hspring07-085bk

Practical Applications of Stochastic Modeling for Disability Insurance

Society of Actuaries

Session 85, Spring Health Meeting

Seattle, WA, June 15 2007


Rick Leavitt, ASA, MAAA Introduction and OverviewVP, Pricing and Consulting ActuarySmith Group

Winter Liu, FSA, MAAA, CFA Adding Random Elements to Consultant the ModelTillinghast

Keith Gant, FSA, MAAA, CFA Application to Individual DIVP, Product ModelingUnum Group

2007 Health Spring Meeting, Session 85: Practical Applications of Stochastic Modeling for Disability Insurance

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 ModelingIntroduce random elements to model. Acknowledge that model does not capture all variables

Fully Robust Modeling


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

Simple Stochastic Modeling in Disability Insurance

52001

Claims Chance Normal0 37% 24%1 37% 40%2 18% 24%3 6% 5%4 2% 0%5 0% 0%6 0% 0%7 0% 0%8 0% 0%9 0% 0%10 0% 0%11 0% 0%

Incident Rate (per 1000)Number of LivesExpected Claims

0%

5%

10%

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45%

0 1 2 3 4 5 6 7 8 9 10 11

Disability Incidence


55002.5



0%

5%

10%

15%

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30%

0 1 2 3 4 5 6 7 8 9 10 11


51000

5



0%

2%

4%

6%

8%

10%

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0 1 2 3 4 5 6 7 8 9 10 11



510000

50

Claims Chance Normal30 0.2% 0.2%35 1.4% 1.5%40 7.0% 6.2%45 18.1% 16.1%50 27.1% 26.0%55 24.7% 26.0%60 14.3% 16.1%65 5.5% 6.2%70 1.4% 1.5%75 0.3% 0.2%


0%

2%

4%

6%

8%

10%

12%

30 40 50 60 70


Incidence Theory

Chance of X claims, given N lives, and Incident Rate P

Binomial Distribution

( )XNX qp

XXNNpNXP −

−=

!!!),,(

Mean: Np Percent Standard DeviationVariance: Np(1-p)

Np1≅

Binomial Distribution approximates Normal Distribution when

Np(1-p) > 5, and (0.1 < p < 0.9 or min (Np) > 10)


Case 1: 10 expected claims: all lives have the same salaryCase 2: 10 expected claims: lives have actual salaries

Spread of Salaries does produce additional variance in the loss

0%

2%

4%

6%

8%

10%

12%

14%

0% 50% 100% 150% 200%

Case 1 Case 2

Incidence plus Claim Size

20,000 Trials

If incidence and claim size are independent than the variances add.

Distribution of Discounted Loss

0%

2%

4%

6%

8%

10%

12%

0K 20K 40K 60K 80K 100K 120K 140K 160K 180K

Table95a: 40 Year Old Male, 90 Day EP, $1000 Net Benefit

Consider Claim Duration: (Not Normal)

$39,954 Mean

$8,615 Median

$996 Mode



Distribution of Loss: 5 Claims

0

0.01

0.02

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0.04

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0.06

0K 20K 40K 60K 80K 100K 120K 140K 160K

Stochastic Simulation: Total Claim Cost



0

0.01

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0K 20K 40K 60K 80K 100K 120K 140K 160K





0

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0.1

0.12

0K 20K 40K 60K 80K 100K 120K 140K 160K


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 ***

Remember that Statistics Class?

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

… or…

N1


Properties of a Normal Distribution

Mean = Median = Mode

Outliers are Prohibitively Rare

Mean minus One Std Dev 15.9%Mean minus Two Std Dev 0.13%Mean minus Three Std Dev 0.0032%Mean minus Four Std Dev 2.87E-07Mean minus Five Std Dev 9.87E-10 <= One in a BillionMean minus Ten Std Dev 1.91E-28 <= Will never happen

Probabilities of being less than …

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

1 26 51 76 101

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 depends on exposure details but varies between 1.2 and 1.5

•Simple Stochastic Simulation adds little new information

NX /

For a $500 million dollar block this translates into expected deviation of 2.5% per quarter, or 1.25% year over year.

but …

…Observed variations exceed this by at least a factor of 3 to 5


Simple Stochastic Simulation for Disability Insurance

Let’s Summarize …

1. Results can be determined analytically without use of the stochastic model

2. 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


1


SOA Health Spring Meeting, SeattleSession 85, June 15, 2007

Keith Gant, FSA, MAAA, CFAVP, Product ModelingUnum Group

2

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


3

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.

