C.1. Grossi  Catastrophe Modeling
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Users of cat model for risk management
 insurers and reinsurers: evaluate their own exposure to catastrophe losses
 reinsurance brokers: assess risk for their clients to send to reinsurers
 capital markets: more accurately price cat bonds
 government: regulators and emergency management agencies

2 main risk management strategies
 risk reduction: nonrenewing policies, limiting coverage, changing rates or deductibles
 risk transfer: insurer purchasing reinsurance, or issuing catastrophe bonds

Why we can't use regular statistical tools
 insufficient historical claim data for catastrophes
 limited data is often inappropriate due to changing factors (building codes, costs of repairs)

4 basic components (modules) of a catastrophe model
 hazard: e.g. EQ hazard is defined by its epicentre location and moment magnitude
 inventory (exposure): portfolio of properties at risk (location, limits and deductibles, ...)
 vulnerability: susceptibility to damage of the structures at risk (quantify hazard physical impact)
 loss: calculate direct (cost to repair and/or replace) and indirect (business interruption, relocation)

Hazard Module Components
 location of potential future events: for EQ this would be regions around faults
 frequency: most critical and uncertain aspect of the model's hazard module
 severity: model the physical parameters that describe the hazard at its source

Inventory Module
 data includes number of properties and their value by LOB, coverage, occupancy, construction type
 inventory data should also reflect regional differences in construction practices and building codes

Vulnerability Module
 engineering judgment: based on expert opinion; simple, but not easy to update and arbitrary
 building response analysis: based on advanced engineering techniques; more accurate, but based on specific buildings so can't be applied to an entire portfolio
 classbased building response analysis: divide risks based on building characteristics; identify the typical building and analyze it in details; evaluate building performance for each class, computing damage ratios by coverage

Loss Module
 linking event parameters directly to expected loss: not easily updated
 instead favour model that first estimates the physical damage from an event, and then use a cost analysis to translate this into monetary loss
 estimation of insured losses are computed by applying policy conditions to total loss estimate

Exceedance Probability
 Occurence EP: Pr(at least one loss > specified amount) in a given time period
 Aggregate EP: Pr(Σ losses > specified amount) in a given time period
 Conditional EP: Pr(single event amount > specified amount) given that the event occurs

Solving for OEP
 E_{i} = disaster event with occurrence probability p_{i} and loss L_{i}
 sort the events in decreasing order by size (E_{1} = largest, E_{2} = second largest, etc.)
 E[L_{i}] = p_{i}*L_{i} so Average Annual Loss = AAL = Σ p_{i}*L_{i}
 OEP(L_{i}) = P(L>L_{i}) = 1  P(L ≤ L_{i}) = 1  ∏(1  p_{j})
 Note that P(L = 0) = P(no event occur) + P($0 event occur)
 OEP graph has Loss in X axis, OEP in Y

Conditions for insurability of a risk
 ability to identify and quantify, or at least partially estimate, the chances of a loss occurring and the extent of losses likely to be incurred
 ability to set premium for each potential customer or class of customers

Considerations in setting rates for catastrophe events
 appears to not be impacted by adverse selection or moral hazard
 uncertainty of losses: wide variation in loss distributions
 state regulation, competition
 highly correlated losses: involves spatially correlated losses or the simultaneous occurrence of many losses from a single event.
 liquidity of assets: need cash to pay cat losses, so losing on investment income

Role of catastrophe modeling in an insurance company’s financial management
 capital allocation: influences calculation of cost of capital, marginal risk, VaR, incremental risk
 ERM: risk of ruin, asset/liability management, solvency/sustainability, return on capital

Relevant actuarial principles re: catastrophe modeling
 Principle 1: a rate is an estimate of the expected value of future costs
 Principle 2: a rate provides for all costs associated with the transfer of risk
 Principle 3: a rate provides for the costs associated with an individual risk transfer
 Principle 4: a rate is reasonable and not excessive, adequate, not unfairly discriminatory if it is an actuarially sound estimate of expected value of all future costs associated w/ individual risk transfer

