1. SoA Stochastic Modeling
for Leading Edge Actuaries 1
Stochastic Modeling for
Leading Edge Actuaries
Overview of the Practical Aspects of
Stochastic Modeling
Ron Harasym MBA, CFA, FSA, FCIA
Vice-President & Chief Risk Officer

2. SoA Stochastic Modeling
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Outline of Presentation
 Stochastic Modeling Defined.
 What Stochastic Modeling is and isn’t.
 Advantages and Limitations of Stochastic Modeling.
 When Stochastic Modeling is Preferred.
 Key steps in Stochastic Modeling
 Conditional Tail Expectation & Examples
 Recommended Practices
 Final Thoughts

3. SoA Stochastic Modeling
for Leading Edge Actuaries 3
Stochastic Modeling - Definition
 Stochastic [Greek stokhastikos: from stokhasts, diviner, from
stokhazesthai, to guess at, from stokhos, aim, goal.]
 A stochastic model by definition has at least one random
variable and deals explicitly with time-variable interaction.
 A stochastic simulation uses a statistical sampling of multiple
replicates, repeated simulations, of the same model.
 Such simulations are also sometimes referred to as Monte
Carlo simulations because of their use of random variables.

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Stochastic Modeling - What it is
 A stochastic model is an imitation of a real world system. An
imprecise technique that provides statistical estimates and not
exact results.
 Stochastic modeling serves as a tool in a company’s risk
measurement toolkit to provide assistance in:
 Valuation, Forecasting, Solvency Testing
 Financial Reporting
 Product Design & Pricing
 Risk Management
 Simulations are used when the systems being modeled are too
complex to be described by a set of mathematical equations for
which a closed form analytic solution is readily attainable.
 Part art, part science, part judgement, part common sense.

5. SoA Stochastic Modeling
for Leading Edge Actuaries 5
Stochastic Modeling - And What it isn’t
 Not a magical solution! One needs to:
 Perform reality checks
 Understand strengths & limitations of the model
 Results are not always intuitively obvious!
 Often requires a different way of looking at problems, issues,
results, and potential solutions.
 Greater exposure to model risk and operational risks.

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Advantages of Stochastic Modeling
 Systems with long time frames can be studied in compressed time.
 Able to assist in decision making and to quantify future outcomes
arising from different actions/strategies before implementation.
 Can attempt to better understand properties of real world systems
such as policyholder behavior.
 Quantification of the benefit from risk diversification.
 Coherent articulation of risk profiles.
 Potential reserve and regulatory capital relief.

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Limitations of Stochastic Modeling
 Requires a considerable investment of time and expertise.
 Technically challenging, computationally demanding.
 Reliance on a few “good” people!
 For any given set of inputs, each scenario gives only a estimate.
 May create a false sense of confidence - a false sense of
precision let alone a false sense of accuracy.
 Relies heavily on data inputs and the identification of variable
interactions.
 It is not possible to include all future events in a model.
 Results may be difficult to interpret.
 Effective communication of results may be even more difficult.
 Garbage in, Garbage out!

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Stochastic modeling is Preferred over
Deterministic Modeling When:
 Product or Line of Business has a “cliff” or “tail” risk profile.
 Risks are dependent and/or there is path dependence.
 When dealing with skewed and/or discontinuous distributions/cost
functions.
 Outcomes are sensitive to initial conditions.
 There is significant volatility in the underlying variables.
 Volatility or skewness of underlying variables is likely to change
over time.
 There are real economic incentives, such as reserve or capital
relief, to perform stochastic modeling.

9. SoA Stochastic Modeling
for Leading Edge Actuaries 9
Is There Really a Starting and Ending Point?
… No!
Output
Historical
Economic Data
Historical
Policyholder
Data
Random Number
Generator
Economic
Scenario
Generator (ESG)
Stochastic ESG
Parameters &
Assumptions
Policyholder
Input Data
Economic
Scenarios
Data Validation
&
ESG Calibration
Random
Numbers
Stochastic
Asset / Liability
Models
Liability Data
Validation
Deterministic &
Stochastic Liability
Assumptions
Deterministic &
Stochastic Asset
Assumptions
Result Tabulation,
Validation, & Review
Reported
Financial Results,
Risk Management
Measures

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for Leading Edge Actuaries 10
Where does one Start? Key Steps Are ...
 Identify the key issues, objectives and potential roadblocks
before considering ways of solving the problem.
 Articulate the process / model in general terms before
proceeding to the specific.
 Develop the model: assumptions, input parameters, data,
output.
 Fit the model: gather and analyze data, estimate input
parameters
 Implement the model.
 Analyze and test sensitivity of the model results & loop back.
 Communicate the results.

