EngSci-GMWB-Complexities-Nov-25-2015 - v2
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Transcript of EngSci-GMWB-Complexities-Nov-25-2015 - v2
Complexities of Variable Annuity Management
Anthony Vaz, PhD, PEng
VP, Models, Methodology & Infrastructure Manulife Financial
2
Outline
1. Terminology & Product Characteristics
2. Valuation Framework for GMWB Products
3. Challenges in Reserve Computation for VA Blocks
4. Expedited Computation Using Analytic Approach
Definitions of Common Terms
4
Variable Annuity: Market linked savings product with insurancefeatures.
GMWB: Guaranteed Minimum Withdrawal Benefit, a verypopular VA rider (optional guarantee).
Valuation Model Assumptions
5
Actuarial Assumptions: Mortality
Deterministic
Lapse Rate Deterministic
Dynamic: lapse rate depends on the moneyness
Investment Portfolio Return Assumptions: Market Indices within Segregated Funds
Expected Growth Rate (risk free rate in Risk Neutral pricing)
Return Volatility
Return Correlations among indices
Fund Mix
6
GMWB Policy Features
Guaranteed Benefits
Death Benefits
Withdrawal Benefits
— Resets (increase withdrawal benefit)
— Bonus (increase withdrawal benefit)
Fees
Insurance Charges (mortality, expense, administrative)
Investment Management Fees (on Segregated Funds)
Surrender Charges (penalty charge for over-limit withdraws)
Rider Charges (addition charge for optional guarantees)
7
GMWB Policy Cash Flows
The survival probability is projected by the mortality rate.
Policyholder dies: Insurer is obliged to pay the balance of death benefit over account value
Policyholder survives: Insurer’s cash inflow:
fees deducted from policy holder’s investment account
Insurer’s cash outflow:
withdrawal made by the policy holder when account value is zero
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Valuation Framework for GMWB Products
Fair Market Value (FMV) of WB Net Liabilities
Price of the Embedded Option: Withdrawal Benefits
Present Value (PV) of WB Guarantee Fees
FMV determined from evolution of
Investment Account
Guaranteed Withdrawal Base/Amount
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Investment Account Modelling
Investment Account Parameters
Initial Premium: V0
Fund Market Returns
r: risk free rate
σ: fund volatility
Fund Management Expense Ratio (MER)
m: as percentage of the account value
GMWB Guarantee Fee
q: as percentage of the account value
Guaranteed Withdrawal Amount (GWA): G
Payout Phase
T1: payout start date
T2: payout end date
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GWA Modelling
Guaranteed Withdrawal Amount Parameters
Bonus Rate during Accumulation Phase: a
Reset during Accumulation Phase
Guarantee withdrawal rate: g
Guaranteed Withdrawal Amount (GWA)
* It is known at the last reset date.
]ValuesReset ,)(1max[ 1T
0 a VgG
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Market & Actuarial Assumptions
Feature Explanation
Interest Rate Forward Curve
Volatility Constant/Deterministic
Fund Mix Up to 12 funds
Guarantee Fee Fixed Percentage of AV or GV
Withdrawal Static (withdraw GWA every month )
Reset Resets can be throughout the product life
Bonus Up to the end of the accumulation phase
Mortality Mortality table
Lapse Dynamic lapse
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Simple Example: Assumptions
Consider a simple GMWB product with
Bonus Rate
Reset at Payout Start Date
Continuous Withdrawals
Constant GWA
No Mortality Risk
No Lapse
Segregated Fund Assumptions:
Single Fund
Constant Interest Rate
Constant Volatility
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T
T
0 ]V,)(1max[1
1
TT
VG
a
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Simple Example: Account Value
Account Value during Accumulation Phase:
10 ,])[( TtdBVdtVqmrdV tttt
Account Value during Payout Phase:
2
21
,][
),min(T ,])[(
TtdtGrVdV
TtdBVdtGVqmrdV
tt
tttt
Account Value at Maturity:
— Positive balance will go to policyholder
— Negative balance will be paid by insurer
: stopping time with V = 0
Bt: Brownian Motion, represents the randomness of market returns
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Simple Example: Cash Flow Case 1
Positive BalanceInsurer:
Asset = Fees Collected between 0 and T2
Liability = 0
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Simple Example: Cash Flow Case 2
Insurer:
Asset = Fees Collected between 0 and
Liability = Withdrawals between and T2
Negative Balance
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Simple Example: Account Value 1
