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2nd Equity Based Guarantee 3 D

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Dynamic policyholder behavior modeling in risk neutral world

Dynamic policyholder behavior modeling in risk neutral world

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  • 1. 2nd Annual Equity Based Guarantee Conference Dynamic Policyholder Behavior Modeling 1330 hours – 1500 hours 5 October 2006 Frank Zhang, CFA, FRM, FSA, MSCF, PRM Vice President Senior Quantitative Derivatives Strategist Head of Structured Derivatives Strategies and Innovations ING USFS Annuity Market Risk Management Frank.Zhang@US.ING.Com 610-425-4222
  • 2. AGENDA Dynamic Policyholder Behavior Critical to VA Pricing and VA Projections (Real World and Risk Neutral) A Bridge between Risk Neutral vs. Real World Valuation of Derivatives Risk Neutral vs. Real World Dynamic Hedging Illustrations 2
  • 3. Life Insurance or Derivatives?VA guarantees blur the boundary between derivatives products and traditional life insurance products: Living or dying! Life Variable Derivatives Insurance Annuities Diversifiable Non-diversifiableLaw of large numbers Derivatives pricing Dynamic Mutual Policyholder Funds Behavior Multiple underlying assets Path Dependency 3
  • 4. Annuity Derivatives Pricing ChallengesDynamic Policyholder Behavior Modeling – Critical and Difficult Dynamic policyholder behavior modeling is critical & difficult• Key driver for pricing but options not always exercised optimally• Mortality risk managed by pool of large numbers but living benefits much more challenging• Behavior very difficult to predict and with little or no experience• Policyholder dynamics causing significant gamma exposure• Capital market risks not diversifiable as insurance risks MBS prepayment vs. annuities dynamic policyholder behavior modeling• MBS prepayments based on real world experience or expectations but validated by active capital market MBS prices, unlike annuities• Risk neutral pricing standard in financial engineering, but transition from actuarial expectations to risk neutral pricing caused confusions about probability distributions and stochastic simulations• MBS markets not usually concerned with nested stochastic projections that mix risk neutral world and risk neutral valuations, unlike annuities• We will show that there are simple connections between the real world and the risk neutral world 4
  • 5. GMWB Pricing Risk Neutral Valuation GMWB is paid only If GMWB is in the money and still In force when AV=0 Persistency and payoff amounts are path dependent Price = sum of all future possible GMWB payoffs on persist contracts 5
  • 6. 6
  • 7. AGENDA Dynamic Policyholder Behavior Critical to VA Pricing and VA Projections (Real World and Risk Neutral) A Bridge between Risk Neutral vs. Real World Valuation of Derivatives Risk Neutral vs. Real World Dynamic Hedging Illustrations 7
  • 8. An Option May Be Priced with Real World Simulations Let’s value a simple European put option. In addition to pricing using risk neutral simulations (@5%), we also price it using real world simulations (@8%). With adjustments, options can be priced with non-risk-neutral simulations!European Put OptionsStrike Maturity RiskFree Volatility Risky Rate1,000 10 5% 20% 8%Spot Black-Scholes Simulated Paths (Annual Steps) Simulated Paths (Annual Steps)Price Put Option 1,000 10,000 100,000 1,000 10,000 100,000 Risk Neutral Simulations Ratio with B-S Price 600 153.0 152.9 152.8 152.9 100.0% 99.9% 100.0% 800 93.6 92.8 93.4 93.6 99.2% 99.8% 100.0% 1,000 58.5 58.4 58.2 58.4 100.0% 99.6% 99.9% 1,200 37.4 37.1 37.4 37.3 99.2% 100.1% 99.9% 1,400 24.4 23.9 24.7 24.4 97.9% 101.2% 100.0% Adjusted Real World Simulations Ratio with B-S Price 600 153.0 151.2 153.7 153.0 98.8% 100.5% 100.0% 800 93.6 92.5 94.5 93.6 98.9% 100.9% 100.0% 1,000 58.5 56.8 59.8 58.5 97.1% 102.3% 100.1% 1,200 37.4 36.2 39.1 37.5 96.8% 104.5% 100.4% 1,400 24.4 22.9 26.0 24.5 93.8% 106.6% 100.4% 8
  • 9. It Is All about the Change of Probability Distributions Shifting the distribution with higher mean moves the “area” under the curve to the right To compensate, we may adjust the outcomes with factors bigger or smaller than 1 9
  • 10. Adjusted Payoffs: Put and Call When Strike =1 10
