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2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006

Canadian Institute of Actuaries. L’Institut canadien des actuaires. 2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006. Stochastic Equity Modeling. Dr. Julia Lynn Wirch-Viinikka AVP Investment Products Pricing AEGON Canada. Agenda.

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2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006

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  1. Canadian Institute of Actuaries L’Institut canadien des actuaires 2006 Seminar for the Appointed Actuary Colloque pour l’actuaire désigné 2006

  2. Stochastic Equity Modeling Dr. Julia Lynn Wirch-Viinikka AVP Investment Products Pricing AEGON Canada

  3. 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? AEGON Canada

  4. 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 AEGON Canada

  5. 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 AEGON Canada

  6. 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 don’t diversify: every policyholder follows the same market scenario at the same time AEGON Canada

  7. What 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 AEGON Canada

  8. 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 AEGON Canada

  9. 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) AEGON Canada

  10. 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 AEGON Canada

  11. 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 AEGON Canada

  12. 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 AEGON Canada

  13. What Equity Risk do you model? • Indices: • Stock Market Indices: • North America: S&P500, 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? AEGON Canada

  14. 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 AEGON Canada

  15. 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 AEGON Canada

  16. 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 AEGON Canada

  17. “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” doesn’t 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 guaranteeif left unhedged) AEGON Canada

  18. Risk-Neutral Valuation • A Random Walk: • μ = expected risk free forward rate • σ = implied volatility, ε = random error Does “Risk-Neutral” = “Market-Consistent”? • If μ and σ are market-consistent, the prices that the model produces are market consistent • Both μ and σ can be functions of time • σ is often considered to be a function of market levels (market volatility increases when market levels fall) AEGON Canada

  19. Real World Model • Random walk for the stochastic model: • Drift rate: long term averages of historical returns for that stock (not the risk-free forward curve) • Volatility: long term average or stochastic (GARCH, jump diffusion, regime-switching lognormal) • 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 AEGON Canada

  20. 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) AEGON Canada

  21. 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 AEGON Canada

  22. 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) AEGON Canada

  23. 2-Regime LogNormal REGIME 1 r1 Low Volatility s1 High Mean m1 REGIME 2 r2 Low Volatility s2 High Mean m2 AEGON Canada

  24. Simple Stochastic Model • 3-year 100% Seg Fund Maturity Guarantee • MER = 3% AEGON Canada

  25. Simple Stochastic Model: Scen 1 3-yr Maturity Guarantee: No death / lapse Initial Deposit = $1; Top-up = $0 AEGON Canada

  26. Simple Stochastic Model: Scen 2 3-yr Maturity Guarantee: No death / lapse Initial Deposit= $1; Top-up = $0.19 AEGON Canada

  27. 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 AEGON Canada

  28. 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 AEGON Canada

  29. + Maximum 75th Percentile Median 25th Percentile Minimum Outliers + + + Box Plots AEGON Canada

  30. Histograms and CTE’s • Histogram of scenario outcomes AEGON Canada

  31. How Many Scenarios are Enough? • Convergence / Sampling error • Variance Reduction Techniques may help • Many techniques work for averages not tails AEGON Canada

  32. 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 AEGON Canada

  33. 1-Tier Stochastic Simulation • Projected Liability Payouts • Can determine t=0 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 AEGON Canada

  34. 2-Tier Stochastic Simulation • Projected Liability Payouts, Reserves, Capital, Net Income …. • Can determine t=3 reserve for each stochastic scenario (CTE70-80%) • Can determine future capital needs and net income projections • Much more time consuming V0 0 T Time AEGON Canada

  35. 2-Tier Stochastic Simulation • Projected Liability Payouts, Reserves, Capital, Net Income …. • Much more time consuming: • 1000 Tier 1 Scenarios • 10 time steps each • 1000*10 points to perform a second tier simulation • 500 scenarios at each point = 5,000,000 Tier 2 scenarios V0 0 T Time AEGON Canada

  36. 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 AEGON Canada

  37. 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) • 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-Stochastic) AEGON Canada

  38. 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. AEGON Canada

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