EXAM MLC

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Raise Your Odds® with Adapt

Exam MLC

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 𝑞𝑞 #|3 $= Probability that 𝑥𝑥 survives 𝑡𝑡 years and dies within the following 𝑢𝑢 years = #𝑝𝑝$⋅ 3𝑞𝑞$D# = #𝑝𝑝$− #D3𝑝𝑝$ = #D3𝑞𝑞$− #𝑞𝑞$ Life Table Functions 𝑑𝑑 G $= 𝑙𝑙$− 𝑙𝑙$DG 𝑝𝑝 # $=𝑙𝑙$D#_𝑙𝑙 $ 𝑞𝑞 # $= # $_𝑙𝑙𝑑𝑑 $ = 𝑙𝑙$− 𝑙𝑙$D# 𝑙𝑙$ 𝑞𝑞 #|3 $= 3 $D#𝑑𝑑_𝑙𝑙 $ = 𝑙𝑙$D#− 𝑙𝑙$D#D3 𝑙𝑙$ Force of Mortality 𝜇𝜇$D#=_𝑆𝑆𝑓𝑓$ 𝑡𝑡 $ 𝑡𝑡 𝜇𝜇$D#= −_d𝑡𝑡𝑑𝑑ln 𝑆𝑆$𝑡𝑡 𝜇𝜇$D#= −_d𝑡𝑡𝑑𝑑ln 𝑝𝑝# $ 𝑓𝑓$𝑡𝑡 = 𝑝𝑝# $⋅ 𝜇𝜇$D# 𝑝𝑝 # $= exp − 𝜇𝜇$DM d𝑠𝑠 # O 𝑞𝑞 # $= M.𝑝𝑝$⋅ 𝜇𝜇$DM d𝑠𝑠 # O 𝑞𝑞 #|3 $= M.𝑝𝑝$⋅ 𝜇𝜇$DM d𝑠𝑠 #D3 # Mortality Laws Constant Force of Mortality 𝜇𝜇$= 𝜇𝜇 𝑝𝑝 # $= 𝑒𝑒RS# Uniform Distribution 𝜇𝜇$=_{𝜔𝜔 − 𝑥𝑥}1 , 0 ≤ 𝑥𝑥 < 𝜔𝜔 𝑝𝑝 # $=𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡_{𝜔𝜔 − 𝑥𝑥} , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥 𝑞𝑞 #|3 $=_{𝜔𝜔 − 𝑥𝑥}𝑢𝑢 , 0 ≤ 𝑡𝑡 + 𝑢𝑢 ≤ 𝜔𝜔 − 𝑥𝑥 Beta Distribution 𝜇𝜇$=_{𝜔𝜔 − 𝑥𝑥}𝛼𝛼 , 0 ≤ 𝑥𝑥 < 𝜔𝜔 𝑝𝑝 # $= 𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡_{𝜔𝜔 − 𝑥𝑥} Y , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥 Gompertz’s Law 𝜇𝜇$= 𝐵𝐵𝑐𝑐$, 𝑐𝑐 > 1 𝑝𝑝 # $= exp −𝐵𝐵𝑐𝑐 $ _𝑐𝑐#_{− 1} ln 𝑐𝑐 Moments Complete Future Lifetime General 𝑒𝑒∘$= # $𝑝𝑝 ] O d𝑡𝑡 Constant Force of Mortality 𝑒𝑒∘$=1_𝜇𝜇 Uniform Distribution 𝑒𝑒∘$=𝜔𝜔 − 𝑥𝑥₂ Beta Distribution 𝑒𝑒∘$=𝜔𝜔 − 𝑥𝑥_{𝛼𝛼 + 1} n-year Temporary Complete Future Lifetime 𝑒𝑒∘_$:G|= # $𝑝𝑝 G O d𝑡𝑡 • Uniform Distribution 𝑒𝑒∘$:G|= 𝑝𝑝G $ 𝑛𝑛 + 𝑞𝑞G $ 𝑛𝑛₂ Curtate Future Lifetime 𝑒𝑒$= 𝑘𝑘 ⋅ ] bcd 𝑞𝑞 b| $= b $𝑝𝑝 ] bcd • Uniform Distribution 𝑒𝑒$= 𝑒𝑒 ∘ $− 0.5 n-year Temporary Curtate Future Lifetime 𝑒𝑒$:G|= 𝑘𝑘 ⋅ GRd bcd 𝑞𝑞 b| $+ 𝑛𝑛 ⋅ 𝑝𝑝G $= b $𝑝𝑝 G bcd • Uniform Distribution 𝑒𝑒_$:G|= 𝑒𝑒∘_$:G|− 0.5 𝑞𝑞G. $ Recursive Formulas 𝑒𝑒∘$= 𝑒𝑒∘$:G|+ 𝑝𝑝G $⋅ 𝑒𝑒∘$DG 𝑒𝑒∘_$:G|= 𝑒𝑒∘_$:f|+ 𝑝𝑝f $⋅ 𝑒𝑒∘$Df:GRf|, 𝑚𝑚 < 𝑛𝑛 𝑒𝑒$= 𝑒𝑒$:G|+ 𝑝𝑝G $⋅ 𝑒𝑒$DG= 𝑒𝑒$:GRd|+ 𝑝𝑝G $ 1 + 𝑒𝑒$DG 𝑒𝑒$= 𝑝𝑝$ 1 + 𝑒𝑒$Dd 𝑒𝑒$:G|= 𝑒𝑒$:f|+ 𝑝𝑝f $⋅ 𝑒𝑒$Df:GRf|, 𝑚𝑚 < 𝑛𝑛 𝑒𝑒$:G|= 𝑒𝑒$:fRd|+ 𝑝𝑝f $ 1 + 𝑒𝑒$Df:GRf| , 𝑚𝑚 < 𝑛𝑛 𝑒𝑒_$:G|= 𝑝𝑝$ 1 + 𝑒𝑒$Dd:GRd| Fractional Ages UDD 0 ≤ 𝑠𝑠 + 𝑡𝑡 ≤ 1 𝑙𝑙$DM= 1 − 𝑠𝑠 ⋅ 𝑙𝑙$+ 𝑠𝑠 ⋅ 𝑙𝑙$Dd 𝑞𝑞 M $= 𝑠𝑠 ⋅ 𝑞𝑞$ 𝑞𝑞 M $D#=_{1 − 𝑡𝑡 ⋅ 𝑞𝑞}𝑠𝑠 ⋅ 𝑞𝑞$ $ 𝜇𝜇$DM=_{1 − 𝑠𝑠 ⋅ 𝑞𝑞}𝑞𝑞$ $ 𝑞𝑞$= 𝑝𝑝M $⋅ 𝜇𝜇$DM Constant Force of Mortality 0 ≤ 𝑠𝑠 + 𝑡𝑡 ≤ 1 𝑙𝑙$DM= 𝑙𝑙$dRM⋅ 𝑙𝑙$Dd M 𝑝𝑝 M $= 𝑝𝑝M $D#= 𝑝𝑝$ M 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.

