# Actuarial Geometry

## Abstract

**:**

## 1. Introduction

## 2. Why Idiosyncratic Insurance Risk Matters

- Technical and axiomatic characterization of risk measures: (Dhaene et al. 2003; Furman and Zitikis 2008; Laeven and Stadje 2013).
- Capital allocation and its relationship with risk measurement: (Dhaene et al. 2003; Venter et al. 2006; Bodoff 2009; Buch and Dorfleitner 2008; Dhaene et al. 2012; Erel et al. 2015; Furman and Zitikis 2008; Powers 2007; Tsanakas 2009).
- The connection between purpose and method in capital allocation: (Dhaene et al. 2008; Zanjani 2010; Bauer and Zanjani 2013b; Goovaerts et al. 2010).
- Questioning the need for capital allocation in pricing: Gründl and Schmeiser 2007.

## 3. Risk Measures, Risk Allocation and the Ubiquitous Gradient

#### 3.1. Definition and Examples of Risk Measures

#### 3.2. Allocation and the Gradient

## 4. Two Motivating Examples

## 5. Lévy process Models of Insurance Losses

#### 5.1. Definition and Basic Properties of Lévy processes

**Definition**

**1.**

- LP1.
- $X(0)=0$ almost surely;
- LP2.
- X has independent increments, so for $0\le {t}_{1}\le \cdots \le {t}_{n+1}$ the variables $X({t}_{j+1})-X({t}_{j})$ are independent;
- LP3.
- X has stationary increments, so $X({t}_{j+1})-X({t}_{j})$ has the same distribution as $X({t}_{j+1}-{t}_{j})$; and
- LP4.
- X is stochastically continuous, so for all $a>0$ and $s\ge 0$$$\underset{t\to s}{lim}\mathrm{Pr}(|X(t)-X(s)|>a)=0.$$

**Example**

**1**(Trivial process)

**.**

**Example**

**2**(Poisson process)

**.**

**Example**

**3**(Compound Poisson process)

**.**

**Example**

**4**(Brownian motion)

**.**

**Example**

**5**(Operational time)

**.**

**Example**

**6**(Subordination)

**.**

**Theorem**

**1**(Lévy-Khintchine)

**.**

#### 5.2. Four Temporal and Volumetric Insurance Loss Models

- IM1.
- $A(x,t)=X(xt)$. This model assumes there is no difference between insuring given insureds for a longer period of time and insuring more insureds for a shorter period.
- IM2.
- $A(x,t)=X(xZ(t))$, for a subordinator $Z(t)$ with $\mathrm{E}(Z(t))=t$. Z is an increasing Lévy process which measures random operational time, rather than calendar time. It allows for systematic time-varying contagion effects, such as weather patterns, inflation and level of economic activity, affecting all insureds. Z could be a deterministic drift or it could combine a deterministic drift with a stochastic component.
- IM3.
- $A(x,t)=X(xCt)$, where C is a mean 1 random variable capturing heterogeneity and non-diversifiable parameter risk across an insured population of size x. C could reflect different underwriting positions by firm, which drive systematic and permanent differences in results. The variable C is sometimes called a mixing variable.
- IM4.
- $A(x,t)=X(xCZ(t))$.

**Definition**

**2.**

- 1.
- If $\upsilon (x,t)\to 0$ as $t\to \infty $ we will call $A(x,t)$ temporally diversifying.
- 2.
- If $\upsilon (x,t)\to 0$ as $x\to \infty $ we will call $A(x,t)$ volumetrically diversifying.
- 3.
- A process which is both temporally and volumetrically diversifying will be called diversifying.

- AM1.
- $A(x,t)=xX(t)$

## 6. Defining the Derivative of a Risk Measure and Directions in the Space of Risks

#### 6.1. Defining the Derivative

#### 6.2. Directions in the Space of Actuarial Random Variables

- The notion that directions, or tangent vectors, live in a separate space called the tangent bundle.
- The identification of tangent vectors as derivatives of curves.
- The idea that Lévy processes, characterized by the additive relation $X(s+t)=X(s)+X(t)$, provide the appropriate analog of rays to use as a basis for insurance risks.

