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# Microscopic Traffic Models, Accidents, and Insurance Losses

Sojung Kim  Leibniz Universität Hannover††thanks: Institute of Actuarial and
Financial Mathematics & House of Insurance, Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany, email<EMAIL_ADDRESS>hannover.de<EMAIL_ADDRESS><EMAIL_ADDRESS>Marcel Kleiber & Stefan Weber  Leibniz
Universität Hannover††thanks: Institute of Actuarial and Financial Mathematics
& House of Insurance, Leibniz Universität Hannover, Welfengarten 1, 30167
Hannover, Germany, email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>hannover.de.

###### Abstract

The paper develops a methodology to enable microscopic models of
transportation systems to be accessible for a statistical study of traffic
accidents. Our approach is intended to permit an understanding not only of
historical losses, but also of incidents that may occur in altered, potential
future systems. Through this, it is possible, from both an engineering and
insurance perspective, to assess changes in the design of vehicles and
transport systems in terms of their impact on functionality and road safety.

Structurally, we characterize the total loss distribution approximatively as a
mean-variance mixture. This also yields valuation procedures that can be used
instead of Monte Carlo simulation. Specifically, we construct an
implementation based on the open-source traffic simulator SUMO and illustrate
the potential of the approach in counterfactual case studies.

Keywords: Microscopic traffic models, car-following models, SUMO, digital
twins, insurance premiums.

##  1 Introduction

Every year, traffic accidents cause substantial damage, both property damage
and injuries and deaths. For example, nearly 43,000 people died in road
traffic accidents in the USA in 2021 ([31]). The frequency and severity of
these accidents depends on the driving behavior of vehicles, on the one hand,
and on the characteristics of traffic systems themselves, on the other.
Improvements in road safety are achieved, for example, by reducing serious
injuries in accidents through the design of vehicles, by car body design,
airbags, seatbelts, etc. However, the frequency and type of accidents can also
be influenced by modifying the transportation system itself and by changes in
driving behavior. From a higher-level perspective, at least two dimensions are
central, and we will examine them in this paper:

  1. (i)

_Engineering._ From an engineering perspective, the focus is on the good
design of vehicles and traffic systems, combining functionality and safety.
Instruments in this respect include traffic rules and their implementation,
the layout of streets, and innovation in vehicle technology. Improvements of
this type may reduce the number of accidents and their severity, but cannot
completely prevent accidents.

  2. (ii)

_Insurance._ Residual risks remain, and accidents cannot be completely
prevented. However, at least in financial terms, the associated losses can be
covered by insurance contracts. The role of actuaries is to develop adequate
contract structures, calculate correct premiums, and implement quantitative
risk management in insurance firms. These tasks require the modeling and
analysis of probability distributions of accident frequencies, corresponding
damages, and insurance losses.

The objective of this paper is to develop a methodology to enable microscopic
models of transportation systems to be accessible for a statistical study of
traffic accidents. Our approach is intended to permit an understanding not
only of historical losses, but also of incidents that may occur in altered,
potential future systems. Through this, it is possible, from both an
engineering and insurance perspective, to assess changes in the design of
vehicles (e.g., the driving behavior of autonomous vehicles) and transport
systems in terms of their impact on functionality and road safety.

To understand current traffic events or structural relationships in the past,
historical data can be used. Historical data can also be applied to test
whether a model framework is appropriate in principle to describe traffic
systems realistically. These data also constitute the essential basis for the
specific pricing of insurance contracts in practice.

But how can we examine risks associated with new technologies and with novel
future strategies for traffic systems? Consider autonomous vehicles, for
example: due to their altered driving behavior, these will reshape existing
traffic patterns, and in turn, accident occurrences and associated losses.
Insurance companies presumably will have to adapt their business models as
well; in the future, premiums for auto insurance may depend upon the driving
configuration of the vehicle rather than the risk profile of the driver.

In order to investigate future developments, we are suggesting to devise
simulation tools in analogy to digital twins of real transport systems, which
allow counterfactual case studies of possible future transport systems. The
digital twin paradigm refers to the triad of a “physical entity, a virtual
counterpart, and the data connections in between” ([23]). In our application,
the physical entity is the (future) real-world transportation system for which
data on losses are not yet available. Its virtual counterpart is the model we
are building. Counterfactual case studies can be used to generate data,
evaluate future driving technologies and their impact on accident losses.
Based on the results, newly developed concepts (e.g., modified traffic rules,
novel insurance coverage and their insurance premiums, etc.) can be adapted in
the real world. The concept of the digital twin makes it possible to
experiment with technologies and policies, and their effects on accident
damage without having to implement risky tests in reality.

Methodologically, this paper combines existing microscopic traffic models with
probabilistic tools from actuarial science and quantitative risk management to
study accident damage and insurance losses in the context of simulations. In
particular, we use the well-established traffic simulator SUMO ([25]) to
realistically model traffic systems. We extend this to include random
accidents and corresponding losses. The losses are modeled as random variables
whose distributions depend on microscopic data. Since insurance contracts
typically cover annual periods, we set up a model for aggregate losses over a
one-year time horizon. We also show that aggregate losses can be approximated
by a mean-variance mixture of Gaussian distributions. This provides an
alternative perspective on the distribution of the aggregate loss and a second
method of evaluation besides crude Monte Carlo sampling. For certain insurance
contracts, we improve the accuracy of the approximation-based valuation by
using a correction term. This was originally developed by [12] for the
efficient pricing of complex financial instruments, there in the context of a
classical Gaussian approximation.

Our digital twin approach enables a comprehensive analysis of risk in
transportation systems: We study the impact of fleet sizes and their driving
configurations on system efficiency and insurance prices. System efficiency is
measured using traditional traffic statistics based on local traffic counts
such as traffic flow, average speed, and density. Insurance claims are
examined in terms of their probability distributions and selected statistical
functionals.

The main contributions of this paper are:

  1. (i)

We develop a powerful methodological framework to generate accident data based
on microscopic traffic models in analogy to the concept of digital twins.

  2. (ii)

Specifically, we construct an implementation based on the state-of-the-art
open-source traffic simulator SUMO and illustrate the potential of the
approach in comprehensive case studies.

  3. (iii)

Structurally, we characterize the total loss distribution approximatively as a
mean-variance mixture. This also yields alternative valuation procedures.

  4. (iv)

Based on Stein’s method, we obtain a correction term in the valuation, derived
from the results of [12], which enables surprisingly accurate pricing of
insurance contracts.

###  1.1 Outline

The paper is organized as follows. Section 1.2 discusses related contributions
in the literature. Section 2 presents the microscopic traffic model that
captures also accidents. Section 3 discusses the evaluation of the losses.
Case studies are presented in Section 4. Section 5 concludes and discusses
further research challenges. The e-companion contains in Appendix A details on
the implemented sampling procedure; additional simulation results, not
presented in Section 4, are documented in Appendix B.

###  1.2 Literature

Our paper combines microscopic traffic models with probabilistic tools from
actuarial science and quantitative risk management to study risks in traffic
systems. The literature can be predominantly classified along the two
dimensions described in the introduction, the engineering perspective and the
actuarial perspective.

The Engineering Perspective. An important field of operations research is the
analysis and optimization of road traffic systems (see, e.g., [18]) with
respect to their efficiency. Traffic models are indispensable tools for this
purpose: Macroscopic models are based on the functional relationships between
macroscopic features such as traffic flow, traffic density, and average speed.
These models allow the study of issues such as the efficient routing of
vehicles under different constraints (see, e.g., [1], [10]). Stochastics can
be used to extend such risk considerations in terms of uncertain travel times
(e.g., [33]). In the context of various applications of transportation
systems, tailored stochastic models provide suitable analytical tools; the
literature is extensive and includes, for example, the efficient routing of
ambulances ([28]) or the allocation of capacity in bike-sharing systems
([15]).

To model transportation systems at a level of higher granularity, microscopic
traffic models are used (see, e.g., [20]). These models typically determine
the acceleration behavior of individual vehicles. Their simulation, i.e., the
computation of trajectories from accelerations, is computationally more
demanding. There are established software solutions that facilitate the
application of microscopic models. In this work, we use SUMO (see the Section
2.3 for an overview). Examples of competing microscopic traffic simulators
include VISSIM ([13]) and Aimsun ([8]). While SUMO is open source software,
these competitors are commercial.

Leveraging the simulator Aimsun, [36] address questions regarding optimal
operation of traffic networks using microscopic traffic models; the authors
develop a stochastic optimization framework based on coupling the Aimsun
simulator with a metamodel to optimize traffic efficiency – here for signal
plans in a city. [37] extend the microscopic traffic model for fuel
consumption to determine energy-efficient traffic management strategies.

In addition to model building (if data are available), whether macroscopic or
microscopic, calibration can be a challenge. [14] propose a Bayesian approach
to calibrating travel demand. [53] discuss the calibration of large-scale
traffic simulators; [38] focus on the calibration of car-following models for
the simulation of a traffic network.

In addition to the traditional focus on efficiency, there is another strand of
literature that evaluates the safety of transportation systems. Up to now,
mainly historical data have been used to examine accident frequency and
severity. For reviews, we refer to [26] and [45]. Statistical modeling
approaches permit inference when sufficient data are available on the level of
the granularity of the analysis. For example, [52] estimate the impact of
microscopic traffic variables on crash risks. [35] analyze the impact of
public traffic policies on crash severity.

In the absence of data, physical models of traffic can be used to generate
artificial data. In our research group, we have shown how perceptual errors
can be added to microscopic traffic models to endogenously model the
occurrence of accidents (cf. [7] and [6]) – a topic that is particularly
relevant for sensors of autonomous vehicles. The models allow characterizing
the trade-off between safety and efficiency of transportation systems.

To deploy autonomous vehicles in the real world, lengthy and expensive testing
phases are required. Acceleration strategies are being developed to shorten
these times (e.g., [54]). These approaches rely on importance sampling
techniques to overcome the rare-event nature of safety-critical situations.
[2] develop simulation-based testing methodologies in order to analyze
autonomous vehicles in relevant scenarios that are constructed using collected
data. [34] create a framework for the black-box assessment of the safety of
autonomous vehicles. They apply their framework on a commercial autonomous
vehicle system. Our work focuses on aggregate losses over relatively long time
horizons. By considering one-year losses via a conditional loss modeling
approach, we bypass the problem of simulating rare events.

The Actuarial Perspective. The ambitious goal of achieving maximum efficiency
and complete safety through engineering design cannot be realized in reality;
accidents can never be completely excluded, even if residual risks can be kept
very small. Insurance is an instrument to deal with infrequent losses. They
make financial transfer payments in the event of claims. We refer to [30] and
[50] for overviews of mathematical and statistical methods in quantitative
risk management and non-life insurance.

An important task of actuaries is pricing; the premiums of motor insurance
contracts are based on historical claims data collected by insurance
companies. Insurance premiums are calculated based on individual
characteristics of the driver (age, driving experience, etc.) and the vehicle
(type, location, etc.). These tariffs are often complemented by bonus-malus
schemes (see, e.g., [11]) to incentivize more careful driving and prevent
insurance fraud.

Novel pricing approaches use telematics technology (see, e.g., [22] for an
overview). This involves collecting GPS data from vehicles, which can be
analyzed and classified. Machine learning techniques are suitable to process
these large amounts of data. We refer to [17] for a methodological overview.
[46] discuss telematics pricing and use generalized additive models to
interpret the impact of telematics variables on accident frequency. More
recently, [21] develop a usage-based auto insurance product in which a base
premium is dynamically updated based on newly available telematics data,
depending on the policyholder’s driving behavior.

Our approach can be understood as complementary to telematics pricing: Instead
of analyzing driving data to determine the driving behavior of individuals, we
specify the behavior of vehicles as a _driving configuration_ and subsequently
generate driving data and insurance claims. Our approach is in particular
suitable, if novel autonomous technologies are studied. To our knowledge,
there is no other work that develops a microfounded model of traffic accidents
that can be leveraged to study insurance pricing.

##  2 Model Components

Our microfounded simulation model for investigating accident losses is based
on two components:

  1. (i)

At its core is a deterministic microscopic traffic model that realistically
characterizes the motion of vehicles in a traffic system typically represented
by a system of ordinary differential equations.

  2. (ii)

This microscopic traffic model is extended to include the possibility of
random accidents. At random accident times, local traffic data are observed
which characterize the probability distribution of the occurring losses. In
our specific implementation of this general conceptual approach, the SUMO
microscopic traffic simulator is applied to realistically represent and
simulate the underlying traffic scenarios.

In this section, we introduce the notation and construct microscopic traffic
models that integrate random accident occurrences and losses. We also explain
in more detail how SUMO is used.

###  2.1 Microscopic Traffic Networks

We consider a road network that is typically embedded into a two-dimensional
area $A\subseteq\mathbb{R}^{2}$. The network may consist of roads, junctions,
roundabouts, intersections, highways, etc. on which vehicles move. The
collection of all vehicles in the network is denoted by $\mathcal{M}$. Each
vehicle $i\in\mathcal{M}$ is assigned an origin-destination pair
$(O^{i},D^{i})\in A$.

We consider a fixed time horizon $T>0$. Vehicles move over time from their
origin to their destination on a (potentially changing) path. We denote by
$x^{i}(t)$ the position of vehicle $i$ at time $t\in[0,T]$, by
$v^{i}(t)=\frac{\mathrm{d}}{\mathrm{d}t}x^{i}(t)$ their velocity, and by
$a^{i}(t)=\frac{\mathrm{d}}{\mathrm{d}t}v^{i}(t)$ their acceleration. We make
the implicit assumption that vehicles are located in $O^{i}$ until some
release time and remain in $D^{i}$ once reached. Thus, we let
$\mathcal{M}(t)=\left\\{i\in\mathcal{M}\colon
x^{i}(t)\notin\\{O^{i},D^{i}\\}\right\\}$ be those vehicles which are
currently _inside_ the network, i.e., they have left their origin but not
reached their destination, yet. If we only consider vehicles which belong to a
certain group of vehicles (also called a _fleet_) $\Phi\subseteq\mathcal{M}$,
we will write $\mathcal{M}^{\Phi}(t)$. This could, for example, be a fleet of
vehicles with the same driving characteristics.

At the core of many microscopic traffic models are _car-following models_
111Prominent examples include the Intelligent Driver Model ([44]), the Optimal
Velocity Model ([3] and [4]), and the Krauß model ([24]).. These determine the
acceleration behavior of an individual vehicle $i$ along its path on the basis
of information on the positions and velocities of the vehicles, typically in a
neighborhood of $i$, and the properties of the system. Often, only the
preceding vehicle on the road is relevant, and the acceleration of $i$ is
constructed such that vehicles move forward while maintaining a minimal
distance. Through specific choices more complex traffic scenarios (e.g.,
intersections, overtaking) can still be represented in such a manner.
Mathematically, car-following models correspond to systems of coupled ordinary
differential equations.

Traffic State. We denote by
$\gamma(t)=\left(x^{i}(t),v^{i}(t),a^{i}(t)\right)_{i\in\mathcal{M}}$ the
state of the traffic system at time $t$. It records the position, velocity,
and acceleration of any vehicle. The evolution of the traffic system over time
is depicted by the (high-dimensional) trajectory $t\mapsto\gamma(t)$.

Macroscopic traffic statistics aggregate these microscopic data. Typical
examples include traffic flow (number of vehicles that pass a certain point
per time unit), traffic density (number of vehicles per length unit), and
average speed. These measures quantify the performance of traffic systems.

Local Traffic Conditions. In order to model the occurrence of accidents
depending on local traffic conditions, we partition $A$ in regions. More
precisely, we partition $A$ into a finite number of disjoint sets
$A_{r}\subseteq A$ such that $A=\bigcup_{r=1}^{R}A_{r}$ and $R\in\mathbb{N}$.
We call the elements $A_{r}$ of the partition a _traffic module_.

We let $\mathcal{M}_{r}(t)=\left\\{i\in\mathcal{M}(t)\colon x^{i}(t)\in
A_{r}\right\\}\subseteq\mathcal{M}(t)$ denote those vehicles that are in
$A_{r}$ at time $t$ (with $\mathcal{M}^{\Phi}_{r}(t)$ defined in analogy). The
local traffic state of the module is
$\gamma_{r}(t)=(x^{i}(t),v^{i}(t),a^{i}(t))_{i\in\mathcal{M}_{r}(t)}$. Key
local traffic characteristics (density, flow, speed, etc.) can then be
expressed as functions of $\gamma_{r}(t)$ and its evolution over small time
windows.