  • 11. But What Bridge Adjustments? Formulas  Let X be a random variable or function of random variable, then T 2 ( N j 0.5 ) EQ[X]=EP[X*Z(T)] r Where and Z (T ) e j 1  Translation: Risk neutral valuation of expected value of random variable X = Risky valuation of expected value of random variable X, multiplied by Z(T)  Here Nj are independent random normal variables in real world P  It is derived from Girsanov’s Theorem in stochastic calculus.  Z is called Radon-Nikodym derivative.  Z is the path dependent bridge adjustment!Example of 10-Year Put Following a Random Sample PathStrike=$1,000; Risk free=5%, Real world gross return = 8.5%; Vol=20%Duration 1 2 3 4 5 6 7 8 9 10Std Random Normal -1.278 0.394 1.155 -0.682 -0.720 3.018 -0.271 -3.193 -2.699 -1.687Risk Neutral AV 798 890 1,155 1,038 926 1,746 1,704 927 557 409Real World AV 826 954 1,283 1,194 1,104 2,154 2,177 1,227 763 581 Payoff Adj Adj Payoff Discount Disc Adj Payoff Sum of Std Random Normals -5.964Risk Neutral 591 1 591 0.6065 358 Theta=(0.085-0.05)/0.20 0.1750Real World 419 2.437 1,021 0.6065 619 Theta squared 0.0306 Adjustment Z 2.437 11
  • 12. How to Apply the Bridge Adjustments? Project random variable X with real world stochastic paths and calculate path-dependent Z(T) accordingly X may be anything such as  Price of the stock index  Put option payoff  Call option payoff  GMAB payoffs with persistency Z(T)’s are path-dependent Z(T)’s are independent of function X Take the average of the product X*Z(T) It works for expected value i.e. mean only (such as option prices) 12
  • 13. GMAB Option Valuation with SimulationsAssuming no fees deducted and the T-year persistency PT is dynamic 13
  • 14. Simulation Example: Real World vs. Risk NeutralGMAB 10 Year MaturityAssuming no fees deducted and 10-year persistency is dynamic Real World Risk NeutralDiscounted Mean 58.50 24.85 58.07 24.39 Scenario Path Unadjuste Bridge Adjusted GMAB GMAB AV10 Persistency AV10 Payoff Persistency Number d Payoff Adj Payoff Payoff Payoff 1 7,744 - 0.253 - 25.5% - 5,457 - 25.8% - 2 581 419 2.437 1,021 32.8% 335 409 591 36.1% 213 3 1,183 - 1.308 - 29.9% - 833 167 40.6% 68 4 32,154 - 0.073 - 24.6% - 22,659 - 24.6% - 5 7,829 - 0.250 - 24.6% - 5,517 - 24.6% - 6 583 417 2.429 1,012 42.0% 425 411 589 45.1% 266 ↕ 4995 ↕ 939 ↕ ↕ ↕ 61 1.602 98 ↕ ↕ ↕ ↕ 32.7% 32 661 339 ↕ ↕ 37.5% 127 4996 6,359 - 0.300 - 24.6% - 4,481 - 24.6% - 4997 878 122 1.699 208 34.7% 72 618 382 39.6% 151 4998 1,776 - 0.917 - 24.6% - 1,252 - 24.7% - 4999 1,873 - 0.875 - 25.4% - 1,320 - 28.7% - 5000 1,161 - 1.330 - 31.0% - 818 182 34.8% 63 Put Strike 1,000 Risk Free 5.0% Volatility 20% Initial AV S0 1,000 Real Rate 8.5% 14
  • 15. Implications from the Bridge Adjustments Real world expected value (with adjustment) = Risk neutral expected value Option can be valued with real world projections, as long as adjustments are made Therefore, for simplicity, we will from now on directly apply the risk neutral model. Real world expected value (without adjustment) ≠ Risk neutral expected value Option valuation projected with real world projection but without adjustments are wrong Therefore, for variable annuities pricing, a method called parallel projection (risk neutral of assets and real world of dynamic policyholder behavior, without bridge adjustments) is wrong! 15
  • 16. AGENDA Dynamic Policyholder Behavior Critical to VA Pricing and VA Projections (Real World and Risk Neutral) A Bridge between Risk Neutral vs. Real World Valuation of Derivatives Risk Neutral vs. Real World Dynamic Hedging Illustrations 16
  • 17. Definition of the Three Different Dynamic PolicyholderModeling Methods in “Annuity Option Pricing” Risk Neutral Real World ConservativeAV Projections Risk Neutral Risk Neutral Risk NeutralDynamic Policyholder Behavior Projections Risk Neutral Real World (Shadow, Parallel) ConservativeEquity Growth Mean r R (usually>r) ConservativeEquity Random Numbers RN same as RN Conservative For annuity “pricing” (not real world projections), all methods simulation variable annuity account value following risk neutral distribution Parallel Shadow Method  This can also be called a “naïve” approach, because it is commonly believed that options should priced in risk neutral while the policyholder behavior is observed and measured in the real world “Conservative Scenario” Method  The persistency follows an independent “conservative” path assuming the equity market at any point in the future always achieve some very low percentile of the possible cumulative returns.  This path is actually very bad but deterministic, resulting potentially higher persistency  The approach is “conservative” against severe market downfalls 17
