Microscopic traffic models, accidents, and insurance losses

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 such a counterfactual analysis, it is possible, from an insurance, but also from an engineering perspective, to assess the impact of changes in the design of vehicles and transport systems in terms of their impact on road safety and functionality. 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.


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 (NHTSA (2022)). 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: (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.
(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" (Jones et al. (2020)). 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 (Lopez et al. (2018)) 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 El Karoui & Jiao (2009) 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: (i) We develop a powerful methodological framework to generate accident data based on microscopic traffic models in analogy to the concept of digital twins.
(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.
(iii) Structurally, we characterize the total loss distribution approximatively as a mean-variance mixture. This also yields alternative valuation procedures.
(iv) Based on Stein's method, we obtain a correction term in the valuation, derived from the results of El Karoui & Jiao (2009), which enables surprisingly accurate pricing of insurance contracts.

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.

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., Gazis (2002)) 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., Acemoglu et al. (2018), Colini-Baldeschi et al. (2020)).
Stochastics can be used to extend such risk considerations in terms of uncertain travel times (e.g., Nikolova & Stier-Moses (2014)). 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 (Maxwell et al. (2010)) or the allocation of capacity in bike-sharing systems (Freund, Henderson & Shmoys (2022)).
To model transportation systems at a level of higher granularity, microscopic traffic models are used (see, e.g., Helbing (2001)). 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 (Fellendorf & Vortisch (2010)) and Aimsun (Casas et al. (2010)). While SUMO is open source software, these competitors are commercial.
Leveraging the simulator Aimsun, Osorio & Bierlaire (2013) 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. Osorio & Nanduri (2015) 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. Flötteröd, Bierlaire & Nagel (2011) propose a Bayesian approach to calibrating travel demand. Zhang, Osorio & Flötteröd (2017) discuss the calibration of largescale traffic simulators; Osorio & Punzo (2019) 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 Lord & Mannering (2010) and Tsoi & Gabler (2015). Statistical modeling approaches permit inference when sufficient data are available on the level of the granularity of the analysis. For example, Yu et al. (2019) estimate the impact of microscopic traffic variables on crash risks. Ortelli, Lapparent & Bierlaire (2021) 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. Berkhahn et al. (2018) and Berkhahn et al. (2022)) -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., Zhao et al. (2018)). These approaches rely on importance sampling techniques to overcome the rareevent nature of safety-critical situations. Arief, Glynn & Zhao (2018) develop simulation-based testing methodologies in order to analyze autonomous vehicles in relevant scenarios that are constructed using collected data. Norden, O'Kelly & Sinha (2019) 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 McNeil, Frey & Embrechts (2015) and Wüthrich (2013) 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., Denuit et al. (2007)) to incentivize more careful driving and prevent insurance fraud.
Novel pricing approaches use telematics technology (see, e.g., Husnjak et al. (2015) 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 Gao, Meng & Wüthrich (2022) for a methodological overview. Verbelen, Antonio & Claeskens (2018) discuss telematics pricing and use generalized additive models to interpret the impact of telematics variables on accident frequency. More recently, Henckaerts & Antonio (2022) 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.

Model Components
Our microfounded simulation model for investigating accident losses is based on two components: (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.
(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.

Microscopic Traffic Networks
We consider a road network that is typically embedded into a two-dimensional area A ⊆ 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 M. Each vehicle 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 their velocity, and by a i (t) = d dt v i (t) their acceleration. We make the implicit assumption that vehicles are located in O i until some release time and remain 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) Φ ⊆ M, we will write M Φ (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 1 . 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 γ(t) = x i (t), v i (t), a i (t) i∈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 → γ(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 ⊆ A such that A = R r=1 A r and R ∈ N. We call the elements A r of the partition a traffic module.
We let M r (t) = i ∈ M(t) : x i (t) ∈ A r ⊆ M(t) denote those vehicles that are in A r at time t (with M Φ r (t) defined in analogy). The local traffic state of the module is γ r (t) = (x i (t), v i (t), a i (t)) i∈Mr(t) . Key local traffic characteristics (density, flow, speed, etc.) can then be expressed as functions of γ r (t) and its evolution over small time windows.

