Policyholder Exercise Behavior for Variable Annuities ... · Policyholder Exercise Behavior for...

Policyholder Exercise Behavior for Variable

Annuities including Guaranteed Minimum

Withdrawal Benefits1

Thorsten Moenig2

Department of Risk Management and Insurance, Georgia State University35 Broad Street, 11th Floor; Atlanta, GA 30303; USA

Email: [email protected]

June 2011

1We gratefully acknowledge sponsorship by the Society of Actuaries.2Joint work with Dr. Daniel Bauer, Georgia State University

/ 23 Overview

1 Introduction

2 A Lifetime Utility Model for Variable Annuities

3 Results

4 Conclusions and Future Research

Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs

/ 23 Introduction

1 Introduction

Motivation

Risk-Neutral Valuation Approach


3 Results



/ 23 Introduction

Motivation

• Variable Annuities:

I Popular long-term investment vehicles

I Tax-deferred growth

I Investment evolves according to underlying (risky) portfolio

I Uncertain payout

• Guaranteed Minimum (Death / Income / Accumulation /Withdrawal) Benefits

I Insurers offer guaranteed payments

I Policyholders can purchase security

I Similar to (combination of) financial options


/ 23 Introduction

Motivation

• Withdrawal uncertainty

I Could mitigate or intensify insurer’s exposure to investment and/ormortality risks

I Interactions non-trivial

I Affects pricing and risk management

• Insurers in trouble

I Disintermediation in 1970s

I Equitable Life closed to new business in 2000

I The Hartford accepted $3.4B in TARP money in June 2009 afterlosing $2.75B in 2008, hurt by investment losses and the cost of VAguarantees


/ 23 IntroductionRisk-Neutral Valuation Approach

• Used in actuarial literature to price variety of options:I Milevsky and Posner (2001): GMDBI Ulm (2006): “Real option to transfer”I Zaglauer and Bauer (2008): Participating life insurance contracts

• To analyze withdrawal behavior for GMWBs:I Milevsky and Salisbury (2006)I Bauer, Kling and Russ (2008)I Optimal stopping problem, akin to pricing American put optionI Exercise / Withdraw if exercise value exceeds continuation valueI Worst-case scenario, calculate correct upper bound

• VA market incomplete: cannot sell – or repurchase – policy at itsrisk-neutral valueI Withdrawing means giving up possible guarantees and tax benefits


/ 23A Lifetime Utility Model for Variable Annuities

1 Introduction


The Model

Bellman Equation

Implementation in a Black-Scholes Framework

Parameter Assumptions

3 Results




The Model

• Consider withdrawal decisions in life-cycle model with outside

investment

• PH maximizes expected lifetime utilityI Consumption and bequestsI Initial wealth W0

I Annual (deterministic) income It

• Invests P0 in VA with finite maturity T, remainder in outsideaccountI Includes GMWB, possibly other guaranteesI Return-of-investment guaranteesI Other types possible, at cost of larger state spaceI All guarantee accounts identical to benefits base, G·

tI Annual guarantee fee φ as percentage of concurrent account value



The Model

• VAs grow tax-deferredI Withdrawals taxed on last-in first-out basisI Early withdrawal tax (10%) if PH withdraws prior to age 59.5

• Restrict all actions to policy anniversary datesI Four state variables

? VA account X−t? Outside account A−t? Benefits base G·t? Tax base Ht

I Three choice variables? Withdrawal amount wt

? Consumption Ct

? Risk allocation in outside account νt



Bellman Equation

Vt(yt) = maxCt ,wt ,νt

uC(Ct)+e−β ·Et [qx+t · uB(bt+1|St+1) + px+t · Vt+1(yt+1|St+1)] , (1)

yt ≡ (A−t ,X−t ,G

·t ,Ht ),

X+t =

(X−t − wt

)+,

A+t = A−t + It + wt − Ct − feeI − feeG − taxes,

feeI = s ·max{

wt −min(gWt ,GW

t )},

feeG = sg · (wt − feeI) · I{x+t<59.5},

taxes = τ ·min{wt − feeI − feeG, (X−t − Ht )

+},

G·t+1 =

(G·t − w

)+: w ≤ gW

t(min

{G·t − w , G·t ·

X+t

X−t

} )+

: w > gWt

,

Ht+1 = Ht −(

wt −(

X−t − Ht

)+)+

,

A−t+1 = A+t ·[νt ·

St+1St− κ ·

(νt ·

St+1St− 1)+]

,

X−t+1 = X+t · e

−φ ·[νX · St+1

St

],

bt+1 = A−t+1 + max{X−t+1,GDt+1},

νt ≥ 0 ,∑

i νt (i) = 1,



Implementation in a Black-Scholes Framework

• Solve by recursive dynamic programming:

(I) Create appropriate state space grid

(II) For t = T : for all grid points (A,X ,G·,H), compute V−T (A,X ,G·,H).