###  2.2 Microscopic Traffic Model With Accident Losses

So far, the evolution of the traffic system is a deterministic function of
time. The advantage of microscopic models is that they enable a detailed
simulation of traffic systems. The driving behavior of the vehicles can be
varied, likewise their number and paths, road conditions, etc. in order to
generate many different scenarios. Such models provide a detailed picture,
similar to digital twins of reality, and can be used to analyze potential
future traffic systems or to understand the impact of new technologies.

We consider a finite collection of different traffic scenarios
$\gamma^{k}:=(\gamma^{k}(t))_{t\in[0,T]}$ with $k\in\\{1,2,\dots,K\\}$ for a
short time horizon $T>0$. The aim is to analyze characteristics of traffic
over the long time horizon $NT$, e.g., one year, for some large
$N\in\mathbb{N}$; this is modeled by a finite sequence of traffic scenarios
$(k_{1},k_{2},\dots,k_{N})\in\\{1,2,\dots,K\\}^{N}$. The $N$ subintervals of
length $T$ are called time buckets. We will be interested in quantities
aggregated or averaged over the whole time horizon $NT$. Examples include the
average traffic flow, the total number of accidents, the aggregate losses due
to accidents, etc. These quantities do not depend on the order of the traffic
scenarios during this time period, but only on their number of occurrences.

We denote by $\mu^{k}$ the number of occurrences of scenario $k$ divided by
$N$, i.e., the relative frequency of this traffic scenario over the considered
time horizon $NT$. The vector $\mu=(\mu^{1},\mu^{2},\dots,\mu^{K})^{\top}$
lies in the simplex
$\Delta^{K-1}=\\{x\in\mathbb{R}^{K}_{+}:\sum_{k=1}^{K}x^{k}=1\\}$. We assume
that $\mu$ is not deterministic, but a random variable. This is to account for
the fact that the relative frequencies of traffic scenarios fluctuate over
different years due to varying weather conditions, random changes in traffic
demand, or other factors. From a mathematical point of view, this construction
leads to a mixture model with exogenous factor $\mu$.

Accident Occurences. We now introduce our traffic accident model that will
permit an analysis of aggregate losses and corresponding insurance contracts.
The likelihood of the occurrence of an accident is modeled as a function of
the traffic scenario. We consider two specifications, a Binomial and a Poisson
model.

  1. (i)

_Binomial Model._ Accidents are rare events. For a given traffic scenario $k$,
we assume that the probability $p^{k}$ of an accident is close to zero. This
probability may, of course, depend on the evolution of the traffic scenario,
i.e., on the path $t\mapsto\gamma^{k}(t)$, and we will discuss concrete
specifications later. Given a realization of $\mu$, accidents are assumed to
be independent across time buckets. This implies that traffic scenario $k$
occurs for $N\mu^{k}$ time buckets corresponding to a duration of $NT\mu^{k}$,
and the number of accidents $C^{k}$ during this period has a conditional
Binomial distribution with parameters $p^{k}$ and $N\mu^{k}$:

$C^{k}\mid\mu\sim{\rm Bin}(p^{k},N\mu^{k}).$

  2. (ii)

_Poisson Model._ An alternative model assumes that accidents occur at random
times with a distribution governed by an intensity $\lambda^{k}/T$ that
depends on the traffic scenario $k$. More specifically, the number of
accidents $C^{k}$ during the period governed by scenario $k$ of duration
$NT\mu^{k}$ is conditionally Poisson distributed with parameter
$\lambda^{k}N\mu^{k}$:

$C^{k}\mid\mu\sim{\rm Poiss}(\lambda^{k}N\mu^{k}).$

Accident Losses. Loss sizes conditional on the occurrence of accidents are
assumed to be independent across traffic scenarios and across time buckets. We
assume that the conditional loss distribution with conditional distribution
function $F^{k}$ depends only on the traffic scenario $k$. We will discuss
examples below. Random total losses over the considered time horizon $NT$ are
equal to

$L\;=\;\sum_{k=1}^{K}\;\sum_{c=1}^{C^{k}}\;X^{k}_{c}$

where the random variables $X^{k}_{c}$, $k=1,2,\dots,K$, $c\in\mathbb{N}$, are
independent and $X^{k}_{c}\sim F^{k}$, $c\in\mathbb{N}$, for any $k$.

Concrete Specifications. Microscopic traffic models are experimental
environments that allow to simulate the behavior of systems where no real data
are yet available. Traffic planning can be supported by such models, and the
impact of new technologies can be tested in a counterfactual analysis. Here,
we specify the general principles how accident occurrences and losses can be
based on microscopic traffic models. An implementation will in this paper be
based on SUMO, see Section 2.3, but could also rely on any other suitable
traffic model.

Initially, $K$ traffic scenarios need to be selected as a basis for the model.
While running any deterministic traffic scenario $k$ over the time window
$[0,T]$, information can be extracted about the traffic states
$\gamma_{r}^{k}$ in each module $r\in\\{1,2,\dots,R\\}$. In SUMO typically not
complete data on the whole paths are extracted, but only selected information
at loop detectors in the network that are part of the implementation.

In reality, the likelihood of accidents typically increases with higher
traffic density and higher velocities, ceteris paribus. Also the distribution
of losses is influenced by quantities of this type. Examples are described in
Section 4. This allows a computation of $p^{k}$ and $\lambda^{k}$ as a
function of the data. Using the data associated with the modules, we may
specify probabilities and intensities for the modules such that
$p^{k}=\sum_{r=1}^{R}p^{k}_{r}$ and
$\lambda^{k}=\sum_{r=1}^{R}\lambda^{k}_{r}$. In the Binomial model,
$p^{k}_{r}/p^{k}$ is the conditional probability that the accident is in
module $r$ given that an accident occurs. In the Poisson model, the
intensities $\lambda^{k}_{r}$, $r=1,2,\dots,R$, determine the accident times
for each module. The resulting sequence of random times in the whole traffic
system possesses the intensity $\lambda^{k}$. Conversely, if we first simulate
random times with intensity $\lambda^{k}$ and then randomly choose a
corresponding module with probability $\lambda^{k}_{r}/\lambda^{k}$ in a
second step, the random times associated with each module $r$ possess
intensity $\lambda^{k}_{r}$. Both procedures produce a number of accidents
$C^{k}$ that occur during the period governed by scenario $k$.

The distributions of losses given a single random event will be chosen as
follows. For each traffic scenario, we consider a collection of distribution
functions $(F^{k,\psi})_{\psi\in\Psi}$ where $\psi$ corresponds to data that
may be extracted from the traffic simulation. In order to do so, we uniformly
simulate a random time in $[0,T]$ and extract at this time the data from
scenario $k$ that determine $\psi$. The resulting distribution $F^{k}$ is a
mixture of the distributions $(F^{k,\psi})_{\psi\in\Psi}$. The mixing
distribution is derived on the basis of the traffic data of scenario $k$ that
are generated from our microscopic traffic model.

Insurance Contracts & Statistical Functionals. The microscopic traffic model
with accidents will be the basis for the generation of aggregate losses. We
study statistical functionals and insurance contracts. The analysis will be
based on Monte Carlo simulations, but we also compare approximation techniques
that we describe in Section 3.

The focus is on functions of aggregate losses $L$. Letting
$h\colon\mathbb{R}\to\mathbb{R}$ be an increasing function, we analyze $h(L)$.
In particular, we investigate the following functions corresponding to three
types of insurance coverage: $h(x)=x$ (full coverage),
$h(x)=\max(x-\theta,0)$, $\theta\geq 0$ (constant deductible),
$h(x)=\min(x,\theta)$, $\theta\geq 0$ (stop loss). In each case, we evaluate
various statistical functionals:

  1. (i)

_Expectation._ $\mathbb{E}(h(L))$,

  2. (ii)

_Variance._ $\mathrm{Var}(h(L))=\mathbb{E}((h(L))^{2}))-\mathbb{E}(h(L))^{2}$,

  3. (iii)

_Skewness._
$\varsigma_{h(L)}=\frac{\mathbb{E}\left[(h(L)-\mathbb{E}(h(L)))^{3}\right]}{(\mathrm{Var}(h(L)))^{3/2}}$,

  4. (iv)

_Value-at-Risk._ $\mathrm{VaR}_{p}(h(L))=\inf\\{x\in\mathbb{R}\colon
P(h(L)\leq x)\geq p\\}$,

  5. (v)

_Expected Shortfall._
$\mathrm{ES}_{p}(h(L))=\frac{1}{1-p}\int_{p}^{1}\mathrm{VaR}_{q}(h(L))\,\mathrm{d}q$.

These functionals allow also the computation of insurance premiums on the
basis of premium principles such as the expectation principle, the variance
principle, or the standard deviation principle.

###  2.3 Traffic Scenarios in SUMO

####  2.3.1 A Brief Overview

A state-of-the-art open source software that allows us to generate traffic
scenarios is SUMO, “Simulation of Urban MObility”. A reference publication on
SUMO is [25]; in addition a detailed user documentation can be found
online222See sumo.dlr.de/docs/index.html.. Freely available since 2001, SUMO
was originally developed by the German Aerospace Center and extended by an
active research community. It allows for a plethora of modeling choices at
different levels and has been successfully applied to tackle many important
research questions addressing333See eclipse.org/sumo/about/., e.g., traffic
light optimization, routing, traffic forecasting, and autonomous driving.

In the following, we give a short overview. At its core, SUMO is a software
which generates a traffic scenario $\gamma=(\gamma(t))_{t\in[0,T]}$ from a
given set of input files:

  1. (i)

_Network File._ In SUMO, a traffic network is described by a directed graph
whose nodes represent intersections and edges roads. All nodes and edges have
attributes including positions, shapes, speed limits, traffic regulation, etc.
As an example, the city of Wildau is represented as a SUMO network in Figure
1.

  2. (ii)

_Route File._ Vehicles are generated on the basis of _traffic demands_ between
origins and destinations. Routes can either be defined for each vehicle as a
_trip_ or as recurring _flows_ along a specific path. If only origin and
destination are provided, the corresponding route is computed444The problem of
allocating traffic demand to routes in a network is referred to as the traffic
assignment problem. A standard reference is [40]. when the vehicle enters the
system.

The _vehicle type_ determines the microscopic characteristics of the vehicle
such as the governing car-following model, driving parameters (e.g., maximal
speed, maximal acceleration, time headway), size, color, etc. By default,
vehicles are passenger cars. Other modes, such as pedestrian, bicycle, or
truck can also be selected.

  3. (iii)

_Additional Files._ Further components are specified in additional files. An
important example are induction loop detectors. These collect time series data
on aggregate traffic statistics by counting the vehicles which pass a certain
position during a short time interval.

The collection of input files determines the traffic evolution, also called
the _SUMO scenario_. The computation can be executed either as a command line
application or with a GUI that visualizes the movement of the vehicles through
the network over time.

Data Extraction. One particularly appealing extension of SUMO is the “Traffic
Control Interface”, TraCI (see [49]). TraCI provides online access to the
microscopic traffic simulation and permits, at each time step, through a
comprehensive list of commands555A detailed description can be found at
sumo.dlr.de/docs/TraCI.html. to retrieve data and to change the states of
objects such as vehicles, roads, traffic lights, etc. Available in standard
programming languages666Our case studies are based on the Python
implementation., TraCI yields easy access to SUMO without the need to modify
the underlying code. We use TraCI to extract microscopic data on positions,
velocities, and accelerations of randomly selected vehicles.

Figure 1: SUMO network of Wildau.

####  2.3.2 Generation of Traffic Scenarios

To represent traffic in a given area over a longer time horizon (e.g.,
$NT=1\,\mathrm{year}$), we generate a diverse collection of traffic scenarios
$\gamma^{1},\dots,\gamma^{K}$ of duration $T$ in SUMO by varying the input
files. Traffic over a longer time horizon is represented by a random
composition of these traffic scenarios. Our general construction has already
been discussed in Section 2.2.

SUMO Scenario. SUMO provides tools that facilitate the creation of input
files, e.g., the graphical network editor _netedit_ 777See
sumo.dlr.de/docs/Netedit/index.html. that visualizes a SUMO scenario and
allows to modify its properties. In practice, network files are typically
imported from other data sources888We refer to
sumo.dlr.de/docs/index.html#network_building for an overview on network
generation.. For example999See
sumo.dlr.de/docs/Networks/Import/OpenStreetMap.html., one can build a real-
world traffic network in SUMO from OpenStreetMap data by selecting an area
from a map. The route file specifies the trips of the vehicles and the
definition of the general vehicle types with their microscopic
characteristics. First, there are several options to generate trips in
SUMO101010We refer to
sumo.dlr.de/docs/Demand/Introduction_to_demand_modelling_in_SUMO.html for an
overview on demand modeling in SUMO.. These can be obtained from empirical
data in the form of traffic counts, imported to SUMO as origin-destination
matrices, or modeled via ad-hoc choices, e.g., using _netedit_. Second, each
vehicle is associated to a vehicle type specified on the basis of a
comprehensive list of attributes111111See
sumo.dlr.de/docs/Definition_of_Vehicles%2C_Vehicle_Types%2C_and_Routes.html..
The corresponding values can be set manually in the route file or accessed and
modified via _netedit_. In the absence of detailed traffic data for
calibration, SUMO offers an _activity-based demand generation_ 121212See
sumo.dlr.de/docs/Demand/Activity-based_Demand_Generation.html. which deduces
traffic demand from general assumptions on the structure of the population
(inhabitants, households, etc.) in the considered area. The tool _activitygen_
131313See sumo.dlr.de/docs/Demand/activitygen.html. automates the process and
produces an artificial route file.

SUMO admits a large variety of modeling choices. Tailored to the needs of the
modeler, a highly detailed SUMO scenario can be constructed. Our case studies
will be based on publicly available SUMO scenarios141414See
sumo.dlr.de/docs/Data/Scenarios.html.; these consist of network and route
files which are calibrated to real-world cities.

Varying Traffic Conditions. The input files need to be constructed in such a
way that they reflect varying traffic conditions over longer time periods.
This includes weather conditions, variation of traffic demand, and other
factors.

[29] review empirical studies on the impact of adverse weather conditions on
traffic. These may induce i) lower traffic demand, ii) higher risk of
accidents, and iii) modified driving behavior. Based on empirical findings,
[41] implement reduced velocities due to rainfall. In [48], the idea of
introducing into SUMO a friction parameter per road is discussed. Traffic
scenarios under adverse weather conditions can be captured by suitable driving
parameters in the route file (e.g., by variation of maximal speed, maximal
acceleration, etc.), and this may be combined with a weather-dependent model
of the occurrence and the severity of accidents.

Traffic demand is traditionally estimated from traffic counts. We refer to [5]
for an overview. With increasing data availability, the estimation can be
enhanced by floating car data (cf., e.g., [32]), i.e., data generated from
vehicles over time as they are driving. Traffic demand varies over time, but
patterns reoccur over longer time horizons (see, e.g., [43]). Demand depends
on the considered traffic network. Weekdays differ from days on the weekend;
peaks in demand occur at common commute times. Rush hours are spatio-temporal
phenomena that can be analyzed in detail (see, e.g., [51]).

To reflect the heterogeneity of traffic scenarios, two options are available
in SUMO: i) A variety of route files is generated that is consistent with the
desired modeling granularity. This process can be automatized via an
additional program, a _route file generator_ , that produces route files with
the desired characteristics. ii) Another option is to select a medium time
horizon (e.g., $T_{\mathrm{SUMO}}=24\,\mathrm{h}$) with a corresponding route
file that depicts varying traffic demand over time. From the generated SUMO
scenario, a selection of small time horizon scenarios (e.g.,
$T=1\,\mathrm{min}$) can be efficiently generated by utilizing SUMO’s option
to save the state of the running simulation at a priori specified times and
load these later.

Besides weather and traffic demand, many other factors influence the traffic
dynamics. [47] discusses the representation of autonomous vehicles in SUMO.
[27] utilize SUMO to study control transition, i.e., selected safety critical
situations where the human driver needs to take over control from an
autonomously driving vehicle. [39] study the impact of wildlife on traffic, an
issue that is also relevant in the context of traffic accidents.