  • 18. Picture of Sample Paths of the Three Different DynamicPolicyholder Modeling Methods To model dynamic policyholder behavior, here are a few different methods to project the “account value” to determine the in-the-moneyness Three Different Paths for Dynamic Policyholder Behavior Modeling 230 210 Real World 190 Risk Neutral ConservativeIndex Value 170 150 130 110 90 70 50 0 2 4 6 8 10 12 14 16 18 20 Year 18
  • 19. Dynamic Policyholder Modeling for GMAB Previously we have shown that  Options can be priced with real world simulations, as long as we also apply the path-dependent “bridge” adjustments.  This real-world simulation plus adjustments approach is equivalent to the risk neutral valuation.  We assume the risk neutral approach is the “correct” approach, but will test others as well. We will next illustrate the three different dynamic policyholder modeling examples for a simple GMAB benefit pricing, with simplified assumptions:  The payoff is like a put option at maturity but dependent on survival to maturity (survivorship or persistency)  The persistency is a function of deterministic death and dynamic lapse  Partial withdrawals are deterministic  Dynamic lapses function generates higher persistency when the contract is more in the money  Non-stochastic and flat interest rates (real rate > risk free rate) and volatilities  The underlying price stochastic process follows Geometric Brownian Motion 19
  • 20. The Comparison Tests The comparison tests will be performed through an illustration of  Stochastic on stochastic projections of a delta-only dynamic hedging program and  Hedge performance attribution Two criteria for a successful VA pricing model:  Price the VA guarantees correctly  The actual hedging performance using the pricing model will lock-in the value (with small tracking errors), no matter what real world path it has followed Two components corresponding to the criteria in dynamic hedging program:  Over- or under- valuation of the VA guarantees (1)  Over- or under- valuation of the Greeks (deltas in these examples) (2)  Total G/L = Premium G/L (1) + Delta G/L (2) 20
  • 21. Dynamic Hedging Performance IllustrationA Stochastic on Stochastic System The ideal hedging strategy is to track the hedge account (yellow line = cash + hedge G/L) with liability (black line) closely all the time weekly following any equity path We will summarize the P/L at the end of 10 year projections14,000,000 Dynamic Hedge Performance With Decrements 200,000 180,00012,000,000 160,00010,000,000 140,000 8,000,000 120,000 Account Value 6,000,000 100,000 80,000 4,000,000 60,000 2,000,000 40,000 - 20,000(2,000,000) - 0 100 200 300 400 500 Option Value Cum Futures G/(L) + PV of Option Premium Account Value 21
  • 22. The Construction of A Stochastic on Stochastic System Weekly time steps in the outer real world loop to project the GMAB contract for its account performance and decrements. The real world economic paths are randomly generated. The the Greeks and option values at each time step are calculated using the C++ add-ins for 2000-scenario risk neutral valuations over 10 years in the inner loops. Make proper adjustment of the delta hedging program positions each week and track the G/L and the cash account forward. The G/L can also be attributed into components for reasons such as different in initial premiums, delta G/L, decrement G/L, tracking errors, MTM earnings volatility, volatility G/L, etc. 22
  • 23. Dynamic Hedging Performance Sorted P/L(=Prem G/L+Δ G/L) Risk Neutral vs. “Conservative” Methods  The “conservative” model over priced the GMAB and resulted initial price gain relative to the risk neutral model  The “conservative” model over estimated deltas and resulted biased P/L that is equivalent to betting for the down market (but to end up with large losses in up market) Sorted Net Gain Difference (-20% Economic Path) Sorted Net Gain Difference (30% Economic Path) Conservative vs. Risk Neutral Models Conservative vs. Risk Neutral Models 2.5 1.0 Millions Millions 2.0 NetGain 0.5 DeltaGains NetGain 1.5 DeltaGains Difference in Net Gains 0.0 Difference in Net Gains 1 74 147 220 293 366 439 512 585 658 731 804 877 950 1.0 -0.5 0.5 -1.0 0.0 1 65 129 193 257 321 385 449 513 577 641 705 769 833 897 961 -0.5 -1.5 -1.0 -2.0 Scenario ScenarioConservative vs. RN Sorted NetGain Prem DeltaGains Conservative vs. RN Sorted NetGain Prem DeltaGains Average 678,368 522,268 156,100 Average (47,895) 522,268 (570,164) 23
