Facade Claddings. *Buildings* **2019**, *9*, 111.
https://doi.org/10.3390/buildings9050111

**AMA Style**

Macedo M, de Brito J, Silva A, Oliveira Cruz C.
Design of an Insurance Policy Model Applied to Natural Stone Facade Claddings. *Buildings*. 2019; 9(5):111.
https://doi.org/10.3390/buildings9050111

**Chicago/Turabian Style**

Macedo, Miguel, Jorge de Brito, Ana Silva, and Carlos Oliveira Cruz.
2019. "Design of an Insurance Policy Model Applied to Natural Stone Facade Claddings" *Buildings* 9, no. 5: 111.
https://doi.org/10.3390/buildings9050111