##  3 Evaluation Methods

The accident losses $L$ can be simulated using Monte Carlo methods. The
simulations may be used to estimate the value of statistical functionals and
to price insurance products. We will briefly describe the Monte Carlo methods.
In addition, on the basis of the Binomial model, we construct a Gaussian
approximation to $\mathbb{E}(h(L))$ where the function $h$ corresponds to the
three types of insurance coverage that we consider: full coverage, constant
deductible, and stop loss. This allows a numerical evaluation similar to [16]
and [12]. The latter paper provides a correction term derived by Stein’s
method that we will exploit in our application.

###  3.1 Monte Carlo Methods

The Monte Carlo simulation of $L$ requires sampling the number of accidents
$C^{k}$ in either the Binomial or Poisson model and sampling the independent
conditional losses $X^{k}_{c}$, $c\in\mathbb{N}$, for each traffic scenario
$k=1,2,\dots,K$ from the corresponding distribution $F^{k}$. These tasks can
be performed separately, but require both a prior evaluation of the
microscopic traffic model.

  1. (i)

_Prior Evaluation of Traffic Model._ For each traffic scenario $k$ a single
run delivers data that are the basis for a computation of, respectively, the
accidents probabilities $p^{k}$ and intensities $\lambda^{k}$ as well as the
corresponding values $p^{k}_{r}$ and $\lambda^{k}_{r}$ on the level of the
modules $r=1,2,\dots,R$.

  2. (ii)

_Number of Accidents._ Sampling from $\mu$ and using the results of the prior
evaluation allows to sample the number of accidents $C^{k}$ for each traffic
scenario $k$ in both the Binomial and Poisson model.

  3. (iii)

_Conditional Accident Losses._ Based on the precomputed values of the accident
probabilities respectively the accident intensities, we simulate for each
traffic scenario $k$ the random locations and times of accidents. These data
can be stored. For these locations and times, traffic data $\psi$ are
extracted from an additional single run of traffic scenario $k$. Given $\psi$,
the losses are generated according to conditional loss distributions
$F^{k,\psi}$ as described in Section 2.2. Details of the implementation are
explained in Section 4.

These Monte Carlo methods can be flexibly applied to all considered
functionals. In the special cases of expectation and variance, Wald’s equation
can simplify the computation, since the losses $L$ are given in the form of a
collective model.

###  3.2 Gaussian Approximation

Another way to compute $\mathbb{E}(h(L))$ for the considered types of
insurance coverage is a Gaussian approximation, possibly improved by a
correction term. A Gaussian approximation can easily be motivated within the
Binomial model. For $N$ sufficiently large, the distribution of $L$ given
$\mu$ is approximately normal, implying that $L$ is a mean-variance-mixture of
Gaussian distributions. This is an important structural insight from this
approximation.

In this section, we condition on $\mu$, i.e., suppose that $\mu$ is fixed and
given. The general results for $\mu$ random are then a corollary by
considering suitable mixtures according to the distribution of $\mu$. The
total random losses in the Binomial model can be rewritten as

$L\;=\;\sum_{k=1}^{K}\;\sum_{c=1}^{N\mu^{k}}\;\mathbf{1}^{k}_{c}\cdot
X^{k}_{c}$

where the random variables $\mathbf{1}^{k}_{c}$, $X^{k}_{c}$, $k=1,2,\dots,K$,
$c\in\mathbb{N}$, are independent, $X^{k}_{c}\sim F^{k}$, and
$\mathbf{1}^{k}_{c}$ are Bernoulli random variables taking the value $1$ with
probability $p^{k}$ and the value $0$ otherwise, $c\in\mathbb{N}$, for any
$k$. Setting151515The estimation of $m^{k}$, ${\left(\sigma^{k}\right)}^{2}$,
and ${\left(\zeta^{k}\right)}^{3}$ requires the simulation of the random
variables $X^{k}_{c}$, $c\in\mathbb{N}$. The independent terms
$\mathbf{1}^{k}_{c}$, $c\in\mathbb{N}$, factor out, are idempotent and have
known expectation $p^{k}$.

$Y^{k}_{c}\;:=\;\mathbf{1}^{k}_{c}\cdot X^{k}_{c},\quad
m^{k}\;:=\;\mathbb{E}(Y^{k}_{c}),\quad{\left(\sigma^{k}\right)}^{2}\;:=\;\mathbb{E}\left([Y^{k}_{c}-m^{k}]^{2}\right),\quad{\left(\zeta^{k}\right)}^{3}:=\;\;\mathbb{E}\left([Y^{k}_{c}-m^{k}]^{3}\right),$

$k=1,2,\dots,K$, $c\in\mathbb{N}$, a classical normal approximation of $L$ is
$\sum_{k=1}^{K}N\mu^{k}m^{k}+Z$ with

$Z\quad\sim\quad\mathcal{N}\left(0,\;\sum_{k=1}^{K}N\mu^{k}{\left(\sigma^{k}\right)}^{2}\right).$

We focus on three types of insurance coverage, $h(x)=x$ (full coverage),
$h(x)=\max(x-\theta,0)$, $\theta\geq 0$ (constant deductible),
$h=\min(x,\theta)$, $\theta\geq 0$ (stop loss), and obtain an approximation
$\mathbb{E}(h(Z))$ of $\mathbb{E}(h(L))$ in each of these cases.

On the basis of Stein’s method161616For an overview on Stein’s method we refer
to [9] and [42]., [12] suggest correction terms in order to improve the
approximation, i.e., the approximation
$\mathbb{E}\left(h\left(\sum_{k=1}^{K}N\mu^{k}m^{k}+Z\right)\right)$ is
replaced by the corrected approximation

$\mathbb{E}\left(h\left(\sum_{k=1}^{K}N\mu^{k}m^{k}+Z\right)\right)\;+\;C_{h}.$

The correction $C_{h}$ depends on the degree of smoothness of the derivatives
of the function $h$ and thus differs171717See Theorem 3.1 and Proposition 3.6
in [12]. for the three types of coverage. We define

$d_{1}\;=\;\sum_{k=1}^{K}N\mu^{k}m^{k},\quad
d_{2}\;=\;\sum_{k=1}^{K}N\mu^{k}{\left(\sigma^{k}\right)}^{2},\quad
d_{3}\;=\;\sum_{k=1}^{K}N\mu^{k}\left(\zeta^{k}\right)^{3},\quad\tilde{h}(x)\;=\;h\left(x+d_{1}\right).$

We obtain the following correction terms:

  1. (i)

_Full Coverage._ In the case of full coverage, the correction term of [12]
disappears. In general, if $h$ is some Lipschitz function with bounded third
derivative, the correction term equals
5cm
$C_{h}\;=\;\frac{d_{3}}{2d_{2}^{2}}\cdot\mathbb{E}\left(\left\\{\frac{Z^{2}}{3d_{2}}-1\right\\}Z\tilde{h}(Z)\right).$

  2. (ii)

_Constant Deductible._ 5cm $C_{h}\;=\;\frac{(\theta-
d_{1})d_{3}}{6d_{2}}\cdot\frac{1}{\sqrt{2\cdot\pi\cdot
d_{2}}}\cdot\exp\left\\{-\frac{(\theta-d_{1})^{2}}{2\cdot d_{2}}\right\\}$

  3. (iii)

_Stop Loss._ A stop loss $x\mapsto\min(x,\theta)$ can be written as the
difference between full coverage $x\mapsto x$ and a constant deductible
$x\mapsto\max(x-\theta,0)$. This implies that the correction term for a
constant deductible appears with a negative sign in this case.

The advantage of the (corrected) Gaussian approximation in comparison to pure
Monte Carlo is that, once the numbers $m^{k}$,
${\left(\sigma^{k}\right)}^{2}$, and $\left(\zeta^{k}\right)^{3}$ have been
computed for each traffic scenario $k=1,2,\dots,K$, no further data need to be
stored or sampled in order to compute $\mathbb{E}(h(L))$. The approximate
representation of the distribution of $L$ as a mean-variance-mixture is a
considerable simplification.

##  4 Application

We illustrate the application of our microscopic traffic model with accidents
on the basis of a publicly available SUMO scenario of a real city.

###  4.1 SUMO Scenario & Accident Data

Wildau is a small German city of approximately 10,000 inhabitants, located
around 30 km south-east of the capital Berlin. A SUMO model of the city was
developed within a study project by the Technical University of Applied
Sciences Wildau and is publicly available181818See github.com/DLR-TS/sumo-
scenarios/tree/main/Wildau..

SUMO Scenario. The implemented road network is visualized in Figure 1. It is
specified using 646 nodes connected by 1,426 edges. The city itself is crossed
by the railway; the tracks are represented by the gray line. Vehicles are
calibrated from real traffic counts. The original scenario has a duration of
$7,010\,\mathrm{s}$. Empty in the beginning, vehicles enter the system with a
peak of approximately $240$ vehicles that drive simultaneously. In total,
$2502$ vehicles are generated.

In the following section, we describe in detail how we adjust this SUMO
scenario to obtain suitable ingredients for our case studies. This yields a
collection of traffic scenarios that allow us to compare the effects of
different driving characteristics and fleet sizes on the total loss and
related insurance premiums.

Varying Traffic Conditions. For different collections of model parameters
representing different traffic systems we generate adjusted SUMO scenarios.
For each choice of parameters we proceed as follows to produce $K=100$ traffic
scenarios of length $T=60\,\mathrm{s}$. Traffic scenarios $k=1,2,\dots,50$
correspond to selected time intervals from the SUMO scenario. Traffic
scenarios $k=51,52,\dots,100$ represent higher traffic volumes. They are
generated by replacing the original route file by a route file that consists
of two copies of the original route file. This simple procedure generates a
larger amount of vehicles along the original paths. The traffic scenarios are
again selected time intervals from the corresponding SUMO scenario.

To represent the full year, we set $N=365\cdot 24\cdot 60=525,600$. We need to
specify the random vector $\mu=(\mu^{1},\dots,\mu^{K})^{\top}$ describing the
number of occurrences of the individual traffic scenarios divided by $N$. For
the purpose of illustration, we specifically assume that two probability
measures $\nu_{g}$, $\nu_{b}$ are given on $\\{1,2,\dots,K\\}$ which
approximately correspond to the relative frequencies of traffic scenarios in
two prototypical years $y=g,b$. In addition, we suppose that the type $y$ of
the current year is random where both values $g$ and $b$ have probability
$1/2$. Given $y$, we generate $\mu=(\mu^{1},\dots,\mu^{K})^{\top}$ from a
multinomial distribution corresponding to $\nu_{y}$. That is, for all time
buckets $n=1,2,\dots,N$, a traffic scenario $k$ is chosen independently from
the distribution $\nu_{y}$ on $\\{1,2,\dots,K\\}$. Dividing the number of
occurrences of a scenario $k$ by $N$, one obtains its random relative
frequency $\mu^{k}$ for any $k=1,2,\dots,K$. In our case study, the
distribution $\nu_{g}$ corresponds to lower traffic densities on average,
while $\nu_{b}$ is associated with higher traffic densities, i.e., we set

$\nu_{g}\;:=\;\left\\{\begin{array}[]{ll}1/75,&k=1,2,\dots,50,\\\
1/150,&k=51,52,\dots,100\end{array}\right.,\quad\nu_{b}\;:=\;\left\\{\begin{array}[]{ll}1/150,&k=1,2,\dots,50,\\\
1/75,&k=51,52,\dots,100.\end{array}\right.$

Accident Data. The German Accident Atlas191919The data are provided by the
_Statistische Ämter des Bundes und der Länder_ under the “Data licence Germany
– attribution – Version 2.0” and can be accessed via
unfallatlas.statistikportal.de/. depicts the locations of all police-reported
accidents _involving personal damage_ that occurred within one year. In 2020,
within the modeled area of Wildau (approximately) 48 accidents were
registered. There are also aggregate statistics for Germany for all police-
reported accidents. In 2020, approximately $11.8\,\mathrm{\%}$ of all road
accidents involved personal damage202020See the online resource:
destatis.de/EN/Themes/Society-Environment/Traffic-Accidents/Tables/accidents-
registered-police.html. . We use an estimate of
$\bar{c}_{\mathrm{year}}=48/11.8\,\mathrm{\%}\approx 407$ accidents for
calibration purposes.

###  4.2 Model Specification

Our goal is to analyze accident losses for a fleet $\Phi$ over the time
horizon of one year.

Fleet Definition. In the Wildau scenario, vehicles are defined using repeated
flows from origins to destinations. Passenger cars (next to trucks and the
train) are defined via 90 different flows. Each passenger car belongs to the
same vehicle type with fixed driving characteristics.

To introduce a fleet $\Phi$ of vehicles whose driving characteristics we can
vary, we define a new vehicle type $\Phi$ and construct corresponding new SUMO
scenarios. Fixing a fraction $\rho^{\Phi}\in[0,1]$ of vehicles belonging to
$\Phi$, we retain approximately $1-\rho^{\Phi}$ of the existing flow
definitions and modify $\rho^{\Phi}$ of the flow definitions suitably in order
to model the fleet. In our case studies, we consider
$\rho^{\Phi}=10\,\mathrm{\%},\leavevmode\nobreak\
50\,\mathrm{\%},\leavevmode\nobreak\ 90\,\mathrm{\%}$.

Driving Configuration. Vehicles in a fleet $\Phi$ are of the same type.
Various characteristics can be varied in SUMO; we focus on _maximal speed_
$v_{\mathrm{max}}$, _maximal acceleration_ $a_{\mathrm{max}}>0$, and _time
headway_ $\zeta>0$. The time headway is the distance which is kept to the
preceding vehicle measured in time, i.e., a velocity weighted safety distance.
We refer to a fixed selection of driving characteristics as a _driving
configuration_.

In our case studies, we will vary the driving configuration for all vehicles
in the fleet $\Phi$ and keep all other vehicles as originally
introduced212121We use the implementation of an Intelligent Driver Model
without any speed deviation that does not include further random effects. We
refer to sumo.dlr.de/docs/Simulation/Randomness.html for any random effects in
SUMO.. A driving configuration of vehicles in fleet $\Phi$ is denoted by
$\xi=(v_{\mathrm{max}},a_{\mathrm{max}},\zeta)$. Specifically, we
consider222222The specific choices are inspired by the following
considerations: The implemented road speed limit in Wildau is 50 km/h which is
approximately 13.9 m/s. Vehicles in the original Wildau scenario have a
maximal acceleration of 0.8 m/s2, while SUMO’s default value is 2.6 m/s2.
Similarly, SUMO’s default time headway is 1.0 s.:

$\displaystyle\xi^{1a}$ $\displaystyle=(5\,\mathrm{m/s},\leavevmode\nobreak\
0.8\,\mathrm{m/s^{2}},\leavevmode\nobreak\
3.0\,\mathrm{s}),\leavevmode\nobreak\
\xi^{2a}=(10\,\mathrm{m/s},\leavevmode\nobreak\
0.8\,\mathrm{m/s^{2}},\leavevmode\nobreak\
2.0\,\mathrm{s}),\leavevmode\nobreak\
\xi^{3a}=(15\,\mathrm{m/s},\leavevmode\nobreak\
0.8\,\mathrm{m/s^{2}},\leavevmode\nobreak\ 1.0\,\mathrm{s}),$
$\displaystyle\xi^{1b}$ $\displaystyle=(5\,\mathrm{m/s},\leavevmode\nobreak\
2.6\,\mathrm{m/s^{2}},\leavevmode\nobreak\
3.0\,\mathrm{s}),\leavevmode\nobreak\
\xi^{2b}=(10\,\mathrm{m/s},\leavevmode\nobreak\
2.6\,\mathrm{m/s^{2}},\leavevmode\nobreak\
2.0\,\mathrm{s}),\leavevmode\nobreak\
\xi^{3b}=(15\,\mathrm{m/s},\leavevmode\nobreak\
2.6\,\mathrm{m/s^{2}},\leavevmode\nobreak\ 1.0\,\mathrm{s}).$

The configurations $1,2,3$ increase in terms of “aggressiveness” from driving
slowly with a large headway to fast with a small headway – with two options
$a$ and $b$ for the maximal acceleration.

Accident Occurrence. The best estimate for the total number of accidents in
Wildau is $\bar{c}_{\mathrm{year}}\approx 407$. From this, we derive a uniform
and a non-uniform accident occurrence model. In both cases, we specify
accident probabilities $p^{1,k}$ and $p^{2,k}$ for the binomial model as well
as accident intensities $\lambda^{1,k}$ and $\lambda^{2,k}$ for the Poisson
model.