**Theorem**

**2**

**.**There is a one-to-one correspondence between Lévy triples and rays (continuous, additive maps) $t\in [0,\infty )\to {\mu}_{t}\in M(\mathbb{R})$. The Lévy triple $(\sigma ,\nu ,\gamma )$ corresponds to the infinitely divisible map $t\mapsto {\mu}_{t}^{(\sigma ,\nu ,\gamma )}$ given by the Lévy process with the same Lévy triple. The map is $t\mapsto {\mu}_{t}^{(\sigma ,\nu ,\gamma )}$ is differentiable and

#### 6.3. Examples

**Example**

**7**(Brownian motion)

**.**

**Example**

**8**(Gamma process)

**.**

**Example**

**9**(Laplace process)

**.**

#### 6.4. Application to Insurance Risk Models IM1-4 and Asset Risk Model AM1

#### 6.5. Higher Order Identification of the Differences Between Insurance and Asset Models

## 7. Empirical Analysis

#### 7.1. Overview

**H1.**- The asymptotic coefficient of variation or volatility as volume grows is strictly positive.
**H2.**- Time and volume are symmetric in the sense that the coefficient of variation of aggregate losses for volume x insured for time t only depends on $xt$.

#### 7.2. Isolating the Mixing Distribution

**Proposition**

**1.**

**Proof.**

#### 7.3. Volumetric Empirics

**Heterogeneity**refers to the level of consistency in terms and conditions and types of insureds within the line, with high heterogeneity indicating a broad range. The two Other Liability lines are catch-all classifications including a wide range of insureds and policies.**Regulation**indicates the extent of rate regulation by state insurance departments.**Limits**refers to the typical policy limit. Personal auto liability limits rarely exceed $300,000 per accident in the US and are characterized as low. Most commercial lines policies have a primary limit of $1M, possibly with excess liability policies above that. Workers compensation policies do not have a limit but the benefit levels are statutorily prescribed by each state.**Cycle**is an indication of the extent of the pricing cycle in each line; it is simply split personal (low) and commercial (high).**Cats**(i.e., catastrophes) covers the extent to which the line is subject to multi-claimant, single occurrence catastrophe losses such as hurricanes, earthquakes, mass tort, securities laddering, terrorism, and so on.

- The loss processes are not volumetrically diversifying, that is the volatility does not decrease to zero with volume.
- Below a range $100M-1B (varying by line) there are material changes in volatility with premium size.
- $100M is a reasonable threshold for large, in the sense that there is less change in volatility beyond $100M.

- Res1.
- Raw loss ratio volatility across all twelve years of data for all companies. This volatility includes a pricing cycle effect, captured by accident year, and a company effect.
- Res2.
- Control for the accident year effect ${\lambda}_{.,y}$. This removes the pricing cycle but it also removes some of the catastrophic loss effect for a year—an issue with the results for homeowners in 2004.
- Res3.
- Control for the company effect ${\lambda}_{c,.}$. This removes spurious loss ratio variation caused by differing expense ratios, distribution costs, profit targets, classes of business, limits, policy size and so forth.
- Res4.
- Control for both company effect and accident year, i.e., perform an unbalanced two-way ANOVA with zero or one observation per cell. This can be done additively, modeling the loss ratio ${\lambda}_{c,y}$ for company c in year y as$${\widehat{\lambda}}_{c,y}={\lambda}_{.,.}+({\lambda}_{c,.}-{\lambda}_{.,.})+({\lambda}_{.,y}-{\lambda}_{.,.}),$$$${\widehat{\lambda}}_{c,y}={\lambda}_{.,.}({\lambda}_{c,.}/{\lambda}_{.,.})({\lambda}_{.,y}/{\lambda}_{.,.}).$$

#### 7.4. Temporal Empirics

## 8. Conclusions

## Conflicts of Interest

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1 | Actuarial Geometry was originally presented to the 2006 Risk Theory Seminar in Richmond, Virginia, Mildenhall (2006). This version is largely based on the original, with some corrections and clarifications, as well as more examples to illustrate the theory. Since 2006 the methodology it described has been successfully applied to a very wide variety of global insurance data in Aon Benfield’s annual Insurance Risk Study, ABI (2007, 2010, 2012, 2013, 2014, 2015), now in its eleventh edition. The findings have remained overwhelmingly consistent. Academically, the importance of the derivative and the gradient allocation method has been re-confirmed in numerous papers since 2006. Applications of Lévy processes to actuarial science and finance have also greatly proliferated. However, the new literature has not touched on the clarification between “direction” in the space of asset return variables and in the space of actuarial variables presented here. |

**Figure 1.**Lévy process and homogeneous embeddings of ${\mathbb{R}}_{+}$ into the space of risks, $\mathbf{L}$. The Lévy process embedding corresponds to the straight line m and the asset embedding to the curved line k.

