# Practical Applications of Stochastic Modeling for Disability Insurance

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1 Practical Applications of Stochastic Modeling for Disability Insurance Society of Actuaries Session 8, Spring Health Meeting Seattle, WA, June 007 Practical Applications of Stochastic Modeling for Disability Insurance Rick Leavitt, ASA, MAAA VP, Pricing and Consulting Actuary Smith Group Winter Liu, FSA, MAAA, CFA Consultant Tillinghast Keith Gant, FSA, MAAA, CFA VP, Product Modeling Unum Group Introduction and Overview Adding Random Elements to the Model Application to Individual DI

2 Definitions and Distinctions Computer Simulation versus Stochastic Modeling versus Scenario Testing Definitions and Distinctions Simulation: Define assumptions and important dynamics. Let the computer produce outcomes Scenario Testing: Consider range of assumptions. Test sensitivity of outcomes on assumptions and dynamics Stochastic Modeling Introduce random elements to model. Acknowledge that model does not capture all variables Fully Robust Modeling

3 Simple Stochastic Modeling in Disability Insurance Number of Claims: Each individual has a chance of claim Claim Size: Variance in outcomes based on who files Claim Duration: Can range from one month to maximum claim duration Disability Incidence Incident Rate (per 000) Number of Lives Expected Claims 00 % 0% Claims Chance Normal 0 7% % 7% 0% 8% % 6% % % 0% 0% 0% 6 0% 0% 7 0% 0% 8 0% 0% 9 0% 0% 0 0% 0% 0% 0% % 0% % 0% % 0% % 0%

4 Disability Incidence Incident Rate (per 000) Number of Lives Expected Claims Claims Chance Normal 0 8% 7% 0% 6% 6% % % % % 6% 7% 7% 6 % % 7 % 0% 8 0% 0% 9 0% 0% 0 0% 0% 0% 0% 00. 0% % 0% % 0% % 0% Disability Incidence Incident Rate (per 000) Number of Lives Expected Claims Claims Chance Normal 0 % % % % 8% 7% % % 8% 6% 8% 8% 6 % 6% 7 0% % 8 7% 7% 9 % % 0 % % % 0% 000 0% 8% 6% % % 0% 8% 6% % % 0%

5 Disability Incidence Incident Rate (per 000) Number of Lives Expected Claims Claims Chance Normal 0 0.% 0.%.%.% 0 7.0% 6.% 8.% 6.% 0 7.% 6.0%.7% 6.0% 60.% 6.% 6.% 6.% 70.%.% 7 0.% 0.% % 0% 8% 6% % % 0% Incidence Theory Chance of X claims, given N lives, and Incident Rate P P( X, N, p) = Binomial Distribution N!! X! ( N X ) p X q N Mean: Np Percent Standard Deviation Variance: Np(-p) Np X Binomial Distribution approximates Normal Distribution when Np(-p) >, and (0. < p < 0.9 or min (Np) > 0)

6 Incidence plus Claim Size % % 0% 8% 6% % % 0,000 Trials Case Case 0% 0% 0% 00% 0% 00% Case : 0 expected claims: all lives have the same salary Case : 0 expected claims: lives have actual salaries Spread of Salaries does produce additional variance in the loss If incidence and claim size are independent than the variances add. Consider Claim Duration: (Not Normal) % Distribution of Discounted Loss 0% 8% 6% Mode Median Mean \$996 \$8,6 \$9,9 % % 0% 0K 0K 0K 60K 80K 00K 0K 0K 60K 80K Table9a: 0 Year Old Male, 90 Day EP, \$000 Net Benefit

7 Stochastic Simulation: Total Claim Cost 0.06 Distribution of Loss: Claims K 0K 0K 60K 80K 00K 0K 0K 60K Table9a: 0 Year Old Male, 90 Day EP, \$000 Net Benefit Stochastic Simulation: Total Claim Cost 0.06 Distribution of Loss: 0 Claims K 0K 0K 60K 80K 00K 0K 0K 60K Table9a: 0 Year Old Male, 90 Day EP, \$000 Net Benefit

8 Stochastic Simulation: Total Claim Cost 0. Distribution of Loss: 0 Claims K 0K 0K 60K 80K 00K 0K 0K 60K Table9a: 0 Year Old Male, 90 Day EP, \$000 Net Benefit Remember that Statistics Class? Central Limit Theorem: The sum of independent identically distributed random variables with finite variance will tend towards a normal distribution as the number of variables increase. *** explains the prevalence of normal distributions *** or The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-normal. Furthermore, this normal distribution will have the same mean as the parent distribution, and, variance equal to the variance of the parent divided by the sample size. Standard Deviation varies as N