  1. (i)

_Uniform Accident Occurrence._ Assuming that accidents occur uniformly over
the year, we obtain a probability per time bucket of an accident in the system
of $p^{1}=\bar{c}_{\mathrm{year}}/N\approx 7.7\cdot 10^{-4}$. This is the
accident probability that we allocate to each traffic scenario $k$. We also
suppose that accidents occur uniformly across all vehicles in the system. This
implies that the probability that any accident occurs in scenario $k$ within
the fleet $\Phi$ is

$p^{\Phi,1,k}\;=\;\rho^{\Phi}\cdot p^{1},\quad k\in\\{1,\dots,K\\}.$

This probability is used in the Binomial model. For the Poisson model we set
$\lambda^{\Phi,1,k}=p^{\Phi,1,k}$, $k=1,2,\dots,K$, since the intensity
approximately equals the probability of an accident per time bucket.

In the case of uniform accident occurrence, we do not consider any spatial
variations of the likelihood of accidents due to different traffic conditions.
This means that we do not distinguish any modules, i.e., we set $R=1$.

  2. (ii)

_Non-Uniform Accident Occurrence._ In reality, the likelihood of accidents
depends on external factors such as weather and local traffic conditions,
e.g., the velocity of vehicles and traffic density. The quantities vary
spatially and over time.

From SUMO runs, we obtain for each traffic scenario $k=1,2,\dots,K$ and each
module $r=1,2,\dots,R$ pairs $(d_{r}^{k},\bar{v}_{r}^{k})$ on the average
density and velocity. These statistics can be computed in SUMO, for example,
from data that are obtained at induction loop detectors which are placed
within the modules; as a proxy for density, we extract the _occupancy_ of the
loop detector, i.e., the fraction of time which it is occupied by a vehicle.
For $r=1,\dots,R$, we choose benchmark values $d^{*}_{r}$ and
$\bar{v}^{*}_{r}$ for the density and velocity and specify occurrence
probabilities and intensities that vary spatially and over time:

$\lambda^{\Phi,2,k}_{r}\;=\;p^{\Phi,2,k}_{r}\;:=\;\frac{p^{\Phi,1,k}}{R}\cdot\frac{\bar{v}_{r}^{k}}{\bar{v}_{r}^{*}}\cdot\frac{d_{r}^{k}}{d_{r}^{*}}\cdot
e^{-(\zeta^{\Phi}-1)},\quad k\in\\{1,\dots,K\\},\quad r\in\\{1,\dots,R\\}.$

The last term refers to deviations of the time headway from SUMO’s default
value of $1.0\,\mathrm{s}$: a larger time headway is associated with less
risky driving. We set $p^{\Phi,2,k}=\sum_{r=1}^{R}p^{\Phi,2,k}_{r}$ and
$\lambda^{\Phi,2,k}=\sum_{r=1}^{R}\lambda^{\Phi,2,k}_{r}$.

In our case studies, we will consider a grid of $R=4$ modules and compute
$d_{r}^{k}$ and $\bar{v}_{r}^{k}$ as averages over measurements from $10$
induction loop detectors that are placed in each module (see also Figure 2).
We use the scenario averages $d^{*}_{r}=\frac{1}{K}\sum_{k=1}^{K}d_{r}^{k}$
and $\bar{v}_{r}^{*}=\frac{1}{K}\sum_{k=1}^{K}\bar{v}_{r}^{k}$. If
$\sum_{k=1}^{K}\mathbb{E}(\mu^{k})\bar{v}_{r}^{k}=\bar{v}_{r}^{*}$,
$\sum_{k=1}^{K}\mathbb{E}(\mu^{k})d_{r}^{k}=d_{r}^{*}$, and $\zeta^{\Phi}=1$,
then we essentially recover on average the case of uniform accident
occurrence.

Figure 2: Partition of Wildau and Placement of Induction Loop Detectors.

Accident Losses. The distributions $F^{k}$ of accident losses associated with
a traffic scenario $k=1,2,\dots,K$ are constructed on the basis of traffic
data that are extracted from the SUMO runs. The general procedure was
described in Section 2.2; here, we explain the specific implementation that we
use in our case studies.

The likelihood of accident occurrence was discussed in the previous section.
$F^{k}$ is the conditional distribution of an accident loss in traffic
scenario $k$ if an accident occurs. Time in traffic module $k$ is enumerated
by $t\in[0,T]$, and we assume that the time $\tau$ of the accident conditional
on its occurrence is uniformly distributed on $[0,T]$, i.e.,

$\tau\;\sim\;\mathrm{Unif}[0,T].$

We choose a module $\mathcal{R}$ at random in which the accident occurs and
assume, respectively, that

$P(\mathcal{R}=r)\;=\;\frac{p^{\Phi,\cdot,k}_{r}}{p^{\Phi,\cdot,k}},\quad\quad
P(\mathcal{R}=r)=\frac{\lambda^{\Phi,\cdot,k}_{r}}{\lambda^{\Phi,\cdot,k}},\quad\quad\quad
r\in\\{1,\dots,R\\}.$

These ratios depend in the case of non-uniform accident occurrence on the
specific fleet, since the properties of the fleet alter the route file that is
used to generate the SUMO scenarios; this is true, although the multiplicative
terms $\rho^{\Phi}$ appear in both the numerator and denominator and cancel
out.

In the chosen module we pick one or more vehicles at random, and extract from
the traffic scenario data for these vehicles. In our concrete implementation,
we simply choose at time $\tau$ a single vehicle $I$ uniformly at random in
module $\mathcal{R}$, i.e., its conditional distribution is

$I\mid\tau,\mathcal{R}\;\sim\;\mathrm{Unif}(\mathcal{M}_{\mathcal{R}}^{\Phi}(\tau)).$

For the purpose of illustrating our approach, the only data we extract are the
velocities $v^{I}$ of the randomly chosen vehicles that are involved in
accidents. We set $\psi=v^{I}$ and assume that the conditional loss
distribution $F^{k,\psi}$ is known232323We assume that $F^{k,0}$ corresponds
to a Dirac measure in $0$; if
$\mathcal{M}_{\mathcal{R}}^{\Phi}(\tau)=\emptyset$, we set $\psi=0$, resulting
in $0$ losses.. If we denote the distribution of $\psi$ by $\mathcal{L}^{k}$
we obtain the distribution $F^{k}$ as a mixture

$F^{k}\;=\;\int F^{k,\psi}\;d\mathcal{L}^{k}.$

In our case studies, we will assume that $F^{k,\psi}=F^{\psi}$ for all $k$;
however, the mixing distribution $\mathcal{L}^{k}$ will dependend on the
traffic scenario $k$. We consider the following examples for $F^{\psi}$:

  1. (i)

_Gamma Distribution._ We define distributions with varying levels of
_dispersion_ 242424A measure for the dispersion of a random variable $X$ is
the coefficient of variation defined by
$c_{v}=\sqrt{\mathrm{Var}(X)}/\mathbb{E}(X)$.. For a given coefficient of
variation $c_{v}\in\\{1/2,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ 2\\}$,
we choose

$F^{\psi}\;=\;\Gamma\left(\frac{1}{c_{v}^{2}},\;\frac{1}{c_{v}^{2}\psi^{2}}\right).$

The expectation of this distribution is $\psi^{2}$ and increases quadratically
with $\psi$, the velocity of the vehicle involved in an accident; this is
consistent with the fact that losses scale with kinetic energy. The variance
of the distribution $F^{\psi}$ equals $c_{v}^{2}\psi^{4}$, hence the
coefficient of variation is indeed $c_{v}$.

  2. (ii)

_Log-Normal Distribution._ We consider log-normal distributions with
expectation $\psi^{2}$ and variance $c_{v}^{2}\psi^{4}$, implying that the
coefficient of variation is again $c_{v}\in\\{1/2,\leavevmode\nobreak\
1,\leavevmode\nobreak\ 2\\}$. This log-normal distribution is obtained as the
distribution of $\exp(Z)$ for a normal random variable $Z$ with expectation
$\ln(\psi^{2}/\sqrt{1+c_{v}^{2}})$ and variance $\ln(1+c_{v}^{2})$, i.e.,

$F^{\psi}\;=\;\mathcal{LN}\left(\ln\left(\frac{\psi^{2}}{\sqrt{1+c_{v}^{2}}}\right),\;\ln(1+c_{v}^{2})\right).$

###  4.3 Case Studies

####  4.3.1 Overview

We illustrate our modeling approach in case studies on multiple levels. A
selection of case studies is discussed in detail in Sections 4.3.2 & 4.3.3.
All numerical results for the following choices are documented online in
tables in Appendix B:

  1. (i)

_Fleet Models._ We analyze six driving configurations with three different
fleet proportions.

  2. (ii)

_Accident Occurrence._ In our traffic system accidents occur uniformly or non-
uniformly in space. Their number is given by a Binomial or a Poisson model
with parameters depending on traffic condition.

  3. (iii)

_Accident Losses._ We study two parametric families of loss distributions with
three different choices for the coefficient of variation.

  4. (iv)

_Insurance Design._ Insurance losses are a function of the total losses; we
distinguish three contract designs.

Denoting aggregate losses by $L$, we evaluate for each type of insurance
coverage $h$ the resulting insurance losses $h(L)$ in terms of their
expectation, variance, and skewness, and the monetary risk measures Value-at-
Risk and Average Value-at-Risk, also called Expected Shortfall. To analyze the
distributions in detail, we provide qq-plots and estimates of cumulative
distribution functions and densities. The main tool to access the random
variable $h(L)$ is Monte Carlo sampling; we provide a pseudo-code how we
obtain samples of $L$ in Algorithm 1 in Appendix A. In Section 4.3.3, we
compare this approach to the Normal mean-variance-mixture approximation
introduced in Section 3.2.

This paper explores the analysis and management of risks that occur in vehicle
fleets in traffic systems. We distinguish to perspectives:

  1. (i)

_The Engineering Perspective._ In these case studies, we fix the accident
occurrence and accident loss distributions and vary the fleet models, i.e.,
driving configurations and fleet proportions. We focus on $\mathbb{E}(L)$,
$\mathrm{Var}(L)$, and complement these with analyses of the performance of
traffic system.

  2. (ii)

_The Actuarial Perspective._ In these case studies, we fix the fleet model and
vary accident occurrence and accident loss distributions as well as the
insurance design. We study the distribution of $L$ and the insurance prices
$\mathbb{E}(h(L))$.

####  4.3.2 The Engineering Perspective

Our micro-modeling approach allows us to study the effects of different
traffic-related controls on total losses $L$; we investigate the effects of
fleet size and traffic configuration. Throughout this section, we consider
non-uniform accident occurrence in the Binomial model with Gamma distributed
accident losses and a coefficient of variation $c_{v}=1$.

Losses. We evaluate expected loss $\mathbb{E}(L)$ and standard deviation
$\mathrm{std}(L)$ for different fleet models. To compare losses for different
fleet sizes, we normalize losses per $100$ expected insured vehicles252525For
each traffic scenario $k$, the number of insured vehicles is the number of
vehicles belonging to $\Phi$ as given in the underlying route file. We refer
to Section B for more details.. The model specification was explained in
Section 4.2 which includes in particular a description of the driving
configurations. The results are documented in Figure 3. The solid lines are
the normalized quantities.

In Figure 3, we see that increasing the aggressiveness of driving increases
both the total and normalized expected loss. An impact of the maximal
acceleration on losses is only substantial for the most aggressive driving
configurations $\xi^{3\cdot}$. Increasing the fleet size increases the
expected loss which is primarily due to the fact that we count losses only
within the fleet and a higher volume is associated with higher losses. More
interesting is the normalized case: apparently higher speeds also increase the
normalized losses.

In Figure 3, we have the corresponding standard deviations. Increasing the
aggressiveness of driving increases the standard deviation of the total loss.
The standard deviations of the normalized losses are decreasing in the fleet
size. The main reason is that fluctuations normalized for a fixed volume are
larger for smaller pools than for larger pools; a rational for this is
provided by the law of large numbers and the central limit theorem.

The frequency and severity of accidents are, of course, increasing in the
aggressiveness of driving. To demonstrate this, we evaluate the expectation
and the standard deviation of the average accident frequency
$\sum_{k=1}^{K}\mu^{k}p^{k}$ (both normalized and unnormalized) and the
average accident severity $\sum_{k=1}^{K}\mu^{k}X_{1}^{k}$, as displayed in
Figure 4 and Figure 5. A larger fleet increases frequency and, in aggressive
scenarios, also the expected accident severity. This is, of course, due to the
specific choice of the driving behavior of the considered fleets in comparison
to the driving behavior of the remaining vehicles and does not necessarily
hold for all traffic systems in general.262626In this section, we briefly
discussed losses for selected special cases. A comprehensive set of detailed
tables for all other cases and statistical functionals is provided online in
Appendix B.

(a) Expectation of the Total Loss

(b) Standard Deviation of the Total Loss

Figure 3: Impact of Fleet Size and Driving Configuration on the Total Loss $L$
(solid lines represent normalized values (left y-axis), dashed lines
unnormalized ones (right y-axis)).

(a) Expectation

(b) Standard Deviation

Figure 4: Impact of Fleet Size and Driving Configuration on Expectation and
Standard Deviation of Average Accident Frequency (solid lines represent
normalized values (left y-axis), dashed lines unnormalized ones (right
y-axis)).

(a) Expectation

(b) Standard Deviation

Figure 5: Impact of Fleet Size and Driving Configuration on Expectation and
Standard Deviation of Average Accident Severity.

Traffic System Performance. Changing the characteristics of the fleet not only
affects the losses. At the same time, this has an impact on the performance of
the traffic system. Using the $40$ induction loop detectors placed in the
traffic system, we evaluate the values of selected traffic statistics (flow,
average speed, and occupancy). Denoting them by
$\chi_{k}^{1},\dots,\chi_{k}^{40}$ for each scenario $k=1,\dots,K$, we compute
averages $\bar{\chi}_{k}=1/40\sum_{i=1}^{40}\chi_{k}^{i}$. Pairing flow-
occupancy values and speed-occupancy values yields empirical fundamental
diagrams on the urban level (see, e.g., [19]). For the purpose of data
exploration, we draw scatter plots of these pairs for the driving
configurations $\xi^{1b},\xi^{2b},\xi^{3b}$ and fleet sizes
$\rho^{\Phi}=0.1,0.9$ in Figure 6 and Figure 7. We recover the classical
u-shape in the flow-occupancy plot. The blue points refer to the low volume
traffic scenarios, the red points to the high volume traffic scenarios, as
introduced in Section 4.1. This is also reflected by the fact that red points
correspond to higher occupancy. Aggressiveness in driving decreases overall
occupancy, increases speed, and increases flow, if the fleet is large.

To better understand the impact of individual driving configurations and fleet
sizes, we study the scenario averages
$\mathbb{E}\left(\sum_{k=1}^{K}\mu_{k}\bar{\chi}_{k}\right)=\frac{1}{K}\sum_{k=1}^{K}\bar{\chi}_{k}$.
The results are displayed in Figure 8. We see that the performance of the
traffic system improves with the aggressiveness of driving in the considered
case studies; flows and average speeds increase, and the occupancy decreases.

(a) $\rho^{\Phi}=0.1$, $\xi^{1b}$

(b) $\rho^{\Phi}=0.1$, $\xi^{2b}$

(c) $\rho^{\Phi}=0.1$, $\xi^{3b}$

(d) $\rho^{\Phi}=0.9$, $\xi^{1b}$

(e) $\rho^{\Phi}=0.9$, $\xi^{2b}$

(f) $\rho^{\Phi}=0.9$, $\xi^{3b}$

Figure 6: Flow-Occupancy Fundamental Diagrams (high traffic volume scenarios
are highlighted in red and low traffic volume scenarios in blue).

(a) $\rho^{\Phi}=0.1$, $\xi^{1b}$

(b) $\rho^{\Phi}=0.1$, $\xi^{2b}$

(c) $\rho^{\Phi}=0.1$, $\xi^{3b}$

(d) $\rho^{\Phi}=0.9$, $\xi^{1b}$

(e) $\rho^{\Phi}=0.9$, $\xi^{2b}$

(f) $\rho^{\Phi}=0.9$, $\xi^{3b}$

Figure 7: Speed-Occupancy Fundamental Diagrams (high traffic volume scenarios
are highlighted in red and low traffic volume scenarios in blue).

(a) Flow

(b) Speed

(c) Occupancy

Figure 8: Impact of Fleet Size and Driving Configuration on Traffic System
Performance.