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Points to Keep in Mind!
 Stochastic modeling is an evolutionary / revolutionary concept.
 There must be a constant feedback loop.
 Learn to “walk” before you “run”.
 Recognize that no one model fits all solutions.
 Be careful of becoming “married to the method”, rather than the
objective.
 Keep it simple, keep it practical, keep it understandable.
 Keep performing validation and reality checks throughout all
modeling steps.
 Strive towards the production of actionable information!

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Conditional Tail Expectation
 Conditional Tail Expectation (CTE) is a conditional expected value
based on downside risk.
 CTE can be defined as the average of outcomes that exceed a
specified percentile.
 The CTE(Q%) is calculated as the arithmetic average of the worst
(100-Q)% results of the stochastic simulation.
 For example: CTE(75%) is the arithmetic average of the worst
25% of the results of the stochastic simulation.
 CTE is considered to be a more robust measure with greater
information content than percentiles.
 The CTE measure can also be “modified”.

13. SoA Stochastic Modeling
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Random Number Generator
 Objective:
 To produce random numbers between 0 and 1
 Issues:
 The Random Number Generator (RNG) is a foundation building block
 Critical, but often ignored/forgotten about!
 Poor RNG can compromise all post modeling sophistication
 Numerous RNGs to choose from
 Desirable Characteristics to check for:
 Robustness independent of the seed number
 Periodicity
 Fast, efficient, & effective algorithm
 Other statistical tests (an internet search will provide many)

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Economic Scenario Generator
 Objective:
 To produce capital market or economic scenarios
 Components:
 Drift, Diffusion, Correlation, …
 Issues:
 Is the focus on the mean, median, or tail events?
 What metric is of concern?
 Economic vs. Statistical model, Arbitrage-Free vs. Equilibrium
 What is our calibration benchmark?
 Numerous ESGs to choose from
 Desirable Characteristics to check for:
 Integrated model (equity, interest rate, inflation, currency)
 Incorporates the principle of parsimony.
 Flexible. A component approach.

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Calibration of the
Economic Scenario Generator
 Stability of the components over time
 Drift Stability versus Diffusion Stability versus Correlation Stability
 Frequency of recalibration
 Historical data period versus forecast horizon
 Selection of lead index
 Selection of starting regime if using a multiple regime model
 Foreign exchange Issues
 Data sources and Caveat Emptor
 Approaches to fitting the data
 Risk-Return relationship
 False sense of precision and subjectivity

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Example: Variable Annuity GMIB Rider
 Product:
 Guaranteed Minimum Income Benefit Rider
 Objective:
 Produce Measures for Financial Reporting
 Calculate Reserve & Capital Requirements
 Nature of the Situation:
 Case #1: MV = $1.00B, GV = $1.40B (in-the-money)
 Case #2: MV = $2.75B, GV = $2.75B (at-the-money)
 Mixture of policyholders
 5% Roll-up rate per annum
 Conservative interest and mortality assumptions at time of product
pricing

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Economic Scenario Generation
 Economic Scenario Generator:
 Equity returns modeled using RSLN2 model
 Fixed Income returns modeled using Cox-Ingersol-Ross model
 Correlated Equity & Fixed Income Returns
 Calibration Method:
 Maximum Likelihood Estimation
 Calibration Issues:
 Data is limited and often inconsistent/incorrect.
 Insufficient effort is often given to data validation.
 Requires complex methods
 Historical data period versus forecast horizon
 Frequency of recalibration
 Simulation:
 1000 scenarios, monthly frequency,
30 year projection horizon

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Present Value vs. Average Interest Rate per Scenario Scatter Plot
Stochastic Base Case: Target Equity Return = 8%, Target Interest Rate = 6%
2%
4%
6%
8%
10%
12%
-$300 -$250 -$200 -$150 -$100 -$50 $0 $50 $100
AverageInterestRateoverProjectionHorizon
2%
4%
6%
8%
10%
12%
Case #1:
MV = $1.0B, GV = $1.4B (in-the-money)