**Figure 2.**Schematic of the Kalkbrener-Meyers example, using the sphere to illustrate the more complex space $\mathbf{L}$.

**Figure 3.**Illustration of the difference between $tZ$ and $\sqrt{t}Z$ for Z a standard normal as $t\to 0$.

**Figure 4.**Theoretical distribution of scaled aggregate losses with no parameter or structure uncertainty and Poisson frequency.

**Figure 5.**Theoretical distribution envelope of scaled aggregate losses with a gamma mixed Poisson frequency with mixing variance $c=0.0625$.

**Figure 6.**The relationship between raw loss ratio volatility, measured as coefficient of variation of loss ratios, and premium volume, using data from accident years 1993–2004. Each inset graph plots the same data on a log/log scale, showing that the volatility continues to decrease materially for premium volumes in the $100Ms. The total line is distorted by changing mix of business by volume; the largest companies are dominated by private passenger auto liability which is the lowest volatility line.

**Figure 7.**Adjusted analysis of variance (ANOVA) for commercial auto, commercial multiperil and homeowners.

**Figure 8.**Adjusted ANOVA for medical malpractice claims made and other liability claims made and occurrence.

**Figure 10.**

**Left**plot shows the loss ratio volatility by line for companies writing $100M or more premium each year based on Schedule P accident year ultimate booked gross loss ratios, from 1993–2004. The graph shows the effect of adjusting the loss ratio for an accident year pricing effect, a company effect, and both effects (i.e., Res1-4). The

**right**hand plot shows the differential impact of the pricing effect and company effect by line. Each bar shows the increase in volatility of the unadjusted loss ratios compared to the adjusted.

**Figure 11.**Commercial auto liability (

**top four plots**) and commercial multiperil volatility (

**bottom four plots**). Note 9/11 loss effect in the lower-left plot. See text for a description of the plots.

**Figure 12.**Homeowners (

**top four plots**) and medical malpractice claims made volatility (

**bottom four plots**). Note the 2004 homowners catastrophe losses. See text for a description of the plots.

**Figure 13.**Other liability claims made (

**top four plots**) and occurrence volatility (

**bottom four plots**). See text for a description of the plots.

**Figure 14.**Private passenger auto liability (

**top four plots**) and workers compensation volatility (

**bottom four plots**). Note vertical scale on private passenger auto loss ratios and the visibly higher volatility of premium than loss in the lower left hand plot. See text for a description of the plots.

**Figure 15.**Coefficient of variation of loss ratio by premium volume for private passenger auto liability, computed using bucketed $xt$ for $t=1,2,4,6,12$.

**Figure 16.**Fit of $\sqrt{\frac{{\textstyle {\sigma}^{2}}}{{\textstyle xt}}+c}$ to volatility by volume, $xt$, for homeowners, private passenger auto and commercial auto.

**Left hand plot**shows data based on a single year $t=1$;

**right hand plot**shows the same data for $t=1,2,4,6,12$.

**Figure 17.**Fit of $\sqrt{\frac{{\textstyle {\sigma}^{2}}}{{\textstyle xt}}+c}$ to volatility by volume, $xt$, for workers compensation, commercial multiperil and other liability occurrence.

**Left hand plot**shows data based on a single year $t=1$;

**right hand plot**shows the same data for $t=1,2,4,6,12$.

Model | Variance | $\mathit{\upsilon}(\mathit{x},\mathit{t})$ | Diversifying | |
---|---|---|---|---|

$\mathit{x}\to \mathsf{\infty}$ | $\mathit{t}\to \mathsf{\infty}$ | |||

IM1: $X(xt)$ | ${\sigma}^{2}xt$ | $\frac{{\textstyle \sigma}}{\sqrt{{\textstyle xt}}}$ | Yes | Yes |

IM2: $X(xZ(t))$ | $xt({\sigma}^{2}+x{\tau}^{2})$ | $\sqrt{\frac{{\textstyle {\sigma}^{2}}}{{\textstyle xt}}+\frac{{\textstyle {\tau}^{2}}}{{\textstyle t}}}$ | No | Yes |

IM3: $X(xCt)$ | $xt({\sigma}^{2}+cxt)$ | $\sqrt{\frac{{\textstyle {\sigma}^{2}}}{{\textstyle xt}}+c}$ | No | No |