Read the 2-year select and ultimate mortality table from the left to the right and then continue downwards. 𝑥𝑥 𝑞𝑞$ 𝑞𝑞$ Dd 𝑞𝑞$ Dh 𝑥𝑥 + 2 30 32 31 33 32 34 33 35 INSURANCE Level Annual Insurance Type of Insurance EPV Whole Life Discrete 𝐴𝐴$= 𝑣𝑣bDd⋅ ] bcO 𝑞𝑞 b| $ Continuous 𝐴𝐴$= 𝑣𝑣#⋅ ] O # $𝑝𝑝 ⋅ 𝜇𝜇$D# d𝑡𝑡 Term Life Discrete 𝐴𝐴d_$:G|_{= 𝐴𝐴} $− 𝐸𝐸G $⋅ 𝐴𝐴$DG Continuous 𝐴𝐴_$∶G| d _{= 𝐴𝐴} $− 𝐸𝐸G $⋅ 𝐴𝐴$DG Deferred Life Discrete 𝐴𝐴 G| $= 𝐴𝐴$− 𝐴𝐴$:G|d = 𝐸𝐸G $⋅ 𝐴𝐴$DG Continuous 𝐴𝐴 G| $= 𝐴𝐴$− 𝐴𝐴_$∶G| d = 𝐸𝐸G $⋅ 𝐴𝐴$DG Pure Endowment Discrete 𝐴𝐴_$:G| d_{= 𝐸𝐸} G $= 𝑣𝑣GG $𝑝𝑝 Continuous N/A Endowment Insurance Discrete 𝐴𝐴_$:G| _{= 𝐴𝐴} $:G| d _{+ 𝐸𝐸} G $ Continuous 𝐴𝐴_$:G| _{= 𝐴𝐴} $:G| d _{+ 𝐸𝐸} G $ EPV under Constant Force of Mortality Discrete Continuous 𝐴𝐴$=_{𝑞𝑞 + 𝑖𝑖}𝑞𝑞 𝐴𝐴$=_{𝜇𝜇 + 𝛿𝛿}𝜇𝜇 𝐴𝐴d_$:G|₌ 𝑞𝑞 𝑞𝑞 + 𝑖𝑖 1 − 𝐸𝐸G $ 𝐴𝐴 $:G| d = 𝜇𝜇 𝜇𝜇 + 𝛿𝛿 1 − 𝐸𝐸G $ 𝐴𝐴 G| $=_{𝑞𝑞 + 𝑖𝑖}𝑞𝑞 ⋅ 𝐸𝐸G $ G| $𝐴𝐴 =_{𝜇𝜇 + 𝛿𝛿}𝜇𝜇 ⋅ 𝐸𝐸G $ 𝐸𝐸 G $= 𝑣𝑣G𝑝𝑝G G $𝐸𝐸 = 𝑒𝑒R(SDo)G EPV under Uniform Distribution Discrete Continuous 𝐴𝐴$= 𝑎𝑎_rR$| 𝜔𝜔 − 𝑥𝑥 𝐴𝐴$= 𝑎𝑎rR$| 𝜔𝜔 − 𝑥𝑥 𝑎𝑎 𝑎𝑎 SURVIVAL DISTRIBUTIONS INSURANCE

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m-thly Insurance 𝐴𝐴$(f)= 𝑣𝑣bDd /f⋅ ] bcO 𝑞𝑞 b f | fd $ Recursive Formulas Discrete 𝐴𝐴$= 𝑣𝑣𝑞𝑞$+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd 𝐴𝐴$= 𝑣𝑣𝑞𝑞$+ 𝑣𝑣h𝑝𝑝$𝑞𝑞$Dd+ 𝑣𝑣hh $𝑝𝑝 ⋅ 𝐴𝐴$Dh 𝐴𝐴d$:G|= 𝑣𝑣𝑞𝑞$+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd:GRd| d 𝐴𝐴$:G|= 𝑣𝑣𝑞𝑞$+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd:GRd| 𝐴𝐴 G| $= 𝑣𝑣𝑝𝑝$⋅GRd| $Dd𝐴𝐴 𝐴𝐴 d_$:G|_{= 𝑣𝑣𝑝𝑝} $⋅ 𝐴𝐴$Dd:GRd| d Continuous 𝐴𝐴$ = 𝐴𝐴d$:d|+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd 𝐴𝐴$= 𝐴𝐴d$:d|+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd:d| d + 𝑣𝑣hh $𝑝𝑝 ⋅ 𝐴𝐴$Dh 𝐴𝐴d$:G|= 𝐴𝐴$:d|d + 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd:GRd| d 𝐴𝐴$:G|= 𝐴𝐴d$:d|+ 𝑣𝑣𝑝𝑝$⋅ 𝐴𝐴$Dd:GRd| 𝐴𝐴 G| $= 𝑣𝑣𝑝𝑝$⋅GRd| $Dd𝐴𝐴 Variances Discrete Var 𝑍𝑍$ = 𝐴𝐴h $− 𝐴𝐴$ h Var 𝑍𝑍$:G| = 𝐴𝐴h $:G|− 𝐴𝐴$:G| h Continuous Var 𝑍𝑍$ = 𝐴𝐴h $− 𝐴𝐴$ h Var 𝑍𝑍$:G| = 𝐴𝐴h $:G|− 𝐴𝐴$:G| h

Note: 𝐴𝐴h _{and 𝐴𝐴}h _{are calculated similar to 𝐴𝐴 and 𝐴𝐴} respectively, but with double the force of interest, 𝛿𝛿. Equivalently, replace 𝑣𝑣 with 𝑣𝑣h_{, or replace 𝑖𝑖 with} 2𝑖𝑖 + 𝑖𝑖h_{. For example, under constant force, 𝐴𝐴}h

$= u

uDhvDvw and 𝐴𝐴h $=_SDhoS .