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 γ k := (γ k (t)) t∈[0,T ] with k ∈ {1, 2, . . . , K} for a short time horizon T > 0. The aim is to analyze characteristics of traffic over the long time horizon N T , e.g., one year, for some large N ∈ N; this is modeled by a finite sequence of traffic scenarios (k 1 , k 2 , . . . , k N ) ∈ {1, 2, . . . , 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 N T . 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 µ 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 N T . The vector µ = (µ 1 , µ 2 , . . . , µ K ) lies in the simplex ∆ K−1 = {x ∈ R K + : K k=1 x k = 1}. We assume that µ 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 µ. 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.
(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 → γ k (t), and we will discuss concrete specifications later. Given a realization of µ, accidents are assumed to be independent across time buckets. This implies that traffic scenario k occurs for N µ k time buckets corresponding to a duration of N T µ k , and the number of accidents C k during this period has a conditional Binomial distribution with parameters p k and N µ k : Model. An alternative model assumes that accidents occur at random times with a distribution governed by an intensity λ 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 N T µ k is conditionally Poisson distributed with parameter λ k N µ 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 N T are equal to where the random variables X k c , k = 1, 2, . . . , K, c ∈ N, are independent and X k c ∼ F k , c ∈ 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 γ k r in each module r ∈ {1, 2, . . . , 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 λ 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 = R r=1 p k r and λ k = R r=1 λ 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 λ k r , r = 1, 2, . . . , R, determine the accident times for each module. The resulting sequence of random times in the whole traffic system possesses the intensity λ k .
Conversely, if we first simulate random times with intensity λ k and then randomly choose a corresponding module with probability λ k r /λ k in a second step, the random times associated with each module r possess intensity λ 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,ψ ) ψ∈Ψ where ψ 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 ψ. The resulting distribution F k is a mixture of the distributions (F k,ψ ) ψ∈Ψ . 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 : R → R be an increasing function, we analyze h(L). In particular, we investigate the following functions corresponding to three In each case, we evaluate various statistical functionals: p VaR q (h(L)) dq. 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. in addition a detailed user documentation can be found online 2 . 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 addressing 3 , e.g., traffic light optimization, routing, traffic forecasting, and autonomous driving.

A Brief Overview
In the following, we give a short overview. At its core, SUMO is a software which generates a traffic scenario γ = (γ(t)) t∈[0,T ] from a given set of input files: (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. 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. 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 Wegener et al. (2008)). TraCI provides online access to the microscopic traffic simulation and permits, at each time step, through a comprehensive list of commands 5 to retrieve data and to change the states of objects such as vehicles, roads, traffic lights, etc.
Available in standard programming languages 6 , 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.

Generation of Traffic Scenarios
To represent traffic in a given area over a longer time horizon (e.g., N T = 1 year), we generate a diverse collection of traffic scenarios γ 1 , . . . , γ K of duration T in SUMO by varying the input 4 The problem of allocating traffic demand to routes in a network is referred to as the traffic assignment problem.
A standard reference is Patriksson (2015). 5 A detailed description can be found at sumo.dlr.de/docs/TraCI.html. 6 Our case studies are based on the Python implementation.
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 7 that visualizes a SUMO scenario and allows to modify its properties. In practice, network files are typically imported from other data sources 8 . For example 9 , 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 SUMO 10 . 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 attributes 11 . 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 12 which deduces traffic demand from general assumptions on the structure of the population (inhabitants, households, etc.) in the considered area. The tool activitygen 13 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 scenarios 14 ; 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.
Maze, Agarwal & Burchett (2006) 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, Phanse, Chaturvedi & Srivastava (2022) implement reduced velocities due to rainfall. In Weber, Driesch & Schramm (2019), 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 Bera & Rao (2011) for an overview. With increasing data availability, the estimation can be enhanced by floating car data (cf., e.g., Nigro, Cipriani & Giudice (2018)), 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., Soriguera (2012)). 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., Xia et al. (2018)).
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 SUMO = 24 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 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. Wagner 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. Pagany (2020) study the impact of wildlife on traffic, an issue that is also relevant in the context of traffic accidents.

