Paper Schedule Reports

指導老師：戴天時 老師 楊曉文 老師

學生： 謝昌宏

Outline

• What’s Guaranteed Minimum Withdrawal Benefit (GMWB)?

• Pricing Method• See Example

2

What’s Guaranteed Minimum Withdrawal Benefit (GMWB)?

Reference Source ：中泰人壽 金富貴外幣變額年金保險

1.Roll-up( 複利增值 )2.Ratchet( 鎖高機制 )3. Break even( 保本 )

1.Roll-up( 複利增值 )2.Ratchet( 鎖高機制 )3. Break even( 保本 )

Pricing Method

• The Bino-trinomial Tree1. 延續 Milevsky and Salisbury(2006)的設計，假設 GMWB 所投資的標的資產符合幾何布朗運動2. 帳戶價值會隨著時間有預期報酬的增加以外，還有公平費用率的收取，若假設公平費用率是連續收取，且保戶不能提前解約，則我們可以將帳戶的隨機過程改為：

Reference Source : The Bino-trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing4

Pricing Method

• Optimized withdrawal rate

[(( * ) ( * ) ( * ))* ]

( )

r tA B C

GMWB Value

A P B P C P e

include Future Annuity Value

Vs.

( )*(1 )t

Full withdrawal Value

W G k G

Optimized withdrawal rule reference Kwork - GUARANTEED MINIMUM WITHDRAWAL BENEFIT IN VARIABLE ANNUITIES

Reference Source : The Bino-trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing5

Pricing Method

• Death Rate - Reduction factorsGAM-94 （ 1994 ）：

The value of AAx refer to “1994 GROUP ANNUITY MORTALITY TABLE AND 1994 GROUP ANNUITY RESERVING TABLE”.

6

)1()1994(1994

AAqq xy

x

y

x

Pricing Method

• Wang Risk TransformGiven a distribution function F, its Wang transform is defined as

where F(x) is the distribution function corresponding to the standard Normal distribution and λ is a parameter called the market price of risk.

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Risk-Neutral Death Rate

Real-world Death Rate

Pricing Method

• SECURITIZATION OF LONGEVITY RISK: PRICING SURVIVOR BONDS WITH WANG TRANSFORM IN THE LEE-CARTER FRAMEWORK

In Belgium, is the appropriate proxy for the market price of an annuity sold to an 65-year-old individual.i = 3.25%, and is the probability that a 65-year-old annuitant does not reach age 65 + t.They get λ65(2005) = −0.4722883 for men and −0.2966378 for women.

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Pricing Method

• Death Rate - Transform Death Rate

• 在此我們令 65 歲時仍生存的人數為基準，來算出各個年齡下的瞬時死亡率，例如：在 2005 年為 65 歲，其未來一年內瞬時死亡率為：

65 65 65

( )65( ) (1 ( )) ( ) ( )

, - 65, 65z

z t dtl y n nq y n l y l y z e

n z z

9

65

65

66 65 1 65

66 65 65

65 1 65 65

65 1 6565 1 65

65

66 ( )65

(2006) (2005) (1 (2005))

(2006) (2005) (2005)

(2005) (1 (2005)) (2005)

(2005) (1 (2005))ln( ) ln(1 (2005))

(2005)

t dt

l l q

l l e l e

l q l e

l qq

l

Pricing Method

• Death Rate - Transform Death Rate其 66 歲時，未來一年內的瞬時死亡率為：

10

65 66

65 66

67 65 2 65

67 65 65

65 2 65 65

65 2 6566 65 2 6

65

66 67( ) ( )65 66

(2007) (2005) (1 (2005))

(2007) (2005) (2005)

(2005) (1 (2005)) (2005)

(2005) (1 (2005))ln( ) ln(1

(2005)

t dt t dt

l l q

l l e l e

l q l e

l qq

l

5 65(2005))

165 64 65

6565

1

64 6565

( 65) (1 ( 65))ln( )

( 65)

= ln(1 ( 65)) , 65 x

xx

x kk

x

x kk

l q

l

q

:

:

:

x Age

Birth Year

Maximum Age

Pricing Method

• Death Rate將一年分成 m 期，每期時間長度為

從 65 歲購買 GMWB 的那一刻往後經過 2 期 的時間，投保人的生存機率為：

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t

t t t65

6565 2

26565 2 65 65

( )( 65 2 ) ( 65) ( 65)

tt

t

t dtl t l e l e

Pricing Method

• When hit the boundary

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Death

Living

0

G

Death

Living

0

G

12

: ( : , : )