Increasing and Decreasing Insurance 𝐼𝐼𝐴𝐴$= 𝐴𝐴$+ 𝐴𝐴d|. $+ 𝐴𝐴h|. $+ ⋯ 𝐼𝐼𝐴𝐴$= 𝑡𝑡𝑣𝑣#⋅ ] O # $𝑝𝑝 ⋅ 𝜇𝜇$D# d𝑡𝑡 𝐼𝐼𝐴𝐴 $:G| d = 𝑡𝑡𝑣𝑣#⋅ G O # $𝑝𝑝 ⋅ 𝜇𝜇$D# d𝑡𝑡 𝐷𝐷𝐴𝐴_$:G| d ₌ G_{𝑛𝑛 − 𝑡𝑡 𝑣𝑣}#_⋅ O # $𝑝𝑝 ⋅ 𝜇𝜇$D# d𝑡𝑡 𝐼𝐼𝐴𝐴 $:G| d + 𝐷𝐷𝐴𝐴 $:G| d = 𝑛𝑛 + 1 ⋅ 𝐴𝐴 $:G| d 𝐼𝐼𝐴𝐴 $:G| d + 𝐷𝐷𝐴𝐴 $:G| d = 𝑛𝑛 + 1 ⋅ 𝐴𝐴 $:G| d 𝐼𝐼𝐴𝐴_$:G| d _{+ 𝐷𝐷𝐴𝐴} $:G| d _{= 𝑛𝑛 ⋅ 𝐴𝐴} $:G| d EPV under Constant Force Discrete Continuous 𝐼𝐼𝐴𝐴 $=_{𝑣𝑣𝑞𝑞}1 _{𝑞𝑞 + 𝑖𝑖}𝑞𝑞 h 𝐼𝐼𝐴𝐴 $= _{𝜇𝜇 + 𝛿𝛿}𝜇𝜇 h EPV under Uniform Distribution Discrete Continuous 𝐼𝐼𝐴𝐴$= 𝐼𝐼𝐼𝐼 _rR$| 𝜔𝜔 − 𝑥𝑥 𝐼𝐼𝐴𝐴$= 𝐼𝐼𝐼𝐼_rR$| 𝜔𝜔 − 𝑥𝑥 𝐼𝐼𝐴𝐴_$:G| d ₌ 𝐼𝐼𝐼𝐼G| 𝜔𝜔 − 𝑥𝑥 𝐼𝐼𝐴𝐴 $:G| d = 𝐼𝐼𝐼𝐼G| 𝜔𝜔 − 𝑥𝑥 𝐷𝐷𝐴𝐴 $:G| d = 𝐷𝐷𝐼𝐼G| 𝜔𝜔 − 𝑥𝑥 𝐷𝐷𝐴𝐴 $:G| d = 𝐷𝐷𝐼𝐼_G| 𝜔𝜔 − 𝑥𝑥 Recursive Formulas 𝐼𝐼𝐴𝐴_$:G| d _{= 𝐴𝐴} $:G| d _{+ 𝑣𝑣𝑝𝑝} $⋅ 𝐼𝐼𝐴𝐴 $Dd:GRd| d d d d Relationship between 𝑨𝑨𝒙𝒙, 𝑨𝑨𝒙𝒙(𝒎𝒎) and 𝑨𝑨𝒙𝒙 (Under UDD Assumption) 𝐴𝐴$=_𝛿𝛿𝑖𝑖𝐴𝐴$ 𝐴𝐴 d $:G|=_𝛿𝛿𝑖𝑖𝐴𝐴 $:G| d 𝐴𝐴 G| $=_𝛿𝛿𝑖𝑖G| $𝐴𝐴 𝐴𝐴$:G|=_𝛿𝛿𝑖𝑖𝐴𝐴 d $:G|+ 𝐴𝐴 $:G| d 𝐴𝐴(f)$ =_𝑖𝑖(f)𝑖𝑖 𝐴𝐴$ 𝐴𝐴 h $=2𝑖𝑖 + 𝑖𝑖 h 2𝛿𝛿 ⋅ 𝐴𝐴h $ ANNUITIES Level Annual Annuities Type of Annuities EPV Whole Life Due; Discrete 𝐼𝐼$= 𝑣𝑣b⋅ 𝑝𝑝b $ ] bcO Immediate; Discrete 𝐼𝐼$= 𝐼𝐼$− 1 Continuous 𝐼𝐼$= 𝑣𝑣#⋅ ] O # $𝑝𝑝 d𝑡𝑡 Temporary Life Due; Discrete 𝐼𝐼$:G|= 𝐼𝐼$− 𝐸𝐸G $⋅ 𝐼𝐼$DG Immediate; Discrete 𝐼𝐼$:G|= 𝐼𝐼$:G|− 1 + 𝐸𝐸G $ Continuous 𝐼𝐼$:G|= 𝐼𝐼$− 𝐸𝐸G $⋅ 𝐼𝐼$DG Deferred Whole Life Due; Discrete 𝐼𝐼$ G| = 𝐼𝐼$− 𝐼𝐼$:G|= 𝐸𝐸G $⋅ 𝐼𝐼$DG Immediate; Discrete 𝐼𝐼$ G| = 𝐼𝐼$− 𝐼𝐼$:G|= 𝐸𝐸G $⋅ 𝐼𝐼$DG Continuous 𝐼𝐼$ G| = 𝐼𝐼$− 𝐼𝐼$:G|= 𝐸𝐸G $⋅ 𝐼𝐼$DG EPV under Constant Force of Mortality Discrete Continuous 𝐼𝐼$=1 + 𝑖𝑖_{𝑞𝑞 + 𝑖𝑖} 𝐼𝐼$=_{𝜇𝜇 + 𝛿𝛿}1 𝐼𝐼$:G|=1 + 𝑖𝑖_{𝑞𝑞 + 𝑖𝑖} 1 − 𝐸𝐸G $ 𝐼𝐼$:G|=_{𝜇𝜇 + 𝛿𝛿}1 1 − 𝐸𝐸G $ 𝐼𝐼$ G| =1 + 𝑖𝑖_{𝑞𝑞 + 𝑖𝑖}⋅ 𝐸𝐸G $ G|𝐼𝐼$=_{𝜇𝜇 + 𝛿𝛿}1 ⋅ 𝐸𝐸G $ 𝐸𝐸 G $= 𝑣𝑣G𝑝𝑝G G $𝐸𝐸 = 𝑒𝑒R(SDo)G Recursive Formulas Discrete 𝐼𝐼$= 1 + 𝑣𝑣𝑝𝑝$⋅ 𝐼𝐼$Dd 𝐼𝐼$:G|= 1 + 𝑣𝑣𝑝𝑝$⋅ 𝐼𝐼$Dd:GRd| 𝐼𝐼$ G| = 𝑣𝑣𝑝𝑝$⋅GRd|𝐼𝐼$Dd Continuous 𝐼𝐼$= 𝐼𝐼$:d|+ 𝑣𝑣𝑝𝑝$⋅ 𝐼𝐼$Dd 𝐼𝐼$:G|= 𝐼𝐼$:d|+ 𝑣𝑣𝑝𝑝$⋅ 𝐼𝐼$Dd:GRd| 𝐼𝐼$ G| = 𝑣𝑣𝑝𝑝$⋅GRd|𝐼𝐼$Dd Relationship between Insurances and Annuities Discrete Continuous 𝐴𝐴$= 1 − 𝑑𝑑𝐼𝐼$ 𝐴𝐴$= 1 − 𝛿𝛿𝐼𝐼$ 𝐴𝐴$:G|= 1 − 𝑑𝑑𝐼𝐼$:G| 𝐴𝐴$:G|= 1 − 𝛿𝛿𝐼𝐼$:G| Variances Discrete Var 𝑌𝑌$ = Var 𝑌𝑌$ = 𝐴𝐴$− 𝐴𝐴$ h h 𝑑𝑑h Var 𝑌𝑌$:G| = Var 𝑌𝑌$:GRd| = 𝐴𝐴$:G|− 𝐴𝐴$:G| h h 𝑑𝑑h Continuous Var 𝑌𝑌$ = 𝐴𝐴$− 𝐴𝐴$ h h 𝛿𝛿h Var 𝑌𝑌$:G| = 𝐴𝐴$:G|− 𝐴𝐴$:G| h h 𝛿𝛿h Increasing and Decreasing Annuities 𝐼𝐼𝐼𝐼$:G| = 𝑡𝑡𝑣𝑣#⋅ G O # $𝑝𝑝 d𝑡𝑡 𝐼𝐼𝐼𝐼$ = _{𝜇𝜇 + 𝛿𝛿}1 h if 𝜇𝜇 is constant 𝐷𝐷𝐼𝐼 $:G| = 𝑛𝑛 − 𝑡𝑡 𝑣𝑣#⋅ G O # $𝑝𝑝 d𝑡𝑡 𝐼𝐼𝐼𝐼_$:G| _{+ 𝐷𝐷𝐼𝐼} $:G| _{= 𝑛𝑛𝐼𝐼} $:G| 𝐼𝐼𝐼𝐼 _$:G| _{+ 𝐷𝐷𝐼𝐼} $:G| _{= 𝑛𝑛 + 1 𝐼𝐼} $:G| Annuities with m-thly Payments UDD Assumption 𝐼𝐼$(f)= 𝛼𝛼 𝑚𝑚 ⋅ 𝐼𝐼$− 𝛽𝛽(𝑚𝑚) 𝐼𝐼_$:G|(f)_{= 𝛼𝛼 𝑚𝑚 ⋅ 𝐼𝐼} $:G|− 𝛽𝛽(𝑚𝑚)(1 − 𝐸𝐸G $) 𝐼𝐼$(f) G| = 𝛼𝛼 𝑚𝑚 ⋅ 𝐼𝐼G| $− 𝛽𝛽 𝑚𝑚 ⋅ 𝐸𝐸G $ Woolhouse’s Formula (3 terms) 𝐼𝐼$(f)≈ 𝐼𝐼$−𝑚𝑚 − 1_2𝑚𝑚 −𝑚𝑚 h_{− 1} 12𝑚𝑚h 𝜇𝜇$+ 𝛿𝛿 𝐼𝐼_$:G|f ≈ 𝐼𝐼$:G|−𝑚𝑚 − 1_2𝑚𝑚 1 − 𝐸𝐸G $ −𝑚𝑚_12𝑚𝑚h− 1_h 𝜇𝜇$+ 𝛿𝛿 − 𝐸𝐸G $ 𝜇𝜇$DG+ 𝛿𝛿 𝐼𝐼$f G| ≈ 𝐼𝐼G| $−𝑚𝑚 − 1_2𝑚𝑚 G𝐸𝐸$ −𝑚𝑚_12𝑚𝑚h− 1_h G𝐸𝐸$𝜇𝜇$DG+ 𝛿𝛿 𝐼𝐼$≈ 𝐼𝐼$−1₂−₁₂1 𝜇𝜇$+ 𝛿𝛿 ANNUITIES

