Financial Modeling of Extreme Events Thomas Weidman CAS Spring Meeting May 19, 2003

Financial Modeling of Extreme Events

Thomas Weidman

CAS Spring Meeting

May 19, 2003


defining and modeling extreme events – insured vs. total financial impact

financial event modeling correlated events: insured + financialcase study: capital management

Defining Extreme Events

Miami HurricaneSan Francisco EQSeptember 11, 2001


0

10

20

30

40

50

60

70

80

San Fran EQ Miami 11-Sep

Insured Damage ($B)


SARSWest Nile VirusSpanish Flu


SARS virus – first outbreak, China Nov 2002

West Nile Virus – first cases in western world 1999

Influenza – first description from 412 B.C. 0

5000

10000

15000

20000

25000

30000

35000

40000

Sar wn Flu

cases

deaths

3-DColumn 3


AsbestosTobaccoShareholders’ Class Actions


Asbestos: $200 billion cost/$100 billion insured

Tobacco: $246 billion settlement with state governments

Tort Costs: $205 billion/$146 billion insured in 2001, a 14% increase over 2000

[source: US Tort Costs-2002 Update, Tillinghast]


Stock Market Credit Markets


Stock Market Returns: (65)% in 1929-33 (37)% in 1973-4 (38)% in 2000-3

Bond Market Returns: (8)% in 1999 (7)% in 19942 worst annual returns

in past 100 years


Pricing InadequacyReserving Inadequacy


Pricing Inadequacy

AY loss ratios 10 points higher than CY loss ratios from 1997 through 2000

Reserving Inadequacy

$48 to $92 billion at December 2001 excl asbestos and environmental (ISO)


Rogue TraderRogue UnderwriterRogue Agent/Broker


Operational Risk:

Risk of direct and indirect loss resulting from failed or inadequate process, systems, or people and from external events

Difficult to quantify, see Basel accords for treatment by banks

Summary of Risk Types and Models

Risk Type:Risk Type:CatastropheNon-catastropheReservesMarketCreditOperational

Risk Model:Risk Model:AIR, RMS, EQEExposure x freq x sevReserve rangesVaR modelsDefault models ?? Basel II?

Quantifying Extreme Events

Historical dataEmpirical distributionsRealistic Disaster ScenariosModelsFitted probability distributionsExtreme value theory

Extreme Value Theory

Based on work describing the extreme behavior of random processes

Extrapolate the tail of a distribution from underlying data

Distributions to fit tails:– Generalized Pareto Distribution (GPD)– Generalized Extreme Value (GEV) Extrapolate the tail

of a distribution from underlying dataProvides a rigorous framework to make

judgments on the possible tail


GEV family of distributions:

Mn = Max{x1,x2,x3,….xn} for n sufficiently large

“What is the maximum loss to be expected in one year?”


Generalized Pareto Distribution (GPD) fits tails of distributions above a threshold

Pr (Y>y+u|y>u) for large u

“What is the expected loss to an excess layer?”


Resources:

The Management of Losses Arising from Extreme Events, GIRO 2002

Kotz and Nadarajah, Extreme Value Distributions

Coles, An Introduction to Statistical Modeling of Extreme Events

Embrechts, etal., Modeling Extremal Events

Modeling Financial Events: VAR

VAR is a method of assessing market risk that uses standard statistical techniques routinely used in other technical fields.

VAR is the maximum loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger.

A bank might have a daily VAR of its trading portfolio of $35 million at the 99% confidence level.

Modeling Financial Events: Credit Risk

Credit Risk ModelsDefault ratesLoss Given Default (LGD)Migration matrices


Default Rates = Frequency of loss = MortalityQuantitative Models for Credit Assessment1. Identify characteristics that differentiate

defaulting firms (e.g., Altman 1968); credit scoring models

2. Use credit market prices to estimate default rates

3. Structural models – use equity option pricing techniques (both equity and debt are options on the value of a firm’s assets)


Loss Given Default = SeverityMany models assume a constant loss given

default Dependent on both exposure volatility and

recovery rate volatilityCorrelated with default rates?


Credit Migration MatricesHistorical changes in credit rating of obligors ‘loss triangles’ for credit ratingsUse S&P or Moody’s dataUseful for portfolio risk assessment, pricing

credit derivatives, capital requirementsDependent on current and future economic

conditions ( recession vs. expansion)

Summary of Risk Types and Models

Risk Type:Risk Type:CatastropheNon-catastropheReservesMarketCreditOperational

Risk Model:Risk Model:AIR, RMS, EQEExposure x freq x sevReserve rangesVaR modelsDefault models ?? Basel II?

Capital Management

Market Share of Industry LossProbable Maximum Loss

(PML)/Aggregate ExposureRisk of Ruin Approach:

Pr (insolvency) < p over time period t

where p is small, e.g., .01 or .001

Capital Management - Issues

Consistent definition across all risk typesCorrelations across risksAllocation/attribution of capital to productAccounting framework: GAAP vs. Fair

Value Matching capital to management

responsibilities, e.g., assets vs. liabilities

Correlated (Extreme) Events

Global warming – storms – virusesLawyers’ fees from tobacco/asbestos winsStock markets – D&O/E&O claimsCredit - Equity pricesPricing – Reserving (e.g., B-F methods)Catastrophes – Demand Surge –

Reinsurance Recoverable


Exposure: cat Non-cat

reserves market credit Ops

risk

Property X X x x x ?

Casualty x X X x X ?

Surety x X x x X ?

Inv Assets x x x X X ?


Cas

cat

Cas

Non-cat

Cas

reserves

Cas

market

Cas

credit

Cas

ops

Prop-cat Low Low Low Low Low ?Prop-non-cat

Low Med Low Low Low ?

Prop reserves

Low Low Med Low Low ?

Prop market

Low Low Low High Med ?

Prop- credit

Low Low Low Med High ?


Generally impossible to model joint distributions of risks (unless multivariate normal)

Therefore:Estimate distributions for each risk typeCombine distributions into a joint

distribution using ‘copulas’


Copulas:Multivariate functions that combine marginal

distributions into a joint distributionUsing a normal copula leads to a simpler

approach for Monte Carlo simulation of correlated variables

CAS papers by Wang (1998) and Meyers (1999)


Past experience lacks credibility

Current state of the art:Sophisticated risk models across all types of risk Integration/Correlation of risk models important

to management, rating agencies and regulatorsMajor role for actuaries

Documents

Financial Modeling of Extreme Events Thomas Weidman CAS Spring Meeting May 19, 2003