Exam MLC
Raise Your Odds® with Adapt
SURVIVAL DISTRIBUTIONS SURVIVAL DISTRIBUTIONS Probability Functions Actuarial Notations # 𝑝𝑝$ = Probability that 𝑥𝑥 survives 𝑡𝑡 years = Pr 𝑇𝑇$ > 𝑡𝑡 = 𝑆𝑆$ 𝑡𝑡 # 𝑞𝑞$ = Probability that 𝑥𝑥 dies within 𝑡𝑡 years = Pr 𝑇𝑇$ ≤ 𝑡𝑡 = 𝐹𝐹$ 𝑡𝑡 # 𝑝𝑝$ + # 𝑞𝑞$ = 1 #|3 𝑞𝑞$ = Probability that 𝑥𝑥 survives 𝑡𝑡 years and dies within the following 𝑢𝑢 years = # 𝑝𝑝$ ⋅ 3 𝑞𝑞$D# = # 𝑝𝑝$ − #D3 𝑝𝑝$ = #D3 𝑞𝑞$ − # 𝑞𝑞$ Life Table Functions G 𝑑𝑑$ = 𝑙𝑙$ − 𝑙𝑙$DG 𝑙𝑙$D# # 𝑝𝑝$ = 𝑙𝑙$ 𝑙𝑙$ − 𝑙𝑙$D# # 𝑑𝑑$ = # 𝑞𝑞$ = 𝑙𝑙$ 𝑙𝑙$ 𝑙𝑙$D# − 𝑙𝑙$D#D3 3 𝑑𝑑$D# = #|3 𝑞𝑞$ = 𝑙𝑙$ 𝑙𝑙$ Force of Mortality 𝑓𝑓$ 𝑡𝑡 𝜇𝜇$D# = 𝑆𝑆$ 𝑡𝑡 𝑑𝑑 𝜇𝜇$D# = − ln 𝑆𝑆$ 𝑡𝑡 d𝑡𝑡 𝑑𝑑 𝜇𝜇$D# = − ln # 𝑝𝑝$ d𝑡𝑡 𝑓𝑓$ 𝑡𝑡 = # 𝑝𝑝$ ⋅ 𝜇𝜇$D# # 𝑝𝑝$
# 𝑞𝑞$
= exp − =
#|3 𝑞𝑞$
=
#
O
#
. M 𝑝𝑝$ #D3
#
O
𝜇𝜇$DM d𝑠𝑠
⋅ 𝜇𝜇$DM d𝑠𝑠
. M 𝑝𝑝$
⋅ 𝜇𝜇$DM d𝑠𝑠
Mortality Laws Constant Force of Mortality 𝜇𝜇$ = 𝜇𝜇 RS# # 𝑝𝑝$ = 𝑒𝑒 Uniform Distribution 1 𝜇𝜇$ = , 0 ≤ 𝑥𝑥 < 𝜔𝜔 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡 , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥 # 𝑝𝑝$ = 𝜔𝜔 − 𝑥𝑥 𝑢𝑢 , 0 ≤ 𝑡𝑡 + 𝑢𝑢 ≤ 𝜔𝜔 − 𝑥𝑥 #|3 𝑞𝑞$ = 𝜔𝜔 − 𝑥𝑥 Beta Distribution 𝛼𝛼 𝜇𝜇$ = , 0 ≤ 𝑥𝑥 < 𝜔𝜔 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡 Y , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥 # 𝑝𝑝$ = 𝜔𝜔 − 𝑥𝑥 Gompertz’s Law 𝜇𝜇$ = 𝐵𝐵𝑐𝑐 $ , 𝑐𝑐 > 1 𝐵𝐵𝑐𝑐 $ 𝑐𝑐 # − 1 # 𝑝𝑝$ = exp − ln 𝑐𝑐 Makeham’s Law 𝜇𝜇$ = 𝐴𝐴 + 𝐵𝐵𝑐𝑐 $ , 𝑐𝑐 > 1 𝐵𝐵𝑐𝑐 $ 𝑐𝑐 # − 1 # 𝑝𝑝$ = exp −𝐴𝐴𝐴𝐴 − ln 𝑐𝑐
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Moments Complete Future Lifetime
∘
𝑒𝑒$ =
General
]
O
# 𝑝𝑝$
1 𝜇𝜇 𝜔𝜔 − 𝑥𝑥 ∘ 𝑒𝑒$ = 2 𝜔𝜔 − 𝑥𝑥 ∘ 𝑒𝑒$ = 𝛼𝛼 + 1 ∘
𝑒𝑒$ =
Constant Force of Mortality Uniform Distribution Beta Distribution
d𝑡𝑡
G
∘
O
# 𝑝𝑝$
d𝑡𝑡
• Uniform Distribution ∘
𝑒𝑒$:G| = G𝑝𝑝$ 𝑛𝑛 + G 𝑞𝑞$
Curtate Future Lifetime 𝑒𝑒$ =
]
bcd
𝑘𝑘 ⋅ b|𝑞𝑞$ =
]
bcd
𝑒𝑒$:G| =
bcd
b 𝑝𝑝$
𝑘𝑘 ⋅ b|𝑞𝑞$ + 𝑛𝑛 ⋅ G 𝑝𝑝$ =
• Uniform Distribution ∘ 𝑒𝑒$:G| = 𝑒𝑒$:G| − 0.5 G.𝑞𝑞$ Recursive Formulas ∘ ∘ ∘ 𝑒𝑒$ = 𝑒𝑒$:G| + G 𝑝𝑝$ ⋅ 𝑒𝑒$DG ∘
∘
∘
bcd
𝑒𝑒$:G| = 𝑒𝑒$:f| + f 𝑝𝑝$ ⋅ 𝑒𝑒$Df:GRf| ,
b 𝑝𝑝$
Term Life
𝑚𝑚 < 𝑛𝑛
𝑒𝑒$:G| = 𝑒𝑒$:fRd| + f 𝑝𝑝$ 1 + 𝑒𝑒$Df:GRf| ,
Fractional Ages UDD 0 ≤ 𝑠𝑠 + 𝑡𝑡 ≤ 1 𝑙𝑙$DM = 1 − 𝑠𝑠 ⋅ 𝑙𝑙$ + 𝑠𝑠 ⋅ 𝑙𝑙$Dd M 𝑞𝑞$ = 𝑠𝑠 ⋅ 𝑞𝑞$ 𝑠𝑠 ⋅ 𝑞𝑞$ M 𝑞𝑞$D# = 1 − 𝑡𝑡 ⋅ 𝑞𝑞$ 𝑞𝑞$ 𝜇𝜇$DM = 1 − 𝑠𝑠 ⋅ 𝑞𝑞$ 𝑞𝑞$ = M𝑝𝑝$ ⋅ 𝜇𝜇$DM
35
34
Whole Life
𝑒𝑒$ = 𝑒𝑒$:G| + G 𝑝𝑝$ ⋅ 𝑒𝑒$DG = 𝑒𝑒$:GRd| + G 𝑝𝑝$ 1 + 𝑒𝑒$DG 𝑒𝑒$ = 𝑝𝑝$ 1 + 𝑒𝑒$Dd 𝑒𝑒$:G| = 𝑒𝑒$:f| + f 𝑝𝑝$ ⋅ 𝑒𝑒$Df:GRf| , 𝑚𝑚 < 𝑛𝑛 𝑒𝑒$:G| = 𝑝𝑝$ 1 + 𝑒𝑒$Dd:GRd|
33
33
INSURANCE INSURANCE Level Annual Insurance Type of EPV Insurance Discrete
𝑛𝑛 2
G
32
32
• Uniform Distribution ∘ 𝑒𝑒$ = 𝑒𝑒$ − 0.5 n-year Temporary Curtate Future Lifetime GRd
30 31
n-year Temporary Complete Future Lifetime 𝑒𝑒$:G| =
Read the 2-year select and ultimate mortality table from the left to the right and then continue downwards. 𝑞𝑞 $ 𝑞𝑞 $ Dd 𝑞𝑞 $ Dh 𝑥𝑥 𝑥𝑥 + 2
𝑚𝑚 < 𝑛𝑛
Constant Force of Mortality 0 ≤ 𝑠𝑠 + 𝑡𝑡 ≤ 1 𝑙𝑙$DM = 𝑙𝑙$ dRM ⋅ 𝑙𝑙$Dd M M M 𝑝𝑝$ = M 𝑝𝑝$D# = 𝑝𝑝$ 𝜇𝜇$DM = − ln 𝑝𝑝$ Select and ultimate mortality A person is ‘selected’ at the age when the policy is first purchased.
