# A Model for Risk Adjustment (IFRS 17) for Surrender Risk in Life Insurance

## Abstract

**:**

## 1. Introduction

#### 1.1. IFRS 17

#### 1.2. Our Objective and Setup

#### 1.3. Risk Measures

#### 1.4. Outline of the Article

## 2. Stochastic Modelling of Surrender Rates

#### 2.1. Introduction

#### 2.2. The Poisson Process

#### 2.3. Discrete Time

#### 2.3.1. The Lognormal Model

#### 2.3.2. The Sticky Model

#### 2.4. Continuous Time

#### 2.5. Term Structure of Surrender Rates

#### 2.6. Realism of the Models

## 3. Convex Ordering of Random Variables and Its Applications

#### 3.1. Definition of Convex Ordering

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

#### 3.2. Comonotonicity

**Definition**

**3.**

**Proposition**

**1.**

#### 3.3. Convex Bounds for Sums of Random Variables

**Theorem**

**2.**

#### 3.4. Applications to Risk Measures

**Theorem**

**3.**

#### 3.5. Application to Sums of Lognormal Variables

## 4. PVFCF for the Total Portfolio

#### 4.1. Approximations by Convex Ordering

#### 4.1.1. The Lognormal Model

#### 4.1.2. The Sticky Model

#### 4.2. Numerical Example

#### 4.2.1. The Lognormal Model

#### 4.2.2. The Sticky Model

## 5. Implementation for Risk Adjustment

#### 5.1. Parametrization

#### 5.2. Risk Adjustment

#### 5.3. Fulfillment of Criteria in IFRS17

- (a)
- risks with low frequency and high severity will result in higher risk adjustments for non-financial risk than risks with high frequency and low severity;
- (b)
- for similar risks, contracts with a longer duration will result in higher risk adjustments for non-financial risk than contracts with a shorter duration;
- (c)
- risks with a wider probability distribution will result in higher risk adjustments for non-financial risk than risks with a narrower distribution;
- (d)
- the less that is known about the current estimate and its trend, the higher will be the risk adjustment for non-financial risk; and
- (e)
- to the extent that emerging experience reduces uncertainty about the amount and timing of cash flows, risk adjustments for non-financial risk will decrease and vice versa.”

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. IFRS17 and Risk Adjustment for Non-Financial Risks

- “The key principles in IFRS 17 are that an entity:…(d) recognises and measures groups of insurance contracts at:(i) a risk-adjusted present value of the future cash flows (the fulfillment cash flows) that incorporates all of the available information about the fulfilment cash flows in a way that is consistent with observable market information; plus (if this value is a liability) or minus (if this value is an asset)(ii) an amount representing the unearned profit in the group of contracts (the contractual service margin).…”

- “...(i) estimates of future cash flows (§§ 33–35);(ii) an adjustment to reflect the time value of money and the financial risks related to the future cash flows, to the extent that the financial risks are not included in the estimates of the future cash flows (paragraph 36); and(iii) a risk adjustment for non-financial risk (§ 37).…”

## Notes

1 | IFRS17 uses the sign convention that (discounted) claims and expenses are positive and (discounted) premiums negative. Here we reverse that, in order to look at it from the company’s point of view, and avoid a lot of minus signs. |

2 | For solvency purposes, more extreme events must be considered. Consequently, in Solvency II standard formula there is a capital requirement for “mass lapse” risk. |

3 | This subsection is courtesy of Lina Balčiūnienė. |

4 | For confidentiality reasons, the data has been modified. However, it has been done in a way that does not distort the statistical testing. |

5 | The Shapiro–Wilk test is tailored to test for normality based on the sample mean and sample variance. However, the specification of the sticky model has a fixed mean, and to be accurate, the Shapiro–Wilk test with a known mean should be used, see Hanusz et al. (2016) for details. The difference in our case is however negligible. |

6 | In fact, an improved right bound proved in Kaas et al. (2000) is ${\sum}_{i=1}^{n}{X}_{i}{\le}_{cx}{\sum}_{i=1}^{n}{F}_{{X}_{i}|\mathsf{\Lambda}}^{-1}\left(U\right)$ if $\mathsf{\Lambda}$ and U are independent. |

7 | For a good overview of approximation methods, see Asmussen et al. (n.d.). |

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**Figure 6.**Quantile function of S (thick green), ${S}_{l}$ (blue), and ${S}_{u}$ (orange), for different p.

**Figure 7.**Partial expectation of S (thick green), ${S}_{l}$ (blue), and ${S}_{u}$ (orange), for different p.

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**MDPI and ACS Style**

Carlehed, M.
A Model for Risk Adjustment (IFRS 17) for Surrender Risk in Life Insurance. *Risks* **2023**, *11*, 62.
https://doi.org/10.3390/risks11030062

**AMA Style**

Carlehed M.
A Model for Risk Adjustment (IFRS 17) for Surrender Risk in Life Insurance. *Risks*. 2023; 11(3):62.
https://doi.org/10.3390/risks11030062

**Chicago/Turabian Style**

Carlehed, Magnus.
2023. "A Model for Risk Adjustment (IFRS 17) for Surrender Risk in Life Insurance" *Risks* 11, no. 3: 62.
https://doi.org/10.3390/risks11030062