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

6/1/20152 Agenda l Just Say No to Correlation l Structural Dependence in Asset and Economic Modeling l Structural Dependence in Liability Modeling

Just Say No to Correlation

6/1/20154 Just Say No to Correlation l Correlation has taken on something of a life of its own l It’s easy to measure l You can use Excel, l People think they know what it means, and have an intuitive sense of ranges

6/1/20155 Just Say No to Correlation l Paul Embrechts, Shaun Wang, and others tell us: â Correlation is simply one measure of Dependence, a more general concept â There are many other such measures l From a Stochastic modeling standpoint, simulating using Correlation surrenders too much control

6/1/20156 Simulating with Correlation l We think we know how to induce correlation between variables in our simulation algorithms l (At least) Two major problems: â Correlation is not the same throughout the simulation space â Known dependency relationships may not be maintained

6/1/20157 Correlation Not Always The Same... l Consider a well-known approach for generating correlated random variables l Using Normal Copulas l Similar to the Iman-Conover algorithm which uses Normal Copulas to generate rank correlation

6/1/20158 Normal Copulas Generate sample from multi-variate Normal with covariance matrix  l Get the CDF value for each point [ these are U(0,1) ] l Invert the U(0,1) points to get target simulated RVs with correlation… l …but what correlation will the target variables have?

6/1/20159 Problem l Correlation in the tails is near 0 - extreme values are nearly un-correlated l Is this your intended result? l Example….

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6/1/ Known Dependencies Not Maintained l Simple example DFA Model for a company l Liabilities: â 4 LOB: Auto, GL, Property, WC l Assets: â Bonds

6/1/ Example DFA Model l Liabilities: â 4 LOB: Auto, GL, Property, WC â Simulation: correlated uniform (0,1] matrix per time period used to generate the variables l Assets: â Bonds â Simulation: yield curve scenarios

6/1/ Example DFA Model - PROBLEMS l Liabilities: â Getting dependence within a year, but what about serial dependence across years? â Could expand the correlation matrix to be [ # variables x # years ] â But what about underwriting cycles? â What about the magnitude of year-over- year changes?

6/1/ Example DFA Model - PROBLEMS l Bottom line: These scenarios (e.g., pricing cycle) could happen… l …but if they do, it’s “random” l …as in we don’t control in what manner and how often they happen, and in conjunction with what other events

6/1/ Example DFA Model - PROBLEMS l Assets: â Including yield curve variation - good thing â What about linkages with liabilities? –Example: inflation will impact severities and yield curve â Naively-built yield curve simulation may actually reduce variability of overall answer !! –Independent asset values will dampen the variability of net income, surplus, etc.

6/1/ Band Aid? l Problem: Resulting scenarios may not be internally consistent l Possible Improvement: a MEGA- CORRELATION matrix (Yield curves and Liabilities)... l …but that just treats the symptoms !! l Still have no guarantee of internal consistency

6/1/ The Real Problem l No Overarching Structural Framework â “All Method, No Model” - LJH l We need a structural model of known relationships and dependencies… l …that has volatility and randomness, but we control how and where it enters… l … and the required internal consistency will be built-in (within constraints)

6/1/ The Real Problem l This represents a significant mindset shift in actuarial modeling for DFA l Moves you away from correlation matrices… l …and towards STOCHASTIC PROCESSES... l …prevalent in asset and economic modeling

Structural Dependence in Asset and Economic Modeling

6/1/ Stochastic Difference Equations l Focus is on Processes, Increments, and Paths l Processes: Time series l Increments: changes from one time period to the next l Paths: simulated evolution of the time series, via randomly generated increments, calibrated to the starting point

6/1/ Stochastic Difference Equations l Generate plausible future scenarios consisting of time series for each of many simulated variables l Preserve internal consistency within each scenario l Introduce volatility in a controlled manner

6/1/ Stochastic Difference Equations l Begin with Driver Variables â “Independent”, Top of the food chain l Generate the simulated time series for these Drivers  Can either generate absolute level or incremental changes, but we need the increments (“  ”) l Example: CPI and Medical CPI

6/1/ Stochastic Difference Equations l The Next “level” of variables have defined functional relationships to the Drivers, plus error terms â “Volatility” or “Noise”  GDP = f(  CPI,  Med CPI) +  dW l dW = “Wiener” term = Standard Normal â How we introduce volatility   = scaling factor for that volatility

6/1/ Stochastic Difference Equations l Each successive level of variables builds upon prior variables up the chain in a CASCADE…

6/1/ CPI Real GDP Growth Yield Curve Equity Index Simple Economic Model Cascade Medical CPI Unemployment

6/1/ Other Process Modeling Terms l Shocks = large incremental changes l Mean Reversion = process tends to correct back toward long term avg l Reversion strength = how quickly it reverts back l Calibration = tuning the parameters â See Madsen and Berger, 1999 DFA Call Paper

Structural Dependence in Liability Modeling

6/1/ A Whole New Framework l Stochastic process modeling is about structure and control â Building in structural relationships we believe exist â Introducing volatility in the increments between periods â Controlling the resulting simulated values through parameters and calibration â Adds another dimension to simulation

6/1/ Insurance Market Model l Following the hierarchical approach of capital markets models l Generate market time series for Product Costs and Price Levels by LOB â Not the same thing !! l Soft market: Costs > Price Levels (“under-pricing”)

6/1/ Individual Company l Individual company product costs are partly a function of the Market Cost level and partly a function of their own book â Undiversifiable and Diversifiable l Individual company price levels behave similarly â Your price is some deviation above or below market â Like the tide

6/1/ Insurance Market Model l What we are evaluating is participation in insurance markets l Market Cost shocks to product â Undiversifiable â Market prices will respond, but over how long? (Reversion strength) â How quickly does company price level respond to market price changes?

6/1/ Market Cost Shock l Examples of a Market Cost shock â Asbestos â Pollution â Construction Defect â Benefit level change in WC â Hurricane Andrew

6/1/ Insurance Market Model l Company-specific Cost shocks to product â Diversifiable â Market Prices will not respond â Company price level may respond, but will be out of step with market l Example: â North Carolina chicken factory that burned down with the doors locked

6/1/ Insurance Market Model l Missing Links â Demand curves by LOB â Strength and nature of structural dependency relationships l This will require fundamental rewrites of our DFA models l Ultimately superior because it supports the scientific method â Requires hypothesis and testing

6/1/ InsureMetrics TM l This is the development of InsureMetrics TM l The insurance kin to econometrics