####  4.3.3 The Actuarial Perspective

From an actuarial perspective, it is relevant to understand the _risk_ that
corresponds to the insurance losses. This requires a more detailed analysis of
the _probability distributions_. To do this, we pick a particular fleet model
and use probabilistic techniques to evaluate the distribution of $h(L)$. From
now on, we consider $\rho^{\Phi}=0.5$ with driving configuration $\xi^{2a}$
and non-uniform accident occurrence.

Distributional Analysis of Losses. We start our investigations with the total
losses $L$. Table 1 shows the evaluation of statistical functionals for
different accident losses in the case of the Binomial model. These numbers
quantify the risk entailed in the total losses. Both the distributional family
and the chosen coefficient of variation for the accident loss model have a
substantial effect on the risk.

| Binomial Model
---|---
| Gamma |  | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
$\mathbb{E}(L)$ | 1577.8 | 1571.5 | 1578.4 |  | 1581.7 | 1576.2 | 1582.7
$\mathrm{Var}(L)$ | 160179.5 | 247943.9 | 626190.6 |  | 162067.5 | 247069.8 | 614713.4
$\varsigma_{L}$ | 0.333 | 0.561 | 0.993 |  | 0.377 | 0.660 | 2.152
$\mathrm{VaR}_{0.9}(L)$ | 2111.9 | 2219.1 | 2634.2 |  | 2112.9 | 2242.4 | 2526.6
$\mathrm{ES}_{0.9}(L)$ | 2331.7 | 2538.8 | 3233.3 |  | 2342.4 | 2557.5 | 3280.3
$\mathrm{VaR}_{0.95}(L)$ | 2275.9 | 2471.3 | 3070.5 |  | 2288.4 | 2468.1 | 2996.2
$\mathrm{ES}_{0.95}(L)$ | 2468.9 | 2748.0 | 3628.9 |  | 2491.4 | 2772.5 | 3838.4
$\mathrm{VaR}_{0.99}(L)$ | 2587.3 | 2891.3 | 3999.6 |  | 2608.8 | 2943.8 | 4228.6
$\mathrm{ES}_{0.99}(L)$ | 2755.9 | 3188.6 | 4490.2 |  | 2809.0 | 3234.4 | 5424.2

  * •

The statistical functionals of the total loss are approximated using $10,000$
independent samples of $L$.

Table 1: Statistical Functionals of $L$ for $\rho^{\Phi}=0.5$ and
$\zeta^{2a}$.

A visual impression of the distributions is provided in Figure 9. We compare
different accident loss models while fixing the coefficient of variation
$c_{v}=2$. We plot the empirical distribution functions as estimates of the
cumulative distribution function (CDF) and a kernel density estimate of the
corresponding densities. Moreover, Figure 9 shows qq-plots for the quantiles
of standardized272727Samples are standardized by subtracting their sample
average and dividing by their sample standard deviation. values of the losses
against quantiles of a standard Normal distribution.

We find that the Binomial and the Poisson model do not differ too much. Yet,
log-normal accident losses produce heavier tails than the corresponding Gamma
losses. The qq-plots reveal that the total losses are not normally
distributed: In particular, the right tails are heavier compared to a Normal
distribution while the left tails are lighter. The latter observation simply
relates to the fact that the original losses are non-negative while the Normal
distribution takes values on the whole real line.

(a) CDF

(b) Density

(c) QQ-Plot

Figure 9: Distribution of the Total Loss for Fixed Coefficient of Variation
$c_{v}=2$.

In Figure 10, we analyze the impact of the coefficient of variation while
fixing the log-normal distribution for the accident losses. We see again that
Binomial and Poisson model do not differ substantially. However, the effect of
the coefficient of variation is clearly visible: increasing $c_{v}$ produces
heavier right tails. Introducing dispersion to accident losses substantially
changes the distribution of the total losses.

(a) CDF

(b) Density

(c) QQ-Plot

Figure 10: Distribution of the Total Loss for Log-Normal Accident Losses and
Varying Coefficient of Variation.

Comparison of Losses and Normal Mean-Variance-Mixture Approximation. In
Section 3.2, we suggested a Normal mean-variance-mixture approximation for the
total loss. To study the quality of this approximation, we generate $10,000$
samples from the approximation; in the following, we focus on the case of
Gamma distributed accident losses with coefficient of variation $c_{v}=1$.

To sample from the approximation, we rely on the following
computations282828For the notation, we refer to Sections 3.2 & 4.2. The
computations are valid for any coefficient of variation $c_{v}$, but we use
only $c_{v}=1$ in the numerical case study. of $m^{k}$ and $(\sigma^{k})^{2}$.
Using $\mathbb{E}(\mathbf{1}_{c}^{k})=p^{k}$ and
$\mathbb{E}(X_{c}^{k})=\mathbb{E}(\mathbb{E}(X_{c}^{k}\mid\psi))=\mathbb{E}(\psi^{2})=\int\psi^{2}d\mathcal{L}^{k}$,
we obtain for the Gamma losses $m^{k}=p^{k}\cdot\int\psi^{2}d\mathcal{L}^{k}$
and $(\sigma^{k})^{2}=p^{k}\cdot
c_{v}^{4}\cdot\int\psi^{4}d\mathcal{L}^{k}\cdot\left(1+\frac{1}{c_{v}^{2}}\right)\frac{1}{c_{v}^{2}}-(m^{k})^{2}$.
The involved moments of $\psi$ are approximated using $10,000$ samples from
the traffic simulation, for each $k=1,\dots,K$. A sample from the Normal mean-
variance-mixture approximation is generated by, first, sampling $\mu$ and,
second (conditional on $\mu$), sampling the normal random variable
$\sum_{k=1}^{K}N\mu^{k}m^{k}+Z$ with
$Z\sim\mathcal{N}\left(0,\;\sum_{k=1}^{K}N\mu^{k}{\left(\sigma^{k}\right)}^{2}\right).$

Figure 11: QQ-Plot of $10,000$ Monte Carlo Samples (y-axis) vs $10,000$
Samples of Mixture Approximation (x-axis) (the red dots mark the
$1\%,\leavevmode\nobreak\ 5\%,\leavevmode\nobreak\ 10\%,\leavevmode\nobreak\
90\%,\leavevmode\nobreak\ 95\%,\leavevmode\nobreak\ 99\%$ quantiles).

Figure 11 shows the qq-plot comparing quantiles of the crude Monte Carlo
simulation with quantiles of the approximation. This demonstrates the quality
of our suggested approximation. It is almost exact between the $5$ % and $95$
% quantile as the values lie on the halfline. It is still very good for the
$1-5$ % and $95-99$ % quantiles and is only less accurate in the extreme tails
where also in the Monte Carlo simulation only few data points are available.
These analyses, on the one hand, confirm the postulated structural model
insight. On the other hand, they also validate the implementation of our crude
Monte Carlo sampling.

Pricing and Evaluation Methods. To conclude our case studies, we study prices
for various insurance contracts. We compare $\mathbb{E}(L)$ (full coverage),
$\mathbb{E}(\max(L-\theta,0))$ (constant deductible), and
$\mathbb{E}(\min(L,\theta))$ (stop loss) for different values of $\theta$.

Figure 12: Insurance Prices.

The results are given in Figure 12. We obtain the typical hockey stick
profiles satisfying the parity
$\mathbb{E}(L)=\mathbb{E}(\max(L-\theta,0))+\mathbb{E}(\min(L,\theta))$. We
note that other insurance contracts can easily be represented in our
framework; also deductibles per accident can be implemented by changing the
accident loss distributions accordingly.

Besides Monte Carlo methods, our Normal mean-variance mixture approximation
and the correction suggested in Section 3.2 provide alternative techniques to
compute the prices $\mathbb{E}(h(L))$ for the different types of coverage $h$.
To compute the correction term $C_{h}$, we need to compute292929For the
notation we refer to Sections 3.2 & 4.2. $(\zeta^{k})^{3}$. For Gamma
distributed losses, we obtain

$(\zeta^{k})^{3}=p^{k}\cdot
c_{v}^{6}\cdot\int\psi^{6}d\mathcal{L}^{k}\cdot\left(2+\frac{1}{c_{v}^{2}}\right)\left(1+\frac{1}{c_{v}^{2}}\right)\frac{1}{c_{v}^{2}}-3m^{k}(\sigma^{k})^{2}-(m^{k})^{3}.$

Since $\mathbb{E}(h(L))=\mathbb{E}(\mathbb{E}(h(L)\mid\mu))$, we may generate
samples of $\mu$ and evaluate the corresponding conditional expectations
$\mathbb{E}(h(L)\mid\mu)$; in the Normal mean-variance mixture approximation,
these are expectations of functions of normally distributed random variables.
We compute these expectations numerically as integrals with respect to
Lebesgue measure using a normal density.

We compare the estimation errors of the different approaches in Figure 13. We
produce $100,000$ samples of $L$ to approximate the “true” value of
$\mathbb{E}(\max(L-\theta,0))$ of coverage with constant deductible, for
different values of $\theta$. Independently, we generate $10,000$ samples and
consider Monte Carlo approximations based on all $10,000$ samples, and based
only on the first $1,000$ samples. We also study the Normal mean-variance
mixture approximation with and without correction using $1,000$ samples of
$\mu$.

(a) Absolute Error

(b) Relative Error

Figure 13: Comparison of Estimation Errors.

While the absolute error generally decreases in the deductible $\theta$ for
all methods (apart from some local effects), the relative error increases for
larger values of $\theta$ which are associated with a lower price of the
contract. At the same time, we observe that, in terms of the relative error,
the Normal mean-variance mixture approximation produces reasonable estimation
results (compared to $1,000$ samples) for moderate values of $\theta$. This is
in line with our previous observations on the quality of our Normal mean-
variance mixture approximation which becomes worse in the extreme tail of $L$.
Interestingly, the estimation error can largely be reduced using the
correction; with the correction, the estimation becomes quite good even for
large values of $\theta$.

##  5 Conclusion

This paper developed a methodology to study accident losses based on an
established microscopic traffic simulator, here SUMO. An adaption of the
digital twin paradigm enabled us to test the impact of fleet sizes and their
driving configuration on system efficiency, accident losses, and insurance
premiums. We illustrated in counterfactual case studies how accident risk can
be successfully analyzed, both from an engineering and an actuarial
perspective. It was shown that – on a one-year horizon – total losses can be
approximated by a mean-variance mixture of normal distributions. This offered
an alternative technique to evaluate the model; the numerical efficiency can
be increased adding a correction term that is derived by Stein’s method. The
proposed methodology can be extended and modified, for example, based on
traffic simulators other than SUMO, and utilized to study future traffic
systems.

Future research should also address the important issues of calibration and
validation. While real data can be used to calibrate models that describe
historical and current transportation systems, simulation models that generate
artificial data are essential to evaluate new technologies in future
transportation systems. An important question is to what extent and how
historical data can be methodically used to calibrate and validate such
simulation models. For example, real microscopic traffic data, e.g., on
accident patterns, collected by means of telematics technologies, could be
applied to optimize models in the future.

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##  Appendix A Electronic Companion: Sampling Procedure

We provide a detailed pseudo-code for the procedure to obtain samples from $L$
in Algorithm 1. In our case studies, we use $M^{\prime}=M=10,000$ samples of
$\psi$ in each scenario $k$ to approximate its distribution $\mathcal{L}^{k}$.
We note that, instead of this bootstrapping approach, one could also pre-
sample sufficiently many values.

Algorithm 1 Sampling of Losses $L$.

Phase 1: Prior Evaluation of Traffic Model

for $k=1,\dots,K$ do

Run SUMO in scenario $k$.

Obtain data to calculate $p^{k}_{r}$ and $\lambda^{k}_{r}$ for $r=1,\dots,R$.

Terminate SUMO.

Set $p^{k}=\sum_{r=1}^{R}p^{k}_{r}$ and
$\lambda^{k}=\sum_{r=1}^{R}\lambda^{k}_{r}$.

end for

Phase 2: Pre-Sampling of $\psi$ conditional on scenario $k=1,\dots,K$.

for $k=1,\dots,K$ do

for $j=1,\dots,M^{\prime}$ do

Sample $\hat{t}_{j}=\mathrm{Unif}(0,T)$.

Sample $\hat{r}_{j}\sim\mathcal{R}$ where $P(\mathcal{R}=r)=p_{r}^{k}/p^{k}$
(or $P(\mathcal{R}=r)=\lambda_{r}^{k}/\lambda^{k}$).

end for

Sort $\hat{t}_{1},\dots,\hat{t}_{M^{\prime}}$ by size (again denoted by
$\hat{t}_{1},\dots,\hat{t}_{M^{\prime}}$).

Start SUMO in scenario $k$.

for $j=1,\dots,M^{\prime}$ do

Continue SUMO until time $\hat{t}_{j}$.

Sample
$\hat{i}_{j}\sim\mathrm{Unif}\left(\mathcal{M}^{\Phi}_{\hat{r}_{j}}\left(\hat{t}_{j}\right)\right)$.

Set $\hat{\psi}^{k}_{j}=v^{\hat{i}_{j}}(\hat{t}_{j})$.

end for

Terminate SUMO.

Store $\hat{\psi}^{k}_{1},\dots,\hat{\psi}^{k}_{M^{\prime}}$.

end for

Phase 3: Sampling of total losses $L$.

for $j=1,\dots,M$ do

Sample $\hat{y}^{j}$ with $P(y=g)=P(y=b)=1/2$.

for $n=1,\dots,N$ do

Sample $\hat{\nu}^{j,n}\sim\nu_{\hat{y}^{j}}$.

end for

for $k=1,\dots,K$ do

Set $(\hat{\mu}^{k})^{j}=1/N\sum_{n=1}^{N}\mathbbm{1}\\{\hat{\nu}^{j,n}=k\\}$.

end for

for $k=1,\dots,K$ do

Sample
$\left(\hat{C}^{k}\right)^{j}\sim\mathrm{Bin}(p^{k},N(\hat{\mu}^{k})^{j})$ (or
$\left(\hat{C}^{k}\right)^{j}\sim\mathrm{Poiss}(\lambda^{k}N(\hat{\mu}^{k})^{j})$).

for $c=1,\dots,\left(\hat{C}^{k}\right)^{j}$ do

Sample $\hat{l}\sim\mathrm{Unif}(\\{1,\dots,M^{\prime}\\})$.

Sample $\hat{X}^{k}_{c}\sim F^{\hat{\psi}^{k}_{\hat{l}}}$.

end for

end for

Set
$\hat{L}^{j}=\sum_{k=1}^{K}\sum_{c=1}^{\left(\hat{C}^{k}\right)^{j}}\hat{X}^{k}_{c}$.

end for

Output: $\hat{L}^{1},\dots,\hat{L}^{M}$.

##  Appendix B Electronic Companion: Tables

We provide detailed tables that contain selected statistical functionals of
the total loss in our different case studies. We evaluate

  1. (i)

_Expectation._ $\mathbb{E}(L)$,

  2. (ii)

_Variance._ $\mathrm{Var}(L)=\mathbb{E}(L^{2})-\mathbb{E}(L)^{2}$,

  3. (iii)

_Skewness._
$\varsigma_{L}=\frac{\mathbb{E}\left[(L-\mathbb{E}(L))^{3}\right]}{(\mathrm{Var}(L))^{3/2}}$,

  4. (iv)

_Value-at-Risk._ $\mathrm{VaR}_{p}(L)=\inf\\{x\in\mathbb{R}\colon P(L\leq
x)\geq p\\}$, $p=0.9,\leavevmode\nobreak\ 0.95,\leavevmode\nobreak\ 0.99$,

  5. (v)

_Expected Shortfall._
$\mathrm{ES}_{p}(L)=\frac{1}{1-p}\int_{p}^{1}\mathrm{VaR}_{q}(L)\,\mathrm{d}q$,
$p=0.9,\leavevmode\nobreak\ 0.95,\leavevmode\nobreak\ 0.99$.

These statistical functionals are presented for both unnormalized and
normalized values of $L$. In total, we provide 18 tables, as shown in the list
of tables below: The first 9 tables contain the statistical functionals for
unnormalized total losses while the second 9 tables contain the results for
normalized total losses.

