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Formula MLC by Coaching Actuaties

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Actuarial Science (CS242)

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Raise Your Odds® with Adapt

Exam MLC

SURVIVAL DISTRIBUTIONS

Probability Functions Actuarial Notations # $𝑝𝑝 =Probability that 𝑥𝑥 survives 𝑡𝑡 years = Pr𝑇𝑇$> 𝑡𝑡 = 𝐴𝐴$𝑡𝑡 # $𝑞𝑞 =Probability that 𝑥𝑥 dies within 𝑡𝑡 years = Pr𝑇𝑇$≤ 𝑡𝑡 = 𝐹𝐹$𝑡𝑡 # $𝑝𝑝 + 𝑞𝑞# $= 1 #|3𝑞𝑞$=Probability that 𝑥𝑥 survives 𝑡𝑡 years and dies within the following 𝑢𝑢 years = #𝑝𝑝$⋅ 3 𝑞𝑞$D# = #𝑝𝑝$− #D3𝑝𝑝$ = #D3𝑞𝑞$− #𝑞𝑞$ Life Table Functions G $𝑑𝑑 = 𝑙𝑙$− 𝑙𝑙$DG # $𝑝𝑝 =

𝑙𝑙$D#

𝑙𝑙$

# $𝑞𝑞 =

# $𝑑𝑑

𝑙𝑙$

=

𝑙𝑙$−𝑙𝑙$D#

𝑙𝑙$

#|3𝑞𝑞$=

3 $D#𝑑𝑑

𝑙𝑙$ =

𝑙𝑙$D#−𝑙𝑙$D#D

𝑙𝑙$

Force of Mortality

𝜇𝜇$D#=

𝑓𝑓$𝑡𝑡

𝐴𝐴$𝑡𝑡

𝜇𝜇$D#= −

𝑑𝑑

d𝑡𝑡ln𝐴𝐴$𝑡𝑡

𝜇𝜇$D#= −

𝑑𝑑

d𝑡𝑡

ln 𝑝𝑝# $

𝑓𝑓$𝑡𝑡 = 𝑝𝑝# $⋅𝜇𝜇$D#

$𝑝𝑝 = exp − 𝜇𝜇$DM d𝑠𝑠null

$𝑞𝑞 = M.𝑝𝑝$⋅𝜇𝜇$DM d𝑠𝑠null

#|3𝑞𝑞$= M.𝑝𝑝$⋅𝜇𝜇$DM d𝑠𝑠

null

Mortality Laws Constant Force of Mortality 𝜇𝜇$= 𝜇𝜇

$𝑝𝑝 = 𝑒𝑒RS#

Uniform Distribution

𝜇𝜇$=

1 𝜔𝜔 −𝑥𝑥

, 0 ≤ 𝑥𝑥 < 𝜔𝜔

# $𝑝𝑝 =

𝜔𝜔 −𝑥𝑥 −𝑡𝑡

𝜔𝜔 −𝑥𝑥 , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 −𝑥𝑥

#|3𝑞𝑞$=

𝑢𝑢

𝜔𝜔 −𝑥𝑥, 0 ≤ 𝑡𝑡 +𝑢𝑢 ≤ 𝜔𝜔 −𝑥𝑥

Beta Distribution

𝜇𝜇$=

𝛼𝛼

𝜔𝜔 −𝑥𝑥, 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#−

ln𝑐𝑐 Makeham’s Law 𝜇𝜇$= 𝐴𝐴 +𝐵𝐵𝑐𝑐$, 𝑐𝑐 > 1

$𝑝𝑝 = exp −𝐴𝐴𝑡𝑡 −

𝐵𝐵𝑐𝑐$𝑐𝑐#−

ln𝑐𝑐

Moments Complete Future Lifetime General 𝑒𝑒

∘ $= # $𝑝𝑝

]