Select mortality is written as 𝑞𝑞 $ D# where 𝑥𝑥 is the ‘selected’ age and 𝑡𝑡 is the number of years after selection. After a certain number of years of ‘select period’, mortality is called the ‘ultimate’ mortality. 𝑞𝑞 $ D# = 𝑞𝑞$D# .
Deferred Life Pure Endowment Endowment Insurance
𝐴𝐴$ =
𝐴𝐴$ =
]
bcO
𝑣𝑣 bDd ⋅ b|𝑞𝑞$
Continuous ]
O
𝑣𝑣 # ⋅ # 𝑝𝑝$ ⋅ 𝜇𝜇$D# d𝑡𝑡
Discrete 𝐴𝐴d$:G| = 𝐴𝐴$ − G 𝐸𝐸$ ⋅ 𝐴𝐴$DG
Continuous 𝐴𝐴 d = 𝐴𝐴$ − G 𝐸𝐸$ ⋅ 𝐴𝐴$DG $∶G|
G|𝐴𝐴$ G|𝐴𝐴$
Discrete = 𝐴𝐴$ − 𝐴𝐴d$:G| = G𝐸𝐸$ ⋅ 𝐴𝐴$DG
Continuous = 𝐴𝐴$ − 𝐴𝐴 d = G𝐸𝐸$ ⋅ 𝐴𝐴$DG $∶G|
Discrete G 𝐴𝐴 d = G 𝐸𝐸$ = 𝑣𝑣 G 𝑝𝑝$ $:G| Continuous N/A Discrete 𝐴𝐴 = 𝐴𝐴d$:G| + G 𝐸𝐸$ $:G|
Continuous 𝐴𝐴 = 𝐴𝐴 d + G 𝐸𝐸$ $:G| $:G|
EPV under Constant Force of Mortality Discrete Continuous 𝜇𝜇 𝑞𝑞 𝐴𝐴$ = 𝐴𝐴$ = 𝜇𝜇 + 𝛿𝛿 𝑞𝑞 + 𝑖𝑖 𝜇𝜇 𝑞𝑞 = 1 − G 𝐸𝐸$ 𝐴𝐴d$:G| = 1 − G 𝐸𝐸$ 𝐴𝐴 d $:G| 𝜇𝜇 + 𝛿𝛿 𝑞𝑞 + 𝑖𝑖 𝑞𝑞 𝜇𝜇 ⋅ 𝐸𝐸 ⋅ 𝐸𝐸 G|𝐴𝐴$ = G|𝐴𝐴$ = 𝑞𝑞 + 𝑖𝑖 G $ 𝜇𝜇 + 𝛿𝛿 G $ G 𝐸𝐸$
= 𝑣𝑣 G 𝑝𝑝G
G 𝐸𝐸$
= 𝑒𝑒 R(SDo)G
EPV under Uniform Distribution Discrete Continuous 𝑎𝑎rR$| 𝑎𝑎rR$| 𝐴𝐴$ = 𝐴𝐴$ = 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 𝑎𝑎G| 𝑎𝑎G| 𝐴𝐴d$:G| = 𝐴𝐴 d = $:G| 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 − 𝑛𝑛 𝜔𝜔 − 𝑥𝑥 − 𝑛𝑛 G G G 𝐸𝐸$ = 𝑣𝑣 ⋅ G 𝐸𝐸$ = 𝑣𝑣 ⋅ 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥
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m-thly Insurance (f)
𝐴𝐴$
=
]
bcO
bDd /f
𝑣𝑣
⋅
Recursive Formulas
(𝒎𝒎)
b d 𝑞𝑞$ | f f
Discrete 𝐴𝐴$ = 𝑣𝑣𝑞𝑞$ + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴$Dd 𝐴𝐴$ = 𝑣𝑣𝑞𝑞$ + 𝑣𝑣 h 𝑝𝑝$ 𝑞𝑞$Dd + 𝑣𝑣 h h𝑝𝑝$ ⋅ 𝐴𝐴$Dh 𝐴𝐴d$:G| = 𝑣𝑣𝑞𝑞$ + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴 d $Dd:GRd| 𝐴𝐴$:G| = 𝑣𝑣𝑞𝑞$ + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴$Dd:GRd| G|𝐴𝐴$ = 𝑣𝑣𝑝𝑝$ ⋅ GRd|𝐴𝐴$Dd d 𝐴𝐴 d $:G| = 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴$Dd:GRd|
Continuous 𝐴𝐴$ = 𝐴𝐴d$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴$Dd h 𝐴𝐴$ = 𝐴𝐴d$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴 d $Dd:d| + 𝑣𝑣 h𝑝𝑝$ ⋅ 𝐴𝐴$Dh d 𝐴𝐴$:G| = 𝐴𝐴d$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴 d $Dd:GRd| 𝐴𝐴$:G| = 𝐴𝐴d$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝐴𝐴$Dd:GRd| G|𝐴𝐴$
= 𝑣𝑣𝑝𝑝$ ⋅ GRd|𝐴𝐴$Dd
Variances
Var 𝑍𝑍$
Discrete = h𝐴𝐴$ − 𝐴𝐴$ h
Relationship between 𝑨𝑨𝒙𝒙 , 𝑨𝑨𝒙𝒙 and 𝑨𝑨𝒙𝒙 (Under UDD Assumption) 𝑖𝑖 𝐴𝐴$ = 𝐴𝐴$ 𝛿𝛿 𝑖𝑖 d 𝐴𝐴 d 𝐴𝐴 $:G| = 𝛿𝛿 $:G| 𝑖𝑖 𝐴𝐴 G|𝐴𝐴$ = 𝛿𝛿 G| $ 𝑖𝑖 d 𝐴𝐴$:G| = 𝐴𝐴 $:G| + 𝐴𝐴 d $:G| 𝛿𝛿 𝑖𝑖 (f) 𝐴𝐴$ = (f) 𝐴𝐴$ 𝑖𝑖 2𝑖𝑖 + 𝑖𝑖 h h h 𝐴𝐴$ = ⋅ 𝐴𝐴$ 2𝛿𝛿 ANNUITIES ANNUITIES Level Annual Annuities Type of EPV Annuities Due; Discrete
h
Var 𝑍𝑍$:G| = h𝐴𝐴$:G| − 𝐴𝐴$:G| Continuous Var 𝑍𝑍$ = h𝐴𝐴$ − 𝐴𝐴$ h h
h
𝑎𝑎$ =
Temporary Life
SDho
Increasing and Decreasing Insurance 𝐼𝐼𝐼𝐼 $ = 𝐴𝐴$ + d|.𝐴𝐴$ + h|.𝐴𝐴$ + ⋯ 𝐼𝐼𝐴𝐴
$
𝐼𝐼𝐴𝐴
𝐷𝐷𝐴𝐴 𝐼𝐼𝐼𝐼
𝐼𝐼𝐴𝐴 𝐼𝐼𝐴𝐴
]
=
O
d $:G|
d $:G|
d $:G| d $:G| d $:G|
=
=
𝑡𝑡𝑡𝑡 # ⋅ # 𝑝𝑝$ ⋅ 𝜇𝜇$D# d𝑡𝑡 G
O
G
O
+ 𝐷𝐷𝐷𝐷
+ 𝐷𝐷𝐴𝐴 + 𝐷𝐷𝐴𝐴
𝑛𝑛 − 𝑡𝑡 𝑣𝑣 # ⋅ # 𝑝𝑝$ ⋅ 𝜇𝜇$D# d𝑡𝑡 d $:G| d $:G| d $:G|
= 𝑛𝑛 + 1 ⋅ 𝐴𝐴 d $:G| = 𝑛𝑛 + 1 ⋅ = 𝑛𝑛 ⋅ 𝐴𝐴 d $:G|
𝐴𝐴 d $:G|
EPV under Constant Force Discrete Continuous 𝜇𝜇 𝑞𝑞 h 1 𝐼𝐼𝐴𝐴 $ = 𝐼𝐼𝐼𝐼 $ = 𝜇𝜇 + 𝛿𝛿 h 𝑣𝑣𝑣𝑣 𝑞𝑞 + 𝑖𝑖
EPV under Uniform Distribution Discrete Continuous 𝐼𝐼𝐼𝐼 rR$| 𝐼𝐼𝑎𝑎 rR$| 𝐼𝐼𝐼𝐼 $ = 𝐼𝐼𝐴𝐴 $ = 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 𝐼𝐼𝐼𝐼 𝐼𝐼𝑎𝑎 G| G| 𝐼𝐼𝐼𝐼 d 𝐼𝐼𝐴𝐴 d $:G| = $:G| = 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥 𝐷𝐷𝐷𝐷 G| 𝐷𝐷𝑎𝑎 G| 𝐷𝐷𝐷𝐷 d 𝐷𝐷𝐴𝐴 d $:G| = $:G| = 𝜔𝜔 − 𝑥𝑥 𝜔𝜔 − 𝑥𝑥