###### List of Tables

  1. 1 Statistical Functionals of $L$ for $\rho^{\Phi}=0.5$ and $\zeta^{2a}$.
  2. 1 Expectation of the Total Loss.
  3. 2 Variance of the Total Loss.
  4. 3 Skewness of the Total Loss.
  5. 4 Value-at-Risk at Level $p=0.9$ of the Total Loss.
  6. 5 Expected Shortfall at Level $p=0.9$ of the Total Loss.
  7. 6 Value-at-Risk at Level $p=0.95$ of the Total Loss.
  8. 7 Expected Shortfall at Level $p=0.95$ of the Total Loss.
  9. 8 Value-at-Risk at Level $p=0.99$ of the Total Loss.
  10. 9 Expected Shortfall at Level $p=0.99$ of the Total Loss.
  11. 10 Expectation of the Normalized Total Loss.
  12. 11 Variance of the Normalized Total Loss.
  13. 12 Skewness of the Normalized Total Loss.
  14. 13 Value-at-Risk at Level $p=0.9$ of the Normalized Total Loss.
  15. 14 Expected Shortfall at Level $p=0.9$ of the Normalized Total Loss.
  16. 15 Value-at-Risk at Level $p=0.95$ of the Normalized Total Loss.
  17. 16 Expected Shortfall at Level $p=0.95$ of the Normalized Total Loss.
  18. 17 Value-at-Risk at Level $p=0.99$ of the Normalied Total Loss.
  19. 18 Expected Shortfall at Level $p=0.99$ of the Normalized Total Loss.

Remarks on the Normalization. Recall that we normalize total losses in order
to compare losses among fleet sizes $\rho^{\Phi}=0.1,\leavevmode\nobreak\
0.5,\leavevmode\nobreak\ 0.9$. More precisely, we normalize $L$ by _100
expected insured vehicles_ as follows:

  * •

In the underlying SUMO scneario, the route files specify a number of vehicles
for each _flow_ belonging to the fleet $\Phi$.

  * •

For each traffic scenario $k$, we denote the sum of these values over all
flows by $n^{k}$. We interpret this as the total number of insured vehicles in
traffic scenario $k$. In our case studies, $n^{k}$ takes two different values
corresponding to the good and bad scenarios:

$n^{k}=\begin{cases}n_{g},&k=1,\dots,50,\\\ n_{b},&k=51,\dots,100.\end{cases}$

  * •

We denote the total number of insured vehicles by $n^{\Phi}$. It is given by
$n^{\Phi}=\sum_{k=1}^{K}\mu^{k}n^{k}$. Note that this number is random as
$\mu$ is random.

  * •

We evaluate the normalized total loss per $100$ expected insured vehicles.
According to our specific choice of $\mu$, it is given by

$\displaystyle\frac{L}{\frac{\mathbb{E}(n^{\Phi})}{100}}$
$\displaystyle=100\cdot\frac{L}{\sum_{k=1}^{K}n^{k}\mathbb{E}(\mu^{k})}=\frac{200}{n_{g}+n_{b}}\cdot
L.$

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 546.7 | 544.7 | 544.2 | 547.0 | 544.2 | 546.5 |  | 545.5 | 546.7 | 546.2 | 543.4 | 547.2 | 542.5
$\xi^{1b}$ | 527.0 | 525.9 | 523.0 | 529.3 | 524.7 | 522.1 |  | 527.0 | 527.4 | 523.0 | 523.2 | 529.5 | 525.7
$\xi^{2a}$ | 1381.6 | 1376.5 | 1374.0 | 1384.3 | 1369.3 | 1367.1 |  | 1376.9 | 1377.7 | 1376.7 | 1376.1 | 1379.5 | 1364.3
$\xi^{2b}$ | 1399.7 | 1389.0 | 1380.1 | 1401.2 | 1387.2 | 1389.0 |  | 1395.5 | 1399.6 | 1392.0 | 1387.6 | 1394.9 | 1381.1
$\xi^{3a}$ | 2023.2 | 2009.8 | 2028.8 | 2027.8 | 2012.6 | 2019.9 |  | 2022.0 | 2026.4 | 2022.8 | 2003.2 | 2019.8 | 2002.1
$\xi^{3b}$ | 2168.9 | 2156.3 | 2148.7 | 2161.5 | 2144.3 | 2150.8 |  | 2163.1 | 2163.5 | 2158.5 | 2150.0 | 2164.1 | 2149.8
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 2173.0 | 2169.6 | 2174.0 | 2172.5 | 2170.7 | 2172.0 |  | 2174.9 | 2175.4 | 2176.2 | 2170.8 | 2165.0 | 2169.4
$\xi^{1b}$ | 2309.2 | 2312.7 | 2317.5 | 2310.9 | 2309.4 | 2311.8 |  | 2310.0 | 2314.6 | 2315.4 | 2309.9 | 2312.8 | 2308.5
$\xi^{2a}$ | 5712.1 | 5731.1 | 5753.4 | 5706.9 | 5719.8 | 5708.2 |  | 5715.9 | 5719.9 | 5707.2 | 5727.0 | 5705.6 | 5728.6
$\xi^{2b}$ | 7235.7 | 7258.7 | 7267.2 | 7260.1 | 7250.1 | 7281.1 |  | 7259.9 | 7262.6 | 7278.0 | 7246.7 | 7249.5 | 7243.4
$\xi^{3a}$ | 10020.9 | 10043.5 | 10034.8 | 10034.3 | 10012.5 | 10034.7 |  | 10037.9 | 10045.6 | 10028.9 | 10040.4 | 10040.6 | 10047.3
$\xi^{3b}$ | 13264.6 | 13275.8 | 13234.9 | 13271.4 | 13260.9 | 13277.3 |  | 13265.9 | 13272.4 | 13236.1 | 13251.3 | 13245.3 | 13241.8
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 3125.0 | 3133.0 | 3123.5 | 3130.7 | 3131.0 | 3119.0 |  | 3124.1 | 3128.5 | 3121.5 | 3128.1 | 3134.5 | 3133.0
$\xi^{1b}$ | 3772.7 | 3773.9 | 3768.3 | 3770.2 | 3778.0 | 3765.6 |  | 3778.6 | 3772.9 | 3770.4 | 3763.3 | 3773.1 | 3771.2
$\xi^{2a}$ | 10054.8 | 10066.1 | 10034.4 | 10069.6 | 10080.3 | 10060.3 |  | 10068.0 | 10065.4 | 10059.5 | 10054.0 | 10072.5 | 10104.8
$\xi^{2b}$ | 12538.0 | 12557.2 | 12527.5 | 12557.4 | 12545.2 | 12516.0 |  | 12554.5 | 12542.0 | 12552.6 | 12535.5 | 12555.9 | 12552.5
$\xi^{3a}$ | 20601.5 | 20620.3 | 20511.0 | 20585.1 | 20600.1 | 20587.4 |  | 20609.0 | 20571.2 | 20549.9 | 20562.7 | 20602.3 | 20603.8
$\xi^{3b}$ | 28596.4 | 28625.0 | 28470.8 | 28605.5 | 28633.4 | 28629.9 |  | 28595.0 | 28587.0 | 28595.0 | 28577.9 | 28647.2 | 28642.7
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 52.3 | 52.2 | 53.3 | 52.5 | 52.1 | 54.4 |  | 51.8 | 52.7 | 52.2 | 52.5 | 53.2 | 53.0
$\xi^{1b}$ | 52.9 | 52.7 | 52.3 | 52.9 | 52.0 | 52.8 |  | 52.6 | 52.3 | 52.6 | 52.4 | 52.0 | 54.1
$\xi^{2a}$ | 304.5 | 306.1 | 295.6 | 305.5 | 307.4 | 304.8 |  | 302.6 | 308.5 | 306.4 | 304.4 | 305.3 | 303.6
$\xi^{2b}$ | 325.5 | 328.6 | 331.2 | 322.7 | 329.7 | 324.6 |  | 327.2 | 325.4 | 327.8 | 327.3 | 325.1 | 322.0
$\xi^{3a}$ | 1003.9 | 1004.6 | 1006.5 | 1000.6 | 1010.8 | 1000.9 |  | 1003.4 | 993.5 | 1022.1 | 1006.5 | 1009.0 | 1008.5
$\xi^{3b}$ | 1208.6 | 1207.7 | 1202.9 | 1215.7 | 1207.0 | 1212.8 |  | 1206.1 | 1209.5 | 1196.5 | 1208.0 | 1211.4 | 1222.5
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 199.1 | 199.5 | 199.9 | 199.7 | 201.4 | 201.2 |  | 199.7 | 198.2 | 197.9 | 199.4 | 198.8 | 200.0
$\xi^{1b}$ | 228.6 | 228.0 | 229.6 | 229.5 | 229.7 | 226.7 |  | 228.3 | 227.7 | 229.3 | 227.2 | 228.5 | 230.2
$\xi^{2a}$ | 1577.8 | 1571.5 | 1578.4 | 1581.7 | 1576.2 | 1582.7 |  | 1579.1 | 1582.9 | 1583.0 | 1579.4 | 1579.2 | 1579.8
$\xi^{2b}$ | 1928.8 | 1912.8 | 1914.6 | 1923.6 | 1917.7 | 1911.4 |  | 1914.0 | 1917.7 | 1937.8 | 1913.3 | 1922.2 | 1911.7
$\xi^{3a}$ | 6946.2 | 6974.9 | 6913.7 | 6928.7 | 6937.4 | 6948.2 |  | 6945.0 | 6913.8 | 6942.0 | 6927.8 | 6943.6 | 6952.9
$\xi^{3b}$ | 9311.7 | 9349.5 | 9349.8 | 9329.4 | 9359.3 | 9343.5 |  | 9340.0 | 9330.0 | 9359.0 | 9334.5 | 9326.2 | 9308.5
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 335.7 | 335.2 | 335.6 | 335.0 | 335.1 | 333.7 |  | 334.7 | 334.7 | 334.3 | 337.0 | 336.5 | 334.9
$\xi^{1b}$ | 381.3 | 383.4 | 383.5 | 381.6 | 380.3 | 383.4 |  | 381.2 | 382.8 | 380.9 | 383.4 | 382.4 | 382.2
$\xi^{2a}$ | 2032.5 | 2020.2 | 2034.2 | 2030.7 | 2029.4 | 2040.2 |  | 2026.1 | 2022.3 | 2027.8 | 2025.6 | 2031.2 | 2035.6
$\xi^{2b}$ | 2562.1 | 2553.8 | 2550.5 | 2559.7 | 2556.1 | 2553.7 |  | 2562.5 | 2556.9 | 2568.5 | 2556.3 | 2567.8 | 2558.7
$\xi^{3a}$ | 12801.2 | 12825.5 | 12775.7 | 12795.2 | 12843.3 | 12815.3 |  | 12811.7 | 12808.1 | 12797.0 | 12818.1 | 12792.4 | 12772.1
$\xi^{3b}$ | 15767.8 | 15782.4 | 15757.3 | 15747.8 | 15789.4 | 15787.4 |  | 15766.4 | 15759.1 | 15736.7 | 15769.6 | 15744.5 | 15806.1

  * •

The expectation of the total loss is approximated using $10,000$ independent
samples of $L$.

Table 1: Expectation of the Total Loss.

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 18893.7 | 27665.8 | 63330.2 | 18797.9 | 28026.1 | 65488.9 |  | 18277.8 | 28087.5 | 63245.1 | 19071.7 | 28092.0 | 62943.3
$\xi^{1b}$ | 18087.2 | 26745.3 | 60657.5 | 18440.3 | 26108.7 | 61767.7 |  | 17887.3 | 26811.1 | 60802.6 | 17589.5 | 27311.8 | 61022.4
$\xi^{2a}$ | 173375.0 | 258237.7 | 600842.9 | 173092.1 | 258920.4 | 598414.0 |  | 172120.9 | 262482.2 | 613084.4 | 173021.5 | 265451.8 | 582242.0
$\xi^{2b}$ | 177214.8 | 265275.6 | 619897.7 | 178629.2 | 271430.9 | 631212.7 |  | 176634.1 | 272308.4 | 613676.7 | 174434.4 | 261582.3 | 590262.8
$\xi^{3a}$ | 485013.5 | 694041.4 | 1621207.6 | 499592.9 | 727156.0 | 1579947.2 |  | 495801.3 | 724757.7 | 1672718.2 | 495659.6 | 710318.7 | 1536261.7
$\xi^{3b}$ | 490118.8 | 722394.1 | 1688625.0 | 491584.1 | 734333.9 | 1711028.6 |  | 488696.3 | 721297.3 | 1687128.3 | 488968.6 | 752454.6 | 1606874.7
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 121040.6 | 157007.0 | 323289.6 | 120936.5 | 166813.1 | 330798.2 |  | 122103.1 | 161366.7 | 326131.8 | 121952.2 | 160241.5 | 307114.2
$\xi^{1b}$ | 181320.5 | 230611.6 | 464392.6 | 179740.3 | 236318.9 | 433770.2 |  | 181861.1 | 233756.7 | 466338.5 | 181687.0 | 239151.9 | 475013.2
$\xi^{2a}$ | 960580.4 | 1331588.1 | 2810898.4 | 976795.7 | 1347996.6 | 2838794.5 |  | 953983.0 | 1324593.2 | 2730096.3 | 975379.6 | 1367343.2 | 2723201.3
$\xi^{2b}$ | 1476836.9 | 1960914.4 | 3775938.1 | 1483347.2 | 1965754.2 | 3854527.9 |  | 1486437.1 | 1935028.3 | 3828259.5 | 1476619.6 | 1924502.1 | 3735207.0
$\xi^{3a}$ | 2993260.8 | 4217438.8 | 8673871.0 | 3068922.2 | 4221409.4 | 8880079.8 |  | 3014765.5 | 4205157.8 | 8801550.9 | 3067141.8 | 4212840.4 | 8536435.4
$\xi^{3b}$ | 4676662.8 | 6224116.1 | 12156646.2 | 4809676.1 | 6329363.3 | 12781771.6 |  | 4762692.2 | 6219709.2 | 12509905.8 | 4676895.4 | 6141764.0 | 12152976.1
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 230663.3 | 292653.8 | 557213.5 | 230110.7 | 299753.8 | 550714.7 |  | 223713.1 | 296432.7 | 556156.1 | 225082.5 | 298376.0 | 550281.9
$\xi^{1b}$ | 420227.2 | 514161.8 | 886441.8 | 419661.1 | 529744.0 | 890496.6 |  | 422378.1 | 506903.9 | 866223.8 | 410689.9 | 518008.1 | 838534.9
$\xi^{2a}$ | 3617972.2 | 4254283.7 | 6785282.6 | 3627251.6 | 4245970.8 | 6861536.3 |  | 3575807.1 | 4150777.2 | 6806228.4 | 3548344.3 | 4209067.1 | 6990333.1
$\xi^{2b}$ | 5148307.6 | 6076761.5 | 9320685.1 | 5301991.9 | 6039432.8 | 9419485.9 |  | 5157667.3 | 5835207.0 | 9384488.3 | 5055167.2 | 6010148.3 | 9398184.7
$\xi^{3a}$ | 13071136.0 | 15680586.9 | 24022504.1 | 13409575.9 | 15713162.4 | 25030090.4 |  | 13116476.7 | 15264724.3 | 25281776.3 | 12989610.2 | 15290800.5 | 25353912.1
$\xi^{3b}$ | 24501429.2 | 28291913.8 | 41813984.7 | 24611290.5 | 28767366.0 | 42224310.3 |  | 24362225.8 | 27467893.6 | 41963940.9 | 24379873.1 | 28203672.3 | 42459073.8
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 1438.0 | 2257.3 | 6100.2 | 1467.8 | 2237.1 | 6720.3 |  | 1440.0 | 2313.5 | 5776.6 | 1471.4 | 2425.6 | 5994.4
$\xi^{1b}$ | 1463.3 | 2378.6 | 5994.7 | 1455.2 | 2321.9 | 5858.0 |  | 1505.6 | 2300.6 | 5844.3 | 1413.0 | 2318.5 | 6845.5
$\xi^{2a}$ | 30953.2 | 49464.1 | 121528.1 | 31352.3 | 48965.0 | 109378.1 |  | 30342.7 | 49461.0 | 127032.7 | 30907.4 | 49753.9 | 111897.9
$\xi^{2b}$ | 33419.3 | 53075.3 | 143017.0 | 32834.7 | 55200.1 | 116193.0 |  | 34025.5 | 54202.3 | 135611.7 | 32757.6 | 54546.8 | 121344.0
$\xi^{3a}$ | 182937.0 | 295334.3 | 722181.9 | 184042.2 | 295803.4 | 738829.0 |  | 184663.6 | 288892.8 | 754295.9 | 184744.1 | 304452.4 | 796616.7
$\xi^{3b}$ | 222882.0 | 353001.2 | 886386.5 | 220849.6 | 360422.9 | 869560.6 |  | 218409.1 | 361988.8 | 862637.6 | 224628.2 | 344226.6 | 977182.7
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 5313.9 | 8478.7 | 21793.0 | 5388.0 | 8619.9 | 24533.1 |  | 5398.9 | 8642.4 | 21592.3 | 5472.4 | 8465.0 | 22170.3
$\xi^{1b}$ | 6306.7 | 9851.5 | 24822.5 | 6414.5 | 10070.0 | 22808.0 |  | 6176.6 | 10229.3 | 24183.2 | 6242.0 | 9831.1 | 25914.1
$\xi^{2a}$ | 160179.5 | 247943.9 | 626190.6 | 162067.5 | 247069.8 | 614713.4 |  | 159388.6 | 250341.3 | 621806.1 | 159415.6 | 248641.8 | 620170.1
$\xi^{2b}$ | 193822.7 | 309370.7 | 772896.4 | 196901.0 | 307477.8 | 776261.6 |  | 189283.6 | 304666.2 | 777809.5 | 196095.0 | 319014.4 | 742031.7
$\xi^{3a}$ | 1212501.9 | 1988777.5 | 4900976.8 | 1218271.7 | 1967587.9 | 4929170.8 |  | 1214816.6 | 1973115.9 | 4820241.3 | 1212656.8 | 1951271.1 | 5212245.7
$\xi^{3b}$ | 2193293.6 | 3252845.0 | 7162873.4 | 2244429.9 | 3255329.8 | 7542890.4 |  | 2213441.4 | 3230865.7 | 7360691.8 | 2272519.9 | 3139292.2 | 7205238.8
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 9418.1 | 14251.7 | 34904.2 | 8856.3 | 14557.9 | 32370.7 |  | 8959.2 | 14462.4 | 35612.6 | 9189.2 | 14733.7 | 35395.1
$\xi^{1b}$ | 10433.6 | 17198.0 | 41040.9 | 10625.6 | 16511.0 | 41640.4 |  | 10743.4 | 16890.3 | 41181.5 | 10870.6 | 17197.3 | 39666.6
$\xi^{2a}$ | 213730.9 | 326337.6 | 793409.8 | 210433.2 | 326920.7 | 848075.4 |  | 209536.2 | 325582.3 | 798698.7 | 213523.4 | 324714.3 | 804142.7
$\xi^{2b}$ | 246957.3 | 400989.1 | 989549.0 | 249269.2 | 399286.6 | 948864.4 |  | 249621.3 | 409990.9 | 984075.1 | 250842.2 | 410979.1 | 1030749.7
$\xi^{3a}$ | 2892886.1 | 4312952.8 | 9670130.8 | 2866821.7 | 4265176.9 | 9459010.8 |  | 2895553.5 | 4295589.2 | 9710683.1 | 2956058.6 | 4316705.3 | 9543766.6
$\xi^{3b}$ | 3320897.8 | 5093373.4 | 11733718.4 | 3380575.1 | 5108875.3 | 11667850.1 |  | 3205683.4 | 5100258.4 | 12004775.6 | 3294870.1 | 5014524.6 | 12241070.1