(3)

PREMIUMS Net Premiums Calculate net premiums using the equivalence principle: 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) Name Type Whole Life Insurance Fully Discrete 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$ 1 − 𝐴𝐴$ Fully Continuous 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$ 1 − 𝐴𝐴$ Endowment Insurance Fully Discrete 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Fully Continuous 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Term Life Insurance Fully Discrete 𝐴𝐴$:G| d 𝑎𝑎$:G| Fully Continuous 𝐴𝐴_$:G| d 𝑎𝑎$:G| Deferred Life Insurance (premiums payable during deferral period) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$:G| Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$:G| Deferred Life Insurance (premiums payable for life) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$ Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$ Deferred Life Annuity (premiums payable during deferral period) Fully Discrete 𝑎𝑎$ G| 𝑎𝑎$:G| Fully Continuous 𝑎𝑎$ G| 𝑎𝑎$:G| Note: Numerator and denominator of net premium formula can be substituted with any other EPV expression depending on premium payment frequency and nature of death benefit (e.g. 𝑚𝑚-thly premiums, continuous premiums, death benefit paid at moment of death). Gross Premiums If gross premiums are calculated using the equivalence principle, then:

𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) + 𝐸𝐸𝐸𝐸𝐸𝐸(expenses) Net Future Loss 𝐿𝐿 O = 𝐸𝐸𝐸𝐸(benefits) − 𝐸𝐸𝐸𝐸(premiums) 𝑏𝑏 = face amount, 𝐸𝐸 = premium Discrete Whole Life 𝐸𝐸 𝐿𝐿O = 𝐴𝐴$ 𝑏𝑏 +𝐸𝐸_𝑑𝑑 −𝐸𝐸_𝑑𝑑 Var 𝐿𝐿O = 𝑏𝑏 +𝐸𝐸_𝑑𝑑 h 𝐴𝐴$− 𝐴𝐴$ h h Endow-ment Insurance 𝐸𝐸 𝐿𝐿O = 𝐴𝐴$:G| 𝑏𝑏 +𝐸𝐸_𝑑𝑑 −𝐸𝐸_𝑑𝑑 Var 𝐿𝐿O = 𝑏𝑏 +𝐸𝐸_𝑑𝑑 h 𝐴𝐴$:G|− 𝐴𝐴$:G|h h Continuous Whole Life 𝐸𝐸 𝐿𝐿O = 𝐴𝐴$ 𝑏𝑏 +𝐸𝐸_𝛿𝛿 −𝐸𝐸_𝛿𝛿 Var 𝐿𝐿O = 𝑏𝑏 +𝐸𝐸_𝛿𝛿 h 𝐴𝐴$− 𝐴𝐴$ h h Endow-ment Insurance 𝐸𝐸 𝐿𝐿O = 𝐴𝐴$:G| 𝑏𝑏 +𝐸𝐸_𝛿𝛿 −𝐸𝐸_𝛿𝛿 Var 𝐿𝐿O = 𝑏𝑏 +𝐸𝐸_𝛿𝛿 h 𝐴𝐴$:G|− 𝐴𝐴$:G|h h Gross Future Loss 𝐿𝐿â O = 𝐸𝐸𝐸𝐸(benefits) + 𝐸𝐸𝐸𝐸(expenses) −𝐸𝐸𝐸𝐸(premiums) Portfolio Percentile Premium Principle Under normal approximation and given the probability of a loss on a portfolio of 𝑛𝑛 policies equals 1 − 𝑝𝑝, solve for the premium per policy such that: 𝐸𝐸 𝐿𝐿O + 𝑧𝑧ã 𝐸𝐸𝑎𝑎𝑉𝑉 𝐿𝐿_𝑛𝑛O = 0 RESERVES Net Premium Reserve Prospective Method 𝐸𝐸 # = 𝐸𝐸𝐸𝐸𝐸𝐸#(future ben.) − 𝐸𝐸𝐸𝐸𝐸𝐸#(future prem.) Retrospective Method 𝐸𝐸 # =𝐸𝐸𝐸𝐸𝐸𝐸O(past prem.) − 𝐸𝐸𝐸𝐸𝐸𝐸_𝐸𝐸 O(past ben.) $ # Recursive Formula 𝐸𝐸 b = bRd𝐸𝐸+ 𝜋𝜋bRd_𝑝𝑝1 + 𝑖𝑖 − 𝑏𝑏b𝑞𝑞$DbRd $DbRd • If 𝑏𝑏b= FA + 𝐸𝐸b (where FA is level) and premiums are level, then: 𝐸𝐸 b = 𝜋𝜋𝑠𝑠b|− FA 𝑞𝑞$DêRd 1 + 𝑖𝑖bRê b êcd Gross Premium Reserve Prospective Method 𝐸𝐸â