Deferred Whole Life
𝑡𝑡𝑡𝑡 # ⋅ # 𝑝𝑝$ ⋅ 𝜇𝜇$D# d𝑡𝑡
]
bcO
𝑣𝑣 b ⋅ b 𝑝𝑝$
Immediate; Discrete 𝑎𝑎$ = 𝑎𝑎$ − 1 Continuous
Whole Life
Var 𝑍𝑍$:G| = 𝐴𝐴$:G| − 𝐴𝐴$:G| Note: h𝐴𝐴 and h𝐴𝐴 are calculated similar to 𝐴𝐴 and 𝐴𝐴 respectively, but with double the force of interest, 𝛿𝛿. Equivalently, replace 𝑣𝑣 with 𝑣𝑣 h , or replace 𝑖𝑖 with 2𝑖𝑖 + 𝑖𝑖 h . For example, under constant force, h𝐴𝐴$ = u S and h𝐴𝐴$ = . w uDhvDv
𝑎𝑎$ =
]
O
#
𝑣𝑣 ⋅ # 𝑝𝑝$ d𝑡𝑡
Immediate; Discrete 𝑎𝑎$:G| = 𝑎𝑎$:G| − 1 + G 𝐸𝐸$ Continuous 𝑎𝑎$:G| = 𝑎𝑎$ − G 𝐸𝐸$ ⋅ 𝑎𝑎$DG Due; Discrete G|𝑎𝑎$ = 𝑎𝑎$ − 𝑎𝑎$:G| = G 𝐸𝐸$ ⋅ 𝑎𝑎$DG Immediate; Discrete G|𝑎𝑎$ = 𝑎𝑎$ − 𝑎𝑎$:G| = G 𝐸𝐸$ ⋅ 𝑎𝑎$DG Continuous G|𝑎𝑎$ = 𝑎𝑎$ − 𝑎𝑎$:G| = G 𝐸𝐸$ ⋅ 𝑎𝑎$DG
= 𝑣𝑣 G 𝑝𝑝G
G 𝐸𝐸$
= 𝑒𝑒 R(SDo)G
Discrete 𝑎𝑎$ = 1 + 𝑣𝑣𝑝𝑝$ ⋅ 𝑎𝑎$Dd 𝑎𝑎$:G| = 1 + 𝑣𝑣𝑝𝑝$ ⋅ 𝑎𝑎$Dd:GRd| G|𝑎𝑎$ = 𝑣𝑣𝑝𝑝$ ⋅ GRd|𝑎𝑎$Dd Continuous 𝑎𝑎$ = 𝑎𝑎$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝑎𝑎$Dd 𝑎𝑎$:G| = 𝑎𝑎$:d| + 𝑣𝑣𝑝𝑝$ ⋅ 𝑎𝑎$Dd:GRd| G|𝑎𝑎$ = 𝑣𝑣𝑝𝑝$ ⋅ GRd|𝑎𝑎$Dd
Relationship between Insurances and Annuities Discrete Continuous 𝐴𝐴$ = 1 − 𝑑𝑑𝑎𝑎$ 𝐴𝐴$ = 1 − 𝛿𝛿𝑎𝑎$ 𝐴𝐴$:G| = 1 − 𝑑𝑑𝑎𝑎$:G| 𝐴𝐴$:G| = 1 − 𝛿𝛿𝑎𝑎$:G|
Variances
Discrete
h
𝐴𝐴$ − 𝐴𝐴$ h 𝑑𝑑 h h h 𝐴𝐴$:G| − 𝐴𝐴$:G| Var 𝑌𝑌$:G| = Var 𝑌𝑌$:GRd| = h 𝑑𝑑 Continuous h 𝐴𝐴$ − 𝐴𝐴$ h Var 𝑌𝑌$ = 𝛿𝛿 h h h 𝐴𝐴$:G| − 𝐴𝐴$:G| Var 𝑌𝑌$:G| = h 𝛿𝛿 Increasing and Decreasing Annuities Var 𝑌𝑌$ = Var 𝑌𝑌$ =
$:G|
𝐼𝐼𝑎𝑎
Due; Discrete 𝑎𝑎$:G| = 𝑎𝑎$ − G 𝐸𝐸$ ⋅ 𝑎𝑎$DG
EPV under Constant Force of Mortality Discrete Continuous 1 + 𝑖𝑖 1 𝑎𝑎$ = 𝑎𝑎$ = 𝑞𝑞 + 𝑖𝑖 𝜇𝜇 + 𝛿𝛿 1 + 𝑖𝑖 1 𝑎𝑎$:G| = 1 − G 𝐸𝐸$ 𝑎𝑎$:G| = 1 − G 𝐸𝐸$ 𝑞𝑞 + 𝑖𝑖 𝜇𝜇 + 𝛿𝛿 1 + 𝑖𝑖 1 ⋅ 𝐸𝐸 ⋅ 𝐸𝐸 G|𝑎𝑎$ = G|𝑎𝑎$ = 𝑞𝑞 + 𝑖𝑖 G $ 𝜇𝜇 + 𝛿𝛿 G $ G 𝐸𝐸$
Recursive Formulas
$
𝐼𝐼𝑎𝑎
𝐷𝐷𝑎𝑎 𝐼𝐼𝑎𝑎 𝐼𝐼𝐼𝐼
=
=
$:G|
$:G| $:G|
G
O
𝑡𝑡𝑡𝑡 # ⋅ # 𝑝𝑝$ d𝑡𝑡
1 𝜇𝜇 + 𝛿𝛿
=
G
O
+ 𝐷𝐷𝑎𝑎
+ 𝐷𝐷𝐷𝐷
h
if 𝜇𝜇 is constant
𝑛𝑛 − 𝑡𝑡 𝑣𝑣 # ⋅ # 𝑝𝑝$ d𝑡𝑡
$:G| $:G|
= 𝑛𝑛𝑎𝑎$:G|
= 𝑛𝑛 + 1 𝑎𝑎$:G|
Annuities with m-thly Payments UDD Assumption (f)
𝑎𝑎$
(f)
= 𝛼𝛼 𝑚𝑚 ⋅ 𝑎𝑎$ − 𝛽𝛽(𝑚𝑚)
𝑎𝑎$:G| = 𝛼𝛼 𝑚𝑚 ⋅ 𝑎𝑎$:G| − 𝛽𝛽(𝑚𝑚)(1 − G𝐸𝐸$ ) (f) G|𝑎𝑎$
= 𝛼𝛼 𝑚𝑚 ⋅ G|𝑎𝑎$ − 𝛽𝛽 𝑚𝑚 ⋅ G 𝐸𝐸$
Woolhouse’s Formula (3 terms) 𝑚𝑚 − 1 𝑚𝑚h − 1 (f) 𝑎𝑎$ ≈ 𝑎𝑎$ − − 𝜇𝜇$ + 𝛿𝛿 12𝑚𝑚h 2𝑚𝑚 𝑚𝑚 − 1 f 1 − G 𝐸𝐸$ 𝑎𝑎$:G| ≈ 𝑎𝑎$:G| − 2𝑚𝑚 𝑚𝑚 h − 1 𝜇𝜇 + 𝛿𝛿 − G𝐸𝐸$ 𝜇𝜇$DG + 𝛿𝛿 − 12𝑚𝑚h $ 𝑚𝑚 −1 f ≈ G|𝑎𝑎$ − 𝐸𝐸 G|𝑎𝑎$ 2𝑚𝑚 G $ 𝑚𝑚 h − 1 𝐸𝐸 𝜇𝜇 + 𝛿𝛿 − 12𝑚𝑚h G $ $DG 1 1 𝜇𝜇 + 𝛿𝛿 𝑎𝑎$ ≈ 𝑎𝑎$ − − 2 12 $
Recursive Formulas d 𝐼𝐼𝐼𝐼 d $:G| = 𝐴𝐴 $:G| + 𝑣𝑣𝑝𝑝$ ⋅ 𝐼𝐼𝐼𝐼 𝐷𝐷𝐷𝐷
d $:G|
=
𝐴𝐴 d $:G|
+ 𝐷𝐷𝐷𝐷
d $Dd:GRd| d $:GRd|
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Copyright © 2016 Coaching Actuaries. All Rights Reserved. 2
PREMIUMS PREMIUMS Net Premiums PREMIUMS Net Premiums Calculate net premiums using the equivalence Calculate net premiums using the equivalence principle: principle: 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) Name Type Name Type Fully Discrete 𝐴𝐴$ Fully Discrete 1 𝑑𝑑𝐴𝐴$ 𝐴𝐴$ = 1 − 𝑑𝑑 = 𝑑𝑑𝐴𝐴$ 𝑎𝑎$ = 𝑎𝑎$ − 𝑑𝑑 = 1 − 𝐴𝐴$ Whole 𝑎𝑎$ Fully Continuous 1 − 𝐴𝐴$ 𝑎𝑎$ Whole Life Insurance Life Insurance 𝐴𝐴$Fully Continuous 1 𝛿𝛿𝐴𝐴$ 𝐴𝐴$ = 1 − 𝛿𝛿 = 𝛿𝛿𝐴𝐴$ 𝑎𝑎$ = 𝑎𝑎$ − 𝛿𝛿 = 1 − 𝐴𝐴$ 𝑎𝑎$ Fully Discrete 𝑎𝑎$ 1 − 𝐴𝐴$ 𝐴𝐴$:G| Fully Discrete 𝑑𝑑𝐴𝐴$:G| 1 𝐴𝐴$:G| = 1 − 𝑑𝑑 = 𝑑𝑑𝐴𝐴$:G| 𝑎𝑎$:G| = 𝑎𝑎$:G| − 𝑑𝑑 = 1 − 𝐴𝐴$:G| Endowment 𝑎𝑎$:G| 𝑎𝑎$:G| 1 − 𝐴𝐴 Endowment Fully Continuous $:G| Insurance Insurance 𝛿𝛿𝐴𝐴$:G| 𝐴𝐴$:G| Fully Continuous 1 𝐴𝐴$:G| = 1 − 𝛿𝛿 = 𝛿𝛿𝐴𝐴$:G| 𝑎𝑎$:G| = 𝑎𝑎$:G| − 𝛿𝛿 = 1 − 𝐴𝐴$:G| 𝑎𝑎$:G| 𝑎𝑎$:G| 1 − 𝐴𝐴 Fully Discrete $:G| d Fully Discrete 𝐴𝐴 d $:G| 𝐴𝐴$:G| 𝑎𝑎$:G| Term 𝑎𝑎$:G| Term Life Insurance Fully Continuous Life Insurance Fully Continuous 𝐴𝐴 d $:G| 𝐴𝐴 d $:G| 𝑎𝑎$:G| 𝑎𝑎$:G| Fully Discrete Deferred Life Fully Discrete Deferred Life Insurance G|𝐴𝐴$ Insurance G|𝐴𝐴$ 𝑎𝑎 $:G| (premiums 𝑎𝑎 $:G| (premiums Fully Continuous payable during Fully Continuous payable during deferral G|𝐴𝐴$ 𝐴𝐴 deferral period) 𝑎𝑎G|$:G|$ period) 𝑎𝑎$:G| Fully Discrete Fully Discrete Deferred Life G|𝐴𝐴$ Deferred Life Insurance G|𝐴𝐴$ 𝑎𝑎$ Insurance 𝑎𝑎$ (premiums Fully Continuous (premiums Fully Continuous payable for G|𝐴𝐴$ payable for life) G|𝐴𝐴$ 𝑎𝑎$ life) 𝑎𝑎$ Fully Discrete Deferred Life Fully Discrete Deferred Life Annuity G|𝑎𝑎$ 𝑎𝑎 Annuity 𝑎𝑎G|$:G|$ (premiums 𝑎𝑎$:G| (premiums Fully Continuous payable during Fully Continuous payable during deferral G|𝑎𝑎$ 𝑎𝑎 deferral period) 𝑎𝑎G|$:G|$ period) 𝑎𝑎$:G| Note: Numerator and denominator of net premium Note: Numerator and denominator of net premium formula can be substituted with any other EPV formula can be substituted with any other EPV expression depending on premium payment expression depending on premium payment frequency and nature of death benefit (e.g. 𝑚𝑚-thly frequency and nature of death benefit (e.g. 𝑚𝑚-thly premiums, continuous premiums, death benefit premiums, continuous premiums, death benefit paid at moment of death). paid at moment of death). Gross Premiums Gross Premiums If gross premiums are calculated using the If gross premiums are calculated using the equivalence principle, then: equivalence principle, then: 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) + 𝐸𝐸𝐸𝐸𝐸𝐸(expenses) 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) Net Future Loss = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) + 𝐸𝐸𝐸𝐸𝐸𝐸(expenses) Net Future Loss O𝐿𝐿 = 𝑃𝑃𝑃𝑃(benefits) − 𝑃𝑃𝑃𝑃(premiums) 𝑃𝑃𝑃𝑃(benefits) 𝑏𝑏O𝐿𝐿==face amount, 𝑃𝑃 − = 𝑃𝑃𝑃𝑃(premiums) premium 𝑏𝑏 = face amount, 𝑃𝑃 = premium
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Whole Whole Life Life
Discrete Discrete 𝑃𝑃 𝑃𝑃 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$ 𝑏𝑏 + 𝑃𝑃 − 𝑃𝑃 𝑑𝑑 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$ 𝑏𝑏h + − 𝑑𝑑 𝑑𝑑 𝑃𝑃 h h𝑑𝑑 Var O𝐿𝐿 = 𝑏𝑏 + 𝑃𝑃 𝐴𝐴 − 𝐴𝐴$ h h $ 𝑑𝑑 Var O𝐿𝐿 = 𝑏𝑏 + 𝐴𝐴$ − 𝐴𝐴$ h 𝑑𝑑 𝑃𝑃 𝑃𝑃 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$:G| 𝑏𝑏 + 𝑃𝑃 − 𝑃𝑃 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$:G|h 𝑏𝑏 + 𝑑𝑑 − 𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑃𝑃 h h
EndowEndowment ment Insurance Var O𝐿𝐿 = 𝑏𝑏 + 𝑃𝑃 h 𝐴𝐴$:G| − 𝐴𝐴$:G| h h Insurance Var O𝐿𝐿 = 𝑏𝑏 + 𝑑𝑑 𝐴𝐴$:G| − 𝐴𝐴$:G| 𝑑𝑑 Continuous Continuous 𝑃𝑃 𝑃𝑃 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$ 𝑏𝑏 + 𝑃𝑃 − 𝑃𝑃 Whole 𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$ 𝑏𝑏h + 𝛿𝛿 − 𝛿𝛿 Whole 𝛿𝛿 𝛿𝛿 𝑃𝑃 Life Var O𝐿𝐿 = 𝑏𝑏 + 𝑃𝑃 h hh𝐴𝐴$ − 𝐴𝐴$ h Life Var O𝐿𝐿 = 𝑏𝑏 + 𝛿𝛿 𝐴𝐴$ − 𝐴𝐴$ h 𝛿𝛿 𝑃𝑃 𝑃𝑃 Endow𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$:G| 𝑏𝑏 + 𝑃𝑃 − 𝑃𝑃 𝛿𝛿 Endow𝐸𝐸 O𝐿𝐿 = 𝐴𝐴$:G|h 𝑏𝑏 + − 𝛿𝛿 ment h ment Var 𝐿𝐿 = 𝑏𝑏 + 𝑃𝑃 h h𝐴𝐴 𝛿𝛿 − 𝛿𝛿𝐴𝐴 Insurance 𝑃𝑃 O $:G| h h $:G| Insurance Var O𝐿𝐿 = 𝑏𝑏 + 𝛿𝛿 𝐴𝐴$:G| − 𝐴𝐴$:G| 𝛿𝛿 Gross Future Loss â Gross Future Loss O𝐿𝐿â = 𝑃𝑃𝑃𝑃(benefits) + 𝑃𝑃𝑃𝑃(expenses) O𝐿𝐿 = 𝑃𝑃𝑃𝑃(benefits) + 𝑃𝑃𝑃𝑃(expenses) −𝑃𝑃𝑃𝑃(premiums) −𝑃𝑃𝑃𝑃(premiums) Portfolio Percentile Premium Principle Portfolio