  * •

The variance of the total loss is approximated using $10,000$ independent
samples of $L$.

Table 2: Variance of the Total Loss.

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.306 | 0.533 | 0.869 | 0.320 | 0.679 | 1.904 |  | 0.327 | 0.454 | 0.858 | 0.386 | 0.536 | 1.728
$\xi^{1b}$ | 0.292 | 0.468 | 0.868 | 0.357 | 0.666 | 1.764 |  | 0.330 | 0.468 | 0.818 | 0.366 | 0.563 | 1.756
$\xi^{2a}$ | 0.409 | 0.609 | 1.095 | 0.451 | 0.733 | 2.822 |  | 0.403 | 0.573 | 1.109 | 0.451 | 0.785 | 2.221
$\xi^{2b}$ | 0.431 | 0.639 | 1.114 | 0.435 | 0.886 | 2.495 |  | 0.359 | 0.598 | 1.073 | 0.448 | 0.650 | 1.987
$\xi^{3a}$ | 0.451 | 0.640 | 1.234 | 0.503 | 0.803 | 2.269 |  | 0.430 | 0.687 | 1.297 | 0.497 | 0.773 | 2.171
$\xi^{3b}$ | 0.463 | 0.584 | 1.142 | 0.453 | 0.865 | 2.763 |  | 0.399 | 0.607 | 1.142 | 0.461 | 0.785 | 2.084
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.192 | 0.254 | 0.492 | 0.130 | 0.323 | 1.454 |  | 0.200 | 0.222 | 0.502 | 0.164 | 0.384 | 0.891
$\xi^{1b}$ | 0.330 | 0.458 | 1.120 | 0.326 | 0.576 | 1.658 |  | 0.350 | 0.532 | 1.032 | 0.341 | 0.679 | 3.431
$\xi^{2a}$ | 0.266 | 0.338 | 0.563 | 0.216 | 0.385 | 1.318 |  | 0.219 | 0.306 | 0.550 | 0.221 | 0.406 | 1.246
$\xi^{2b}$ | 0.208 | 0.323 | 0.515 | 0.196 | 0.290 | 0.942 |  | 0.209 | 0.257 | 0.484 | 0.184 | 0.307 | 0.919
$\xi^{3a}$ | 0.211 | 0.285 | 0.566 | 0.237 | 0.358 | 1.206 |  | 0.182 | 0.314 | 0.609 | 0.229 | 0.358 | 1.146
$\xi^{3b}$ | 0.194 | 0.264 | 0.496 | 0.187 | 0.323 | 1.166 |  | 0.235 | 0.242 | 0.532 | 0.193 | 0.263 | 0.972
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.265 | 0.380 | 0.795 | 0.261 | 0.402 | 1.359 |  | 0.257 | 0.397 | 0.746 | 0.292 | 0.436 | 1.418
$\xi^{1b}$ | 0.249 | 0.386 | 0.836 | 0.218 | 0.392 | 2.090 |  | 0.234 | 0.351 | 0.687 | 0.237 | 0.393 | 1.154
$\xi^{2a}$ | 0.143 | 0.209 | 0.391 | 0.145 | 0.212 | 0.755 |  | 0.150 | 0.230 | 0.382 | 0.125 | 0.195 | 0.979
$\xi^{2b}$ | 0.155 | 0.195 | 0.386 | 0.132 | 0.200 | 0.650 |  | 0.123 | 0.177 | 0.394 | 0.125 | 0.209 | 0.724
$\xi^{3a}$ | 0.142 | 0.194 | 0.370 | 0.119 | 0.231 | 0.652 |  | 0.125 | 0.191 | 0.384 | 0.147 | 0.209 | 0.791
$\xi^{3b}$ | 0.104 | 0.163 | 0.335 | 0.104 | 0.170 | 0.541 |  | 0.102 | 0.140 | 0.292 | 0.113 | 0.171 | 0.483
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.893 | 1.482 | 2.832 | 0.934 | 1.875 | 7.599 |  | 0.948 | 1.516 | 2.779 | 0.938 | 1.916 | 7.127
$\xi^{1b}$ | 0.916 | 1.463 | 2.841 | 0.931 | 2.016 | 5.193 |  | 0.970 | 1.385 | 2.789 | 0.908 | 1.875 | 6.440
$\xi^{2a}$ | 0.770 | 1.255 | 2.486 | 0.768 | 1.372 | 3.274 |  | 0.747 | 1.182 | 2.367 | 0.805 | 1.647 | 3.572
$\xi^{2b}$ | 0.778 | 1.070 | 2.370 | 0.739 | 1.503 | 3.182 |  | 0.717 | 1.181 | 2.276 | 0.753 | 1.600 | 3.484
$\xi^{3a}$ | 0.560 | 0.941 | 1.727 | 0.646 | 1.216 | 6.279 |  | 0.565 | 0.927 | 1.663 | 0.612 | 1.291 | 7.033
$\xi^{3b}$ | 0.602 | 0.848 | 1.639 | 0.550 | 1.169 | 2.945 |  | 0.501 | 0.857 | 1.578 | 0.591 | 1.013 | 5.332
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.502 | 0.733 | 1.364 | 0.510 | 0.931 | 5.395 |  | 0.488 | 0.761 | 1.502 | 0.508 | 0.975 | 2.945
$\xi^{1b}$ | 0.412 | 0.676 | 1.274 | 0.476 | 0.913 | 2.348 |  | 0.410 | 0.674 | 1.249 | 0.469 | 0.936 | 3.525
$\xi^{2a}$ | 0.333 | 0.561 | 0.993 | 0.377 | 0.660 | 2.152 |  | 0.299 | 0.522 | 1.045 | 0.417 | 0.711 | 2.155
$\xi^{2b}$ | 0.333 | 0.489 | 0.891 | 0.343 | 0.620 | 2.241 |  | 0.292 | 0.466 | 0.934 | 0.305 | 0.631 | 1.921
$\xi^{3a}$ | 0.209 | 0.362 | 0.663 | 0.222 | 0.507 | 1.711 |  | 0.199 | 0.429 | 0.664 | 0.188 | 0.446 | 2.234
$\xi^{3b}$ | 0.170 | 0.291 | 0.548 | 0.182 | 0.324 | 1.787 |  | 0.222 | 0.343 | 0.626 | 0.194 | 0.354 | 1.358
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 0.375 | 0.595 | 0.986 | 0.380 | 0.702 | 1.992 |  | 0.340 | 0.588 | 1.078 | 0.388 | 0.716 | 2.678
$\xi^{1b}$ | 0.386 | 0.562 | 0.975 | 0.393 | 0.689 | 2.316 |  | 0.374 | 0.556 | 1.006 | 0.359 | 0.751 | 2.101
$\xi^{2a}$ | 0.298 | 0.440 | 0.813 | 0.319 | 0.626 | 2.733 |  | 0.274 | 0.509 | 0.918 | 0.306 | 0.627 | 2.219
$\xi^{2b}$ | 0.251 | 0.420 | 0.759 | 0.270 | 0.517 | 1.567 |  | 0.260 | 0.363 | 0.743 | 0.315 | 0.585 | 2.460
$\xi^{3a}$ | 0.203 | 0.310 | 0.502 | 0.194 | 0.325 | 1.069 |  | 0.161 | 0.266 | 0.469 | 0.239 | 0.359 | 1.044
$\xi^{3b}$ | 0.145 | 0.235 | 0.437 | 0.138 | 0.316 | 1.084 |  | 0.096 | 0.218 | 0.440 | 0.129 | 0.280 | 1.006

  * •

The skewness of the total loss is approximated using $10,000$ independent
samples of $L$.

Table 3: Skewness of the Total Loss.

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 726.8 | 766.2 | 882.9 | 727.7 | 763.1 | 861.4 |  | 725.3 | 772.8 | 887.0 | 723.0 | 771.9 | 851.7
$\xi^{1b}$ | 707.7 | 743.7 | 848.1 | 709.3 | 737.7 | 829.6 |  | 706.0 | 743.0 | 857.3 | 699.0 | 748.5 | 827.4
$\xi^{2a}$ | 1933.3 | 2052.4 | 2413.8 | 1937.5 | 2041.5 | 2288.0 |  | 1926.0 | 2058.1 | 2429.1 | 1928.5 | 2065.1 | 2267.1
$\xi^{2b}$ | 1958.5 | 2072.9 | 2437.3 | 1964.7 | 2070.4 | 2317.7 |  | 1958.2 | 2099.0 | 2451.4 | 1934.8 | 2083.1 | 2314.8
$\xi^{3a}$ | 2952.1 | 3134.7 | 3703.4 | 2987.1 | 3151.4 | 3531.9 |  | 2970.2 | 3154.5 | 3774.6 | 2949.0 | 3137.8 | 3483.6
$\xi^{3b}$ | 3112.3 | 3283.0 | 3887.7 | 3097.2 | 3288.6 | 3659.2 |  | 3096.9 | 3317.0 | 3921.2 | 3070.4 | 3328.9 | 3674.9
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 2639.3 | 2686.3 | 2929.1 | 2632.9 | 2708.7 | 2895.6 |  | 2637.8 | 2714.0 | 2927.2 | 2635.1 | 2687.8 | 2886.9
$\xi^{1b}$ | 2889.2 | 2953.6 | 3169.3 | 2880.1 | 2944.2 | 3121.2 |  | 2888.1 | 2957.7 | 3188.6 | 2891.1 | 2952.3 | 3114.5
$\xi^{2a}$ | 7028.2 | 7253.9 | 7995.6 | 7033.8 | 7242.1 | 7807.5 |  | 7029.6 | 7254.4 | 7909.1 | 7051.0 | 7260.4 | 7853.7
$\xi^{2b}$ | 8878.5 | 9109.5 | 9838.1 | 8901.2 | 9130.7 | 9830.7 |  | 8910.9 | 9109.8 | 9876.5 | 8890.4 | 9091.0 | 9780.9
$\xi^{3a}$ | 12338.0 | 12765.5 | 13936.0 | 12345.0 | 12762.5 | 13804.0 |  | 12337.5 | 12802.0 | 13882.5 | 12382.5 | 12750.5 | 13753.0
$\xi^{3b}$ | 16184.0 | 16600.0 | 17849.0 | 16194.5 | 16581.0 | 17812.0 |  | 16181.5 | 16578.0 | 17881.0 | 16130.5 | 16525.0 | 17747.0
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 3766.3 | 3847.8 | 4095.0 | 3777.0 | 3843.9 | 4045.0 |  | 3759.6 | 3856.4 | 4095.4 | 3762.1 | 3866.1 | 4065.3
$\xi^{1b}$ | 4643.1 | 4744.1 | 4968.7 | 4633.4 | 4743.4 | 4900.4 |  | 4641.6 | 4735.9 | 5007.1 | 4622.5 | 4740.3 | 4905.8
$\xi^{2a}$ | 12584.5 | 12848.5 | 13568.0 | 12590.5 | 12830.0 | 13436.5 |  | 12552.5 | 12775.0 | 13579.5 | 12541.5 | 12822.0 | 13505.5
$\xi^{2b}$ | 15531.0 | 15888.0 | 16692.0 | 15583.5 | 15805.0 | 16638.5 |  | 15568.0 | 15769.5 | 16635.5 | 15505.0 | 15797.5 | 16529.5
$\xi^{3a}$ | 25402.0 | 25952.0 | 27072.5 | 25422.0 | 25873.5 | 27115.0 |  | 25397.5 | 25802.5 | 27306.5 | 25371.0 | 25876.5 | 27061.0
$\xi^{3b}$ | 35096.5 | 35639.0 | 37232.0 | 35087.5 | 35789.5 | 37168.5 |  | 35042.0 | 35566.0 | 37240.0 | 35099.5 | 35698.5 | 37191.5
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 103.7 | 115.1 | 147.7 | 105.1 | 113.8 | 125.9 |  | 102.1 | 116.3 | 145.2 | 105.6 | 116.1 | 124.6
$\xi^{1b}$ | 105.0 | 119.7 | 147.2 | 104.8 | 113.7 | 124.5 |  | 105.2 | 118.3 | 144.5 | 103.0 | 114.6 | 125.3
$\xi^{2a}$ | 541.5 | 608.5 | 730.7 | 542.4 | 606.4 | 657.7 |  | 537.3 | 606.1 | 765.5 | 541.3 | 592.2 | 656.2
$\xi^{2b}$ | 569.6 | 644.4 | 809.9 | 568.4 | 631.2 | 686.1 |  | 577.3 | 633.0 | 808.1 | 573.4 | 634.3 | 691.4
$\xi^{3a}$ | 1576.8 | 1734.2 | 2138.4 | 1571.1 | 1715.8 | 1917.8 |  | 1582.2 | 1726.8 | 2173.7 | 1582.0 | 1725.8 | 1899.4
$\xi^{3b}$ | 1826.7 | 2013.5 | 2439.2 | 1835.4 | 2000.0 | 2240.9 |  | 1832.4 | 2007.2 | 2436.1 | 1841.4 | 1984.4 | 2258.1
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 295.1 | 322.5 | 396.1 | 295.2 | 324.0 | 360.6 |  | 297.5 | 321.2 | 392.9 | 296.8 | 319.7 | 365.7
$\xi^{1b}$ | 334.8 | 361.3 | 438.3 | 336.4 | 363.2 | 405.9 |  | 332.5 | 363.2 | 437.0 | 331.8 | 358.9 | 409.1
$\xi^{2a}$ | 2111.9 | 2219.1 | 2634.2 | 2112.9 | 2242.4 | 2526.6 |  | 2101.2 | 2244.4 | 2623.3 | 2098.1 | 2226.8 | 2509.3
$\xi^{2b}$ | 2512.1 | 2640.6 | 3081.1 | 2515.9 | 2653.5 | 2972.0 |  | 2484.1 | 2638.4 | 3128.9 | 2497.4 | 2669.7 | 2968.7
$\xi^{3a}$ | 8385.3 | 8836.3 | 9837.0 | 8360.5 | 8773.8 | 9648.6 |  | 8382.1 | 8764.2 | 9810.4 | 8361.8 | 8757.2 | 9736.3
$\xi^{3b}$ | 11227.5 | 11706.0 | 12932.0 | 11281.5 | 11695.0 | 12703.0 |  | 11290.5 | 11670.0 | 12979.0 | 11312.5 | 11630.0 | 12613.0
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 465.9 | 493.1 | 593.0 | 459.5 | 494.5 | 549.9 |  | 457.4 | 493.9 | 588.5 | 463.1 | 499.8 | 555.4
$\xi^{1b}$ | 516.2 | 557.2 | 661.0 | 517.7 | 549.2 | 621.4 |  | 517.8 | 555.6 | 654.1 | 523.6 | 554.4 | 624.3
$\xi^{2a}$ | 2645.5 | 2778.7 | 3241.0 | 2629.0 | 2781.6 | 3113.7 |  | 2623.5 | 2779.6 | 3242.5 | 2632.3 | 2784.2 | 3112.9
$\xi^{2b}$ | 3206.6 | 3388.2 | 3904.8 | 3213.4 | 3387.9 | 3760.7 |  | 3216.9 | 3408.2 | 3889.2 | 3209.2 | 3400.6 | 3760.1
$\xi^{3a}$ | 15023.5 | 15581.0 | 16960.0 | 15046.0 | 15562.5 | 16712.0 |  | 15090.0 | 15568.0 | 16927.0 | 15121.0 | 15542.5 | 16791.0
$\xi^{3b}$ | 18160.0 | 18762.0 | 20237.0 | 18102.0 | 18736.0 | 20122.0 |  | 18088.5 | 18705.5 | 20215.0 | 18106.5 | 18662.5 | 20252.5

