# A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies

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## Abstract

**:**

## 1. Introduction

“Where practicable, insurance and reinsurance undertakings shall derive the solvency capital requirement directly from the probability distribution forecast generated by the internal model of those undertakings, using the Value-at-Risk measure set out in Article 101(3).”

## 2. Cash Flow Projection Models and LSMC

#### 2.1. Cash Flow Projection Models

#### 2.1.1. Full Balance Sheet Projections

- Assets
- Liabilities
- Capital Market
- Management Actions

#### 2.1.2. Pricing Machine

- inflows of premiums (mostly pre-determined);
- outflow of costs (mostly pre-determined); and
- outflow of benefits and dividends (volatile, dependent on the capital market development).

#### 2.2. LSMC—From American Option Pricing to Capital Requirements

#### The Basic Idea behind LSMC

#### 2.3. Calculating Capital Requirements

#### 2.3.1. Nested Valuation Problem

#### 2.3.2. Least-Squares Monte Carlo Solution

## 3. Least-Squares Monte Carlo Model for Life Insurance Companies

- a detailed description of the simulation setting and the required task;
- a concept for a calibration procedure for the proxy function;
- a validation procedure for the obtained proxy function; and
- the actual application of the LSMC model to forecast the full loss distribution.

#### 3.1. Simulation Setting

#### 3.1.1. Filtered Probability Space

#### 3.1.2. Solvency Capital Requirement

#### 3.1.3. Available Capital

#### 3.1.4. Fitting Points

#### 3.1.5. Practical Implementation

#### 3.2. Proxy Function Calibration

#### 3.2.1. Two Approximations

#### 3.2.2. Convergence

**Proposition**

**1.**

**Proposition**

**2.**

#### 3.2.3. Adaptive Algorithm to Build up the Proxy Function

#### 3.2.4. Initialization

#### 3.2.5. Iterative Procedure

#### 3.2.6. Refinements

#### 3.3. Proxy Function Validation

#### 3.3.1. Validation Points

- Points known to be in the Capital Region, that is scenarios producing a risk capital close to the value-at-risk from previous risk capital calculations;
- Quasi random points from the entire fitting space;
- One-dimensional risks leading to a 1-in-200 loss in the one-dimensional distribution of this risk factor, that is points which have only one coordinate changed and which ensure a good interpretability;
- Two-dimensional or three-dimensional stresses for risk factors with high interdependency, for example interest rate and lapse; and
- Points with the same inner scenarios which can be used to more accurately measure a risk capital in scenarios which do not have the ESG relevant risk factors changed.

#### 3.3.2. Practical Implementation

#### 3.3.3. Out-of-Sample Test

#### 3.4. Full Distribution Forecast

#### 3.4.1. Solvency Capital Requirement

#### 3.4.2. Convergence

**Corollary**

**1.**

**Corollary**

**2.**

#### 3.4.3. Practical Implementation

## 4. Numerical Illustration of Convergence

#### 4.1. Simulation Setting

#### 4.2. Proxy Function Calibration

#### 4.3. Proxy Function Validation

#### 4.4. Full Distribution Forecast

## 5. Numerical Comparison with Nested Stochastics

#### 5.1. Computation Time

#### 5.2. Accuracy

## 6. Conclusions, Further Aspects, and Potential Improvements

#### 6.1. Simulation Setting

#### 6.2. Proxy Function Calibration

#### 6.3. Proxy Function Validation

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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Component | Risk Factor Description |
---|---|

${X}_{1}$ | Risk-free interest rates movement |

${X}_{2}$ | Change in interest rate volatility |

${X}_{3}$ | Change in equity volatility |

${X}_{4}$ | Shock on volatility adjustment (if used by the company) |

${X}_{5}$ | Credit default |

${X}_{6}$ | Credit spread widening |

${X}_{7}$ | Currency exchange rate risk |

${X}_{8}$ | Shock on equity market value |

${X}_{9}$ | Shock on property market value |

${X}_{10}$ | Lapse stress on best estimate assumptions |

${X}_{11}$ | Mortality catastrophe stress with a one-off increase in mortality |

${X}_{12}$ | Mortality trend volatility stress |

${X}_{13}$ | Mortality level stress on best estimate assumptions |

${X}_{14}$ | Longevity trend volatility stress on best estimate assumptions |

${X}_{15}$ | Longevity level stress on best estimate assumptions |

${X}_{16}$ | Morbidity stress on best estimate assumptions |

${X}_{17}$ | Expenses stress on best estimate assumptions |

**Table 2.**Construction sequence of the proxy function in the adaptive algorithm with the final coefficients. Furthermore, AIC and out-of-sample mean squared errors after each iteration.

