![Page 1: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/1.jpg)
Structural Dependence Structural Dependence and Stochastic Processesand Stochastic Processes
Don Mango
American Re-Insurance
2001 CAS DFA Seminar
![Page 2: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/2.jpg)
04/19/23 2
AgendaAgenda
Just Say No to Correlation Structural Dependence in Asset and
Economic Modeling Structural Dependence in Liability
Modeling
![Page 3: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/3.jpg)
Just Say No to Just Say No to CorrelationCorrelation
![Page 4: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/4.jpg)
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
![Page 5: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/5.jpg)
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
![Page 6: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/6.jpg)
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
![Page 7: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/7.jpg)
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
![Page 8: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/8.jpg)
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?
![Page 9: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/9.jpg)
04/19/23 9
ProblemProblem
Correlation in the tails is near 0 - extreme values are nearly un-correlated
Is this your intended result? Example….
![Page 10: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/10.jpg)
04/19/23 10
![Page 11: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/11.jpg)
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
![Page 12: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/12.jpg)
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
![Page 13: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/13.jpg)
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?
![Page 14: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/14.jpg)
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
![Page 15: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/15.jpg)
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.
![Page 16: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/16.jpg)
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
![Page 17: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/17.jpg)
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)
![Page 18: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/18.jpg)
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
![Page 19: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/19.jpg)
Structural Dependence in Structural Dependence in Asset and Economic Asset and Economic
ModelingModeling
![Page 20: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/20.jpg)
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
![Page 21: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/21.jpg)
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
![Page 22: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/22.jpg)
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
![Page 23: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/23.jpg)
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
![Page 24: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/24.jpg)
04/19/23 24
Stochastic Difference EquationsStochastic Difference Equations
Each successive level of variables builds upon prior variables up the chain in a CASCADE…
![Page 25: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/25.jpg)
04/19/23 25
CPI
Real GDP Growth
Yield Curve Equity Index
Simple Economic Model CascadeSimple Economic Model Cascade
Medical CPI
Unemployment
![Page 26: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/26.jpg)
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
![Page 27: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/27.jpg)
Structural Dependence in Structural Dependence in Liability ModelingLiability Modeling
![Page 28: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/28.jpg)
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
![Page 29: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/29.jpg)
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”)
![Page 30: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/30.jpg)
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
![Page 31: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/31.jpg)
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?
![Page 32: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/32.jpg)
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
![Page 33: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/33.jpg)
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
![Page 34: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/34.jpg)
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
![Page 35: Structural Dependence and Stochastic Processes](https://reader036.vdocuments.us/reader036/viewer/2022062314/56812d64550346895d9272be/html5/thumbnails/35.jpg)
04/19/23 35
InsureMetricsInsureMetricsTMTM
This is the development of InsureMetricsTM
The insurance kin to econometrics