9 .% Properties of a Normal Distribution.0%.%.0% Mean = Median = Mode Outliers are Prohibitively Rare Probabilities of being less than Mean minus One Std Dev.9%.% Mean minus Two Std Dev 0.% Mean minus Three Std Dev 0.00%.0% Mean minus Four Std Dev.87E-07 Mean minus Five Std Dev 9.87E-0 <= One in a Billion Mean 0.% minus Ten Std Dev.9E-8 <= Will never happen 0.0% Application to Disability Insurance For a large, and diverse block: Random variations produce expected distributions that are normal with standard deviation that varies as X / N X depends on exposure details but varies between. and. Simple Stochastic Simulation adds little new information For a \$00 million dollar block this translates into expected deviation of.% per quarter, or.% year over year. but Observed variations exceed this by at least a factor of to

10 Simple Stochastic Simulation for Disability Insurance Let s Summarize. Results can be determined analytically without use of the stochastic model. Results do not accurately model observations Simple model does not include: Changes in process (underwriting and claims) Changes in the external environment Beware the Stochastic Modeling Conundrum Simple Stochastic Modeling will underestimate volatility When adding complexity, it is important to review qualitative behavior to ensure reasonability of the model. However, qualitative behavior is primary model output It is easy to bake expectations into model outcome

11 Practical Applications of Stochastic Modeling for Disability Insurance SOA Health Spring Meeting, Seattle Session 8, June, 007 Keith Gant, FSA, MAAA, CFA VP, Product Modeling Unum Group Outline of Presentation Challenges in Modeling DI business Solution A new Framework Implementation of the Framework Stochastic DI model Stochastic DI models and Interest Rate Scenarios Uses of Stochastic models Sample results

12 Challenges in Modeling DI Business Complications in validating to short term financial results Reporting lags Timing of reserve changes and start of benefit payments Reopens, settlements Number of combinations of product provisions (EP, BP, Benefit patterns, COLA types, Reinsurance, etc.) Makes grouping into homogeneous model points difficult. Solution A New Framework Goals Link experience analysis and models with financial and operational metrics. Focus on the drivers of reserve change. Policy Statuses Active Unreported claim Open (without payment) Open (with payment) Closed (without payment) Closed (with payment) Termination (lapse, death, settlement, expiry) Policy movements among statuses

13 Policy Movements From State To State From State To State Active Active Open with Payment Open with Payment Lapsed Closed with Payment Unreported (Disability) Death Death Settlement Expiry BP expiry Unreported Unreported Closed without Payment Closed without Payment Closed without Payment Closed with Payment Closed with Payment Open without Payment Open without Payment Open with Payment Open with Payment Lapsed Unreported (Disability) Open without Payment Open without Payment Active Closed without Payment Closed with Payment Closed with Payment Closed with Payment Open with Payment Open with Payment Death Settlement Lapsed Unreported (Disability) Active Policy Movements -- Example Active Month 0 Disability Unreported Month Claim is reported Open Without Payment Month 0 Return to Active Month 6 Payments Begin Mo 6: Recovery Closed With Payment Mo 6: Reopen Open With Payment Mo 80: Recovery 6

14 Stochastic DI model Stochastic approach greatly simplifies implementation of the multi-state model. Mechanics Movements are driven by random numbers. Each month, a random number is generated and compared with the movement probabilities to determine what the policy does in that month. A policy may be in only one state in each month. If a disability occurs in the month, two more random numbers are generated to determine the type of disability (total accident, total sickness, presumptive, residual, MNAD), and the reporting month. Comparison to a deterministic model. Multiple iterations are run, providing a distribution of results. The mean over all iterations provides a single deterministic-like value. 7 Stochastic DI model (cont.) Beyond Statistical (random) Fluctuation Addressed in Winter s presentation. The convergence issue A function of the size and composition of the block. A function of the item of interest (e.g., premium vs. profits, or PV s vs. quarterly values). Seriatim model Avoids modeling effort and possible inaccuracy of groupings. Handle each policy s characteristics (EP/BP/Benefit pattern/cola/reinsurance/etc.) directly rather than aggregating. Fast runtimes are possible Calculations for a policy go down a single path. Calculations stop when the policy terminates. 8

15 Stochastic DI models and Interest Rate Scenarios Linking to Interest Rate Scenarios The volatility on the liability side adds to that on the asset side. Are DI liability cash flows interest-sensitive? Expenses (inflation rate) CPI-linked COLA Impact on ALM: Shortening of liability price-sensitivity duration. Claim incidence and recovery? DI lapse rates? LTD renewal pricing strategy Dynamics can be similar to annuity crediting strategy. Modeling future LTD premium and/or persistency as a function of interest rates gives strongly interest-sensitive CF s. 9 Uses of Stochastic Models Uses of mean (deterministic-like) projected values CFT, RAS, Pricing, Financial Plan Uses of distributions of projected values Analyzing recent experience and financials (Is it random fluctuation or a real shift?) LTD Credibility ALM Valuing a block of business Range of potential outcomes Determining capital levels Stop-loss reinsurance Risk Management, VaR Setting reserves (CTE) 0