# = 𝐸𝐸𝐸𝐸𝐸𝐸#(f. ben.) + 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp.) − 𝐸𝐸𝐸𝐸𝐸𝐸#(f. prem.)

Retrospective Method 𝐸𝐸â # = [𝐸𝐸𝐸𝐸𝐸𝐸O(p. prem.) − 𝐸𝐸𝐸𝐸𝐸𝐸O(p. ben.) − 𝐸𝐸𝐸𝐸𝐸𝐸Op. exp.) / 𝐸𝐸# $ Recursive Formula 𝐸𝐸â # = bRd𝐸𝐸+ 𝐺𝐺bRd− 𝑒𝑒bRd 1 + 𝑖𝑖 − 𝑞𝑞$DbRd𝑏𝑏b+ 𝐸𝐸b /𝑝𝑝$DbRd Expense Reserve 𝐸𝐸ì # = 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp.) − 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp. loadings) exp. loadings = gross premium – net premium 𝐸𝐸ì # = 𝐸𝐸# â− 𝐸𝐸# Modified Reserve Full preliminary term (FPT): one-year term insurance followed by an insurance issued to life one year older. • FPT net premium First-year valuation premium: 𝐸𝐸$:d|d = 𝑏𝑏𝑏𝑏𝑞𝑞$ Renewal valuation premium: 𝐸𝐸$Dd=𝑏𝑏𝐴𝐴_𝑎𝑎$Dd $Dd • FPT reserve 𝐸𝐸 # $ñóò=#Rd $Dd𝐸𝐸 Treat reserves after first year as if the policy were issued one year later. Reserve between Premium Dates 𝐸𝐸 = b𝐸𝐸+ 𝜋𝜋b 1 + 𝑖𝑖M_𝑝𝑝− 𝑏𝑏bDd⋅ 𝑞𝑞M $Db⋅ 𝑏𝑏dRM $Db M bDM for 0 < 𝑠𝑠 < 1 Thiele’s Differential Equation d d𝑡𝑡 𝐸𝐸# = 𝛿𝛿##𝐸𝐸+ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸#− 𝐸𝐸# 𝜇𝜇$ D# 𝐺𝐺 = gross premium, 𝑒𝑒 = level expense, 𝑏𝑏 = face amount, 𝐸𝐸 = settlement expense Euler’s Method • From 𝑡𝑡 + ℎ to 𝑡𝑡: 𝐸𝐸 # =#Dö𝐸𝐸− ℎ 𝐺𝐺_{1 + ℎ 𝜇𝜇}#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# $ D#+ 𝛿𝛿 • From 𝑡𝑡 to 𝑡𝑡 − ℎ: 𝐸𝐸 #Rö = 𝐸𝐸# 1 − ℎ 𝜇𝜇$ D#+ 𝛿𝛿 −ℎ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# Policy Alterations To calculate face amount or duration of new altered contract, use equivalence principle: 𝐶𝐶𝐸𝐸 # + 𝐸𝐸𝐸𝐸𝐸𝐸# future prem. = 𝐸𝐸𝐸𝐸𝐸𝐸# future ben. Surrenders • Paid-up term policy (extended term) 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d + PE⋅ 𝐸𝐸GR# $D# for endowment insurance, where PE = pure endowment amt. • Reduced paid-up policy 𝑊𝑊 # $=_𝐴𝐴#𝐶𝐶𝐸𝐸$ $D# 𝐶𝐶𝐸𝐸 = cash surrender value, 𝑊𝑊 = face amount MARKOV CHAINS Discrete Probabilities 𝑝𝑝$vê # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 is in state 𝑗𝑗 (where 𝑗𝑗 may equal 𝑖𝑖) at time 𝑥𝑥 + 𝑡𝑡 𝑝𝑝$vv # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 remains in state 𝑖𝑖 until time 𝑥𝑥 + 𝑡𝑡 𝐏𝐏𝒕𝒕_{: transition matrix} Homogeneous Markov chain: Only one transition matrix needed for all periods Non-homogeneous Markov chain: One transition matrix needed for each period Perform matrix multiplication to calculate 𝑝𝑝# $vê. PREMIUMS Net Premiums Calculate net premiums using the equivalence principle: 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) Name Type Whole Life Insurance Fully Discrete 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$ 1 − 𝐴𝐴$ Fully Continuous 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$ 1 − 𝐴𝐴$ Endowment Insurance Fully Discrete 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Fully Continuous 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Term Life Insurance Fully Discrete 𝐴𝐴$:G| d 𝑎𝑎$:G| Fully Continuous 𝐴𝐴_$:G| d 𝑎𝑎$:G| Deferred Life Insurance (premiums payable during deferral period) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$:G| Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$:G| Deferred Life Insurance (premiums payable for life) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$ Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$ Deferred Life Annuity (premiums payable during deferral period) Fully Discrete 𝑎𝑎$ G| 𝑎𝑎$:G| Fully Continuous 𝑎𝑎$ G| 𝑎𝑎$:G| Note: Numerator and denominator of net premium formula can be substituted with any other EPV expression depending on premium payment frequency and nature of death benefit (e.g. 𝑚𝑚-thly premiums, continuous premiums, death benefit paid at moment of death). Gross Premiums If gross premiums are calculated using the equivalence principle, then:

Retrospective Method 𝐸𝐸â # = [𝐸𝐸𝐸𝐸𝐸𝐸O(p. prem.) − 𝐸𝐸𝐸𝐸𝐸𝐸O(p. ben.) − 𝐸𝐸𝐸𝐸𝐸𝐸Op. exp.) / 𝐸𝐸# $ Recursive Formula 𝐸𝐸â # = bRd𝐸𝐸+ 𝐺𝐺bRd− 𝑒𝑒bRd 1 + 𝑖𝑖 − 𝑞𝑞$DbRd𝑏𝑏b+ 𝐸𝐸b /𝑝𝑝$DbRd Expense Reserve 𝐸𝐸ì # = 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp.) − 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp. loadings) exp. loadings = gross premium – net premium 𝐸𝐸ì # = 𝐸𝐸# â− 𝐸𝐸# Modified Reserve Full preliminary term (FPT): one-year term insurance followed by an insurance issued to life one year older. • FPT net premium First-year valuation premium: 𝐸𝐸$:d|d = 𝑏𝑏𝑏𝑏𝑞𝑞$ Renewal valuation premium: 𝐸𝐸$Dd=𝑏𝑏𝐴𝐴_𝑎𝑎$Dd $Dd • FPT reserve 𝐸𝐸 # $ñóò=#Rd $Dd𝐸𝐸 Treat reserves after first year as if the policy were issued one year later. Reserve between Premium Dates 𝐸𝐸 = b𝐸𝐸+ 𝜋𝜋b 1 + 𝑖𝑖M_𝑝𝑝− 𝑏𝑏bDd⋅ 𝑞𝑞M $Db⋅ 𝑏𝑏dRM $Db M bDM for 0 < 𝑠𝑠 < 1 Thiele’s Differential Equation d d𝑡𝑡 𝐸𝐸# = 𝛿𝛿##𝐸𝐸+ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸#− 𝐸𝐸# 𝜇𝜇$ D# 𝐺𝐺 = gross premium, 𝑒𝑒 = level expense, 𝑏𝑏 = face amount, 𝐸𝐸 = settlement expense Euler’s Method • From 𝑡𝑡 + ℎ to 𝑡𝑡: 𝐸𝐸 # =#Dö𝐸𝐸− ℎ 𝐺𝐺_{1 + ℎ 𝜇𝜇}#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# $ D#+ 𝛿𝛿 • From 𝑡𝑡 to 𝑡𝑡 − ℎ: 𝐸𝐸 #Rö = 𝐸𝐸# 1 − ℎ 𝜇𝜇$ D#+ 𝛿𝛿 −ℎ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# Policy Alterations To calculate face amount or duration of new altered contract, use equivalence principle: 𝐶𝐶𝐸𝐸 # + 𝐸𝐸𝐸𝐸𝐸𝐸# future prem. = 𝐸𝐸𝐸𝐸𝐸𝐸# future ben. Surrenders • Paid-up term policy (extended term) 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d + PE⋅ 𝐸𝐸GR# $D# for endowment insurance, where PE = pure endowment amt. • Reduced paid-up policy 𝑊𝑊 # $=_𝐴𝐴#𝐶𝐶𝐸𝐸$ $D# 𝐶𝐶𝐸𝐸 = cash surrender value, 𝑊𝑊 = face amount MARKOV CHAINS Discrete Probabilities 𝑝𝑝$vê # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 is in state 𝑗𝑗 (where 𝑗𝑗 may equal 𝑖𝑖) at time 𝑥𝑥 + 𝑡𝑡 𝑝𝑝$vv # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 remains in state 𝑖𝑖 until time 𝑥𝑥 + 𝑡𝑡 𝐏𝐏𝒕𝒕_{: transition matrix} Homogeneous Markov chain: Only one transition matrix needed for all periods Non-homogeneous Markov chain: One transition matrix needed for each period Perform matrix multiplication to calculate 𝑝𝑝# $vê. PREMIUMS Net Premiums Calculate net premiums using the equivalence principle: 𝐸𝐸𝐸𝐸𝐸𝐸(premiums) = 𝐸𝐸𝐸𝐸𝐸𝐸(benefits) Name Type Whole Life Insurance Fully Discrete 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$ 1 − 𝐴𝐴$ Fully Continuous 𝐴𝐴$ 𝑎𝑎$= 1 𝑎𝑎$− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$ 1 − 𝐴𝐴$ Endowment Insurance Fully Discrete 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝑑𝑑 = 𝑑𝑑𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Fully Continuous 𝐴𝐴$:G| 𝑎𝑎$:G|= 1 𝑎𝑎$:G|− 𝛿𝛿 = 𝛿𝛿𝐴𝐴$:G| 1 − 𝐴𝐴$:G| Term Life Insurance Fully Discrete 𝐴𝐴$:G| d 𝑎𝑎$:G| Fully Continuous 𝐴𝐴$:G| d 𝑎𝑎$:G| Deferred Life Insurance (premiums payable during deferral period) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$:G| Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$:G| Deferred Life Insurance (premiums payable for life) Fully Discrete 𝐴𝐴$ G| 𝑎𝑎$ Fully Continuous 𝐴𝐴$ G| 𝑎𝑎$ Deferred Life Annuity (premiums payable during deferral period) Fully Discrete 𝑎𝑎$ G| 𝑎𝑎$:G| Fully Continuous 𝑎𝑎$ G| 𝑎𝑎$:G| Note: Numerator and denominator of net premium formula can be substituted with any other EPV expression depending on premium payment frequency and nature of death benefit (e.g. 𝑚𝑚-thly premiums, continuous premiums, death benefit paid at moment of death). Gross Premiums If gross premiums are calculated using the equivalence principle, then:

Retrospective Method 𝐸𝐸â # = [𝐸𝐸𝐸𝐸𝐸𝐸O(p. prem.) − 𝐸𝐸𝐸𝐸𝐸𝐸O(p. ben.) − 𝐸𝐸𝐸𝐸𝐸𝐸Op. exp.) / 𝐸𝐸# $ Recursive Formula 𝐸𝐸â # = bRd𝐸𝐸+ 𝐺𝐺bRd− 𝑒𝑒bRd 1 + 𝑖𝑖 − 𝑞𝑞$DbRd𝑏𝑏b+ 𝐸𝐸b /𝑝𝑝$DbRd Expense Reserve 𝐸𝐸ì # = 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp.) − 𝐸𝐸𝐸𝐸𝐸𝐸#(f. exp. loadings) exp. loadings = gross premium – net premium 𝐸𝐸ì # = 𝐸𝐸# â− 𝐸𝐸# Modified Reserve Full preliminary term (FPT): one-year term insurance followed by an insurance issued to life one year older. • FPT net premium First-year valuation premium: 𝐸𝐸$:d|d = 𝑏𝑏𝑏𝑏𝑞𝑞$ Renewal valuation premium: 𝐸𝐸$Dd=𝑏𝑏𝐴𝐴_𝑎𝑎$Dd $Dd • FPT reserve 𝐸𝐸 # $ñóò=#Rd $Dd𝐸𝐸 Treat reserves after first year as if the policy were issued one year later. Reserve between Premium Dates 𝐸𝐸 = b𝐸𝐸+ 𝜋𝜋b 1 + 𝑖𝑖M− 𝑏𝑏bDd⋅ 𝑞𝑞M $Db⋅ 𝑏𝑏dRM 𝑝𝑝$Db M bDM for 0 < 𝑠𝑠 < 1 Thiele’s Differential Equation d d𝑡𝑡 𝐸𝐸# = 𝛿𝛿##𝐸𝐸+ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸#− 𝐸𝐸# 𝜇𝜇$ D# 𝐺𝐺 = gross premium, 𝑒𝑒 = level expense, 𝑏𝑏 = face amount, 𝐸𝐸 = settlement expense Euler’s Method • From 𝑡𝑡 + ℎ to 𝑡𝑡: 𝐸𝐸 # =#Dö𝐸𝐸− ℎ 𝐺𝐺_{1 + ℎ 𝜇𝜇}#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# $ D#+ 𝛿𝛿 • From 𝑡𝑡 to 𝑡𝑡 − ℎ: 𝐸𝐸 #Rö = 𝐸𝐸# 1 − ℎ 𝜇𝜇$ D#+ 𝛿𝛿 −ℎ 𝐺𝐺#− 𝑒𝑒#− 𝑏𝑏#+ 𝐸𝐸# 𝜇𝜇$ D# Policy Alterations To calculate face amount or duration of new altered contract, use equivalence principle: 𝐶𝐶𝐸𝐸 # + 𝐸𝐸𝐸𝐸𝐸𝐸# future prem. = 𝐸𝐸𝐸𝐸𝐸𝐸# future ben. Surrenders • Paid-up term policy (extended term) 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d 𝐶𝐶𝐸𝐸 # $= 𝐴𝐴$D#:G| d + PE⋅ 𝐸𝐸GR# $D# for endowment insurance, where PE = pure endowment amt. • Reduced paid-up policy 𝑊𝑊 # $=_𝐴𝐴#𝐶𝐶𝐸𝐸$ $D# 𝐶𝐶𝐸𝐸 = cash surrender value, 𝑊𝑊 = face amount MARKOV CHAINS Discrete Probabilities 𝑝𝑝$vê # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 is in state 𝑗𝑗 (where 𝑗𝑗 may equal 𝑖𝑖) at time 𝑥𝑥 + 𝑡𝑡 𝑝𝑝$vv # : probability that a life in state 𝑖𝑖 at time 𝑥𝑥 remains in state 𝑖𝑖 until time 𝑥𝑥 + 𝑡𝑡 𝐏𝐏𝒕𝒕_{: transition matrix} Homogeneous Markov chain: Only one transition matrix needed for all periods Non-homogeneous Markov chain: One transition matrix needed for each period Perform matrix multiplication to calculate 𝑝𝑝# $vê. PREMIUMS RESERVES MARKOV CHAINS