d𝑡𝑡

Constant Force of Mortality 𝑒𝑒

∘ $=

1 𝜇𝜇

Uniform Distribution 𝑒𝑒

∘ $=

𝜔𝜔 −𝑥𝑥

2

Beta Distribution 𝑒𝑒

∘ $=

𝜔𝜔 −𝑥𝑥

𝛼𝛼 +

n -year Temporary Complete Future Lifetime 𝑒𝑒

∘ $:G|= # $𝑝𝑝

G O

d𝑡𝑡

ïUniform Distribution 𝑒𝑒

∘ $:G|= 𝑝𝑝G $𝑛𝑛+ 𝑞𝑞G $

𝑛𝑛

2

Curtate Future Lifetime

𝑒𝑒$= 𝑘𝑘 ⋅

]

bcd

b|𝑞𝑞$= b𝑝𝑝$

]

bcd ïUniform Distribution 𝑒𝑒$= 𝑒𝑒∘$−0. n -year Temporary Curtate Future Lifetime

𝑒𝑒$:G|= 𝑘𝑘 ⋅

GRd

bcd

b|𝑞𝑞$+𝑛𝑛 ⋅ 𝑝𝑝G $= b𝑝𝑝$

bcd ïUniform Distribution 𝑒𝑒$:G|= 𝑒𝑒

∘ $:G|−0 𝑞𝑞G.$ Recursive Formulas 𝑒𝑒

∘ $= 𝑒𝑒

∘ $:G|+ 𝑝𝑝G $⋅𝑒𝑒

∘ $DG 𝑒𝑒

∘ $:G|= 𝑒𝑒

∘ $:f|+ 𝑝𝑝f $⋅𝑒𝑒

𝑠𝑠 ⋅𝑞𝑞$

1−𝑡𝑡 ⋅𝑞𝑞$

𝜇𝜇$DM=

𝑞𝑞$

1−𝑠𝑠 ⋅𝑞𝑞$

𝑞𝑞$= 𝑝𝑝M $⋅𝜇𝜇$DM

Constant Force of Mortality 0 ≤ 𝑠𝑠 +𝑡𝑡 ≤ 1 𝑙𝑙$DM=𝑙𝑙$dRM⋅𝑙𝑙$DdM M $𝑝𝑝 = 𝑝𝑝M $D#= 𝑝𝑝$M 𝜇𝜇$DM= − ln𝑝𝑝$ 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.

Aft er a certain number of years of ‘select period’, mortality is called the ‘ultimate’ mortality. 𝑞𝑞$D#= 𝑞𝑞$D#.

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 + 30 32 31 33 32 34 33 35

INSURANCE

Level Annual Insurance Type of Insurance EPV

Whole Life

Discrete

𝐴𝐴$= 𝑣𝑣bDd⋅

]

bcO

b|𝑞𝑞$

Continuous

𝐴𝐴$= 𝑣𝑣#⋅

]

$𝑝𝑝 ⋅𝜇𝜇$D# d𝑡𝑡

Term Life

Discrete 𝐴𝐴$:G|d = 𝐴𝐴$− 𝐸𝐸G $⋅𝐴𝐴$DG Continuous 𝐴𝐴 $∶G| d = 𝐴𝐴$− 𝐸𝐸G $⋅𝐴𝐴$DG

Deferred Life

Discrete G|𝐴𝐴$= 𝐴𝐴$−𝐴𝐴$:G|d = 𝐸𝐸G $⋅𝐴𝐴$DG Continuous G|𝐴𝐴$= 𝐴𝐴$−𝐴𝐴$∶G| d = 𝐸𝐸G $⋅𝐴𝐴$DG