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 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 Frey, Popp & Weber (2008) and El Karoui & Jiao (2009). The latter paper provides a correction term derived by Stein's method that we will exploit in our application.

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 ∈ N, for each traffic scenario k = 1, 2, . . . , K from the corresponding distribution F k . These tasks can be performed separately, but require both a prior evaluation of the microscopic traffic model.
(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 λ k as well as the corresponding values p k r and λ k r on the level of the modules r = 1, 2, . . . , R.
(ii) Number of Accidents. Sampling from µ 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.
(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 ψ are extracted from an additional single run of traffic scenario k. Given ψ, the losses are generated according to conditional loss distributions F k,ψ 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.

Gaussian Approximation
Another way to compute 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 µ 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 µ, i.e., suppose that µ is fixed and given. The general results for µ random are then a corollary by considering suitable mixtures according to the distribution of µ. The total random losses in the Binomial model can be rewritten as where the random variables 1 k c , X k c , k = 1, 2, . . . , K, c ∈ N, are independent, X k c ∼ F k , and 1 k c are Bernoulli random variables taking the value 1 with probability p k and the value 0 otherwise, c ∈ N, for any k. Setting 15 We focus on three types of insurance coverage, h(x) = x (full coverage), h(x) = max(x − θ, 0), θ ≥ 0 (constant deductible), h = min(x, θ), θ ≥ 0 (stop loss), and obtain an approximation E(h(Z)) of E(h(L)) in each of these cases.
On the basis of Stein's method 16 , El Karoui & Jiao (2009) suggest correction terms in order to improve the approximation, i.e., the approximation E h K k=1 N µ k m k + Z is replaced 15 The estimation of m k , σ k 2 , and ζ k 3 requires the simulation of the random variables X k c , c ∈ N. The independent terms 1 k c , c ∈ N, factor out, are idempotent and have known expectation p k . 16 For an overview on Stein's method we refer to Chen, Goldstein &Shao (2011) andRoss (2011). by the corrected approximation The correction C h depends on the degree of smoothness of the derivatives of the function h and thus differs 17 for the three types of coverage. We define We obtain the following correction terms: In the case of full coverage, the correction term of El Karoui & Jiao (2009) disappears. In general, if h is some Lipschitz function with bounded third derivative, the correction term equals can be written as the difference between full coverage x → x and a constant deductible x → max(x − θ, 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 , σ k 2 , and ζ k 3 have been computed for each traffic scenario k = 1, 2, . . . , K, no further data need to be stored or sampled in order to compute E(h(L)). The approximate representation of the distribution of L as a mean-variance-mixture is a considerable simplification.

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.

SUMO Scenario & Accident Data
Wildau is a small German city of approximately 10,000 inhabitants, located around 30 km southeast 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 available 18 .
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 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 s. Traffic scenarios k = 1, 2, . . . , 50 correspond to selected time intervals from the SUMO scenario. Traffic scenarios k = 51, 52, . . . , 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.
Accident Data. The German Accident Atlas 19 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 % of all road accidents involved personal damage 20 . We use an estimate ofc year = 48/11.8 % ≈ 407 accidents for calibration purposes.