Percentile Premium Principle Under normal approximation and given the Under normal approximation and given the probability of a loss on a portfolio of 𝑛𝑛 policies probability of a loss on a portfolio of 𝑛𝑛 policies equals 1 − 𝑝𝑝, solve for the premium per policy equals 1 − 𝑝𝑝, solve for the premium per policy such that: such that: 𝑉𝑉𝑉𝑉𝑉𝑉 O𝐿𝐿 𝐸𝐸 O𝐿𝐿 + 𝑧𝑧ã 𝑉𝑉𝑉𝑉𝑉𝑉 O𝐿𝐿 = 0 𝑛𝑛 = 0 𝐸𝐸 O𝐿𝐿 + 𝑧𝑧ã 𝑛𝑛 RESERVES RESERVES RESERVES Net Premium Reserve Net Premium Reserve Prospective Method Prospective Method # 𝑉𝑉 = 𝐸𝐸𝐸𝐸𝑉𝑉# (future ben.) − 𝐸𝐸𝐸𝐸𝑉𝑉# (future prem.) 𝑉𝑉 = 𝐸𝐸𝐸𝐸𝑉𝑉 (future ben.) − 𝐸𝐸𝐸𝐸𝑉𝑉 (future prem.) # # # Retrospective Method Retrospective Method 𝐸𝐸𝐸𝐸𝑉𝑉O (past prem.) − 𝐸𝐸𝐸𝐸𝑉𝑉O (past ben.) # 𝑉𝑉 = 𝐸𝐸𝐸𝐸𝑉𝑉O (past prem.) − 𝐸𝐸𝐸𝐸𝑉𝑉O (past ben.) # 𝐸𝐸$ # 𝑉𝑉 = # 𝐸𝐸$ Recursive Formula Recursive Formula bRd𝑉𝑉 + 𝜋𝜋bRd 1 + 𝑖𝑖 − 𝑏𝑏b 𝑞𝑞$DbRd b 𝑉𝑉 = bRd𝑉𝑉 + 𝜋𝜋bRd 1 + 𝑖𝑖 − 𝑏𝑏b 𝑞𝑞$DbRd 𝑝𝑝$DbRd b 𝑉𝑉 = 𝑝𝑝$DbRd • If 𝑏𝑏b = FA + b 𝑉𝑉 (where FA is level) and • premiums are level, then: If 𝑏𝑏b = FA + b 𝑉𝑉 (where FA is level) and b premiums are level, then: b 𝑞𝑞$DêRd 1 + 𝑖𝑖 bRê b 𝑉𝑉 = 𝜋𝜋𝑠𝑠b| − FA 𝑉𝑉 = 𝜋𝜋𝑠𝑠 − FA 𝑞𝑞 1 + 𝑖𝑖 bRê b| b êcd $DêRd
êcd Gross Premium Reserve Gross Premium Reserve Prospective Method â Prospective Method # 𝑉𝑉 = 𝐸𝐸𝐸𝐸𝑉𝑉# (f. ben.) + 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp.) − 𝐸𝐸𝐸𝐸𝑉𝑉# (f. prem.)
# 𝑉𝑉
â
= 𝐸𝐸𝐸𝐸𝑉𝑉# (f. ben.) + 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp.) − 𝐸𝐸𝐸𝐸𝑉𝑉# (f. prem.)
Retrospective Method â Retrospective Method # 𝑉𝑉 â = [𝐸𝐸𝐸𝐸𝑉𝑉O (p. prem.) − 𝐸𝐸𝐸𝐸𝑉𝑉O (p. ben.) # 𝑉𝑉 = [𝐸𝐸𝐸𝐸𝑉𝑉O (p. prem.) − 𝐸𝐸𝐸𝐸𝑉𝑉O (p. ben.) − 𝐸𝐸𝐸𝐸𝑉𝑉O p. exp.) / # 𝐸𝐸$ − 𝐸𝐸𝐸𝐸𝑉𝑉O p. exp.) / # 𝐸𝐸$ Recursive Formula â Recursive Formula # 𝑉𝑉 â = bRd𝑉𝑉 + 𝐺𝐺bRd − 𝑒𝑒bRd 1 + 𝑖𝑖 + 𝑖𝑖 # 𝑉𝑉 = bRd𝑉𝑉 + 𝐺𝐺bRd−−𝑞𝑞𝑒𝑒bRd 1 $DbRd 𝑏𝑏b + 𝐸𝐸b /𝑝𝑝$DbRd Expense Reserve − 𝑞𝑞$DbRd 𝑏𝑏b + 𝐸𝐸b /𝑝𝑝$DbRd ì Expense Reserve # 𝑉𝑉 ì = 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp.) − 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp. loadings) 𝑉𝑉 = 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp.) − 𝐸𝐸𝐸𝐸𝑉𝑉# (f. exp. loadings) # exp. loadings = gross premium – net premium exp. loadings = gross premium – net premium â ì # 𝑉𝑉 ì = # 𝑉𝑉 â − # 𝑉𝑉 # 𝑉𝑉 = # 𝑉𝑉 − # 𝑉𝑉
Modified Reserve Modified Reserve Full preliminary term (FPT): one-year term Full preliminary term (FPT): one-year term insurance followed by an insurance issued to life insurance followed by an insurance issued to life one year older. one year older. • FPT net premium d • FPT net premium First-year valuation premium: 𝑃𝑃$:d| = 𝑏𝑏𝑏𝑏𝑞𝑞$ d =𝑏𝑏𝐴𝐴 𝑏𝑏𝑏𝑏𝑞𝑞$ First-year valuation premium: 𝑃𝑃$:d| $Dd Renewal valuation premium: 𝑃𝑃$Dd = 𝑏𝑏𝐴𝐴$Dd Renewal valuation premium: 𝑃𝑃$Dd = 𝑎𝑎$Dd 𝑎𝑎$Dd • FPT reserve • FPT reserve ñóò # 𝑉𝑉$ñóò = #Rd𝑉𝑉$Dd = #Rd𝑉𝑉$Dd # 𝑉𝑉$ Treat reserves after first year as if the policy were Treat reserves after first year as if the policy were issued one year later. issued one year later. Reserve between Premium Dates Reserve between Premium Dates M dRM b 𝑉𝑉 + 𝜋𝜋b 1 + 𝑖𝑖 M − 𝑏𝑏bDd ⋅ M 𝑞𝑞$Db ⋅ 𝑣𝑣 dRM − 𝑏𝑏bDd ⋅ M𝑞𝑞$Db ⋅ 𝑣𝑣 bDM 𝑉𝑉 = b 𝑉𝑉 + 𝜋𝜋b 1 + 𝑖𝑖 M 𝑝𝑝$Db bDM 𝑉𝑉 = M 𝑝𝑝$Db for 0 < 𝑠𝑠 < 1 for 0 < 𝑠𝑠 < 1 Thiele’s Differential Equation Thiele’s Differential Equation d d 𝑉𝑉 = 𝛿𝛿 𝑉𝑉 + 𝐺𝐺 − 𝑒𝑒 − 𝑏𝑏 + 𝐸𝐸 − 𝑉𝑉 𝜇𝜇 d𝑡𝑡 ## 𝑉𝑉 = 𝛿𝛿## ## 𝑉𝑉 + 𝐺𝐺## − 𝑒𝑒## − 𝑏𝑏## + 