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

3

Terminology & Product Characteristics

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

8

Valuation Framework for GMWB Products

9

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

10

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

11

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

12

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

13

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

12

T

T

0 ]V,)(1max[1

1

TT

VG

a

14

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

15

Simple Example: Cash Flow Case 1

Positive BalanceInsurer:

Asset = Fees Collected between 0 and T2

Liability = 0

16

Simple Example: Cash Flow Case 2

Insurer:

Asset = Fees Collected between 0 and

Liability = Withdrawals between and T2

Negative Balance

17

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

18

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

19

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

20

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

21

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

22

Challenges in Reserve Computation

Definitions of Common Terms

23

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.

2424

Cash Flows for a VA Unhedged Block

Claims

Withdrawal benefit

Death benefit

Maturity benefit

Fees

Rider fee

Risk charge

Actuarial Margins

Net Liability CF

25

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

26

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

28

Expedited Computation Using Analytic Approach

29

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

30

Analytic Approach: Regression Function

Explanatory Variables

• Account Value

• Withdrawal Benefit Guarantee

• Death Benefit GuaranteeRegression

Multi-dimensional

Function

sFitting Target

• ∆FMV

Cluster

Sampling

31

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.

32

Thank You