  * •

The Value-at-Risk at level $p=0.9$ of the total loss is approximated using
$10,000$ independent samples of $L$.

Table 4: Value-at-Risk at Level $p=0.9$ of the Total Loss.

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 801.9 | 866.5 | 1055.4 | 803.7 | 871.6 | 1096.5 |  | 797.8 | 868.0 | 1058.7 | 805.2 | 875.7 | 1079.0
$\xi^{1b}$ | 778.2 | 840.5 | 1025.3 | 786.6 | 844.8 | 1058.5 |  | 777.6 | 842.5 | 1022.9 | 773.4 | 853.6 | 1054.4
$\xi^{2a}$ | 2175.9 | 2378.3 | 2998.7 | 2184.2 | 2384.9 | 3029.9 |  | 2163.1 | 2372.4 | 3029.2 | 2170.4 | 2418.7 | 3010.3
$\xi^{2b}$ | 2205.7 | 2400.3 | 3026.6 | 2213.2 | 2441.4 | 3095.9 |  | 2186.3 | 2429.5 | 3030.8 | 2188.0 | 2411.8 | 3062.6
$\xi^{3a}$ | 3356.2 | 3661.7 | 4733.3 | 3395.1 | 3726.8 | 4760.5 |  | 3367.4 | 3718.1 | 4821.5 | 3361.2 | 3716.6 | 4715.2
$\xi^{3b}$ | 3513.6 | 3819.4 | 4899.6 | 3513.2 | 3879.1 | 4976.4 |  | 3494.5 | 3840.0 | 4904.8 | 3485.0 | 3924.3 | 4907.7
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 2799.2 | 2888.9 | 3270.3 | 2785.5 | 2935.4 | 3319.3 |  | 2799.9 | 2906.4 | 3278.1 | 2791.8 | 2899.6 | 3280.8
$\xi^{1b}$ | 3087.8 | 3225.4 | 3702.7 | 3090.7 | 3249.3 | 3646.8 |  | 3100.8 | 3248.5 | 3705.4 | 3098.0 | 3259.8 | 3668.1
$\xi^{2a}$ | 7510.7 | 7881.5 | 9017.4 | 7499.5 | 7906.8 | 9128.0 |  | 7490.1 | 7866.2 | 8908.5 | 7530.2 | 7925.5 | 9048.0
$\xi^{2b}$ | 9413.0 | 9842.3 | 11013.9 | 9434.8 | 9832.0 | 11238.5 |  | 9449.8 | 9817.9 | 11033.4 | 9422.1 | 9819.5 | 11135.3
$\xi^{3a}$ | 13182.2 | 13850.3 | 15809.1 | 13228.0 | 13870.7 | 16164.8 |  | 13168.1 | 13888.1 | 15843.7 | 13243.7 | 13869.9 | 16010.4
$\xi^{3b}$ | 17154.5 | 17851.0 | 19962.8 | 17194.5 | 17938.8 | 20513.8 |  | 17219.5 | 17833.8 | 20094.8 | 17124.6 | 17832.9 | 20268.2
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 3993.4 | 4143.9 | 4599.1 | 3996.1 | 4155.3 | 4613.5 |  | 3975.6 | 4152.8 | 4595.2 | 3989.6 | 4171.8 | 4612.8
$\xi^{1b}$ | 4912.4 | 5106.7 | 5633.6 | 4898.1 | 5119.9 | 5614.8 |  | 4915.7 | 5085.0 | 5590.8 | 4884.6 | 5102.2 | 5595.5
$\xi^{2a}$ | 13210.3 | 13668.8 | 14950.9 | 13273.1 | 13698.8 | 15088.9 |  | 13245.1 | 13701.9 | 14949.2 | 13177.9 | 13682.6 | 15224.2
$\xi^{2b}$ | 16347.2 | 16845.4 | 18250.6 | 16353.2 | 16839.6 | 18467.3 |  | 16314.3 | 16711.2 | 18310.7 | 16260.2 | 16831.1 | 18364.0
$\xi^{3a}$ | 26696.6 | 27567.5 | 29709.6 | 26704.5 | 27646.9 | 30247.4 |  | 26664.9 | 27421.2 | 30053.0 | 26627.2 | 27437.3 | 30369.4
$\xi^{3b}$ | 36557.6 | 37674.3 | 40459.2 | 36582.8 | 37775.4 | 40933.9 |  | 36529.7 | 37550.3 | 40509.2 | 36598.9 | 37750.6 | 40952.3
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 130.6 | 156.5 | 241.4 | 131.8 | 156.6 | 228.5 |  | 130.3 | 158.3 | 234.4 | 132.1 | 163.7 | 220.1
$\xi^{1b}$ | 131.5 | 160.8 | 239.2 | 131.0 | 158.5 | 220.7 |  | 133.3 | 158.4 | 235.4 | 130.1 | 160.0 | 230.1
$\xi^{2a}$ | 658.2 | 782.7 | 1109.9 | 662.3 | 788.1 | 1063.8 |  | 652.1 | 785.0 | 1135.4 | 661.0 | 791.9 | 1066.1
$\xi^{2b}$ | 692.2 | 815.4 | 1214.7 | 687.5 | 834.8 | 1100.9 |  | 695.5 | 820.5 | 1180.9 | 690.6 | 830.1 | 1110.8
$\xi^{3a}$ | 1838.5 | 2128.6 | 2906.0 | 1846.6 | 2164.8 | 2827.8 |  | 1847.1 | 2098.3 | 2995.7 | 1859.1 | 2182.8 | 2885.5
$\xi^{3b}$ | 2139.4 | 2420.7 | 3305.6 | 2126.6 | 2455.3 | 3272.6 |  | 2107.2 | 2437.5 | 3259.1 | 2137.0 | 2419.6 | 3328.8
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 339.7 | 383.8 | 523.7 | 342.2 | 390.8 | 527.5 |  | 340.6 | 386.0 | 520.0 | 342.7 | 388.3 | 528.1
$\xi^{1b}$ | 379.3 | 424.8 | 567.2 | 384.5 | 435.2 | 560.5 |  | 378.3 | 428.6 | 565.6 | 379.4 | 432.1 | 577.9
$\xi^{2a}$ | 2331.7 | 2538.8 | 3233.3 | 2342.4 | 2557.5 | 3280.3 |  | 2324.1 | 2555.6 | 3233.9 | 2336.6 | 2565.0 | 3267.6
$\xi^{2b}$ | 2753.2 | 2979.5 | 3705.3 | 2756.6 | 3011.0 | 3791.3 |  | 2730.9 | 2976.9 | 3753.3 | 2738.7 | 3030.6 | 3741.5
$\xi^{3a}$ | 8967.6 | 9655.6 | 11313.2 | 8955.7 | 9653.9 | 11593.3 |  | 8958.3 | 9595.8 | 11302.3 | 8922.0 | 9616.2 | 11665.8
$\xi^{3b}$ | 11979.9 | 12682.9 | 14571.5 | 12049.5 | 12751.3 | 14897.9 |  | 12063.2 | 12699.1 | 14705.5 | 12071.2 | 12689.7 | 14794.7
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 519.2 | 568.8 | 726.0 | 513.6 | 575.8 | 724.3 |  | 511.8 | 569.1 | 731.6 | 518.6 | 578.1 | 737.0
$\xi^{1b}$ | 574.4 | 640.2 | 806.4 | 576.4 | 634.2 | 821.1 |  | 576.8 | 636.2 | 805.5 | 579.1 | 645.1 | 813.9
$\xi^{2a}$ | 2897.3 | 3111.9 | 3856.7 | 2888.5 | 3159.2 | 3974.0 |  | 2875.6 | 3129.3 | 3868.2 | 2889.7 | 3156.2 | 3931.6
$\xi^{2b}$ | 3477.8 | 3761.0 | 4556.0 | 3481.6 | 3775.2 | 4613.1 |  | 3494.4 | 3763.1 | 4572.9 | 3489.5 | 3812.7 | 4679.3
$\xi^{3a}$ | 15904.3 | 16738.7 | 18784.8 | 15905.5 | 16719.6 | 19046.2 |  | 15883.1 | 16660.3 | 18772.6 | 15990.8 | 16718.3 | 19092.6
$\xi^{3b}$ | 19082.9 | 19943.9 | 22286.1 | 19081.7 | 19990.4 | 22609.8 |  | 18973.6 | 19903.4 | 22353.6 | 19040.1 | 19888.6 | 22843.3

  * •

The Expected Shortfall at level $p=0.9$ of the total loss is approximated
using $10,000$ independent samples of $L$.

Table 5: Expected Shortfall at Level $p=0.9$ of the Total Loss.

| Binomial Model |  | Poisson Model
---|---|---|---
| Gamma | Log-Normal |  | Gamma | Log-Normal
| $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ |  | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$ | $c_{v}=0.5$ | $c_{v}=1.0$ | $c_{v}=2.0$
| Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 781.9 | 840.4 | 1007.2 | 787.2 | 838.1 | 1012.7 |  | 779.0 | 844.5 | 1019.1 | 787.2 | 850.7 | 997.9
$\xi^{1b}$ | 760.3 | 818.7 | 982.2 | 769.6 | 814.9 | 978.3 |  | 760.0 | 820.8 | 978.6 | 757.8 | 823.5 | 977.2
$\xi^{2a}$ | 2119.8 | 2292.5 | 2840.8 | 2124.7 | 2275.7 | 2773.4 |  | 2097.7 | 2273.9 | 2865.1 | 2107.4 | 2305.6 | 2735.3
$\xi^{2b}$ | 2143.4 | 2317.1 | 2834.7 | 2152.1 | 2340.0 | 2803.0 |  | 2127.7 | 2344.1 | 2849.9 | 2118.2 | 2337.5 | 2805.7
$\xi^{3a}$ | 3251.4 | 3529.7 | 4454.2 | 3309.8 | 3561.7 | 4312.2 |  | 3262.0 | 3560.0 | 4545.6 | 3260.0 | 3560.8 | 4310.9
$\xi^{3b}$ | 3416.7 | 3691.4 | 4645.1 | 3414.2 | 3695.1 | 4506.5 |  | 3418.6 | 3704.2 | 4610.5 | 3374.7 | 3775.3 | 4425.9
$\mathbf{\rho^{\Phi}=0.5:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 2763.8 | 2834.9 | 3194.6 | 2752.6 | 2887.8 | 3166.0 |  | 2759.9 | 2857.7 | 3180.4 | 2757.9 | 2836.4 | 3154.5
$\xi^{1b}$ | 3035.9 | 3156.2 | 3517.7 | 3038.5 | 3165.7 | 3442.4 |  | 3063.8 | 3164.7 | 3533.1 | 3051.4 | 3165.4 | 3449.1
$\xi^{2a}$ | 7392.6 | 7719.9 | 8776.3 | 7392.8 | 7746.9 | 8611.8 |  | 7388.3 | 7723.1 | 8614.5 | 7427.2 | 7746.2 | 8626.5
$\xi^{2b}$ | 9288.1 | 9640.5 | 10722.0 | 9310.3 | 9654.7 | 10796.5 |  | 9298.9 | 9675.0 | 10743.0 | 9307.5 | 9662.4 | 10692.0
$\xi^{3a}$ | 13018.0 | 13556.5 | 15357.5 | 13035.5 | 13626.0 | 15409.5 |  | 12969.0 | 13618.0 | 15313.5 | 13044.5 | 13588.0 | 15245.5
$\xi^{3b}$ | 16932.0 | 17548.0 | 19415.5 | 16980.5 | 17664.0 | 19639.0 |  | 16992.0 | 17557.5 | 19533.5 | 16878.5 | 17556.5 | 19507.5
$\mathbf{\rho^{\Phi}=0.9:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 3938.9 | 4059.4 | 4430.8 | 3939.9 | 4069.6 | 4396.3 |  | 3921.6 | 4085.6 | 4449.4 | 3931.8 | 4094.0 | 4414.9
$\xi^{1b}$ | 4853.6 | 5023.9 | 5449.5 | 4833.3 | 5012.8 | 5355.8 |  | 4847.5 | 5015.1 | 5410.6 | 4819.0 | 5009.6 | 5345.6
$\xi^{2a}$ | 13059.5 | 13482.0 | 14591.5 | 13115.0 | 13535.0 | 14544.5 |  | 13117.5 | 13480.5 | 14580.5 | 13015.0 | 13508.5 | 14641.5
$\xi^{2b}$ | 16158.5 | 16606.0 | 17873.5 | 16176.0 | 16587.0 | 17866.0 |  | 16176.0 | 16490.0 | 17951.0 | 16087.5 | 16617.5 | 17774.5
$\xi^{3a}$ | 26393.5 | 27251.0 | 29085.5 | 26415.0 | 27246.5 | 29354.5 |  | 26378.5 | 27064.0 | 29458.0 | 26324.5 | 27095.0 | 29377.5
$\xi^{3b}$ | 36241.5 | 37249.0 | 39731.0 | 36210.5 | 37290.5 | 39735.0 |  | 36181.0 | 37099.5 | 39811.0 | 36260.0 | 37316.0 | 39890.0
| Non-Uniform Accident Occurrence
$\mathbf{\rho^{\Phi}=0.1:}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$\xi^{1a}$ | 123.5 | 144.5 | 210.9 | 124.6 | 141.3 | 183.1 |  | 123.4 | 146.4 | 206.3 | 125.3 | 148.4 | 177.3
$\xi^{1b}$ | 125.3 | 148.6 | 207.6 | 122.9 | 141.9 | 176.0 |  | 125.4 | 148.0 | 206.1 | 123.4 | 145.3 | 181.7
$\xi^{2a}$ | 626.3 | 730.9 | 989.5 | 634.1 | 733.8 | 917.8 |  | 618.9 | 730.9 | 1026.8 | 634.2 | 726.4 | 915.9
$\xi^{2b}$ | 659.5 | 772.7 | 1094.8 | 658.2 | 763.1 | 942.9 |  | 663.5 | 765.0 | 1054.2 | 654.6 | 772.0 | 932.8