Pure Endowment

Discrete 𝐴𝐴$:G| d= 𝐸𝐸G $= 𝑣𝑣GG $𝑝𝑝 Continuous N/A

Endowment Insurance

Discrete 𝐴𝐴$:G| = 𝐴𝐴$:G|d + 𝐸𝐸G $ Continuous 𝐴𝐴$:G| = 𝐴𝐴 d $:G|+ 𝐸𝐸G $

EPV under Constant Force of Mortality Discrete Continuous 𝐴𝐴$=

𝑞𝑞

𝑞𝑞 +𝑖𝑖 𝐴𝐴$=

𝜇𝜇

𝜇𝜇 +𝛿𝛿

𝐴𝐴$:G|d =

𝑞𝑞

𝑞𝑞 +𝑖𝑖

1− 𝐸𝐸G $ 𝐴𝐴 $:G| d =

𝜇𝜇

𝜇𝜇 +𝛿𝛿 1− 𝐸𝐸G $

G|𝐴𝐴$=

𝑞𝑞

𝑞𝑞 +𝑖𝑖⋅ 𝐸𝐸G $ G|𝐴𝐴$=

𝜇𝜇

𝜇𝜇 +𝛿𝛿⋅ 𝐸𝐸G $

G $𝐸𝐸= 𝑣𝑣G𝑝𝑝G G $𝐸𝐸 = 𝑒𝑒R(SDo)G

EPV under Uniform Distribution Discrete Continuous 𝐴𝐴$=

𝑎𝑎rR$| 𝜔𝜔 −𝑥𝑥 𝐴𝐴$=

𝑎𝑎rR$| 𝜔𝜔 −𝑥𝑥 𝐴𝐴$:G|d =

𝑎𝑎G|

𝜔𝜔 −𝑥𝑥 𝐴𝐴 $:G|

d = 𝑎𝑎G|
𝜔𝜔 −𝑥𝑥

G $𝐸𝐸 = 𝑣𝑣G⋅

𝜔𝜔 −𝑥𝑥 −𝑛𝑛

𝜔𝜔 −𝑥𝑥

G $𝐸𝐸= 𝑣𝑣G⋅

𝜔𝜔 −𝑥𝑥 −𝑛𝑛

𝜔𝜔 −𝑥𝑥

SURVIVAL DISTRIBUTIONS

INSURANCE

m -thly Insurance

𝐴𝐴$(f)= 𝑣𝑣bDd/f⋅

]

bcO

b 𝑞𝑞 f |

Variances Discrete Var𝑍𝑍$ = 𝐴𝐴h$−𝐴𝐴$h Var𝑍𝑍$:G| = 𝐴𝐴h$:G|−𝐴𝐴$:G|h Continuous Var𝑍𝑍$ = 𝐴𝐴h$−𝐴𝐴$h Var𝑍𝑍$:G| = 𝐴𝐴h$:G|−𝐴𝐴$:G|h Note: h𝐴𝐴 and h𝐴𝐴 are calculated similar to 𝐴𝐴 and 𝐴𝐴 respectively, but with double the force of interest, 𝛿𝛿. Equivalently, replace 𝑣𝑣 with 𝑣𝑣h, or replace 𝑖𝑖 with 2𝑖𝑖 +𝑖𝑖h. For example, under constant force, h𝐴𝐴$= u uDhvDvw and 𝐴𝐴

h $= S SDho. Increasing and Decreasing Insurance 𝐼𝐼𝐴𝐴$= 𝐴𝐴$+ 𝐴𝐴d|. $+ 𝐴𝐴h|.$+⋯

𝐼𝐼𝐴𝐴$= 𝑡𝑡𝑣𝑣#⋅

]