4

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


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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

6

Active

Open WithPayment

UnreportedOpen Without

Payment

Closed With Payment

Month 40

DisabilityMonth 43

Claim is reported

Month 46

Payments Begin

Mo 56: Recovery

Mo 62: Reopen

Mo 80: Recovery

Month 104

Return to Active

Policy Movements -- Example


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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 thatmonth. A policy may be in only one state in each month.

• If a disability occurs in the month, two more random numbers aregenerated 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.

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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.


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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.

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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)


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Sample ResultsDistribution of PV of Stat Profit

DI block of 280,000 policies

50%

70%

90%

110%

130%

150%

0 100 200 300 400 500Sorted Iteration

PV

of S

tat P

rofit

(% o

f mea

n)

Statistical Volatility Statistical + Secular Volatility

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Sample ResultsPercentile Distribution of Quarterly Profit

DI block of 280,000 policies

-200%

-100%

0%

100%

200%

300%

400%

1Q 07 2Q 07 3Q 07 4Q 07 1Q 08 2Q 08 3Q 08 4Q 08

Quarter

Stat

Pro

fit (%

of m

ean

in q

tr) Max

90%75%Median25%10%Min


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Practical Applications of Stochastic Modeling for Disability Insurance – Introduce Random Elements to Your Model

Winter Liu, FSA, MAAA, CFA

June 15, 2007

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Strength and Limit of Stochastic Modeling

StrengthProvide a more complete picture of possible outcomesAllow embedded options to triggerHelp develop stress tests for extreme events

LimitComplex and expensiveDoes not help with best (mean) estimatesGarbage in, garbage out. A fancier garbage bin though.


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Source of Deviation / Volatility

Deviation due to sample sizeLaw of large number does not hold

Deviation due to specific company management / risk exposureUnderwritingReputationEconomy

Deviation due to shockMortality (pandemic)Incidence / recovery (depression)

Deviation due to assumption misestimationDo NOT count on stochastic modeling

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Deviation due to Sample Size

Our favorite riskLarge numberSmall claimIndependentLaw of large number

Kill someone vs. kill a piece of someoneTraditional models take the Law for grantedFull-blown multi-state models roll the dice on individual policy

AlternativeFull-blown multi-state models are expensiveMonte Carlo simulation


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Volatility due to Sample Size – Monte Carlo Sample

Example: 10,000 identical policies with a 0.005 annual incidencerate.

Traditional: q = E(q) = 0.005 for every projection yearMonte Carlo: — Roll a die (generate a random number) on each policy— An incidence if (random number < 0.005)— 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

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Monte Carlo Sample – 10,000 Identical Policies

0.9027 1.2234 0.9335 0.8689 0.9052 1.1400 51.2077 1.2004 0.7702 0.9079 0.7637 0.9800 41.1485 1.0811 0.9945 0.8891 0.9652 1.0800 31.2109 0.8786 1.1983 0.9702 1.1855 0.9200 20.8013 0.9814 1.0576 1.0520 1.3064 0.9800 1

654321ScenarioPeriod =>

0.143 0.137 0.146 0.148 0.144 0.147 St Dev1.007 0.991 1.001 1.003 0.997 0.997 Average0.614 0.593 0.609 0.586 0.603 0.640 Min0.821 0.818 0.813 0.828 0.824 0.820 10%1.004 0.980 0.995 0.991 0.985 1.000 50%1.191 1.165 1.182 1.193 1.185 1.200 90%1.455 1.411 1.483 1.455 1.447 1.460 Max



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Volatility due to Sample Size – Monte Carlo Sample

Example 2: 10,000 policies; 1,000 policies account for 90% of face amount

Example 3: 1,000 identical policies

Example 4: 1,000 policies; 100 policies account for 90% of face amount

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Monte Carlo Sample – 10,000 Policies w/ Various Size

1.5595 1.0485 1.2017 0.8471 1.1895 50.8917 1.3128 1.0194 0.9090 0.8316 41.3188 0.9072 0.7973 1.2361 0.7368 31.1072 1.2711 1.2947 0.9190 0.7053 21.2015 0.9714 0.9033 0.8360 1.0632 1