  • 24. Dynamic Hedging Performance Sorted P/L(=Prem G/L+Δ G/L) Risk Neutral vs. Naïve Shadow Parallel Methods  The “naive” model under priced the GMAB and resulted initial price lose relative to the risk neutral model  The “naive” model under estimated deltas and resulted biased P/L that is equivalent to betting the up market (but to end up with large losses in down market) Sorted Net Gain Difference (-20% Economic Path) Sorted Net Gain Difference (30% Economic Path) Naive vs. RN Models Naive vs. RN Models 0.4 0.6 Millions Millions 0.2 0.4 0.0 1 65 129 193 257 321 385 449 513 577 641 705 769 833 897 961 0.2 Difference in Net Gains Difference in Net Gains -0.2 0.0 1 60 119 178 237 296 355 414 473 532 591 650 709 768 827 886 945 -0.4 -0.2 -0.6 NetGain -0.4 -0.8 DeltaGains NetGain DeltaGains -1.0 -0.6 -1.2 -0.8 Scenario ScenarioNaive vs. RN Sorted NetGain Prem DeltaGains Naive vs. RN Sorted NetGain Prem DeltaGains Average (111,806) (501,517) 389,711 Average (562,872) (501,517) (61,355) 24
  • 25. Dynamic Hedging Performance ComparisonAverage P/L Following Varying Economic Paths The risk neutral model is as tight as BS model, but both have some random simulation errors The “naive” and “conservative” models perform better in opposite economic scenarios Net Hedging Gain % of Premium 20% 10 Year Put BS RN Conservative 10% Naïve 0% -10% -20% -30% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% Economic Path Mean 25
  • 26. Dynamic Hedging Performance ComparisonError Range Around the Mean Following Varying Economic Paths Range (Mean+/- Std) of Net Gains % Premium Range (Mean+/- Std) of Net Gains % Premium RN Model50% Conservative Model40% RN + 1SD 50% 40% Conservative + 1SD30% RN Mean20% 30% Conservative Mean RN - 1SD10% 20% Conservative - 1SD 0% 10%-10% 0%-20% -10% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30%-30% -20%-40% -30%-50% -40% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% -50% Economic Path Mean Economic Path Mean Range (Mean+/- Std) of Net Gains % Premium 50% Naive Model  The risk neutral model are relatively 40% tight around the means, independent Naive + 1SD 30% of the economic scenarios Naive Mean 20% Naive - 1SD 10% 0%  The “naive” and “conservative”-10% models errors are bigger and-20% directionally dependent on economic-30% scenarios-40%-50% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% Economic Path Mean 26
  • 27. Dynamic Hedging Performance Comparison Attribution of Gains  The risk neutral model has consistently the smallest G/L across all (real world) economic scenarios (from very bad to very good).  Two main drivers: Option premiums and delta G/L  The “conservative” model’s G/Ls are mostly due to the excess premium collected. However, after removing the excess premiums, this model generates large losses in up markets  The opposite is true for the “naive” model’s G/L, which collects too little option premiums and gains in the up market is not enough to cover the deficiency in option premium GMAB10 Dynamic Hedging Policyholder Modeling Comparision - Attribution of Gains Net Gain = PV Asset10 - PV Liab10 Net Gain % of PremiumEconomic Mean Growth => -20% -10% 0% 10% 20% 30% -20% -10% 0% 10% 20% 30% PremiumConservative 2,462,156 363,239 506,230 269,058 (186,220) (192,741) (108,044) 14.8% 20.6% 10.9% -7.6% -7.8% -4.4%RN 2,442,914 (75,463) (50,106) (46,145) (32,737) (10,874) (17,986) -3.1% -2.1% -1.9% -1.3% -0.4% -0.7%Naïve 2,121,199 (591,812) (623,780) (496,832) (249,397) (111,269) (82,041) -27.9% -29.4% -23.4% -11.8% -5.2% -3.9% Attribution of Net Gains of Conservative/Naïve Relative to RN Net Gain Attribution % of RN PremiumEconomic Mean Growth => -20% -10% 0% 10% 20% 30% -20% -10% 0% 10% 20% 30%Conservative - RN 2,442,914 438,703 556,336 315,203 (153,483) (181,867) (90,058) 18.0% 22.8% 12.9% -6.3% -7.4% -3.7%Due to Premium Diff 2,442,914 19,242 19,242 19,242 19,242 19,242 19,242 0.8% 0.8% 0.8% 0.8% 0.8% 0.8%Due to Delta Diff 2,442,914 419,461 537,093 295,961 (172,725) (201,109) (109,300) 17.2% 22.0% 12.1% -7.1% -8.2% -4.5%Naive - RN 2,442,914 (516,349) (573,675) (450,687) (216,661) (100,396) (64,055) -21.1% -23.5% -18.4% -8.9% -4.1% -2.6%Due to Premium Diff 2,442,914 (321,714) (321,714) (321,714) (321,714) (321,714) (321,714) -13.2% -13.2% -13.2% -13.2% -13.2% -13.2%Due to Delta Diff 2,442,914 (194,635) (251,960) (128,973) 105,054 221,318 257,659 -8.0% -10.3% -5.3% 4.3% 9.1% 10.5% 27