Simple Rate Making Model
 Premium = AAL + Risk Load + Expense Load,
 AAL = ∑p_{i}L_{i} and we can use σ(EP curve) as risk load
 σ = √ [ ∑(L_{i}^{2}p_{i})  AAL^{2} ]
 determining whether to provide coverage P(Loss > nz + A) < p_{1}

Differentiating Risk
 structure attributes: features of insured risk related to the physical performance of a building in an extreme event; e.g. construction, building code, year built, occupancy
 location attributes: reflect the degree to which structure are subject to damages from hazards as a function of where they are built (e.g. flood plain, earthquake faults)

Regulation vs Catastrophe Modeling
 regulators have not been supportive of having modelgenerated information introduced
 difficult to evaluate the models since they require subject matter experts
 modeling firms are not willing to share proprietary information
 despite models being scientifically rational, it’s also a black box that can be used to inflate prices
 still, regulators formed commissions made up of technical experts to certify model reasonableness

California Earthquake Authority (CEA)
 after 1996 Northridge EQ caused $b in losses, California legislature created CEA to avoid insurers leaving the market
 constraints on CEA ratemaking: actuarially sound & scientific info must be consistent with available geophysical data and current knowledge of scientific community

CEA challenges
 recurrence rates: frequency used was twice as high as historical
 time dependent probabilities: EQs are not time independent, except they used Poisson
 damage estimates: damage curves used were based on just 1 event
 underinsurance factor: modeled losses are expressed as % of TIV
 demand surge: it's hard to quantify
 policy sublimits: claims data wasn't detailed enough
 rating plan deviation: grouped high and low risk for affordability reasons (potential adverse sln)
 retrofit discount: 5+% discount to policyholder retrofitting their homes (not based on data)

Open Issues for Using Cat Models to Determine Rates
 regulatory acceptance: regulators don't have the technical expertise to assess models
 public acceptance: acceptance is low, mostly because it results in higher rates
 actuarial acceptance: need to get familiar with models, since it's outside of usual expertise
 modeltomodel variance: can be significant, should use one or more submodels

Actuarial Standards Board requirement re: cat models
 determine appropriate reliance on experts
 have a basic understanding of the model
 evaluate whether the model is appropriate for the intended application
 determine that appropriate validation has occurred
 determine the appropriate use of the model

Types of Uncertainty
 aleatory: inherent randomness of natural hazard events; can't be reduced by collecting more data
 epistemic: uncertainty due to lack of information or knowledge of the hazard (can be reduced)

Sources of Uncertainty
 limited scientific knowledge
 lack of historical data, including specific gaps on GIS data, market values
 cross disciplinary nature, as it's based on many assumptions from engineers, actuaries, seismology experts
 laboratory testing of structural material has been limited to certain types of materials only

Representing and Quantifying Uncertainty
 logic trees: assign alternative parameter values for each model component, and compute the resulting weighted average result (can use credibility as weights); tractable and useful as a communication tool, but weights are often based on expert opinion (and thus biased)
 simulation techniques: most widely used quantitative approach as it can be modelled as an extremely complex process; simulate each parameter value, then compute many sample runs
 blended approach: used to create exceedance probability curve; instead of manually selecting the tree branches, use the simulation

Catastrophe Modeling  Bottomup approach
 provides the most robust means to quantify portfolio risk
 losses are first calculated at the location level based on 3 thresholds: exposure, limit, deductible
 next all location losses are aggregated by policy, and then by portfolio
 could also aggregate by zip code to limit exposure in high risk zones

Special issues regarding Portfolio Risk
 data quality: can reduce or increase degree of epistemic uncertainty depending on accuracy of data for construction type, building age, soil data, etc.
 uncertainty modeling: can't only rely on μ and σ, we need the entire loss distribution
 impact of correlation: diversification vs high concentration

Considerations when adding a policy to a portfolio
 magnitude of the risk
 correlation with existing portfolio
 highest price the risk is willing to pay

2 critical questions for Portfolio Managers re: cat risk
 what is the average annual loss?
 what is the likelihood that the insurer will go insolvent?