Model Specification
Our goal is to analyze accident losses for a fleet Φ 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 Φ of vehicles whose driving characteristics we can vary, we define a new vehicle type Φ and construct corresponding new SUMO scenarios. Fixing a fraction ρ Φ ∈ [0, 1] of vehicles belonging to Φ, we retain approximately 1 − ρ Φ of the existing flow definitions and modify ρ Φ of the flow definitions suitably in order to model the fleet. In our case studies, we consider ρ Φ = 10 %, 50 %, 90 %.
Driving Configuration. Vehicles in a fleet Φ are of the same type. Various characteristics can be varied in SUMO; we focus on maximal speed v max , maximal acceleration a max > 0, and time headway ζ > 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 Φ and keep all other vehicles as originally introduced 21 . A driving configuration of vehicles in fleet Φ is denoted by ξ = (v max , a max , ζ). Specifically, we consider 22 : 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 c year ≈ 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 λ 1,k and λ 2,k for the Poisson model.
(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 =c year /N ≈ 7.7 · 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 Φ is This probability is used in the Binomial model. For the Poisson model we set λ Φ,1,k = p Φ,1,k , k = 1, 2, . . . , 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. 21 We 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.  (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, . . . , K and each module r = 1, 2, . . . , R pairs (d k r ,v k r ) 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, . . . , R, we choose benchmark values d * r andv * r for the density and velocity and specify occurrence probabilities and intensities that vary spatially and over time: The last term refers to deviations of the time headway from SUMO's default value of 1.0 s: a larger time headway is associated with less risky driving. We set p Φ,2,k = R r=1 p Φ,2,k r and λ Φ,2,k = R r=1 λ Φ,2,k r .
In our case studies, we will consider a grid of R = 4 modules and compute d k r andv k r 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 * , and ζ Φ = 1, then we essentially recover on average the case of uniform accident occurrence. Accident Losses. The distributions F k of accident losses associated with a traffic scenario k = 1, 2, . . . , 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 ∈ [0, T ], and we assume that the time τ of the accident conditional on its occurrence is uniformly distributed on [0, T ], i.e., We choose a module R at random in which the accident occurs and assume, respectively, that 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 ρ Φ 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 τ a single vehicle I uniformly at random in module R, i.e., its conditional distribution is 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 ψ = v I and assume that the conditional loss distribution F k,ψ is known 23 . If we denote the distribution of ψ by L k we obtain the distribution F k as a mixture In our case studies, we will assume that F k,ψ = F ψ for all k; however, the mixing distribution L k will dependend on the traffic scenario k. We consider the following examples for F ψ : (i) Gamma Distribution. We define distributions with varying levels of dispersion 24 . For a given coefficient of variation c v ∈ {1/2, 1, 2}, we choose The expectation of this distribution is ψ 2 and increases quadratically with ψ, 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 ψ equals c 2 v ψ 4 , hence the coefficient of variation is indeed c v .
(ii) Log-Normal Distribution. We consider log-normal distributions with expectation ψ 2 and variance c 2 v ψ 4 , implying that the coefficient of variation is again c v ∈ {1/2, 1, 2}. This log-normal distribution is obtained as the distribution of exp(Z) for a normal random 23 We assume that F k,0 corresponds to a Dirac measure in 0; if M Φ R (τ ) = ∅, we set ψ = 0, resulting in 0 losses. 24 A measure for the dispersion of a random variable X is the coefficient of variation defined by cv = Var(X)/E(X).

Overview
We illustrate our modeling approach in case studies on multiple levels. 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: (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 E(L), Var(L), and complement these with analyses of the performance of traffic system.
(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 E(h(L)).

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 E(L) and standard deviation std(L) for different fleet models. To compare losses for different fleet sizes, we normalize losses per 100 expected insured vehicles 25 . 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 3a, 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 ξ 3· . 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 3b, 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 K k=1 µ k p k (both normalized and unnormalized) and the average accident severity K k=1 µ k X k 1 , 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. 26 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 χ 1 k , . . . , χ 40 k for each scenario k = 1, . . . , K, we compute averagesχ k = 1/40 40 i=1 χ i k . Pairing flow-occupancy values and speed-occupancy values yields empirical fundamental diagrams on the urban level (see, e.g., (2008)). For the purpose of data exploration, we draw scatter plots of these pairs for the driving configurations ξ 1b , ξ 2b , ξ 3b and fleet sizes ρ Φ = 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.

Geroliminis & Daganzo
Driving Configuration 10,000 15,000 . . To better understand the impact of individual driving configurations and fleet sizes, we study 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.

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 ρ Φ = 0.5 with driving configuration ξ 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.
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 9c shows qq-plots for the quantiles of standardized 27 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 27 Samples are standardized by subtracting their sample average and dividing by their sample standard deviation.  Driving Configuration Driving Configuration Speed (m/s)   The statistical functionals of the total loss are approximated using 10, 000 independent samples of L. 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.