𝐸𝐸## − ## 𝑉𝑉 𝜇𝜇 $$ D# D# d𝑡𝑡 𝐺𝐺 = gross premium, 𝑒𝑒 = level expense, 𝐺𝐺 = gross premium, 𝑒𝑒 = level expense, 𝑏𝑏 = face amount, 𝐸𝐸 = settlement expense 𝑏𝑏 = face amount, 𝐸𝐸 = settlement expense Euler’s Method •Euler’s Method From 𝑡𝑡 + ℎ to 𝑡𝑡: • From 𝑡𝑡 +𝑉𝑉ℎ to 𝑡𝑡: #Dö − ℎ 𝐺𝐺# − 𝑒𝑒# − 𝑏𝑏# + 𝐸𝐸# 𝜇𝜇 $ D# # 𝑉𝑉 = #Dö 𝑉𝑉 − ℎ 𝐺𝐺# − 𝑒𝑒# − 𝑏𝑏# + 𝐸𝐸# 𝜇𝜇 $ D# 1 + ℎ 𝜇𝜇 $ D# + 𝛿𝛿 = 𝑉𝑉 # 1 + ℎ 𝜇𝜇 $ D# + 𝛿𝛿 • From 𝑡𝑡 to 𝑡𝑡 − ℎ: • From 𝑡𝑡 to 𝑡𝑡 − ℎ: #Rö 𝑉𝑉 = # 𝑉𝑉 1 − ℎ 𝜇𝜇 $ D# + 𝛿𝛿 = 𝑉𝑉 1 − ℎ 𝜇𝜇 + 𝛿𝛿 𝑉𝑉 # #Rö 𝐺𝐺# − 𝑒𝑒# − 𝑏𝑏$# D# + 𝐸𝐸# 𝜇𝜇 $ D# −ℎ 𝐺𝐺 − 𝑒𝑒 − 𝑏𝑏 + 𝐸𝐸# 𝜇𝜇 $ D# −ℎ # # # Policy Alterations Policy Alterations To calculate face amount or duration of new To calculate face amount or duration of new altered contract, use equivalence principle: altered contract, use equivalence principle: # 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝑉𝑉# future prem. = 𝐸𝐸𝐸𝐸𝑉𝑉# future ben. 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝑉𝑉 future prem. = 𝐸𝐸𝐸𝐸𝑉𝑉 future ben. # # # Surrenders Surrenders • Paid-up term policy (extended term) d • Paid-up term policy (extended term) # 𝐶𝐶𝐶𝐶$ = 𝐴𝐴 d $D#:G| # 𝐶𝐶𝐶𝐶$ = 𝐴𝐴 d $D#:G| 𝐶𝐶𝐶𝐶 = 𝐴𝐴 # $ $D#:G| + PE⋅ GR# 𝐸𝐸$D# for endowment d # 𝐶𝐶𝐶𝐶$ = 𝐴𝐴$D#:G| + PE⋅ GR# 𝐸𝐸$D# for endowment insurance, where PE = pure endowment amt. insurance, where PE = pure endowment amt. • Reduced paid-up policy • Reduced paid-up policy # 𝐶𝐶𝐶𝐶$ # 𝑊𝑊$ = # 𝐶𝐶𝐶𝐶$ 𝐴𝐴 $D# = 𝑊𝑊 # $ 𝐴𝐴$D# 𝐶𝐶𝐶𝐶 = cash surrender value, 𝑊𝑊 = face amount 𝐶𝐶𝐶𝐶 = cash surrender value, 𝑊𝑊 = face amount MARKOV CHAINS MARKOV CHAINS MARKOV CHAINS Discrete Probabilities vê Discrete Probabilities # 𝑝𝑝$vê : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 is in # 𝑝𝑝$ : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 is in state 𝑗𝑗 (where 𝑗𝑗 may equal 𝑖𝑖) at time 𝑥𝑥 + 𝑡𝑡 vv state 𝑗𝑗 (where 𝑗𝑗 may equal 𝑖𝑖) at time 𝑥𝑥 + 𝑡𝑡 # 𝑝𝑝$vv : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 # 𝑝𝑝$ : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 remains in state 𝑖𝑖 until time 𝑥𝑥 + 𝑡𝑡 remains in state 𝑖𝑖 until time 𝑥𝑥 + 𝑡𝑡 𝐏𝐏 𝒕𝒕 : transition matrix 𝒕𝒕 𝐏𝐏 : transition matrix Homogeneous Markov chain: Only one transition Homogeneous Markov chain: Only one transition matrix needed for all periods matrix needed for all periods Non-homogeneous Markov chain: One transition Non-homogeneous Markov chain: One transition matrix needed for each period vê matrix needed for each period Perform matrix multiplication to calculate # 𝑝𝑝$vê . Perform matrix multiplication to calculate # 𝑝𝑝$ .
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 3
Continuous Probabilities vv # 𝑝𝑝$
#
= exp −
Discrete Insurances 𝐴𝐴 =
vê
𝜇𝜇$DM d𝑠𝑠
O ê°v
#
vê
vê
=
Euler’s Method vê #Dö 𝑝𝑝$
vê # 𝑝𝑝$
≈
vê
bê
vb # 𝑝𝑝$
bcO b°ê
+ℎ
vb # 𝑝𝑝$
bcO b°ê
⋅
vê
𝐴𝐴$ =
O
−
vê # 𝑝𝑝$
⋅
êb 𝜇𝜇$D#
bê
𝑒𝑒 Ro# # 𝑝𝑝$vb ⋅ 𝜇𝜇$D# d𝑡𝑡
b°ê
Annuity pays benefit as long as one remains in state j: vê 𝑎𝑎$ vê
]
=
𝑎𝑎$ =
𝑎𝑎$vv =
O ]
vê 𝑒𝑒 Ro# # 𝑝𝑝$ d𝑡𝑡
G
−
vê
𝜇𝜇$D# 𝑏𝑏#
êcO ê°v
v
vê
+ # 𝑉𝑉
ê
G
êcd ê
êcO ê°v
vê
𝜇𝜇$D# 𝑏𝑏#
vê
+ # 𝑉𝑉
v
ê
− # 𝑉𝑉
MULTIPLE DECREMENT MODELS MULTIPLE DECREMENT MODELS Probabilities =
ê # 𝑞𝑞$
=
ê
G
êcd #Rd bcO
ê # 𝑞𝑞$
§ b 𝑝𝑝$
𝑞𝑞$Db
§
#|3 𝑞𝑞$ = # 𝑝𝑝$
Life Table Formulas §
𝑑𝑑$ = §
f
êcd §
§ b 𝑝𝑝$ § § 𝑙𝑙$ b 𝑝𝑝$
𝑙𝑙$Db = 𝑙𝑙$ ê
𝑑𝑑$Db =
#D3Rd bc#
§ b 𝑝𝑝$
ê
𝑞𝑞$Db
ê
𝑑𝑑$
§
§
= 𝑙𝑙$ − b 𝑑𝑑$ ê
𝑞𝑞$Db