16 Sample Results PV of Stat Profit (% of mean) 0% 0% 0% 90% 70% 0% Distribution of PV of Stat Profit DI block of 80,000 policies Sorted Iteration Statistical Volatility Statistical + Secular Volatility Sample Results Percentile Distribution of Quarterly Profit DI block of 80,000 policies 00% Stat Profit (% of mean in qtr) 00% 00% 00% 0% Q 07 Q 07 Q 07 Q 07 Q 08 Q 08 Q 08 Q 08-00% Max 90% 7% Median % 0% Min -00% Quarter

17 Practical Applications of Stochastic Modeling for Disability Insurance Introduce Random Elements to Your Model Winter Liu, FSA, MAAA, CFA June, 007 Strength and Limit of Stochastic Modeling Strength Provide a more complete picture of possible outcomes Allow embedded options to trigger Help develop stress tests for extreme events Limit Complex and expensive Does not help with best (mean) estimates Garbage in, garbage out. A fancier garbage bin though.

18 Source of Deviation / Volatility Deviation due to sample size Law of large number does not hold Deviation due to specific company management / risk exposure Underwriting Reputation Economy Deviation due to shock Mortality (pandemic) Incidence / recovery (depression) Deviation due to assumption misestimation Do NOT count on stochastic modeling Deviation due to Sample Size Our favorite risk Large number Small claim Independent Law of large number Kill someone vs. kill a piece of someone Traditional models take the Law for granted Full-blown multi-state models roll the dice on individual policy Alternative Full-blown multi-state models are expensive Monte Carlo simulation

19 Volatility due to Sample Size Monte Carlo Sample Example: 0,000 identical policies with a 0.00 annual incidence rate. Traditional: q = E(q) = 0.00 for every projection year Monte Carlo: Roll a die (generate a random number) on each policy An incidence if (random number < 0.00) q = sum of incidence / total # of policies Record actual-to-expected (A/E) ratio of q / E(q) Repeat N years and M scenarios Apply incidence (A/E) ratios to traditional model (Incorporate other decrements if appropriate) Single decrement Monte Carlo alternative: stochastic LE Monte Carlo Sample 0,000 Identical Policies Period => Scenario 6 Max % % % Min Average St Dev Period => Scenario

20 Volatility due to Sample Size Monte Carlo Sample Example : 0,000 policies;,000 policies account for 90% of face amount Example :,000 identical policies Example :,000 policies; 00 policies account for 90% of face amount 7 Monte Carlo Sample 0,000 Policies w/ Various Size Period => Scenario Max % % % Min Average St Dev Period => Scenario

21 Monte Carlo Sample,000 Identical Policies Period => Scenario Max % % % Min Average St Dev Period => Scenario Monte Carlo Sample,000 Policies w/ Various Size Period => Scenario Max % % % Min Average St Dev Period => Scenario

22 First Projection Period A/E Summary 0,000 0,000v,000,000v Max % % % Min Average St Dev More with Monte Carlo Metric to evaluate scenarios PV Cumulative decrements Model can be expanded to consider product cash flows Estimate impact on earning Estimate duration range Dynamically model experience refund / repricing

23 Deviation due to specific company management / risk exposure Calibrate to historical volatility Important for both traditional model and multi-state model Sample size (statistical) volatility is not large enough to explain experience Total variance = statistical variance + company variance Choice of time series Random walk ARIMA Other mean-reversion model Volatility due to specific company management / risk exposure Time Series Sample Example: Historical incidence volatility σ h = % Statistical incidence volatility σ s = 8% Historical incidence rates show certain pattern Company incidence volatility σ c = sqrt (σ h - σ s ) = 9% Choice of time series st -order Moving-Average Y t = μ + ε t + θε t- st -order Auto-Regression Y t = φ Y t- + δ + ε t nd -order Auto-Regression Y t = φ Y t- + φ Y t- + δ + ε t Generate time series scenarios with σ c = 9%

24 MA() vs. AR() vs. AR() 0.0% 0.0% Incidence A/E Ratio 00.0% 80.0% 60.0% 0.0% Actual A/E MA() AR() AR() 0.0% 0.0% Deviation due to Shock More difficult to model Limited experience Mortality: Spanish Influenza Incidence / recovery: Great Depression High severity Stress test could be a better choice Jump model Also Jump Diffusion model proposed by Merton A modified random walk framework, with jumps modeled by Poisson distribution Example: % per year increase in incidence rate for 0 years vs. a 0% jump every 0 years. Works better for small jumps and difficult to parameterize 6

25 Final Thoughts Do not rely entirely on stochastic models More scenarios is not necessarily better Assumption may eliminate risk Avoid the normal trap Watch for correlation Stochastic asset modeling 7

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