Dynamic policyholder behavior modeling in risk neutral world

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-diversifiable
Law of large numbers Derivatives pricing
Dynamic
Mutual
Policyholder
Funds
Behavior
Multiple underlying assets Path Dependency
3

4. Annuity Derivatives Pricing Challenges
Dynamic 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 Options
Strike Maturity RiskFree Volatility Risky Rate
1,000 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 Path
Strike=$1,000; Risk free=5%, Real world gross return = 8.5%; Vol=20%
Duration 1 2 3 4 5 6 7 8 9 10
Std Random Normal -1.278 0.394 1.155 -0.682 -0.720 3.018 -0.271 -3.193 -2.699 -1.687
Risk Neutral AV 798 890 1,155 1,038 926 1,746 1,704 927 557 409
Real 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 -5.964
Risk Neutral 591 1 591 0.6065 358 Theta=(0.085-0.05)/0.20 0.1750
Real 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 Simulations
Assuming no fees deducted and the T-year persistency PT is dynamic
13

14. Simulation Example: Real World vs. Risk Neutral
GMAB 10 Year Maturity
Assuming no fees deducted and 10-year persistency is dynamic
Real World Risk Neutral
Discounted 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 Policyholder
Modeling Methods in “Annuity Option Pricing”
Risk Neutral Real World Conservative
AV Projections Risk Neutral Risk Neutral Risk Neutral
Dynamic Policyholder Behavior Projections Risk Neutral Real World (Shadow, Parallel) Conservative
Equity Growth Mean r R (usually>r) Conservative
Equity 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 Dynamic
Policyholder 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
Conservative
Index 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 Illustration
A 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 projections
14,000,000
Dynamic Hedge Performance With Decrements 200,000
180,000
12,000,000
160,000
10,000,000
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 Scenario
Conservative 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 Scenario
Naive 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 Comparison
Average 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 Comparison
Error Range Around the Mean Following Varying Economic Paths
Range (Mean+/- Std) of Net Gains % Premium
Range (Mean+/- Std) of Net Gains % Premium
RN Model
50% Conservative Model
40% RN + 1SD 50%
40% Conservative + 1SD
30% RN Mean
20% 30% Conservative Mean
RN - 1SD
10% 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 Premium
Economic Mean Growth => -20% -10% 0% 10% 20% 30% -20% -10% 0% 10% 20% 30%
Premium
Conservative 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 Premium
Economic 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 Process
Beginning 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 behavior
Interests 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 & Comments
Integrating financial engineering and actuarial science …

33. Appendix: Extensions and Importance Sampling Simulations
Extension 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
j
Comparison 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