* * * *x xT TrT r T

Option Value age x long of a period T

G G e e G e e Discount factor Conditional probability of living

See Exapmle

• Find BTT Middle Point

ln (0) [ln ( ) ln (0)] => BW E W t W

2

ln ( ) [( ) ] ( )2

d W t r dt dB t

CRR steps is odd:

(ln ( ) ln ( ))1.38

2

0.5 1.5

int *2

W t d W t lIndex

t

Index

Middle Po l Index t

Reference Source : The Bino-trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing

4.605

4.499

4.548

4.972

4.124

CRR

Boundary

3.912

2*CellHeigh

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See Example

• First year

4.605(100)

Boundary

2*CellHeigh

CellHeigh

4.97(144.41)

4.55(94.48)

4.12(61.82)

5.18(178.54)

4.76(116.81)

4.34(76.42)

3.91(50)

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See Example

• Withdrawal G

4.605(100)

Boundary

4.97(144.41)

4.55(94.48)

4.12(61.82)

5.18(178.54)

4.76(116.81)

4.34(76.42)

3.91(50)

4.86(128.54)

4.2(66.81)

3.27(26.42)

015

See Example

• Second year

4.605(100)

Boundary

4.97(144.41)

4.55(94.48)

4.12(61.82)

5.18(178.54)

4.76(116.81)

4.34(76.42)

3.91(50)

4.86(128.54)

4.2(66.81)

3.27(26.42)

0

178.54

116.81

76.42

50

32.71

21.4

1614

See Example• Calculate final value

4.605(100)

Boundary

4.97(144.41)

4.55(94.48)

4.12(61.82)

5.18(178.54)

4.76(116.81)

4.34(76.42)

3.91(50)

4.86(128.54)

4.2(66.81)

3.27(26.42)

0

178.54

116.81

76.42

50

32.71

21.4

1714

50

See Example

• Forecast probability of death(Age>=65)

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See Example

• Forecast probability of death(Age>=65)

19

Ex: q65(2005)=q65(1994)x(1-AA65)(2005-1994)=0.019016 q66(2006)=q66(2004) x(1-AA66)(2006-1994)=0.0207688

)1()1994(1994

AAqq xy

x

y

x

See Example

• Calculate risk-neutral (n=1,2,3,…)1.calculate ( 總生存率 ), x>=65

2.

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1 1( 1) (1 ( 1)) ( )x x xl y q y l y

*65nq

( )xl y

65 6665

65

65 6765

65

(2005) (2006)1 0.017485

(2005)

(2005) (2007)2 0.0356861

(2005)

...

l lq

l

l lq

l

See Example

• Calculate risk-neutral 3.

21

65nq

λ65(2005) = −0.4722883 for men*

65nq

*65nq

*65

*65

1 0.004926

2 0.0114413

...

q

q

See Example

• Calculate risk-neutral conditional death force

Conditional Survival Probability:

22

*65 1 65

*66 2 65 65

= ln(1 (2005)) 0.00494

= ln(1 (2005)) 0.006569

...

q

q

65

66

0.997534131, 0.5

0.996720697

t

t

e t

e

1

64 6565

= ln(1 ( 65)) , 65 xx

x x kk

q

See Example

• Backward induction - CRR

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Boundary

178.54

116.81

76.42

50

32.71

21.4

14

50

66

66

((178.54* ) (116.81* ))* *

((178.54* ) (116.81* ))* *(1 )

tr t

tr t

Value Pu Pd e e

Pu Pd e e

Pu

Pd

66

66

((50* ) (50* ))* *

((32.71* ) (21.4* ))* *(1 )

tr t

tr t

Value Pu Pd e e

Pu Pd e e

Survival value

Death value

See Example

• Backward induction – first term

24

66

66

((130.67* ) (85.49* ) (55.93* ))

* * ((144.41* ) (94.48* )

(61.81* ))* *(1 )

tr t

tr t

Value Pu Pm Pd

e e Pu Pm

Pd e e G

Survival value

Death value

130.67(144.41)

85.49(94.48)

55.93(61.81)

Pu

Pm

Pd

Vs. (we choice the higher)

(1- )*(178.54 - )Full withdrawal value k G G

(178.54)

See Example

• Backward induction – hit the boundary

25

66 )( * *rValue G G e e

130.67(144.41)

85.49(94.48)

55.93(61.81)

Pu

Pm

Pd

(178.54)

Thanks for your attation

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