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 computations 28 of m k and (σ k ) 2 .
A sample from the Normal mean-variance-mixture approximation is generated by, first, sampling µ and, second (conditional on µ), sampling the normal random variable K k=1 N µ k m k + Z with Z ∼ N 0, K k=1 N µ k σ k 2 . 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 E(L) (full coverage), E(max(L − θ, 0)) (constant deductible), and E(min(L, θ)) (stop loss) for different values of θ. The results are given in Figure 12. We obtain the typical hockey stick profiles satisfying the parity E(L) = E(max(L−θ, 0))+E(min(L, θ)).
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 E(h(L)) for the different types of coverage h. To compute the correction term C h , we need to compute 29 (ζ k ) 3 . For Gamma distributed losses, we obtain Since E(h(L)) = E(E(h(L) | µ)), we may generate samples of µ and evaluate the corresponding conditional expectations E(h(L) | µ); in the Normal mean-variance mixture approximation, these are expectations of functions of normally distributed random variables. We compute these 29 For the notation we refer to Sections 3.2 & 4.2. 0 500 1,000 1,500 2,000 2,500 3,000 0 500 1,000 Figure 12: Insurance Prices.
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 E(max(L − θ, 0)) of coverage with constant deductible, for different values of θ. 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 µ.
While the absolute error generally decreases in the deductible θ for all methods (apart from some local effects), the relative error increases for larger values of θ 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 θ. 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 θ.

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 = M = 10, 000 samples of ψ in each scenario k to approximate its distribution 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, . . . , K do Run SUMO in scenario k.
Obtain data to calculate p k r and λ k r for r = 1, . . . , R. Terminate SUMO. Set p k = R r=1 p k r and λ k = R r=1 λ k r . end for Phase 2: Pre-Sampling of ψ conditional on scenario k = 1, . . . , K.
end for for k = 1, . . . , K do SampleX k c ∼ Fψ k l . end for end for

B Electronic Companion: Tables
We provide detailed tables that contain selected statistical functionals of the total loss in our different case studies. We evaluate 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. Remarks on the Normalization. Recall that we normalize total losses in order to compare losses among fleet sizes ρ Φ = 0.1, 0.5, 0.9. More precisely, we normalize L by 100 expected insured vehicles as follows:

List of Tables
• In the underlying SUMO scneario, the route files specify a number of vehicles for each flow belonging to the fleet Φ.
• 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 =    n g , k = 1, . . . , 50, n b , k = 51, . . . , 100.
• We denote the total number of insured vehicles by n Φ . It is given by n Φ = K k=1 µ k n k . Note that this number is random as µ is random.
• We evaluate the normalized total loss per 100 expected insured vehicles. According to our specific choice of µ, it is given by   Non-Uniform Accident Occurrence ρ Φ = 0. The expectation of the total loss is approximated using 10, 000 independent samples of L. The variance of the total loss is approximated using 10, 000 independent samples of L.  The skewness of the total loss is approximated using 10, 000 independent samples of L.     The Value-at-Risk at level p = 0.9 of the total loss is approximated using 10, 000 independent samples of L.   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.     The Value-at-Risk at level p = 0.95 of the total loss is approximated using 10, 000 independent samples of L.  The Expected Shortfall at level p = 0.95 of the total loss is approximated using 10, 000 independent samples of L.     The Value-at-Risk at level p = 0.99 of the total loss is approximated using 10, 000 independent samples of L.  The Expected Shortfall at level p = 0.99 of the total loss is approximated using 10, 000 independent samples of L.  The expectation of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.  The variance of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.  The skewness of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.  The Value-at-Risk at level p = 0.9 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.  The Expected Shortfall at level p = 0.9 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.  The Value-at-Risk at level p = 0.95 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.   The Expected Shortfall at level p = 0.95 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.   The Value-at-Risk at level p = 0.99 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.   The Expected Shortfall at level p = 0.99 of the normalized total loss is approximated using 10, 000 independent samples of L. Total losses are normalized by 100 expected insured vehicles.