Structural Dependence and Stochastic Processes Don Mango American Re-Insurance 2001 CAS DFA Seminar

Structural Dependence Structural Dependence and Stochastic Processesand Stochastic Processes

Don Mango

American Re-Insurance

2001 CAS DFA Seminar

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AgendaAgenda

Just Say No to Correlation Structural Dependence in Asset and

Economic Modeling Structural Dependence in Liability

Modeling

Just Say No to Just Say No to CorrelationCorrelation

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Just Say No to CorrelationJust Say No to Correlation

Correlation has taken on something of a life of its own

It’s easy to measure You can use Excel, or @Risk People think they know what it means,

and have an intuitive sense of ranges

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Just Say No to CorrelationJust Say No to Correlation

Paul Embrechts, Shaun Wang, and others tell us: Correlation is simply one measure of

Dependence, a more general concept There are many other such measures

From a Stochastic modeling standpoint, simulating using Correlation surrenders too much control

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Simulating with CorrelationSimulating with Correlation

We think we know how to induce correlation between variables in our simulation algorithms

(At least) Two major problems: Correlation is not the same throughout the

simulation space Known dependency relationships may not

be maintained

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Correlation Not Always The Same...Correlation Not Always The Same...

Consider a well-known approach for generating correlated random variables

Using Normal Copulas Similar to the Iman-Conover algorithm

(in @Risk) which uses Normal Copulas to generate rank correlation

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Normal CopulasNormal Copulas

Generate sample from multi-variate

Normal with covariance matrix Get the CDF value for each point

[ these are U(0,1) ] Invert the U(0,1) points to get target

simulated RVs with correlation… …but what correlation will the target

variables have?

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ProblemProblem

Correlation in the tails is near 0 - extreme values are nearly un-correlated

Is this your intended result? Example….

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Known Dependencies Not Known Dependencies Not MaintainedMaintained

Simple example DFA Model for a company

Liabilities: 4 LOB: Auto, GL, Property, WC

Assets: Bonds

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Example DFA ModelExample DFA Model

Liabilities: 4 LOB: Auto, GL, Property, WC Simulation: correlated uniform (0,1] matrix

per time period used to generate the variables

Assets: Bonds Simulation: yield curve scenarios

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Example DFA Model - PROBLEMSExample DFA Model - PROBLEMS

Liabilities: Getting dependence within a year, but

what about serial dependence across years?

Could expand the correlation matrix to be [ # variables x # years ]

But what about underwriting cycles? What about the magnitude of year-over-

year changes?

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Bottom line: These scenarios (e.g., pricing cycle) could happen…

…but if they do, it’s “random” …as in we don’t control in what

manner and how often they happen, and in conjunction with what other events

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Assets: Including yield curve variation - good thing What about linkages with liabilities?

– Example: inflation will impact severities and yield curve

Naively-built yield curve simulation may actually reduce variability of overall answer !!

– Independent asset values will dampen the variability of net income, surplus, etc.

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Band Aid?Band Aid?

Problem: Resulting scenarios may not be internally consistent

Possible Improvement: a MEGA-CORRELATION matrix (Yield curves and Liabilities)...

…but that just treats the symptoms !! Still have no guarantee of internal

consistency

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The Real ProblemThe Real Problem No Overarching Structural Framework

“All Method, No Model” - LJH We need a structural model of known

relationships and dependencies… …that has volatility and randomness, but

we control how and where it enters… … and the required internal consistency

will be built-in (within constraints)

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The Real ProblemThe Real Problem

This represents a significant mindset shift in actuarial modeling for DFA

Moves you away from correlation matrices…

…and towards STOCHASTIC PROCESSES...

…prevalent in asset and economic modeling

Structural Dependence in Structural Dependence in Asset and Economic Asset and Economic

ModelingModeling

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Stochastic Difference EquationsStochastic Difference Equations

Focus is on Processes, Increments, and Paths

Processes: Time series Increments: changes from one time

period to the next Paths: simulated evolution of the time

series, via randomly generated increments, calibrated to the starting point

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Generate plausible future scenarios consisting of time series for each of many simulated variables

Preserve internal consistency within each scenario

Introduce volatility in a controlled manner

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Begin with Driver Variables “Independent”, Top of the food chain

Generate the simulated time series for these Drivers Can either generate absolute level or

incremental changes, but we need the

increments (“”)

Example: CPI and Medical CPI

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The Next “level” of variables have defined functional relationships to the Drivers, plus error terms “Volatility” or “Noise”

GDP = f(CPI, Med CPI) + dW dW = “Wiener” term = Standard Normal

How we introduce volatility = scaling factor for that volatility

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Each successive level of variables builds upon prior variables up the chain in a CASCADE…

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CPI

Real GDP Growth

Yield Curve Equity Index

Simple Economic Model CascadeSimple Economic Model Cascade

Medical CPI

Unemployment

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Other Process Modeling TermsOther Process Modeling Terms

Shocks = large incremental changes Mean Reversion = process tends to

correct back toward long term avg Reversion strength = how quickly it

reverts back Calibration = tuning the parameters

See Madsen and Berger, 1999 DFA Call Paper

Structural Dependence in Structural Dependence in Liability ModelingLiability Modeling

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A Whole New FrameworkA Whole New Framework

Stochastic process modeling is about structure and control Building in structural relationships we

believe exist Introducing volatility in the increments

between periods Controlling the resulting simulated values

through parameters and calibration Adds another dimension to simulation

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Insurance Market ModelInsurance Market Model

Following the hierarchical approach of capital markets models

Generate market time series for Product Costs and Price Levels by LOB Not the same thing !!

Soft market: Costs > Price Levels (“under-pricing”)

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Individual CompanyIndividual Company

Individual company product costs are partly a function of the Market Cost level and partly a function of their own book Undiversifiable and Diversifiable

Individual company price levels behave similarly Your price is some deviation above or below

market Like the tide

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What we are evaluating is participation in insurance markets

Market Cost shocks to product Undiversifiable Market prices will respond, but over how

long? (Reversion strength) How quickly does company price level

respond to market price changes?

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Market Cost ShockMarket Cost Shock

Examples of a Market Cost shock Asbestos Pollution Construction Defect Benefit level change in WC Hurricane Andrew

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Company-specific Cost shocks to product Diversifiable Market Prices will not respond Company price level may respond, but will

be out of step with market Example:

North Carolina chicken factory that burned down with the doors locked

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Missing Links Demand curves by LOB Strength and nature of structural

dependency relationships This will require fundamental rewrites of

our DFA models Ultimately superior because it supports

the scientific method Requires hypothesis and testing

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InsureMetricsInsureMetricsTMTM

This is the development of InsureMetricsTM

The insurance kin to econometrics

Documents

Structural Dependence and Stochastic Processes Don Mango American Re-Insurance 2001 CAS DFA Seminar