ê
ê
𝜇𝜇$D# 𝑏𝑏# d𝑡𝑡
1
ê
(ê) 𝜇𝜇$D#
êcd
www.coachingactuaries.com
(ê)
d # 𝑞𝑞$ d𝑡𝑡
§ # 𝑝𝑝$ G
#
O
= exp − #
=
•(ê) # 𝑝𝑝$
O
O
(ê)
𝜇𝜇$DM d𝑠𝑠
• ê
d # 𝑝𝑝$ d𝑡𝑡
v
® ¶ß ©
§
=−
d • ê ln # 𝑝𝑝$ d𝑡𝑡
= M𝑝𝑝$ ¶ß , 0 ≤ 𝑠𝑠 ≤ 1 UDD in Associated Single Decrement Tables (UDDASDT) For 2 decrements: • h 𝑡𝑡 h 𝑞𝑞$ (d) • d 𝑡𝑡 − , 0 ≤ 𝑡𝑡 ≤ 1 # 𝑞𝑞$ = 𝑞𝑞$ 2 For 3 decrements: • h • ™ • h • ™ 𝑡𝑡 h 𝑞𝑞$ + 𝑞𝑞$ 𝑡𝑡 ™ 𝑞𝑞$ 𝑞𝑞$ d • d 𝑡𝑡 − + , # 𝑞𝑞$ = 𝑞𝑞$ 2 3
0 ≤ 𝑡𝑡 ≤ 1 MULTIPLE LIVES MULTIPLE LIVES Joint Life 𝑇𝑇$¨ = min 𝑇𝑇$ , 𝑇𝑇¨ # 𝑝𝑝$¨
+ # 𝑞𝑞$¨ = 1
= #𝑝𝑝$¨ − #D3 𝑝𝑝$¨ #D3 𝑝𝑝$¨ ∘
𝑒𝑒$¨ =
𝑒𝑒$¨ =
=
#D3 𝑞𝑞$¨
− # 𝑞𝑞$¨
= # 𝑝𝑝$¨ ⋅ 3 𝑝𝑝$D#:¨D# ]
O ]
bcd
# 𝑝𝑝$¨
d𝑡𝑡
b 𝑝𝑝$¨
+ # 𝑞𝑞$¨ = 1
O
∘
𝜇𝜇$DM + 𝜇𝜇¨DM d𝑠𝑠
= # 𝑝𝑝$¨ − #D3 𝑝𝑝$¨ = ]
# 𝑝𝑝$¨
O ]
d𝑡𝑡
#D3 𝑞𝑞$¨
− # 𝑞𝑞$¨
b 𝑝𝑝$¨
bcd
𝐴𝐴$¨ = 1 − 𝛿𝛿𝑎𝑎$¨
Independent Lives # 𝑞𝑞$¨
= # 𝑞𝑞$ ⋅ # 𝑞𝑞¨ # 𝑞𝑞$
𝜇𝜇$¨ 𝑡𝑡 =
∙ # 𝑝𝑝¨ 𝜇𝜇¨D# + # 𝑞𝑞¨ ∙ # 𝑝𝑝$ 𝜇𝜇$D# # 𝑝𝑝$¨
Relationship between (𝒙𝒙𝒙𝒙) Status and (𝒙𝒙𝒙𝒙) Status 𝑇𝑇$¨ + 𝑇𝑇$¨ = 𝑇𝑇$ + 𝑇𝑇¨ # 𝑝𝑝$¨ + # 𝑝𝑝$¨ = # 𝑝𝑝$ ∘ ∘ ∘ ∘ 𝑒𝑒$¨ + 𝑒𝑒$¨ = 𝑒𝑒$ + 𝑒𝑒¨
𝑒𝑒$¨ + 𝑒𝑒$¨ = 𝑒𝑒$ + 𝑒𝑒¨
(§)
• ê
# 𝑝𝑝$¨
𝑒𝑒$¨ =
(§)
•(ê) (ê)
1
= exp −
𝑒𝑒$¨ =
M 𝑝𝑝$ 𝜇𝜇$DM d𝑠𝑠
= # 𝑝𝑝$
#
# 𝑝𝑝$¨
#|3 𝑞𝑞$¨
𝜇𝜇$DM d𝑠𝑠
#
Independent Lives # 𝑝𝑝$¨ = # 𝑝𝑝$ ⋅ # 𝑝𝑝¨ 𝜇𝜇$D#:¨D# = 𝜇𝜇$D# + 𝜇𝜇¨D# Last Survivor 𝑇𝑇$¨ = max 𝑇𝑇$ , 𝑇𝑇¨
𝜇𝜇$DM d𝑠𝑠
#|3 𝑞𝑞$¨ = # 𝑝𝑝$¨ ⋅ 3 𝑞𝑞$D#:¨D#
ê
ê
(§) M 𝑝𝑝$
O
• ê M 𝑝𝑝$
3 𝑞𝑞$D# =
(ê)
# 𝑝𝑝$ UDD in Multiple-Decrement Tables (UDDMDT)
G
#
𝜇𝜇$D# = −
− # 𝑉𝑉
Euler’s Method v v = # 𝑉𝑉 v 1 − 𝛿𝛿# ℎ + ℎ𝐵𝐵# #Rö 𝑉𝑉
êcd
= exp −
•(ê)
# 𝑞𝑞$
𝐵𝐵# : difference between benefit and premium in state 𝑖𝑖 vê 𝑏𝑏# : benefit for transitioning from state 𝑖𝑖 to 𝑗𝑗 +ℎ
(ê)
𝑞𝑞$DbRd 𝑏𝑏b
Fractional Ages UDD in the multiple decrement table: (ê) (ê) 0 ≤ 𝑠𝑠 ≤ 1 M 𝑞𝑞$ = 𝑠𝑠𝑞𝑞$ , Constant forces of decrement: ê 𝑞𝑞$ ê § M 1 − 𝑝𝑝$ M 𝑞𝑞$ = § 𝑞𝑞$ Associated Single Decrement Tables The associated single decrements are independent.
for constant force, where 𝜇𝜇 v • is the
Thiele’s Differential Equation d v 𝑉𝑉 v = 𝛿𝛿# # 𝑉𝑉 v − 𝐵𝐵# d𝑡𝑡 #
§ # 𝑞𝑞$
(§) # 𝑝𝑝$
vê
bcO d
=
•(ê) # 𝑝𝑝$
sum of forces of interest out of state 𝑖𝑖
(§) 𝜇𝜇$D#
𝑣𝑣 b b 𝑝𝑝$
S ¢ • Do
§
=
(ê)
Premiums and Reserves Insurance pays benefit upon transition to state j: ]
ê
G
𝑣𝑣 # # 𝑝𝑝$
𝜇𝜇$D# =
êb
bê 𝜇𝜇$D#
O
ê # 𝑞𝑞$
⋅ 𝜇𝜇$D# − # 𝑝𝑝$ ⋅ 𝜇𝜇$D# G
(§)
𝑣𝑣 b bRd𝑝𝑝$
Forces of Decrement
Kolmogorov’s Forward Equations d vê 𝑝𝑝 = Rate of entry into state 𝑗𝑗 d𝑡𝑡 # $ − Rate of leaving state 𝑗𝑗 G
]
𝐴𝐴 =
êê
vv M 𝑝𝑝$ ⋅ 𝜇𝜇$DM ⋅ #RM 𝑝𝑝$DM d𝑠𝑠
O
bcd
Continuous Insurances
For permanent disability model: # 𝑝𝑝$ =
]
+ # 𝑝𝑝¨
∘
∘
∘
∘
Cov 𝑇𝑇$¨ , 𝑇𝑇$¨ = Cov 𝑇𝑇$ , 𝑇𝑇¨ + 𝑒𝑒$ − 𝑒𝑒$¨ 𝑒𝑒¨ − 𝑒𝑒$¨
Cov 𝑇𝑇$ , 𝑇𝑇¨ = 0 if 𝑇𝑇$ and 𝑇𝑇¨ are independent 𝐴𝐴$¨ + 𝐴𝐴$¨ = 𝐴𝐴$ + 𝐴𝐴¨ 𝑎𝑎$¨ + 𝑎𝑎$¨ = 𝑎𝑎$ + 𝑎𝑎¨ G 𝐸𝐸$¨
+ G 𝐸𝐸$¨ = G𝐸𝐸$ + G𝐸𝐸¨
d G 𝑞𝑞$¨
=
Contingent Probabilities
d G 𝑞𝑞$¨ d G 𝑞𝑞$¨
h G 𝑞𝑞$¨
h G 𝑞𝑞$¨ h G 𝑞𝑞$¨
d G 𝑞𝑞$¨ d G 𝑞𝑞$¨ d G 𝑞𝑞$¨
=
+
=
=
+
G
O
G
# 𝑝𝑝$¨
O
# 𝑝𝑝$¨
O
# 𝑝𝑝$
d G 𝑞𝑞$¨ G G
O
∙ 𝜇𝜇¨D# 𝑑𝑑𝑑𝑑
= G 𝑞𝑞$¨
# 𝑝𝑝¨
h G 𝑞𝑞$¨
∙ 𝜇𝜇$D# 𝑑𝑑𝑑𝑑
1 − # 𝑝𝑝¨ ∙ 𝜇𝜇$D# 𝑑𝑑𝑑𝑑
1 − # 𝑝𝑝$ ∙ 𝜇𝜇¨D# 𝑑𝑑𝑑𝑑
= G 𝑞𝑞$¨
h + G 𝑞𝑞$¨ = G 𝑞𝑞$
h + G 𝑞𝑞$¨ = G 𝑞𝑞¨
h = G 𝑞𝑞$¨ + G 𝑞𝑞$ G 𝑝𝑝¨ Contingent Insurance 𝐴𝐴d$¨ + 𝐴𝐴 d $¨ = 𝐴𝐴$¨
𝐴𝐴h$¨ + 𝐴𝐴 h $¨ = 𝐴𝐴$¨ 𝐴𝐴d$¨ + 𝐴𝐴h$¨ = 𝐴𝐴$
𝐴𝐴d$¨ − 𝐴𝐴 h $¨ = 𝐴𝐴$ − 𝐴𝐴$¨ = 𝐴𝐴$¨ − 𝐴𝐴¨ Reversionary Annuities 𝑎𝑎$|¨ = 𝑎𝑎¨ − 𝑎𝑎$¨
𝐴𝐴$¨ = 1 − 𝛿𝛿𝑎𝑎$¨
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 4
PENSION MATHEMATICS PENSION MATHEMATICS Replacement Ratio, R
