paper schedule reports
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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. What’s Guaranteed Minimum Withdrawal Benefit ( GMWB )?. 1.Roll-up( 複利增值 ) 2.Ratchet( 鎖高機制 ) 3. Break even( 保本 ). - PowerPoint PPT PresentationTRANSCRIPT
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.
7
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.
8
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:
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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
13
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)
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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
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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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