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Continuous Probabilities 𝑝𝑝$vv # = exp − 𝜇𝜇$DMvê ê°v d𝑠𝑠 # O For permanent disability model: 𝑝𝑝$vê # = M𝑝𝑝$vv⋅ 𝜇𝜇$DMvê ⋅#RM𝑝𝑝$DMêê d𝑠𝑠 # O Kolmogorov’s Forward Equations d d𝑡𝑡 #𝑝𝑝$vê= Rate of entry into state 𝑗𝑗 − Rate of leaving state 𝑗𝑗 = #𝑝𝑝$vb⋅𝜇𝜇bê$D#− 𝑝𝑝# $vê⋅𝜇𝜇$D#êb G bcO b°ê Euler’s Method 𝑝𝑝$vê #Dö ≈ 𝑝𝑝# $vê+ ℎ #𝑝𝑝$vb⋅𝜇𝜇$D#bê − 𝑝𝑝# $vê⋅𝜇𝜇$D#êb G bcO b°ê Premiums and Reserves Insurance pays benefit upon transition to state j: 𝐴𝐴$vê= 𝑒𝑒Ro# #𝑝𝑝$vb⋅𝜇𝜇$D#bê b°ê d𝑡𝑡 ] O Annuity pays benefit as long as one remains in state j: 𝑎𝑎$vê= 𝑒𝑒Ro##𝑝𝑝$vê d𝑡𝑡 ] O 𝑎𝑎$vê= 𝑣𝑣bb𝑝𝑝$vê ] bcO 𝑎𝑎$vv=_S¢ •d_Do for constant force, where 𝜇𝜇v • is the sum of forces of interest out of state 𝑖𝑖 Thiele’s Differential Equation d d𝑡𝑡 #𝑉𝑉v = 𝛿𝛿##𝑉𝑉v − 𝐵𝐵#v − 𝜇𝜇$D#vê 𝑏𝑏#vê + 𝑉𝑉# ê − 𝑉𝑉# v G êcO ê°v 𝐵𝐵#v: difference between benefit and premium in state 𝑖𝑖 𝑏𝑏#vê: benefit for transitioning from state 𝑖𝑖 to 𝑗𝑗 Euler’s Method 𝑉𝑉 #Rö v = 𝑉𝑉# v 1 − 𝛿𝛿#ℎ + ℎ𝐵𝐵#v +ℎ 𝜇𝜇$D#vê 𝑏𝑏#vê + 𝑉𝑉# ê − 𝑉𝑉# v G êcO ê°v MULTIPLE DECREMENT MODELS Probabilities 𝑞𝑞$§ # = #𝑞𝑞$ê G êcd 𝑞𝑞$ê # = b𝑝𝑝$§ #Rd bcO 𝑞𝑞$Dbê 𝑞𝑞$ê #|3 = 𝑝𝑝# $§ 3𝑞𝑞$D#ê = b𝑝𝑝$§𝑞𝑞$Dbê #D3Rd bc# Life Table Formulas f Discrete Insurances 𝐴𝐴 = 𝑣𝑣b _𝑝𝑝 $(§) bRd ] bcd 𝑞𝑞_$DbRd(ê) _𝑏𝑏 b(ê) ê Continuous Insurances 𝐴𝐴 = 𝑣𝑣# _𝑝𝑝 $§ # ] O 𝜇𝜇$D# ê _𝑏𝑏 #ê G êcd d𝑡𝑡 Forces of Decrement 𝑞𝑞$ê # = M𝑝𝑝$(§) # O 𝜇𝜇$DM (ê)_d𝑠𝑠 𝜇𝜇$D#(ê) = _𝑝𝑝1 $§ # d 𝑞𝑞# $ê d𝑡𝑡 𝜇𝜇$D#(§) = 𝜇𝜇$D#(ê) G êcd 𝑝𝑝$(§) # = exp − 𝜇𝜇$DM(§) # O d𝑠𝑠 Fractional Ages UDD in the multiple decrement table: 𝑞𝑞$(ê) M = 𝑠𝑠𝑞𝑞$(ê), 0 ≤ 𝑠𝑠 ≤ 1 Constant forces of decrement: 𝑞𝑞$ê M =𝑞𝑞$ ê 𝑞𝑞$§ 1 − 𝑝𝑝$§ M Associated Single Decrement Tables The associated single decrements are independent. 𝑝𝑝$•(ê) # = exp − 𝜇𝜇$DM(ê) # O d𝑠𝑠 𝑞𝑞$•(ê) # = M𝑝𝑝$•(ê)𝜇𝜇$DM(ê) # O d𝑠𝑠 𝑝𝑝$•(ê) # G êcd = 𝑝𝑝# $(§) 𝜇𝜇$D#ê = − 1 𝑝𝑝$• ê # d 𝑝𝑝# $• ê d𝑡𝑡 = − d d𝑡𝑡ln 𝑝𝑝# $• ê UDD in Multiple-Decrement Tables (UDDMDT) 𝑝𝑝$• ê M = M𝑝𝑝$§ ¶ß® ¶ß©_{, 0 ≤ 𝑠𝑠 ≤ 1} UDD in Associated Single Decrement Tables (UDDASDT) For 2 decrements: 𝑞𝑞$(d) # = 𝑞𝑞$• d 𝑡𝑡 −𝑡𝑡 h_𝑞𝑞 $ • h 2 , 0 ≤ 𝑡𝑡 ≤ 1 For 3 decrements: 𝑞𝑞$d # = 𝑞𝑞$• d 𝑡𝑡 −𝑡𝑡 h _𝑞𝑞 $• h + 𝑞𝑞$• ™ 2 + 𝑡𝑡™_𝑞𝑞 $• h𝑞𝑞$• ™ 3 , 0 ≤ 𝑡𝑡 ≤ 1 MULTIPLE LIVES Joint Life 𝑇𝑇$¨= min 𝑇𝑇$, 𝑇𝑇¨ 𝑝𝑝$¨ # + 𝑞𝑞# $¨= 1 𝑞𝑞$¨ #|3 = 𝑝𝑝# $¨⋅ 𝑞𝑞3 $D#:¨D# = 𝑝𝑝# $¨−#D3𝑝𝑝$¨ =#D3𝑞𝑞$¨− 𝑞𝑞# $¨ 𝑝𝑝$¨ #D3 = 𝑝𝑝# $¨⋅ 𝑝𝑝3 $D#:¨D# 𝑒𝑒∘$¨= #𝑝𝑝$¨ ] O d𝑡𝑡 ] Independent Lives 𝑝𝑝$¨ # = 𝑝𝑝# $ ⋅ 𝑝𝑝# ¨ 𝜇𝜇$D#:¨D#= 𝜇𝜇$D#+ 𝜇𝜇¨D# 𝑝𝑝$¨ # = exp − 𝜇𝜇$DM+ 𝜇𝜇¨DM # O d𝑠𝑠 Last Survivor 𝑇𝑇$¨= max 𝑇𝑇$, 𝑇𝑇¨ 𝑝𝑝$¨ # + 𝑞𝑞# $¨= 1 𝑞𝑞$¨ #|3 = 𝑝𝑝# $¨−#D3𝑝𝑝$¨=#D3𝑞𝑞$¨− 𝑞𝑞# $¨ 𝑒𝑒∘$¨= #𝑝𝑝$¨ ] O d𝑡𝑡 𝑒𝑒$¨= b𝑝𝑝$¨ ] bcd 𝐴𝐴$¨= 1 − 𝛿𝛿𝑎𝑎$¨ Independent Lives 𝑞𝑞$¨ # = 𝑞𝑞# $ ⋅ 𝑞𝑞# ¨ 𝜇𝜇$¨ 𝑡𝑡 = #𝑞𝑞$ ∙ 𝑝𝑝# ¨𝜇𝜇¨D#_𝑝𝑝+ 𝑞𝑞# ¨ ∙ 𝑝𝑝# $𝜇𝜇$D# $¨ # Relationship between (𝒙𝒙𝒙𝒙) Status and (𝒙𝒙𝒙𝒙) Status 𝑇𝑇$¨+ 𝑇𝑇$¨= 𝑇𝑇$+ 𝑇𝑇¨ 𝑝𝑝$¨ # + 𝑝𝑝# $¨= 𝑝𝑝# $ + 𝑝𝑝# ¨ 𝑒𝑒∘$¨+ 𝑒𝑒∘$¨= 𝑒𝑒∘$+ 𝑒𝑒∘¨ 𝑒𝑒$¨+ 𝑒𝑒$¨= 𝑒𝑒$+ 𝑒𝑒¨ 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¨ 𝐸𝐸$¨ G + 𝐸𝐸G $¨= 𝐸𝐸G $ + 𝐸𝐸G ¨ Contingent Probabilities 𝑞𝑞$¨d G = #𝑝𝑝$¨∙ 𝜇𝜇$D#𝑑𝑑𝑡𝑡 G O 𝑞𝑞$¨ d G = #𝑝𝑝$¨∙ 𝜇𝜇¨D#𝑑𝑑𝑡𝑡 G O 𝑞𝑞$¨d + 𝑞𝑞G $¨ d= G G𝑞𝑞$¨ 𝑞𝑞$¨h G = #𝑝𝑝$ 1 − 𝑝𝑝# ¨ ∙ 𝜇𝜇$D#𝑑𝑑𝑡𝑡 G O 𝑞𝑞$¨ h G = #𝑝𝑝¨ 1 − 𝑝𝑝# $ ∙ 𝜇𝜇¨D#𝑑𝑑𝑡𝑡 G O 𝑞𝑞$¨h + 𝑞𝑞G $¨ h= G G𝑞𝑞$¨ 𝑞𝑞$¨d + G G𝑞𝑞$¨h = 𝑞𝑞G $ 𝑞𝑞$¨ d+ G G𝑞𝑞$¨ h= 𝑞𝑞G ¨ 𝑞𝑞$¨d = G G𝑞𝑞$¨ h+ G𝑞𝑞$ G𝑝𝑝¨ Contingent Insurance 𝐴𝐴$¨d + 𝐴𝐴$¨ d= 𝐴𝐴$¨ 𝐴𝐴$¨h + 𝐴𝐴$¨ h= 𝐴𝐴$¨ 𝐴𝐴$¨d + 𝐴𝐴$¨h = 𝐴𝐴$ 𝐴𝐴$¨d − 𝐴𝐴$¨ h= 𝐴𝐴$− 𝐴𝐴$¨= 𝐴𝐴$¨− 𝐴𝐴¨ Reversionary Annuities 𝑎𝑎$|¨= 𝑎𝑎¨− 𝑎𝑎$¨

