2006 Seminar for the Appointed Actuary

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Canadian Institute of Actuaries
LInstitut canadien des actuaires

• 2006 Seminar for the Appointed Actuary
• Colloque pour lactuaire désigné 2006

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Stochastic Equity Modeling

• Dr. Julia Lynn Wirch-Viinikka
• AVP Investment Products Pricing

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Agenda

• Equity Risk where is it?
• Stochastic Modeling what is it?
• What options do we have for modeling equity risk?
• How do we start?
• How do we improve our model?
• How do we illustrate our results?
• How complicated can it get?

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Where is Your Equity Risk?

• Assets backing Liabilities (LTC)
• Surplus
• Liabilities
• Seg Fund, VA and UL Guarantees
• Equity Linked Products
• Fee Income based on Fund Value
• Hedging Mismatch
• Tracking error, basis risk

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Why Manage Equity Risk?

• Regulatory requirement
• Equity Limits, MCCSR requirements
• DCAT testing
• Valuation
• Reserves and Capital Requirements
• CGAAP, IFRS, US GAAP results
• Income and Surplus volatility
• Risk Management objectives

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Market risk vs. Insurance risk

• Traditional insurance risks, such as mortality
  and longevity are less risky when pooled
  together each individual follows their own
  scenario and the insurance company pays off on
  the average
• Capital market risks dont diversify every
  policyholder follows the same market scenario at
  the same time

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Which Risks can be Managed?

• Risks that are identifiable and well understood
• Risks that are monitored and controlled
• Risks where there is the knowledge and expertise
  to effectively manage them.
• Where the reward is sufficient for the remaining
  risk
• Where financial instruments and methods are
  available to hedge or control risk

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What is a Model

• Imitation/simplification of a real world system
• Tool that provides statistical estimates and not
  exact results
• Computational, statistical or judgment-based
• Helpful for product design and pricing,
  valuation, forecasting, risk management,
  financial reporting, and performance management.
• Understand how your liability value changes over
  time, when your liability value needs to be
  calculated stochastically

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What is a Stochastic Model?

• A model that involves probability or randomness
• Random inputs (Normal, Lognormal, Uniform)
• Generally run many times (1000, 10000)
• Representative sampling (Yvonne Chueh)
• Distribution of outputs
• Estimates of statistics (mean, ile, std.dev)
• Error estimates (direct or bootstrapping)

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What is Model Risk?

• Model risk the possibility of loss or error
  resulting from the use of models.
• Model misspecification
• Assumption misspecification
• Inappropriate use or application
• Inadequate testing, validation, and documentation
• Lack of knowledge or understanding, user and/or
  management
• Error and negligence

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How do you Model Equity Risk?

• Flat Return(8) with an Extreme MfAD(-30)
• Set of deterministic scenarios (stress tests)
• Purchase sets of stochastic scenarios
• Stochastic Scenarios
• Normal/Lognormal Returns
• Autocorrelated Returns (time series)
• Regime Switching LogNormal (RSLN)
• One correlation matrix
• Different correlation matrices for each regime
• Other stochastic model (Wilkie, Smith, Lognormal,
  Stoch Volatility, empirical)
• Risk Neutral or Real World

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Yield Curve vs. Equity

• Are they related?
• Direct relation shows zero correlation
• However
• Bond Funds and Equity Indices show 30-60
  correlation
• Duration analysis can explain 90 of bond fund
  returns
• an( int int-1) Bond Fund Return (t-1,t)
• One way to connect Yield Curves with Equity
  Returns
• Leads to interest rates driving equity returns

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What Equity Risk do you model?

• Indices
• Stock Market Indices
• North America SP500, TSX, NASDAQ
• Europe FTSE, DAX
• Industry specific? Company specific?
• Public Equity / Private Equity
• Do you model
• Hedge Funds? Pass-through products?
• Real Estate? REITs?
• Credit Spreads/Counterparty Risk?
• Currency Risk?

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Is Equity Related to other Returns?

• NO ?? Independent
• Correlation Matrix (Normal/Lognormal)
• Regime Switching Assumptions
• Time Series, Volatility Jumps
• Macro-Economic Drivers (Wilkie Model)
• Does it matter?
• It depends on what you are trying to do

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Scenario Generators

• Issues
• Is the focus on the mean, median, or tail events?
• Economic vs. Risk Neutral model
• Calibration (current/historical data)
• Numerous Scenario Generators to choose from
• Desirable Characteristics to check for
• Integrated model
• Incorporates the principle of parsimony
• Flexible. A component approach.
• Beware Often there is a false sense of precision

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Why risk-neutral?