$𝑝𝑝 ⋅𝜇𝜇$D# d𝑡𝑡

𝐼𝐼𝐴𝐴 $:G| d = 𝑡𝑡𝑣𝑣#⋅

G O

$𝑝𝑝 ⋅𝜇𝜇$D# d𝑡𝑡

𝐷𝐷𝐴𝐴 $:G| d = 𝑛𝑛 −𝑡𝑡𝑣𝑣#⋅

G O

$𝑝𝑝⋅𝜇𝜇$D# d𝑡𝑡

EPV under Constant Force Discrete Continuous 𝐼𝐼𝐴𝐴$=

1 𝑣𝑣𝑞𝑞

𝑞𝑞

𝑞𝑞 +𝑖𝑖

h 𝐼𝐼𝐴𝐴$=

𝜇𝜇

𝜇𝜇 +𝛿𝛿h

EPV under Uniform Distribution Discrete Continuous 𝐼𝐼𝐴𝐴$=

𝐼𝐼𝑎𝑎rR$| 𝜔𝜔 −𝑥𝑥

𝐼𝐼𝐴𝐴$=

𝐼𝐼𝑎𝑎rR$| 𝜔𝜔 −𝑥𝑥 𝐼𝐼𝐴𝐴 $:G| d =

𝐼𝐼𝑎𝑎G|

𝜔𝜔 −𝑥𝑥

𝐼𝐼𝐴𝐴 $:G| d =

𝐼𝐼𝑎𝑎G|

𝜔𝜔 −𝑥𝑥

𝐷𝐷𝐴𝐴 $:G| d =

𝐷𝐷𝑎𝑎G|

𝜔𝜔 −𝑥𝑥 𝐷𝐷𝐴𝐴 $:G|

d =𝐷𝐷𝑎𝑎G|

**Relationship between ** 𝑨𝑨𝒙𝒙 **, ** 𝑨𝑨𝒙𝒙(𝒎𝒎) ** and ** 𝑨𝑨𝒙𝒙 (Under UDD Assumption) 𝐴𝐴$=

𝑖𝑖

𝛿𝛿

𝐴𝐴$

𝐴𝐴 $:G| d =

𝑖𝑖

𝛿𝛿𝐴𝐴 $:G|

G|𝐴𝐴$=

𝑖𝑖

𝛿𝛿G|𝐴𝐴$

𝐴𝐴$:G|=

𝑖𝑖

𝛿𝛿𝐴𝐴 $:G|

d +𝐴𝐴
    $:G|

                    d

𝐴𝐴$(f)=

𝑖𝑖

𝑖𝑖(f)𝐴𝐴$ h𝐴𝐴$= 2𝑖𝑖 +𝑖𝑖h 2𝛿𝛿 ⋅ 𝐴𝐴

h $

ANNUITIES

Level Annual Annuities Type of Annuities EPV

Whole Life

Due; Discrete

𝑎𝑎$= 𝑣𝑣b⋅ 𝑝𝑝b$

]

bcO Immediate; Discrete 𝑎𝑎$= 𝑎𝑎$− Continuous 𝑎𝑎$= 𝑣𝑣#⋅

] O

$𝑝𝑝 d𝑡𝑡

Temporary Life

Due; Discrete 𝑎𝑎$:G|= 𝑎𝑎$− 𝐸𝐸G $⋅𝑎𝑎$DG Immediate; Discrete 𝑎𝑎$:G|= 𝑎𝑎$:G|−1+ 𝐸𝐸$G Continuous 𝑎𝑎$:G|= 𝑎𝑎$− 𝐸𝐸G $⋅𝑎𝑎$DG