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Present Value vs. Average Equity Return per Scenario Scatter Plot
Stochastic Base Case: Target Equity Return = 8%, Target Interest Rate = 6%
-5%
0%
5%
10%
15%
20%
25%
-$300 -$250 -$200 -$150 -$100 -$50 $0 $50 $100
AverageEquityReturnoverProjectionHorizon
Case #1:
MV = $1.0B, GV = $1.4B (in-the-money)

20. SoA Stochastic Modeling
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CTE Percentile
100% -$481.9 -$481.9
95% -$247.9 -$164.0
90% -$187.6 -$97.5
85% -$151.4 -$64.2
80% -$126.7 -$42.2
75% -$107.3 -$18.3
70% -$90.6 $1.7
65% -$76.4 $15.1
60% -$64.3 $24.3
55% -$54.0 $32.8
50% -$45.0 $38.0
45% -$37.2 $44.6
40% -$30.1 $50.1
35% -$23.8 $54.0
30% -$18.1 $56.9
25% -$13.0 $60.1
20% -$8.3 $65.2
15% -$3.8 $70.8
10% $0.5 $76.2
5% $4.8 $88.3
0% $9.7 $137.6
CTE & Percentiles: GMIB Case #2
Stochastic Simulation Results
Present Value of GMIB Rider Cash Flows
Assumptions:
Expected equity return = 8% per annum
Expected long term interest yield = 6%
Number of Scenarios = 1,000
Expected Value or Average
Median or 50th Percentile
Maximum Value
Maximum
Value
Minimum
Value

21. SoA Stochastic Modeling
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PV of GMIB Rider Cash Flow s: Distribution of Stochastic Results
0%
5%
10%
15%
20%
25%
30%
35%
-$500 -$450 -$400 -$350 -$300 -$250 -$200 -$150 -$100 -$50 $0 $50 $100 $150
Probability
CTE(0%)
50th Percentile
Assume Reserve is
set at CTE(70%)CTE(95%)
Capital
$0
Reserve
Total Gross Calculated
Requirement
Maximum
CTE & Percentiles: GMIB Case #2

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Sensitivity/Stress Testing
 Quantifies the impact of an immediate change in an
assumption or variable.
 Useful for validation of the model with respect to individual
assumptions
 Also a check on the modeled variable interactions
 Allows one to identify and thereby direct more effort on key
assumptions or variables.

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GMIB CTE Measures: Liability Assumption Sensitivity Testing
$0
$50
$100
$150
$200
$250
$300 BaseCase
RiderCharge-10bps
CurrentPricing
Spread-10bps
Pre-AnnMortDecr
10%
Post-AnnMortDecr
10%
LapseRatex2
LapseRatex0.5
AnnuitizationRatex2
AnnuitizaionRate
x0.5
CTE(95%)
CTE(90%)
CTE(80%)
CTE(70%)
CTE(60%)
CTE(0%)
Base
Case
Case #1:
MV = $1.0B, GV = $1.4B (in-the-money)

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GMIB CTE Measures: Investment Assumption Sensitivity Testing
$0
$50
$100
$150
$200
$250
$300 BaseCase
EquityReturn=10%
EquityReturn=9%
EquityReturn=7%
EquityReturn=6%
LTYield=8%
LTYield=7%
LTYield=5%
LTYield=4%
CTE(95%)
CTE(90%)
CTE(80%)
CTE(70%)
CTE(60%)
CTE(0%)
Base
Case
Case #1:
MV = $1.0B, GV = $1.4B (in-the-money)

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Recommended Practices:
 Keep focused on the business objectives.
 No one model fits all. Best to understand fundamentals.
 Cultivate “best practices”.
 Keep it simple and practical.
 Focus on accuracy first, precision second.
 Add complexity on a cost/benefit basis.
 Don’t ignore data validation and model validation procedures.
 Continually perform reality checks.
 Constantly loop back through the process.

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Other Issues to Wrestle With:
 Some model set-ups generate more volatility in results than
others. How do we choose between them?
 How do we perform calibration and parameter estimation?
 How do we capture the correlations between markets.
 How many scenarios do we use?
 How do we model policyholder behaviour?
 How do we incorporate hedging in the model?

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Finally, Where Are We Going … ???
 Will stochastic modeling change the way the insurance industry
conducts business?
 What will be the impact of the recent acceptance/application of
stochastic modeling within the next 1, 5, 10+ years?
 How will stochastic modeling alter/impact pricing, product
development, and valuation / risk management practices &
procedures?
 Even closer to home, how will stochastic modeling impact the
educational experience and skill requirements of current and
future actuaries?

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Wrap-Up
Questions & Answers!