Financial Modeling of Variable Annuities

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2 Financial Modeling of Variable Annuities Robert Chen June,

3 Agenda Building blocks of a variable annuity model A Stochastic within Stochastic Model Rational policyholder behaviour Discussion / Questions 2

4 Building blocks of a VA model

5 Variable Annuities Shareholders Fund Capital Support Non-Unit fund Non-unit reserves Target Surplus Hedging instruments Non-unit fund Payout Cost of Guarantee Release of Reserves Investment Returns Unit fund Fund 1 Unit fund Payout Policyholder Unit benefits Premium Fund 2 Fund 3... Unit charges Shareholder Profit Fund n Expenses 4

6 Variable Annuities Shareholders Fund Capital Support Investment Returns Premium... Non-Unit fund Non-unit reserves Target Surplus Hedging instruments Fund 1 Fund 2 Fund 3 Fund n Unit fund Non-unit fund Payout Determine suitable charges Unit fund Policyholder Payout for main contract Unit benefits for each guarantee Unit charges Cost of Guarantee Release of Reserves Shareholder Profit Expenses 5

7 Variable Annuities Capital Support Non-Unit fund Non-unit reserves Target Surplus Hedging instruments Shareholders Fund Set up suitable reserves At policy issue Each year in the future Cost of Ability to meet any guarantee Guarantee Non-unit payments fund as they fall due Payout Following statutory rules Release of Reserves Investment Returns Unit fund Fund 1 Unit fund Payout Policyholder Unit benefits Premium Fund 2 Fund 3... Unit charges Shareholder Profit Fund n Expenses 6

8 Variable Annuities Shareholders Fund Capital Support Non-Unit fund Non-unit reserves Target Surplus Hedging instruments Non-unit fund Payout Cost of Guarantee Release of Reserves Investment Returns Premium Unit fund Fund 1 Fund 2 Fund 3... Fund n Unit fund Payout Policyholder Unit benefits Shareholder Design and evaluate hedging Profit schemes to mitigate the Unit charges guarantees Expenses 7

9 Variable Annuities Features Insurance coverage Capital protection GMDB, GMIB, GMAB, GMWB Fund linked return Choice among different funds A range of guarantees have been developed, initially in the U.S., typically expressed as optional rider benefits: GMDB - Guaranteed minimum death benefits, either return on premium or increasing over time (roll up or ratchet) GMIB Guaranteed minimum income (annuity) benefits on retirement, determined using guaranteed annuity factors applied to a base which may be premiums paid or increasing over time GMAB Guaranteed minimum accumulation benefits, effectively guaranteed surrender or maturity values at specified dates, typically premiums paid, sometimes with options to renew (reset) GMWB Guaranteed minimum withdrawal benefits, subject to annual limits, initially return of premium but often with ability to reset to fund value periodically, e.g. every 7 years. Latest variants guarantee withdrawals for life 8

10 Variable Annuities Valuation of Guarantees Guarantees are like put options Payout = MAX(strike price - account value, 0) For GMAV, maybe also GMIB Payout =MAX(guaranteed value - account value, 0) Can use Black-Scholes formula for valuation Strike price Payoff Price of Underlying Otherwise, e.g. some forms of GMWB: Timing is uncertain Level is uncertain Payout may be path dependent Needs a Monte Carlo stochastic approach 9

11 The Stochastic Modeling Process Economic generator Table of scenarios Investment returns, yields Inflation Cash flow projections Repeated for each scenario Reporting (across scenarios) Analysis 10

12 The Stochastic Modeling Process Economic generator Table of scenarios Future investment earnings assumptions, varying stochastically Investment returns, yields Inflation Cash flow projections Repeated for each scenario Reporting (across scenarios) Analysis 11

13 Stochastic Generators Project economic entities over multiple paths (scenarios) Equity returns Interest rates Inflation Etc 12

14 Stochastic Generators 13 Output from a stochastic economic generator

15 Economic generator Table of scenarios Needs assumptions on: The Stochastic Modeling Process Policyholder behavior Switching between funds Resets (ratchets) Level of partial withdrawals Surrender Expenses, rates of mortality, tax, etc Future investment earnings assumptions, varying stochastically Investment returns, yields Inflation Cash flow projections Repeated for each scenario Reporting (across scenarios) Analysis 14

16 Needs assumptions on: The Stochastic Modeling Process Policyholder behavior Switching between funds Resets (ratchets) Level of partial withdrawals Surrender Expenses, rates of mortality, tax, etc Templates showing Mean, 90% percentile, 65% conditional tail expectation (CTE), etc. Economic generator Table of scenarios Future investment earnings assumptions, varying stochastically Investment returns, yields Inflation Cash flow projections Repeated for each scenario Reporting (across scenarios) Analysis 15

