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Structural Dependence and Stochastic Processes. Don Mango American Re-Insurance 2001 CAS DFA Seminar. Agenda. Just Say No to Correlation Structural Dependence in Asset and Economic Modeling Structural Dependence in Liability Modeling. Just Say No to Correlation. - PowerPoint PPT Presentation
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Structural Dependence Structural Dependence and Stochastic Processesand Stochastic Processes
Don Mango
American Re-Insurance
2001 CAS DFA Seminar
04/19/23 2
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
04/19/23 4
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
04/19/23 5
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
04/19/23 6
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
04/19/23 7
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
04/19/23 8
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?
04/19/23 9
ProblemProblem
Correlation in the tails is near 0 - extreme values are nearly un-correlated
Is this your intended result? Example….
04/19/23 10
04/19/23 11
Known Dependencies Not Known Dependencies Not MaintainedMaintained
Simple example DFA Model for a company
Liabilities: 4 LOB: Auto, GL, Property, WC
Assets: Bonds
04/19/23 12
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
04/19/23 13
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?
04/19/23 14
Example DFA Model - PROBLEMSExample DFA Model - PROBLEMS
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
04/19/23 15
Example DFA Model - PROBLEMSExample DFA Model - PROBLEMS
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.
04/19/23 16
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
04/19/23 17
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)
04/19/23 18
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
04/19/23 20
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
04/19/23 21
Stochastic Difference EquationsStochastic Difference Equations
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
04/19/23 22
Stochastic Difference EquationsStochastic Difference Equations
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
04/19/23 23
Stochastic Difference EquationsStochastic Difference Equations
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
04/19/23 24
Stochastic Difference EquationsStochastic Difference Equations
Each successive level of variables builds upon prior variables up the chain in a CASCADE…
04/19/23 25
CPI
Real GDP Growth
Yield Curve Equity Index
Simple Economic Model CascadeSimple Economic Model Cascade
Medical CPI
Unemployment
04/19/23 26
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
04/19/23 28
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
04/19/23 29
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”)
04/19/23 30
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
04/19/23 31
Insurance Market ModelInsurance Market Model
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?
04/19/23 32
Market Cost ShockMarket Cost Shock
Examples of a Market Cost shock Asbestos Pollution Construction Defect Benefit level change in WC Hurricane Andrew
04/19/23 33
Insurance Market ModelInsurance Market Model
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
04/19/23 34
Insurance Market ModelInsurance Market Model
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
04/19/23 35
InsureMetricsInsureMetricsTMTM
This is the development of InsureMetricsTM
The insurance kin to econometrics