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)?
1.Roll-up(複利增值)
2.Ratchet(鎖高機制)
3. Break even(保本)
Reference Source：中泰人壽 金富貴外幣變額年金保險
Pricing Method
• The Bino-trinomial Tree
1. 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：
dW (t )  (r   )W (t )dt   W (t )dB1 (t )
dW (t )
 (r   )dt   dB1 (t )
W (t )
d ln W (t )  [(r   ) 
2
2
d ln W (t )  [(r (t )   ) 
]dt   dB1 (t )
2
2
]dt   [  dB2 (t )  1   2 dZ (t )]
Reference Source : The2Bino-trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing
4
Pricing Method
• Optimized withdrawal rate
GMWB Value
 [(( A * PA )  ( B * PB )  (C * PC ))* e rt ]
(include Future Annuity Value)
Vs.
Full withdrawal Value
 (Wt  G)*(1  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 Pricing
5
Pricing Method
• Death Rate - Reduction factors
GAM-94（1994）：
y
1994
x
x
q q
 (1 AAx)
( y 1994 )
The value of AAx refer to “1994 GROUP ANNUITY
MORTALITY TABLE AND 1994 GROUP ANNUITY
RESERVING TABLE”.
6
Pricing Method
• Wang Risk Transform
Given a distribution function F, its Wang transform is
defined as Risk-Neutral
Real-world Death Rate
Death Rate
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
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
l65 ( y  n)  (1  nq65 ( y  n))  lz ( y )  l65 ( y  z )e
z  ( t ) dt
 65
, n  z - 65, z  65
• 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：
l66 (2006)  l65 (2005)  (1  1 q65 (2005))
l66 (2006)  l65 (2005)e
66  ( t ) dt
 65
 l65 (2005)e  65
 l65 (2005)  (1  1 q65 (2005))  l65 (2005)e  65
 65   ln(
l65 (2005)  (1  1 q65 (2005))
)   ln(1  1 q65 (2005))
l65 (2005)
9
Pricing Method
• Death Rate - Transform Death Rate
其66歲時，未來一年內的瞬時死亡率為：
l67 (2007)  l65 (2005)  (1  2 q65 (2005))
l67 (2007)  l65 (2005)e
66  ( t ) dt  67  ( t ) dt
 65
66
 l65 (2005)e  65  66
 l65 (2005)  (1  2 q65 (2005))  l65 (2005)e  65  66
 66   ln(
l65 (2005)  (1  2 q65 (2005))
)  65   ln(1  2 q65 (2005))  65
l65 (2005)
x 1
l65 (  65)  (1  x 64 q65 (  65))
 x   ln(
)   k
l65 (  65)
k  65
x : Age
x 1
 : Birth Year
=  ln(1  x 64 q65 (  65)) 
k , 65  x  
 : Maximum Age
k  65

10
Pricing Method
• Death Rate
將一年分成m期，每期時間長度為
65
t
t
t
t
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：
l65 2 t (  65  2 t )  l65 (  65)e
65 2 t  (t ) dt
 65
 l65 (  65)e 65 2
11
t
Pricing Method
• When hit the boundary
Option Value : (age : x, long of a period : T )
G  G * e rT * e xT  G * e r 2T * e x1T
Discount factor
Conditional probability of living
Living
Living
G
Death
Death
G
0
0
12
See Exapmle
CRR
• Find BTT Middle Point
d ln W (t )  [(r   ) 
2
2
]dt   dB(t )
ln W (0)  E[ln W (t )  ln W (0)] => B
CRR steps is odd:
(ln W (t )  d ln W (t ))  l
 1.38
2 t
 Index   0.5  1.5
Index 
Middle Po int  l  Index * 2 t
4.972
4.605
2*Cell
Heigh
4.548
4.499
4.124
3.912
Boundary
Reference Source : The Bino-trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing
13
See Example
• First year
5.18(178.54)
4.97(144.41)
2*CellHeigh
4.605(100)
4.55(94.48)
4.76(116.81)
4.34(76.42)
4.12(61.82)
3.91(50)
CellHeigh
Boundary
14
See Example
• Withdrawal G
5.18(178.54)
4.97(144.41)
4.86(128.54)
4.605(100)
4.55(94.48)
4.76(116.81)
4.2(66.81)
4.34(76.42)
4.12(61.82)
3.27(26.42)
3.91(50)
Boundary
0
15
See Example
• Second year
5.18(178.54)
178.54
4.97(144.41)
4.86(128.54)
4.605(100)
4.76(116.81)
116.81
4.34(76.42)
76.42
4.55(94.48)
4.12(61.82)
4.2(66.81)
3.91(50)
50
Boundary
3.27(26.42)
0
32.71
21.4
14
16
See Example
• Calculate final value
5.18(178.54)
178.54
4.97(144.41)
4.86(128.54)
4.605(100)
4.76(116.81)
116.81
4.34(76.42)
76.42
4.55(94.48)
4.12(61.82)
4.2(66.81)
3.91(50)
3.27(26.42)
50
Boundary
32.71
50
0
21.4
14
17
See Example
• Forecast probability of death(Age>=65)
18
See Example
• Forecast probability of death(Age>=65)
y
1994
x
x
q q
 (1 AAx)
( y 1994 )
Ex: q65(2005)=q65(1994)x(1-AA65)(2005-1994)=0.019016
q66(2006)=q66(2004) x(1-AA66)(2006-1994)=0.0207688
19
See Example
• Calculate risk-neutral nq65*(n=1,2,3,…)
1.calculate
lx ( y ) (總生存率),
x>=65
lx1 ( y 1)  (1  qx1 ( y 1))  lx ( y)
2. 1q65  l65 (2005)  l66 (2006)  0.017485
l65 (2005)
l65 (2005)  l67 (2007)
2q65 
 0.0356861
l65 (2005)
...
20
See Example
•
*
nq
Calculate risk-neutral 65
λ65(2005) = −0.4722883 for men
3.
*
65
nq
nq65
1q65*  0.004926
2q65*  0.0114413
...
21
See Example
• Calculate risk-neutral conditional death force
x 1
 x =  ln(1  x 64 q65 (  65))   k , 65  x  
k 65
65* =  ln(1  1 q65 (2005))  0.00494
66* =  ln(1  2 q65 (2005))  65  0.006569
...
Conditional Survival Probability:
e
 65 t
e 66
t

0.997534131, t  0.5

0.996720697
22
See Example
• Backward induction - CRR
Pu
178.54
Survival value
Value  ((178.54* Pu )  (116.81* Pd ))* e r t * e 66

t
Pd
116.81
((178.54* Pu )  (116.81* Pd ))* e r t *(1  e 66 t )
Death value
76.42
Value  ((50* Pu)  (50* Pd ))* e r t * e 66

((32.71* Pu )  (21.4* Pd ))* e
r t
*(1  e
t
 66 t
50
)
Boundary
32.71
50
21.4
14
23
See Example
• Backward induction – first term
(178.54)
130.67(144.41)
Pu
Survival value
Value  ((130.67 * Pu )  (85.49* Pm)  (55.93* Pd ))
*e  r t * e  66
t

((144.41* Pu )  (94.48* Pm) 
(61.81* Pd )) * e  r t *(1  e  66 t )  G
Pm
85.49(94.48)
Pd
55.93(61.81)
Death value
Vs. (we choice the higher)
Full withdrawal value  (1- k )*(178.54 - G)  G
24
See Example
• Backward induction – hit the boundary
(178.54)
130.67(144.41)
Pu
Pm
85.49(94.48)
Pd
Value  G  (G * e r * e 66 )
55.93(61.81)
25
Thanks for your attation
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