0
50
100
150
200
250
300
350
400
450
500
0 2 4 6 8
10
12
14
16
18
20
22
24
Year
Ac
co
un
t V
alu
e
Withdrawal Rate = 1/15
No Withdrawal
Withdrawal Phase
T1
Accumulation Phase
T2
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Simple Example: Account Value 2
-150
-100
-50
0
50
100
150
0 2 4 6 8
10
12
14
16
18
20
22
24
Year
Ac
co
un
t V
alu
e
Withdrawal Rate = 1/15
No Withdrawal
Withdrawal Phase
T1
Accumulation Phase
T2
Stopping time
Ne
ga
tive
Ba
lan
ce
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FMV of WB Net Liability
FMV of WB Net Liab. = PV of Liab. – PV of Asset
= Option Price – PV of Future Guarantee Fees
Unknown Variables:
: Stopping Time with V = 0
G: GWA
Vs: Account Value
Parameters:
— T2: Payout End Date
— r: Risk Free Rate
— q: GMWB Guarantee Fee
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FMV of WB Net Liab.: Option Price
Guaranteed Withdrawal Amount G is known at T1:
Next Step: to Find the Conditional Distribution of the Stopping Time
Option Price = PV of the Amount of Negative Balance in Invest. Account
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4. FMV of WB Net Liab.: PV of Fees
Unknown Variables:
Vs: Account Value
Parameters:
— T2: Payout End Date
— r: Risk Free Rate
— q: GMWB Guarantee Fee
Definitions of Common Terms
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IFRS: International Financial Reporting Standards.
Reserve: The amount of funds that an insurer must set aside asa liability, to meet future policy obligations. Can be consideredas the PV of a conservative estimate of future net liabilities.
IFRS Reserve Method
IFRS Booked Reserve = BE Reserve + PfADs Reserve + Adjustments
Best Estimate Reserve
• expected PV of future liabilities based on cash flows projected with BestEstimate actuarial and economic assumptions.
PfADs Reserve
• increment in reserve by considering Provisions for Adverse Deviations.
Adjustments
• changes made to certain items such as rider fee accrual balance and bondfair value.
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Cash Flows for a VA Unhedged Block
Claims
Withdrawal benefit
Death benefit
Maturity benefit
Fees
Rider fee
Risk charge
Actuarial Margins
Net Liability CF
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Cash Flows for a VA Hedged Block
Claims
Withdrawal benefit
Death benefit
Maturity benefit
Fees
Rider fee
Risk charge
Actuarial Margins
Hedge Settlement CF
P&L from hedging instruments
Net Liability CF
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27
1 2 30 4 5 6 7
Time
Outer-loop Scenarios
(Real World)
Inner-loop Scenarios
(Risk Neutral)
Scenario 1
Scenario n
Scenario
1000
Stochastic Cash Flow Projection (Hedged VA Block)
Stochastic on Stochastic Process
(SOS)
Cash Flow Projection (Hedged VA Block)
Outer-loop
Projection
Inner-loop
Valuation
Hedging SensitivitiesFeesClaims
Hedge Settlement CF
Net Liability CF
Stochastic on Stochastic Process
(SOS)
CFs are calculated at each outer-loop node
Calculates FMVs and sensitivities
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Expedited Computation Using Analytic Approach
Objective:
Speed up the IFRS reserve calculation without changing the outer loop
projection.
Proposed Method:
At each time step, calculate “delta” for a small subset of 1000 outer loop
scenarios and apply regression method to estimate “delta” for the rest outer
loop scenarios.
Current Method Analytic
Approach
# of Scenarios in Outer Loop 1000 50
# of Time Steps in Outer Loop 480 = 40 years × 12 months
# of FMV Valuation at Each Node 2 2
Total # of FMV Valuation 958,002 47,902
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Analytic Approach: Regression Function
Explanatory Variables
• Account Value
• Withdrawal Benefit Guarantee
• Death Benefit GuaranteeRegression
Multi-dimensional
Function
sFitting Target
• ∆FMV
Cluster
Sampling
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Concluding Remarks
The analytic approach speeds up the SOS calculation by 20 times, while
preserving the accuracy of the reserve numbers.
The algorithm used in the underlying predictive analytic algorithm is
proprietary; however, it is based on a synthesis of cluster sampling and
regression kernel techniques that were adapted from the framework of a SVM
neural network.
A SoS Analytic tool based on this algorithm was put into production for official
reporting, which reduced quarterly valuation time from 18 days to 2 days.
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