0.247 0.260 0.247 0.243 0.244 St Dev1.011 1.000 1.010 1.000 1.005 Average0.377 0.309 0.330 0.403 0.295 Min0.698 0.682 0.712 0.698 0.705 10%0.991 0.975 0.990 0.983 0.989 50%1.343 1.335 1.340 1.311 1.326 90%1.781 1.921 1.917 2.119 1.884 Max



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Monte Carlo Sample – 1,000 Identical Policies

0.4065 0.6079 0.4044 0.6048 1.6000 52.0346 0.8105 0.6061 1.6032 0.4000 41.2295 1.0194 1.6178 1.4056 0.8000 30.2039 0.6098 0.8097 1.6064 0.8000 20.4073 1.0132 1.0081 1.2024 0.4000 1


0.454 0.444 0.451 0.465 0.470 St Dev1.008 0.987 1.003 1.012 1.013 Average

-----Min0.409 0.407 0.404 0.402 0.400 10%1.019 1.013 1.009 1.005 1.000 50%1.631 1.623 1.616 1.610 1.600 90%2.444 2.857 2.825 2.632 2.600 Max


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Monte Carlo Sample – 1,000 Policies w/ Various Size

1.6243 1.8240 0.2144 0.9600 2.6316 50.6431 1.3837 1.2692 0.3168 0.6316 40.2136 0.2133 0.6380 1.6869 0.3158 31.4085 0.5402 3.8156 0.5285 0.8421 20.5313 0.2123 0.7403 0.6326 0.3158 1


0.774 0.806 0.802 0.791 0.800 St Dev1.003 1.003 1.035 1.046 1.040 Average

-----Min0.217 0.215 0.216 0.214 0.316 10%0.656 0.644 0.742 0.739 0.737 50%1.954 1.929 2.041 2.227 2.105 90%4.570 4.920 4.594 5.274 5.789 Max



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First Projection Period A/E Summary

10,000 10,000v 1,000 1,000v

Max 1.460 1.884 2.600 5.789 90% 1.200 1.326 1.600 2.105 50% 1.000 0.989 1.000 0.737 10% 0.820 0.705 0.400 0.316 Min 0.640 0.295 - -Average 0.997 1.005 1.013 1.040 St Dev 0.147 0.244 0.470 0.800

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More with Monte Carlo

Metric to evaluate scenariosPVCumulative decrements

Model can be expanded to consider product cash flowsEstimate impact on earningEstimate duration rangeDynamically model experience refund / repricing


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Deviation due to specific company management / risk exposure

Calibrate to historical volatilityImportant for both traditional model and multi-state modelSample size (statistical) volatility is not large enough to explain experienceTotal variance = statistical variance + company variance

Choice of time seriesRandom walkARIMAOther mean-reversion model

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Volatility due to specific company management / risk exposure – Time Series Sample

Example:

Historical incidence volatility σh = 12%

Statistical incidence volatility σs = 8%Historical incidence rates show certain pattern

Company incidence volatility σc = sqrt (σh2 - σs

2) = 9%

Choice of time series1st-order Moving-Average Yt = μ + εt + θεt-1

1st-order Auto-Regression Yt = φ1Yt-1 + δ + εt

2nd-order Auto-Regression Yt = φ1Yt-1 + φ2Yt-2 + δ + εt

Generate time series scenarios with σc = 9%


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MA(1) vs. AR(1) vs. AR(2)

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

140.0%

1 2 3 4 5 6 7 8 9

Inci

denc

e A

/E R

atio

Actual A/E

MA(1)

AR(1)

AR(2)

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Deviation due to Shock

More difficult to modelLimited experience— Mortality: Spanish Influenza— Incidence / recovery: Great DepressionHigh severityStress test could be a better choice

Jump modelAlso “Jump Diffusion” model proposed by MertonA modified random walk framework, with jumps modeled by Poisson distributionExample: 1% per year increase in incidence rate for 10 years vs.a 10% jump every 10 years.Works better for small jumps and difficult to parameterize


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Final Thoughts

Do not rely entirely on stochastic models

More scenarios is not necessarily better

Assumption may eliminate riskAvoid the “normal” trap

Watch for correlationStochastic asset modeling


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