• Financial derivatives value depends on the value
  of another financial instrument
• Their prices do not depend on the particular
  risk-preferences of the purchaser
• so we can assume any risk-preferences
• Mathematically convenient to assume purchaser is
  risk-neutral
• If you project market movements along a
  risk-neutral random walk and discount asset
  payoffs at the risk-free rate, you will obtain
  the fair value of that asset

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Fair Value

• Two portfolios with identical payoffs must have
  the same price
• ?Arbitrage - opportunity for profit buy the
  less expensive portfolio and sell the more
  expensive portfolio
• FOR INSURANCE Liabilities
• No Arbitrage doesnt work perfectly the market
  cannot freely buy and sell the insurance
  liability
• Risk-neutral pricing tells you what it would cost
  to buy the same payoffs in the market. (not
  necessarily a good estimate of the expected cost
  of the guarantee if left unhedged)

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Risk-Neutral Valuation

• Brownian Motion
• µ expected risk free forward rate
• s implied volatility, e random error
• Does Risk-Neutral Market-Consistent?
• If µ and s are market-consistent, the prices that
  the model produces are market consistent
• Both µ and s can be functions of time
• s is often considered to be a function of market
  levels (market volatility increases when market
  levels fall)

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Real World Model

• Brownian Motion for the stochastic model
• Drift rate historical long-term avg returns
  (not the risk-free forward curve)
• Volatility long term average or stochastic
  (GARCH, jump diffusion, RSLN)
• Goal to reflect a reasonable distribution of
  potential future returns
• Fewer expected payoffs of the embedded option
  than under risk-neutral valuation (on average,
  the stock market has a better return than
  risk-free investments)
• Higher variability of profit by scenario
• The bad tail can be very bad

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Rule of Thumb

• Tail risk
• Use real-world valuation to measure tail risk
• Average cost
• Use real world inputs when you are willing to
  accept the average result with a high amount of
  variability
• Use risk neutral when you want results (e.g. a
  price or a profit measure) which you can be very
  confident can be realized (through hedging)

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Who uses your Equity Models?

• Hedging (Financial Engineering)
• Market-consistent pricing - RN
• Risk Management, Valuation and Pricing (Actuarial
  Modeling)
• Tail exposures RW
• Volatility - RW
• Averages RW/RN
• Static Hedging - RN
• Dynamic Hedging RW/RN

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Regime Switching Models

• Discrete time (e.g. daily, monthly)
• Any model with different parameters in each
  regime (Normal, AR(1), ARCH.)
• 2-Regime Lognormal Monthly estimation software
  free from SOA website
• Very simple stoch vol model
• Tractable, intuitive, 2 Regimes are usually
  enough for monthly data - 6 parameters m1, m2,
  s1, s2, p12, p21
• Regime 1 Low Vol, High Mean, High Persistence
  (small p12)
• Regime 2 High Vol, Low Mean, Low Persistence
  (large p21)

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2-Regime LogNormal

• REGIME 1 r1
• Low Volatility s1
• High Mean m1

REGIME 2 r2 High Volatility s2 Low Mean m2
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Simple Stochastic Model

• 3-year 100 Seg Fund Maturity Guarantee
• MER 3

Regime 1 Regime 2
Fund LN1(11,16) LN2(-8, 20)
P(Switch) p124 p2122
Time in Regime Time in Regime
Regime 1 84.6
Regime 2 15.4
Mean Std.Dev
8.4 18.1
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Simple Stochastic Model Scen 1

• 3-yr Maturity Guarantee No death / lapse
• Initial Deposit 1 Top-up 0

Time 0 1 2 3
Uniform RAND()0.934 0.641 0.135 0.053
Regime 2 2 1 1
Normal NORMSINV(RAND()) -0.1635 0.7642 0.9195
Return (1it) expmut sigmasqrt(t)RN 0.89 1.26 1.29
Fund Fund(t-1)Return 0.89 1.13 1.46
Fund Less MER FundLessMER(t-1)Return (1-MER) 0.87 1.06 1.33
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Simple Stochastic Model Scen 2