IM4: $X(xCZ(t))$ |
$$\begin{array}{c}{x}^{2}{t}^{2}\left(\frac{{\textstyle (c+1){\tau}^{2}}}{{\textstyle t}}+c\right)\hfill \\ \hfill +{\sigma}^{2}xt\end{array}$$
| $\sqrt{\frac{{\textstyle {\sigma}^{2}}}{{\textstyle xt}}+\frac{{\textstyle {\tau}^{\prime 2}}}{{\textstyle t}}+c}$ | No | No |

AM1: $xX(t)$ | ${x}^{2}{\sigma}^{2}t$ | $\sigma /\sqrt{t}$ | Const. | Yes |

Characterization of Ray | Required Structure on ${\mathbb{R}}^{\mathit{n}}$ |
---|---|

$\alpha $ is the shortest distance between $\alpha (0)$ and $\alpha (1)$ | Notion of distance in ${\mathbb{R}}^{n}$, differentiable manifold |

${\alpha}^{\u2033}(t)=0$, constant velocity, no acceleration | Very complicated on a general manifold. |

$\alpha (t)=t\mathbf{x}$, $\mathbf{x}\in {\mathbb{R}}^{n}$. | Vector space structure |

$\alpha (s+t)=\alpha (s)+\alpha (t)$ | Can add in domain and range, semigroup structure only. |

Insurance Line | Heterogeneity | Regulation | Limits | Cycle | Cats |
---|---|---|---|---|---|

Personal Auto | Low | High | Low | Low | No |

Commercial Auto | Moderate | Moderate | Moderate | High | No |

Workers Compensation | Moderate | High | Statutory | High | Possible |

Medical Malpractice | Moderate | Moderate | Moderate | High | No |

Commercial Multi-Peril | Moderate | Moderate | Moderate | High | Moderate |

Other Liability Occurrence | High | Low | High | High | Yes |

Homeowners Multi-Peril | Moderate | High | Low | Low | High |

Other Liability Claims Made | High | Low | High | High | Possible |

Abbreviation | Parameters | Distribution | Fitting Method |
---|---|---|---|

Wald | 2 | Wald (inverse Gaussian) | Maximum likelihood |

EV | 2 | Frechet-Tippet extreme value | Method of moments |

Gamma | 2 | Gamma | Method of moments |

LN | 2 | Lognormal | Maximum likelihood |

SLN | 3 | Shifted lognormal | Method of moments |

Dimension | Actuarial Geometry | Solvency II |
---|---|---|

Time horizon | to ultimate | one year |

Catastrophe risk | included | excluded |

Size of company | large | average |

**Table 6.**Coefficient of variation of gross loss ratio, Source: Aon Benfield Insurance Risk Study, 7th Edition, used with permission.

Line | 1st Edition | 7th Edition | Change |
---|---|---|---|

Private Passenger Auto | 14% | 14% | 0% |

Commercial Auto | 24% | 24% | 0% |

Workers’ Compensation | 26% | 27% | 1% |

Commercial Multi Peril | 32% | 34% | 2% |

Medical Malpractice: Claims-Made | 33% | 42% | 9% |

Medical Malpractice: Occurrence | 35% | 35% | 0% |

Other Liability: Occurrence | 36% | 38% | 2% |

Special Liability | 39% | 39% | 0% |

Other Liability: Claims-Made | 39% | 41% | 2% |

Reinsurance Liability | 42% | 67% | 25% |

Products Liability: Occurrence | 43% | 47% | 4% |

International | 45% | 72% | 27% |

Homeowners | 47% | 48% | 1% |

Reinsurance: Property | 65% | 85% | 20% |

Reinsurance: Financial | 81% | 93% | 12% |

Products Liability: Claims-Made | 102% | 100% | −2% |

© 2017 by the author. 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/).

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**MDPI and ACS Style**

Mildenhall, S.J.
Actuarial Geometry. *Risks* **2017**, *5*, 31.
https://doi.org/10.3390/risks5020031

**AMA Style**

Mildenhall SJ.
Actuarial Geometry. *Risks*. 2017; 5(2):31.
https://doi.org/10.3390/risks5020031

**Chicago/Turabian Style**

Mildenhall, Stephen J.
2017. "Actuarial Geometry" *Risks* 5, no. 2: 31.
https://doi.org/10.3390/risks5020031