(III) For t = T − 1,T − 2, . . . ,1: Given V−t+1, calculate V−

t (A,X ,G·,H)

recursively for each (A,X ,G·,H) on the grid using anapproximation of the integral in (1)

I Discretize return space and evaluate via Green’s function

I Gauss-Hermite quadrature

(IV) For t = 0: For the given starting values A0 = W0 − P0, X0 = P0,G·

0 = G·1 = P0 and H0 = H1 = P0, compute

V−0 (W0 − P0,P0,P0,P0) recursively from Equation (1)



Parameter Assumptions

• Policyholder is 55 years old, T = 15 years to maturity

• P0 = 100K ; W0 = 2 · P0 = 200K ; It = 40K

• CRRA(γ = 3) utilities; B = 1; β = r

• τ = 30%, κ = 15%

• Guarantee fee φ = 50 bps

• Surrender fee s = 5%,

gWt =

{0 : t ≤ 5

20,000 : t > 5

• r = 4%, µ = 8%, σ = 15%I Merton ratio: µ−r

σ2·γ = 0.08−0.040.152·3 ≈ 0.5926

• νX = 100% equity exposure in VA


/ 23 Results

1 Introduction


3 Results

Withdrawal Behavior

Pricing and Sensitivities



/ 23 Results

Withdrawal Behavior

• Little withdrawal activity (approx. 12K per PH on average)I No withdrawals during accumulation periodI No premature withdrawals in 67% of casesI PH empties guarantee account in 6% of casesI < 1% chance of excessive withdrawal during contract phase

• Two main reasons to withdraw prematurely:I VA account below tax base (approx. 7K on average)

? Nuanced patterns? Interaction of in-the-moneyness of guarantee, tax considerations and

excess withdrawal charge

I VA account much greater than outside account (approx. 5K onaverage)? To reduce overall risk exposure (Merton ratio)


/ 23 Results

Withdrawal Behavior

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x 104

0

2

4

6

8

10

12

14

15x 10

4

Fig. 1: t=14, At = 180K , G.

t = 100K, H

t = 100K.

Xt−

wt

wt

max(Guarantee,Xt−)


/ 23 Results

Withdrawal Behavior

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x 104

0

2

4

6

8

10

12

14

15x 10

4 Fig. 3: t=10, At = 180K , G.

t = 100K, H

t = 100K.

Xt−

wt

wt

max(Guarantee,Xt−)


/ 23 Results

Withdrawal Behavior

0 1 2 3 4 5 6 7 8 9 10 11 12 13

x 105

0

1

2

3

4

5

6

7

8x 10

4 Fig. 7: t=10, At = 20K , G.

t = 100K, H

t = 100K.

Xt−

wt,c

t

wt

Ct A

t− + I

t = 60K


/ 23 ResultsPricing and Sensitivities

• Guarantee fee of φ = 50 bps sufficient to cover expected costs• In-the-moneyness appears to be OK proxy for pricing

I Different source to withdrawals

• Eliminating excess withdrawal fee increases net profits (win-win)

Base Case w/d if X−t ≤ G·t s = 0%

EQ[Fees] 6, 252 5, 925 5, 907EQ[Excess-Fee] 19 0 0EQ[Guarantee] 4, 558 4, 761 2, 136

%(Guarantee > 0) 24.34% 33.97% 31.94%

E[agg. w/d] 12, 084 14, 180 16, 374E[agg. w/d & t ≤ 6] 0 0 4, 374

E[agg. w/d & X−t ≤ Ht ] 6, 953 14, 119 6, 047E[agg. w/d & X−t > Ht ] 5, 030 0 5, 816


/ 23 ResultsPricing and Sensitivities

• Withdrawal patterns highly sensitive to volatility

• Considering taxes important

BC: σ = 15% σ = 20% σ = 25% No Taxes

EQ[Fees] 6, 252 6, 047 5, 152 4, 734EQ[Excess-Fee] 19 62 1, 006 64EQ[Guarantee] 4, 558 8, 384 11, 533 4, 746

%(Guarantee > 0) 24.34% 36.48% 45.57% 28.16%

E[agg. w/d] 12, 084 29, 914 85, 879 88, 791E[agg. w/d & t ≤ 6] 0 54 13, 356 82

E[agg. w/d & X−t ≤ Ht ] 6, 953 17, 615 27, 877 19, 500E[agg. w/d & X−t > Ht ] 5, 030 10, 674 31, 221 69, 033


/ 23 Conclusions and Future Research

1 Introduction


3 Results




• Develop lifetime-utility model to analyze withdrawal behavior for

VA with guarantees

• Numerically solve policyholder’s decision making problem inBlack-Scholes frameworkI Return-of-investment GMWB

• Infrequent withdrawals

• PH withdraws when VA account is below tax baseI Interaction of in-the-moneyness of guarantee, tax considerations

and excess w/d fee

• PH withdraws when VA account is largeI To lower overall risk exposure



• Extend policyholder environment

I Unemployment Risk

I Subjective mortalities

• Withdrawal patterns highly sensitive w.r.t. volatility σ

I Stochastic volatility framework

• Alternatives to EUT

I Epstein-Zin preferences

I Correlation Aversion


/ 23

THANK YOU!


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