𝑅𝑅 =
1st year pension after retirement salary in the final year of work
Salary Rate Assumption
• Salaries increase continuously
𝑠𝑠¨ salary rate at age 𝑦𝑦 = 𝑠𝑠$ salary rate at age 𝑥𝑥
Salary Scale Assumption
• Salaries increase at discrete intervals 𝑠𝑠¨ salary earned between age 𝑦𝑦 and 𝑦𝑦 + 1 = 𝑠𝑠$ salary earned between age 𝑥𝑥 and 𝑥𝑥 + 1
Final average salary over the last 3 years (e.g. retire at age 65)
1 𝑠𝑠≤h + 𝑠𝑠≤™ + 𝑠𝑠≤≥ =3 ⋅ Salary between age 𝑥𝑥 and 𝑥𝑥 + 1 𝑠𝑠$
Salary rate to salary scale: 𝑠𝑠$ =
d
O
𝑠𝑠$D# d𝑡𝑡
Salary scale to salary rate: 𝑠𝑠$ = 𝑠𝑠$RO.µ
Normal Contribution
𝐶𝐶# = 𝑣𝑣 d𝑝𝑝$OO #Dd𝑉𝑉 − # 𝑉𝑉 + EPV(mid-year exits benefits)
• TUC if the actuarial liability is calculated with the traditional unit method • PUC if the actuarial liability is calculated with the projected unit method. Under constant and independent of salary accrual rate with no exit benefits: • TUC: O𝑉𝑉
∂ß∑∏ GDd ∂ß
G
− 1 PUC: O𝑉𝑉
d
G
INTEREST RATE RISK INTEREST RATE RISK Replicating Cash Flows Spot rate, 𝑦𝑦# : effective interest rate paid by a zerocoupon bond maturing at time 𝑡𝑡 𝑣𝑣 𝑡𝑡 : Present value of 1 paid at time 𝑡𝑡 1 𝑣𝑣 𝑡𝑡 = 1 + 𝑦𝑦# # Forward rate, 𝑓𝑓 𝑡𝑡, 𝑡𝑡 + 𝑘𝑘 : yield paid at time 0 by a zero-coupon bond bought at time 𝑡𝑡 and maturing for 1 at time 𝑡𝑡 + 𝑘𝑘 1 + 𝑦𝑦#Db #Db 𝑣𝑣 𝑡𝑡 b 1 + 𝑓𝑓 𝑡𝑡, 𝑡𝑡 + 𝑘𝑘 = = 𝑣𝑣 𝑡𝑡 + 𝑘𝑘 1 + 𝑦𝑦# # Variance of loss per policy 𝐿𝐿ã 𝐸𝐸 Var 𝐿𝐿d 𝐼𝐼 Var = Var 𝐸𝐸 𝐿𝐿v 𝐼𝐼 + 𝑛𝑛 𝑛𝑛 PROFIT TESTS PROFIT TESTS Asset Shares b 𝐴𝐴𝐴𝐴 = bRd𝐴𝐴𝐴𝐴 + 𝐺𝐺bRd − 𝑒𝑒bRd 1 + 𝑖𝑖 π
(π)
−𝑞𝑞$DbRd 𝑏𝑏b + 𝐸𝐸b π 𝑞𝑞$DbRd
∫ − 𝑞𝑞$DbRd ∫ 𝑞𝑞$DbRd
b CV
(∫)
+ 𝐸𝐸b
/
− 1 − 𝐺𝐺 = gross premium, 𝑒𝑒 = level expenses, 𝑏𝑏 = face amount, 𝐸𝐸 ê = settlement expenses paid on decrement 𝑗𝑗, 𝐶𝐶𝐶𝐶 = cash value Profits for Traditional Products Profit Vector, Prb Profit per policy in force at the beginning of each year Prb = bRd𝑉𝑉 + 𝐺𝐺bRd − 𝑒𝑒bRd 1 + 𝑖𝑖 π −𝑞𝑞$DbRd 𝑏𝑏b + (§) −𝑝𝑝$DbRd b 𝑉𝑉
π 𝐸𝐸b
−
∫ 𝑞𝑞$DbRd
Profit Signature, Πb Profit per policy issued Πb = Prb ⋅ bRd𝑝𝑝$ , 𝑘𝑘 ≥ 1 Πb = Prb , 𝑘𝑘 = 0
Change in reserve Δb 𝑉𝑉 = 1 + 𝑖𝑖
bRd𝑉𝑉
b CV
+
∫ 𝐸𝐸b
Profit Margin The ratio of the NPV to the (expected) present value of future premiums.
Discounted Payback Period (DPP) Solve for lowest 𝑚𝑚 such that
f
bcO
Πb 𝑣𝑣 b = 0.
Universal Life General AV# = AV#Rd + 𝑃𝑃# − 𝑒𝑒# − COI# 1 + 𝑖𝑖 COI# = 𝑣𝑣u 𝑞𝑞$D#Rd DB# − AV#
Type A (Death Benefit = Face Amount) AV#Rd + 𝑃𝑃# − 𝑒𝑒# 1 + 𝑖𝑖 − 𝑞𝑞$D#Rd FA AV# = 1 − 𝑞𝑞$D#Rd
Type B (Death Benefit = Face Amount + AV√ ) AV# = AV#Rd + 𝑃𝑃# − 𝑒𝑒# 1 + 𝑖𝑖 − 𝑞𝑞$D#Rd FA
Corridor Factor, γ AV#Rd + 𝑃𝑃# − 𝑒𝑒# 1 + 𝑖𝑖 AV# = 1 + 𝑞𝑞$D#Rd 𝛾𝛾 − 1
If 𝛾𝛾 ⋅ AV# > death benefit, set death benefit = 𝛾𝛾 ⋅ AV# .
Note: For all types, replace 𝑞𝑞$D#Rd with 𝑞𝑞$D#Rd 1 + 𝑖𝑖 𝑣𝑣u if 𝑖𝑖 ≠ 𝑖𝑖u Gain by Source Total Profit = bRd𝑉𝑉 + 𝐺𝐺b − 𝑒𝑒b 1 + 𝑖𝑖
−𝑞𝑞$DbRd 𝑏𝑏b + 𝐸𝐸b − 𝑝𝑝$DbRd b 𝑉𝑉
Total Gain = Actual Profit − Expected Profit
Components of Gain (∗ = assumed, ′ = actual): Interest: 𝑖𝑖 • − 𝑖𝑖 ∗ bRd𝑉𝑉 + 𝐺𝐺b − 𝑒𝑒b Expense: 𝑒𝑒b∗ − 𝑒𝑒b• 1 + 𝑖𝑖 + 𝑞𝑞$DbRd 𝐸𝐸b∗ − 𝐸𝐸b• ∗ • − 𝑞𝑞$DbRd 𝑏𝑏b + 𝐸𝐸b − b 𝑉𝑉 Mortality: 𝑞𝑞$DbRd ∫ ∗
∫ …
Lapse: 𝑞𝑞$DbRd − 𝑞𝑞$DbRd
kCV
∫
+ 𝐸𝐸b
− b 𝑉𝑉
(§)
− 𝑝𝑝$DbRd b 𝑉𝑉
IRR: GbcO Πb 𝑣𝑣 b = 0 b NPV = ] bcO Πb 𝑣𝑣ø , where 𝑟𝑟 = discount/hurdle rate Partial NPV NPV 𝑡𝑡 =
#
bcO
Πb 𝑣𝑣øb ,
where 𝑟𝑟 = discount/hurdle rate
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