17 Variable Annuities - GMWB Specific Requirements in US Capital requirement 90% CTE of the discounted value of minus the least surplus (i.e. the worst deficit, if any) at any time during the projection Using real world scenarios Statutory reserve (draft) Sufficient to meet the 75% CTE of the above amount Reserve for financial reporting (GAAP) Calculate mean of the discounted value of future guarantee payments E(PV(gtee payts),t) Calculate the level of fund charge (x%) needed to give a value at policy issue equal to value of g tee payments E(PV(charges),t) i.e. x% * E(PV(charge),0) = E(PV(gtee payts),0) Reserve(t) = MAX(0, E(PV(gtee payts),t) x% * E(PV(charges),t) Using risk neutral scenarios 16

18 Stochastic within stochastic

19 Stochastic within Stochastic Model Valuation of guarantees Closed form formulae Stochastic projection Uses an analytical formula (e.g. Black Scholes) to value the guarantee Used for simple guarantees Normally requires simplifying assumptions (e.g. normal distribution) Not able to allow for policyholder behaviour Uses Monte Carol simulation methods to value the guarantee Used where closed form formulae are not suitable Normally required for pathdependent guarantees (e.g. guarantees with ratchets) Slower, more rigorous 18

20 Valuation of guarantees Simple stochastic simulation Sim 1 Sim 2 Sim 3 Sim 4 Sim n

21 Valuation of guarantees Stochastic within stochastic simulation Sim 1 Sim 2 Sim 3 Sim 4 Sim n

22 Valuation of guarantees Sim 1 Sim 2 Real world simulations (policyholder behaviour) Sim Valuation Cost of Loops guarantees at the needed end of at each time year point 21

23 Valuation of guarantees Sim 1 Sim 2 Stochastic projection at each point in time to calculate value of g tee Real world simulations (policyholder behaviour) Sim Valuation Cost of Loops guarantees at the needed end of at each time year point 22

24 Valuation of guarantees Sim 1 Sim 2 Real world simulations (policyholder behaviour) Sim Valuation Cost of Loops guarantees at the needed end of at each time year point 23

25 Valuation of guarantees Sim 1 Sim 2 Stochastic projection at each point in time to calculate value of g tee Real world simulations (policyholder behaviour) Sim Valuation Cost of Loops guarantees at the needed end of at each time year point 24

26 Rational Policyholder Behavior

27 Dynamic Policyholder Behavior Typical shape of a dynamic variable annuity surrender rate function Expressed as a multiple of a base table of rates Relatively high surrender rates, because guarantee has little value Base Surrender Rate Multiplier Typical Dynamic Behavior Function As In-the-moneyness increases, surrender rates fall, as the value of the guarantee becomes significant to more policyholders Some policyholders will need to surrender, regardless of economic conditions Base rates here assumed to be the average across all economic conditions In-the-Moneyness Ratio For a GMIB, In-the-Moneyness (ITM) = Guaranteed value * guaranteed annuity rate Account value * market annuity rate ITM is driven by a combination of interest rates and market returns 26

28 Rational Policyholder Behavior Rational behavior is driven by a variety of factors surrounding the policyholder, not just in-the-moneyness Typical Dynamic Behavior Function Base Surrender Rate Multiplier Possibly Rational Possibly Rational In-the-Moneyness Ratio 27

29 Degrees of Rational Policyholder Behavior In a market consistent valuation, we should consider using an approach that reflects the approach adopted in capital markets: If it is cost effective to exercise an option, it will be exercised 100% rational 3 Typical Dynamic Dynamic and Rational Behavior Behaviors Function 100% Rational Behavior Dynamic Behavior Base Surrender Rate Multiplier In-the-Moneyness Ratio Passive Behavior (0% Rational) Threshold determined to maximize value 28

30 Dynamic versus Rational Policyholder Behavior Where does dynamic behavior fit on the scale of 0% to 100% rationality? Value with dynamic behavior based on historic information Guarantee Value * 1, (500) (1,000) (1,500) Example: Value of a GMIB rider, charging 50bps Effect of Rational Behavior 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Degree of Rationality On surrender rates The more rational the behavior, the more expensive the guarantee In this example, the compound effect of rational rates over successive years flattens the curve Is the dynamic approach sufficiently rational? 29

31 Final thoughts

32 Final thoughts A model has 2 parts Formulae / coding Assumptions Increase in computing power has made sophisticated analysis possible Temptation to take a 1-step approach to advanced analysis Modelling is a journey: take your time, and you will find it more rewarding E.g. start with a rough model, and refine it Assumptions become more complex as the model becomes more complex E.g. volatility is a new assumption in stochastic models how do we set these assumptions, what is appropriate? Important to model things correctly, but more important to understand your model Phased approach to implementation Testing of sensitivities 31

33 Thank you Robert Chen A soft copy of this presentation will be available at 32

34 2007 Deloitte Touche Tohmatsu. All rights reserved. Member of Deloitte Touche Tohmatsu 33