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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¨ 𝑠𝑠$= salary rate at age 𝑦𝑦 salary rate at age 𝑥𝑥 Salary Scale Assumption • Salaries increase at discrete intervals 𝑠𝑠¨ 𝑠𝑠$= salary earned between age 𝑦𝑦 and 𝑦𝑦 + 1 salary earned between age 𝑥𝑥 and 𝑥𝑥 + 1 Final average salary over the last 3 years (e.g. retire at age 65) = 1 3 𝑠𝑠≤h+ 𝑠𝑠≤™+ 𝑠𝑠≤≥ 𝑠𝑠$ ⋅ Salary between age 𝑥𝑥 and 𝑥𝑥 + 1 Salary rate to salary scale: 𝑠𝑠$= 𝑠𝑠$D# d O d𝑡𝑡 Salary scale to salary rate: 𝑠𝑠$= 𝑠𝑠$RO.µ Normal Contribution 𝐶𝐶#= 𝑣𝑣 𝑝𝑝d $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: 𝑉𝑉 ∂ß∑∏ ∂ß GDd G − 1 O PUC: 𝑉𝑉O _Gd INTEREST RATE RISK Replicating Cash Flows Spot rate, 𝑦𝑦# : effective interest rate paid by a zero-coupon bond maturing at time 𝑡𝑡 𝑣𝑣 𝑡𝑡 : Present value of 1 paid at time 𝑡𝑡 𝑣𝑣 𝑡𝑡 = _{1 + 𝑦𝑦}1 # # 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 + 𝑓𝑓 𝑡𝑡, 𝑡𝑡 + 𝑘𝑘 b=_{𝑣𝑣 𝑡𝑡 + 𝑘𝑘}𝑣𝑣 𝑡𝑡 = 1 + 𝑦𝑦_{1 + 𝑦𝑦}#Db#Db # # Variance of loss per policy

Var 𝐿𝐿_𝑛𝑛ã = Var 𝐸𝐸 𝐿𝐿v_{𝐼𝐼 +}𝐸𝐸 Var 𝐿𝐿d𝐼𝐼 𝑛𝑛 PROFIT TESTS Asset Shares 𝐴𝐴𝐴𝐴 b = bRd𝐴𝐴𝐴𝐴+ 𝐺𝐺bRd− 𝑒𝑒bRd 1 + 𝑖𝑖 −𝑞𝑞$DbRdπ 𝑏𝑏b+ 𝐸𝐸b(π) − 𝑞𝑞$DbRd∫ bCV+ 𝐸𝐸b(∫) / 1 − 𝑞𝑞$DbRdπ − 𝑞𝑞$DbRd∫

𝐺𝐺 = gross premium, 𝑒𝑒 = level expenses, 𝑏𝑏 = face amount, 𝐸𝐸ê _{= settlement expenses paid on} decrement 𝑗𝑗, 𝐶𝐶𝑉𝑉 = 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 −𝑞𝑞$DbRdπ 𝑏𝑏b+ 𝐸𝐸bπ − 𝑞𝑞$DbRd∫ bCV+ 𝐸𝐸b∫ −𝑝𝑝$DbRd(§) b𝑉𝑉 Profit Signature, Πb Profit per policy issued Πb= Prb⋅bRd𝑝𝑝$ , 𝑘𝑘 ≥ 1 Πb= Prb , 𝑘𝑘 = 0 Change in reserve Δb𝑉𝑉 = 1 + 𝑖𝑖 bRd𝑉𝑉− 𝑝𝑝$DbRd(§) b𝑉𝑉 IRR: GbcOΠb𝑣𝑣b= 0 NPV = ]bcOΠb𝑣𝑣øb, where 𝑟𝑟 = discount/hurdle rate Partial NPV NPV 𝑡𝑡 = Πb𝑣𝑣øb # bcO , where 𝑟𝑟 = discount/hurdle rate Profit Margin The ratio of the NPV to the (expected) present value of future premiums. Discounted Payback Period (DPP) Solve for lowest 𝑚𝑚 such that Πb𝑣𝑣b f bcO = 0. Universal Life General AV#= AV#Rd+ 𝑃𝑃#− 𝑒𝑒#− COI# 1 + 𝑖𝑖 COI#= 𝑣𝑣u𝑞𝑞$D#RdDB#− AV# Type A (Death Benefit = Face Amount) AV#= AV#Rd+ 𝑃𝑃#− 𝑒𝑒_{1 − 𝑞𝑞}# 1 + 𝑖𝑖 − 𝑞𝑞$D#RdFA $D#Rd Type B (Death Benefit = Face Amount + AV√) AV#= AV#Rd+ 𝑃𝑃#− 𝑒𝑒# 1 + 𝑖𝑖 − 𝑞𝑞$D#RdFA Corridor Factor, γ AV#= AV_{1 + 𝑞𝑞}#Rd+ 𝑃𝑃#− 𝑒𝑒# 1 + 𝑖𝑖 $D#Rd𝛾𝛾 − 1 If 𝛾𝛾 ⋅ AV#> death benefit, set death benefit = 𝛾𝛾 ⋅ AV#. Note: For all types, replace 𝑞𝑞$D#Rd with 𝑞𝑞$D#Rd1 + 𝑖𝑖 𝑣𝑣u if 𝑖𝑖 ≠ 𝑖𝑖u Gain by Source Total Profit = bRd𝑉𝑉+ 𝐺𝐺b− 𝑒𝑒b 1 + 𝑖𝑖 −𝑞𝑞$DbRd𝑏𝑏b+ 𝐸𝐸b − 𝑝𝑝$DbRdb𝑉𝑉 Total Gain = Actual Profit − Expected Profit Components of Gain (∗ = assumed, ′ = actual): Interest: 𝑖𝑖•_{− 𝑖𝑖}∗ _𝑉𝑉 bRd + 𝐺𝐺b− 𝑒𝑒b Expense: 𝑒𝑒b∗− 𝑒𝑒b• 1 + 𝑖𝑖 + 𝑞𝑞$DbRd 𝐸𝐸b∗− 𝐸𝐸b• Mortality: 𝑞𝑞$DbRd∗ − 𝑞𝑞$DbRd• 𝑏𝑏b+ 𝐸𝐸b− 𝑉𝑉b Lapse: 𝑞𝑞_$DbRd∫ ∗ _{− 𝑞𝑞} $DbRd∫ … CV k + 𝐸𝐸b∫ − 𝑉𝑉b

PENSION MATHEMATICS INTEREST RATE RISK