• 3-yr Maturity Guarantee No death / lapse
• Initial Deposit 1 Top-up 0.19

Time 0 1 2 3
Uniform RAND()0.649 0.039 0.827 0.154
Regime 1 2 2 1
Normal NORMSINV(RAND()) 0.1635 -0.7642 0.3195
Return (1it) expmut sigmasqrt(t)RN 0.95 0.79 1.17
Fund Fund(t-1)Return 0.95 0.76 0.88
Fund Less MER FundLessMER(t-1)Return (1-MER) 0.93 0.71 0.81
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More Advanced Stochastic Models

• Other Modeling Considerations
• Death and Lapse (dynamic lapse?)
• Death Benefits and Living Benefits
• Ratchets and Resets
• Policyholder Behaviour
• Commissions / Surrender Charges / DAC
• Reserves / Capital
• Net Income / Tax / Distributable Earnings
• Discount Rates for Present Values
• Illustrating Results
• Hedging Strategies

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Summary Statistics

• Mean, Standard Deviation, Skewness, Kurtosis,
• Percentiles (Quantiles)
• Confidence intervals http//www.fenews.com/fen47/
  topics_act_analysis/topics-act-analysis.htm
• CTE 95 Mean of worst 5 of results
• Variance Estimate Hancock and Manistre NAAJ
  9(2) 129-156

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Box Plots
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Histograms and CTEs

• Histogram of scenario outcomes

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How Many Scenarios are Enough?

• Convergence / Sampling error
• Variance Reduction Techniques may help
• Many techniques work for averages not tails

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Are you taking a Holistic Approach?

• ERM Approach takes advantage of synergies across
  products
• Consistent set of RW and/or RN scenarios used for
  all lines of business
• Projections aggregated by scenario across lines
  of business
• Yield curve and equity return assumptions must be
  consistent
• More difficult if two Tier Stochastic simulation
  is required

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1-Tier Stochastic Simulation

• Projected Liability Payouts

• Can determine t0 reserve(CTE70-80 and TBSR
  (CTE95)
• Can determine liability payout projections
• Can not accurately determine future reserve and
  capital projections (approximations NPATH,
  Black-Scholes)

V0
0
T
Time
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2-Tier Stochastic Simulation

• Projected Liability Payouts, Reserves, Capital,
  Net Income .

• Can determine t3 reserve for each stochastic
  scenario (CTE70-80)
• Can determine future capital needs and net income
  projections
• Much more time consuming

V0
0
T
Time
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2-Tier Stochastic Simulation

• Projected Liability Payouts, Reserves, Capital,
  Net Income .

• Much more time consuming
• 1000 Tier 1 Scenarios
• 10 time steps each
• 100010 points to perform a second tier
  simulation
• 500 scenarios at each point 5,000,000 Tier 2
  scenarios

V0
0
T
Time
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Insurance Options

• Embedded options in insurance liabilities are
  different from financial options
• Sub-optimal exercise behavior
• FPDA can pay surrender charges and get a new
  contract if new money rates rise
• Evidence PHs are inefficient in using this
  option
• Some PH will not surrender their contracts no
  matter how uncompetitive their renewal rate
• Segregated Funds (VA/VL) GMAB should invest in
  the most aggressive funds available
• CAPM more risk implies more return
• Evidence PHs invest in conservative and balanced
  funds

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Stochastic Modeling Challenges

• Option payoffs that depend on policyholder
  behavior will reflect
• Historical behavior patterns
• Actuarial judgment
• Path-dependent behavior (ie. lower lapses for in
  the money guarantees) can be modeled
• Introduces uncertainty to valuation results
• Practitioners have argued about the proper way
  to model behavior in a risk-neutral framework
• (library.soa.org/library-pdf/RRN0608.pdf by M.
  Evans)

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Stochastic Modeling Challenges (Continued)

• Long-term nature of liabilities
• Expected market forward rates past 30 years is
  needed for valuation
• Instruments that will hedge the yield curve past
  30 years or equity risks past 10 years are
  illiquid or unavailable
• Computational Requirements
• Distributed processing (AXIS, MatLab, .)
• 2-Tier Stochastic Analysis (Stochastic-in-Stochast
  ic)

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Conclusions

• Equity risk is not like traditional insurance
  risk.
• Stochastic Modeling is a tool that can help us
  understand complex dynamic processes.
• Start simple and build.
• Test uncertain assumptions.
• Develop expertise.