Iteration | Exponents of Risk Factors | Regression Results | Out-of-Sample MSE | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{k}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{1}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{2}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{3}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{4}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{5}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{6}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{7}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{8}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{9}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{10}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{11}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{12}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{13}}$ | ${\mathit{r}}_{\mathit{k}\mathbf{,}\mathit{14}}$ | ${\widehat{\mathit{\beta}}}_{\mathit{k}}^{\mathbf{\left(}\mathit{N}\mathbf{\right)}}$ | AIC | (1) | (2) |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8048.89 | 405,919.54 | 279,972.87 | 41,918.48 |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 5796.16 | 347,121.57 | 22,744.22 | 32,479.31 |

2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 80.08 | 342,819.19 | 9863.42 | 5112.25 |

3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −28.03 | 337,759.88 | 4048.53 | 4413.39 |

4 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 26.61 | 336,372.65 | 3003.24 | 4339.50 |

5 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 108.73 | 335,245.70 | 2534.24 | 4329.35 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 22.68 | 334,903.08 | 2334.00 | 4314.37 |

7 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −614.79 | 334,579.21 | 2257.41 | 4308.28 |

8 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 297.50 | 334,299.51 | 1341.89 | 1970.71 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 32.93 | 334,043.75 | 1376.76 | 1945.95 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 71.60 | 333,840.00 | 1084.23 | 1501.77 |

11 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 57.32 | 333,769.40 | 1134.52 | 1843.88 |

12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1636.91 | 333,523.57 | 638.87 | 748.50 |

13 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −58.12 | 333,420.22 | 637.63 | 700.10 |

14 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15.67 | 333,367.94 | 645.54 | 699.57 |

15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 29.69 | 333,316.80 | 327.84 | 172.71 |

16 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 42.48 | 333,265.87 | 313.90 | 172.91 |

17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 13.64 | 333,222.65 | 321.78 | 169.47 |

18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6.81 | 333,182.38 | 280.15 | 167.30 |

19 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −5.66 | 333,144.02 | 292.49 | 166.74 |

20 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −9.91 | 333,118.13 | 285.39 | 167.79 |

21 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −440.05 | 333,093.91 | 259.30 | 178.81 |

22 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 36.14 | 333,083.85 | 268.45 | 169.96 |

23 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −35.76 | 333,074.41 | 256.88 | 166.85 |

24 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.85 | 333,066.67 | 270.83 | 163.39 |

25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | −82.22 | 333,059.15 | 264.46 | 163.59 |

26 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | −2.32 | 333,047.07 | 248.44 | 163.48 |

27 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | −2.21 | 333,039.92 | 254.61 | 163.62 |

28 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 3.69 | 333,033.27 | 251.67 | 163.51 |

29 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −48.47 | 333,026.83 | 259.06 | 163.76 |

30 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 126.06 | 333,020.64 | 255.68 | 162.53 |

31 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −20.94 | 333,015.48 | 247.72 | 169.26 |

32 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.30 | 333,010.56 | 282.84 | 264.54 |

33 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.98 | 333,001.42 | 290.84 | 273.88 |

34 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 51.52 | 332,996.57 | 285.59 | 270.99 |

35 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | −93.15 | 332,991.83 | 299.74 | 269.59 |

36 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | −3.99 | 332,987.49 | 294.01 | 269.92 |

37 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 9.48 | 332,983.89 | 244.46 | 193.39 |

38 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 17.52 | 332,981.05 | 246.77 | 193.69 |

39 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | −42.29 | 332,978.40 | 252.84 | 193.19 |

40 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 76.70 | 332,975.77 | 239.20 | 186.54 |

41 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −279.08 | 332,971.77 | 203.89 | 133.62 |

42 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2.57 | 332,969.67 | 208.28 | 133.57 |

43 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.62 | 332,967.85 | 218.11 | 146.76 |

44 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.71 | 332,965.46 | 210.39 | 148.32 |

45 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 13.36 | 332,963.97 | 208.86 | 148.27 |

46 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 11.10 | 332,962.49 | 198.29 | 148.32 |

47 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −5.85 | 332,961.35 | 226.05 | 193.90 |

48 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 5.69 | 332,960.33 | 222.46 | 193.81 |

49 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | −56.01 | 332,959.53 | 207.33 | 198.33 |

50 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 70.44 | 332,958.77 | 209.11 | 197.42 |

51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1390.37 | 332,958.18 | 217.77 | 203.07 |

52 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −1.06 | 332,957.82 | 219.21 | 203.00 |

53 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −64.50 | 332,957.51 | 192.10 | 159.88 |

54 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 17.64 | 332,953.46 | 165.97 | 143.94 |

55 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 17.25 | 332,953.31 | 169.81 | 137.14 |

56 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 37.24 | 332,951.99 | 172.65 | 137.00 |

57 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2.00 | 332,951.87 | 172.90 | 137.59 |

58 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | −26.38 | 332,951.38 | 171.80 | 137.55 |

59 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6.21 | 332,951.14 | 182.94 | 149.27 |

60 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | −66.58 | 332,951.14 | 182.17 | 148.10 |

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**MDPI and ACS Style**

Krah, A.-S.; Nikolić, Z.; Korn, R.
A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies. *Risks* **2018**, *6*, 62.
https://doi.org/10.3390/risks6020062

**AMA Style**

Krah A-S, Nikolić Z, Korn R.
A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies. *Risks*. 2018; 6(2):62.
https://doi.org/10.3390/risks6020062

**Chicago/Turabian Style**

Krah, Anne-Sophie, Zoran Nikolić, and Ralf Korn.
2018. "A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies" *Risks* 6, no. 2: 62.
https://doi.org/10.3390/risks6020062