  • 28. Path Dependency and Adaptive Attribution Analysis Variable annuity liability value is very path dependent• Complicated nature of benefits means that it must be dynamically replicated (but as statically as possible).• Due to uncertainty of assumptions, multiple underlying assets, and changing business volumes, VA liability is almost impossible to completely statically locked in without further adjustments. Liability option value roll-forward valuation analysis• Very detailed and extensive liability roll forward analysis is needed to account for all changes in the option values of the block of business.• Useful to understand all components of the liability option value changes, to understand trends and behavior, to catch outliers, and to direct potential future improvements. Asset and liability hedging performance attribution analysis• Hedging is not perfect• Useful to deepen the understanding and gain the insights of the dynamic hedging program performance, to understand the key drivers / assumptions of a dynamic hedging program, to catch the outliers, and to direct potential future improvements.• Important feedback to product design and dynamic policyholder behavior assumptions so that with regular updating the assumptions, hedging is never too far from where it should be 28
  • 29. Adaptive Learning:Through Liability Roll-forward ProcessBeginning Ending Period Period Expected vs. Actual Liability Liability Option Option Value and Greeks Option Value Value Changes in market levels, interest rates, and volatilities New/add-on/backdated premiums Time decay, fees, asset classification Deaths and lapses, withdrawals Transfers of assets between mutual funds Model changes, and other assumption updates, etc. 29
  • 30. Adaptive Learning:Through Hedge Performance Attribution Process Net hedging G/L from: Market risks & actuarial risks Tracking errors Gamma/volatility/interest G/L Policyholder behaviorInterests on cash pool & other actuarial elements Trading costs, etc. 30
  • 31. Conclusions Once the dynamic policyholder behavior formula is set, it is deterministic and can be hedged. The risk neutral modeling is the way to price and hedging variable annuities. Pricing (need to charge enough option premium) and dynamic hedging (need to have correct Greeks to hedge) should be based on the best estimated formula and evaluated in risk neutral world. The conservatism or margin of profitability should set separately from this formula with additional charge, etc. Stochastic on stochastic dynamic hedging projection system is very helpful to study hedging strategies and the financial impact. A sophisticated dynamic hedging projection system requires solid understanding of the derivatives theory and practices. Hedge performance attribution is the key to such understanding, including but not limited to decrement G/L and assumption change G/L. While most existing dynamic policyholder behavior modeling is not perfect, the combination of dynamic hedging and hedging performance attribution will automatically readjust the hedging positions over time to adapt to the changes that not only affect the dynamic policyholder behavior but also other elements in the hedging program. 31
  • 32. Your Questions & CommentsIntegrating financial engineering and actuarial science …
  • 33. Appendix: Extensions and Importance Sampling SimulationsExtension to Non-Flat Interest Rate and Volatility Term Structures Θ can be easily expanded to deal with term structure of interest rate and T volatilities 2 ( N j 0.5 ) j rj j j Where j and Z (T ) e j 1 jComparison with Importance Sampling Both can shift the distributions to different in-the-moneyness zone. They are very similar in this simple case of changing “means”. Importance sampling can change the distributions in more general ways. Changes by importance sampling is supposed to improve the simulation efficiency by reducing the resulting variance in the simulations, under very general probability distributions. The bridge adjustment is more powerful “stochastic process” with a “time dimension”. The bridge adjustment (Radon-Nikodym derivative) is the key link between real world and risk neutral simulations and derivatives pricing in general, under the Brownian motions stochastic process. 33