Practical Applications of Stochastic Modeling for Disability Insurance

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 Overview
VP, Pricing and Consulting Actuary
Smith Group
Winter Liu, FSA, MAAA, CFA Adding Random Elements to
Consultant the Model
Tillinghast
Keith Gant, FSA, MAAA, CFA Application to Individual DI
VP, Product Modeling
Unum 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 Modeling
Introduce 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
5
200
1
Claims Chance Normal
0 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 Lives
Expected Claims
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0 1 2 3 4 5 6 7 8 9 10 11
Disability Incidence

5
500
2.5
0 8% 7%
1 20% 16%
2 26% 24%
3 21% 24%
4 13% 16%
5 7% 7%
6 3% 2%
7 1% 0%
8 0% 0%
9 0% 0%
10 0% 0%
11 0% 0%
Number of Lives
Expected Claims
0%
5%
10%
15%
20%
25%
30%
0 1 2 3 4 5 6 7 8 9 10 11
5
1000
5
0 1% 1%
1 3% 4%
2 8% 7%
3 14% 12%
4 18% 16%
5 18% 18%
6 15% 16%
7 10% 12%
8 7% 7%
9 4% 4%
10 2% 1%
11 1% 0%
Number of Lives
Expected Claims
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
0 1 2 3 4 5 6 7 8 9 10 11

5
10000
50
30 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%
Number of Lives
Expected Claims
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
XXN
N
pNXP −
−
=
!!
!
),,(
Mean: Np Percent Standard Deviation
Variance: Np(1-p)
Np
1≅
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 salary
Case 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)
$39,954Mean
$8,615Median
$996Mode

Distribution of Loss: 5 Claims
0
0.01
0.02
0.03
0.04
0.05
0.06
0K 20K 40K 60K 80K 100K 120K 140K 160K
Stochastic Simulation: Total Claim Cost
0
0.01
0.02
0.03
0.04
0.05
0.06
0K 20K 40K 60K 80K 100K 120K 140K 160K

0
0.02
0.04
0.06
0.08
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-07
Mean minus Five Std Dev 9.87E-10 <= One in a Billion
Mean 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, Seattle
Session 85, June 15, 2007
Keith Gant, FSA, MAAA, CFA
VP, Product Modeling
Unum 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

5
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 With
Payment
Unreported
Open Without
Payment
Closed With Payment
Month 40
Disability
Month 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

7
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.
8
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.

9
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.
10
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)

11
Sample Results
Distribution of PV of Stat Profit
DI block of 280,000 policies
50%
70%
90%
110%
130%
150%
0 100 200 300 400 500
Sorted Iteration
PVofStatProfit(%ofmean)
Statistical Volatility Statistical + Secular Volatility
12
Sample Results
Percentile 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
StatProfit(%ofmeaninqtr)
Max
90%
75%
Median
25%
10%
Min

1<copyright>
Practical Applications of Stochastic Modeling
for Disability Insurance
– Introduce Random Elements to Your Model
Winter Liu, FSA, MAAA, CFA
June 15, 2007
2<copyright>
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.

3<copyright>
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
4<copyright>
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

5<copyright>
Volatility due to Sample Size – Monte Carlo Sample
Example: 10,000 identical policies with a 0.005 annual incidence
rate.
Traditional: q = E(q) = 0.005 for every projection year
Monte 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
6<copyright>
Monte Carlo Sample – 10,000 Identical Policies
0.90271.22340.93350.86890.90521.14005
1.20771.20040.77020.90790.76370.98004
1.14851.08110.99450.88910.96521.08003
1.21090.87861.19830.97021.18550.92002
0.80130.98141.05761.05201.30640.98001
654321Scenario
Period =>
0.1430.1370.1460.1480.1440.147St Dev
1.0070.9911.0011.0030.9970.997Average
0.6140.5930.6090.5860.6030.640Min
0.8210.8180.8130.8280.8240.82010%
1.0040.9800.9950.9910.9851.00050%
1.1911.1651.1821.1931.1851.20090%
1.4551.4111.4831.4551.4471.460Max
654321Scenario
Period =>

7<copyright>
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
8<copyright>
Monte Carlo Sample – 10,000 Policies w/ Various Size
1.55951.04851.20170.84711.18955
0.89171.31281.01940.90900.83164
1.31880.90720.79731.23610.73683
1.10721.27111.29470.91900.70532
1.20150.97140.90330.83601.06321
54321Scenario
Period =>
0.2470.2600.2470.2430.244St Dev
1.0111.0001.0101.0001.005Average
0.3770.3090.3300.4030.295Min
0.6980.6820.7120.6980.70510%
0.9910.9750.9900.9830.98950%
1.3431.3351.3401.3111.32690%
1.7811.9211.9172.1191.884Max
54321Scenario
Period =>

9<copyright>
Monte Carlo Sample – 1,000 Identical Policies
0.40650.60790.40440.60481.60005
2.03460.81050.60611.60320.40004
1.22951.01941.61781.40560.80003
0.20390.60980.80971.60640.80002
0.40731.01321.00811.20240.40001
54321Scenario
Period =>
0.4540.4440.4510.4650.470St Dev
1.0080.9871.0031.0121.013Average
-----Min
0.4090.4070.4040.4020.40010%
1.0191.0131.0091.0051.00050%
1.6311.6231.6161.6101.60090%
2.4442.8572.8252.6322.600Max
54321Scenario
Period =>
10<copyright>
Monte Carlo Sample – 1,000 Policies w/ Various Size
1.62431.82400.21440.96002.63165
0.64311.38371.26920.31680.63164
0.21360.21330.63801.68690.31583
1.40850.54023.81560.52850.84212
0.53130.21230.74030.63260.31581
54321Scenario
Period =>
0.7740.8060.8020.7910.800St Dev
1.0031.0031.0351.0461.040Average
-----Min
0.2170.2150.2160.2140.31610%
0.6560.6440.7420.7390.73750%
1.9541.9292.0412.2272.10590%
4.5704.9204.5945.2745.789Max
54321Scenario
Period =>

11<copyright>
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
12<copyright>
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

13<copyright>
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
14<copyright>
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 (σh
2 - σs
2) = 9%
Choice of time series
1st-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%

15<copyright>
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
IncidenceA/ERatio
Actual A/E
MA(1)
AR(1)
AR(2)
16<copyright>
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: 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

17<copyright>
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

Practical Applications of Stochastic Modeling for Disability Insurance

Recommended

Recommended

More Related Content

Similar to Practical Applications of Stochastic Modeling for Disability Insurance

Similar to Practical Applications of Stochastic Modeling for Disability Insurance (20)

Practical Applications of Stochastic Modeling for Disability Insurance