Deferred Whole Life

EPV under Constant Force of Mortality Discrete Continuous 𝑎𝑎$=

1+𝑖𝑖

𝑞𝑞 +𝑖𝑖

𝑎𝑎$=

1 𝜇𝜇 +𝛿𝛿

𝑎𝑎$:G|=

1+𝑖𝑖

𝑞𝑞 +𝑖𝑖

1− 𝐸𝐸G $ 𝑎𝑎$:G|=

1 𝜇𝜇 +𝛿𝛿

1− 𝐸𝐸G $

G|𝑎𝑎$=

1+𝑖𝑖

𝑞𝑞 +𝑖𝑖⋅ 𝐸𝐸G $ G|𝑎𝑎$=

1 𝜇𝜇 +𝛿𝛿⋅ 𝐸𝐸G $

G $𝐸𝐸 = 𝑣𝑣G𝑝𝑝G G $𝐸𝐸 = 𝑒𝑒R(SDo)G

h𝐴𝐴$−𝐴𝐴$h 𝑑𝑑h

Var𝑌𝑌$:G| = Var𝑌𝑌$:GRd| =

h𝐴𝐴$:G|−𝐴𝐴$:G|h 𝑑𝑑h Continuous Var𝑌𝑌$ =

h𝐴𝐴$−𝐴𝐴$h 𝛿𝛿h

Var𝑌𝑌$:G| =

h𝐴𝐴$:G|−𝐴𝐴$:G|h 𝛿𝛿h Increasing and Decreasing Annuities 𝐼𝐼𝑎𝑎$:G| = 𝑡𝑡𝑣𝑣#⋅

$𝑝𝑝 d𝑡𝑡

𝐼𝐼𝑎𝑎$ =

1

𝜇𝜇 +𝛿𝛿h if 𝜇𝜇 is constant 𝐷𝐷𝑎𝑎$:G| = 𝑛𝑛 −𝑡𝑡𝑣𝑣#⋅

G O

$𝑝𝑝 d𝑡𝑡

𝑎𝑎$(f)≈ 𝑎𝑎$−

𝑚𝑚 −

2𝑚𝑚 −

𝑚𝑚h− 12 𝑚𝑚h 𝜇𝜇$+𝛿𝛿 𝑎𝑎$:G|f ≈ 𝑎𝑎$:G|−

𝑚𝑚 −

2𝑚𝑚

1− 𝐸𝐸$G

−

𝑚𝑚h− 12 𝑚𝑚h 𝜇𝜇$+𝛿𝛿 − 𝐸𝐸$G 𝜇𝜇$DG+𝛿𝛿 G|𝑎𝑎$f≈ 𝑎𝑎G|$−

𝑚𝑚 −

2𝑚𝑚

𝐸𝐸$G

−

𝑚𝑚h− 12 𝑚𝑚h 𝐸𝐸$G 𝜇𝜇$DG+𝛿𝛿 𝑎𝑎$≈ 𝑎𝑎$−

1 2 −

1 12 𝜇𝜇$+𝛿𝛿

ANNUITIES

Continuous Probabilities

#𝑝𝑝$vv=exp − 𝜇𝜇$DMvê ê°v

d𝑠𝑠

null

For permanent disability model:

#𝑝𝑝$vê= M𝑝𝑝$vv⋅𝜇𝜇$DMvê ⋅ 𝑝𝑝#RM$DMêê d𝑠𝑠

null

O Kolmogorov’s Forward Equations d d𝑡𝑡 #𝑝𝑝$vê= Rate of entry into state 𝑗𝑗 −Rate of leaving state 𝑗𝑗

                                                        = #𝑝𝑝$vb⋅𝜇𝜇$D#bê − 𝑝𝑝#$vê⋅𝜇𝜇$D#êb

bcOb°ê

Euler’s Method

#Dö𝑝𝑝$vê≈ 𝑝𝑝#$vê+ℎ #𝑝𝑝$vb⋅𝜇𝜇$D#bê − 𝑝𝑝#$vê⋅𝜇𝜇$D#êb

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

]

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¢ •dDo 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

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 #Rö𝑉𝑉v= 𝑉𝑉# v 1−𝛿𝛿#ℎ +ℎ𝐵𝐵#v

                                                                            +ℎ 𝜇𝜇$D#vê 𝑏𝑏#vê+ 𝑉𝑉# ê− 𝑉𝑉# v

êcO ê°v

MULTIPLE DECREMENT MODELS

Probabilities

#𝑞𝑞$§= #𝑞𝑞$ê

êcd

#𝑞𝑞$ê = b𝑝𝑝$§

#Rd

bcO

𝑞𝑞$Dbê

#|3𝑞𝑞$ê = 𝑝𝑝#$§ 3 𝑞𝑞$D#ê = b𝑝𝑝$§𝑞𝑞$Dbê

#D3Rd

bc# Life Table Formulas

𝑑𝑑$§= 𝑑𝑑$ê

êcd 𝑙𝑙$Db§ = 𝑙𝑙$§b𝑝𝑝$§= 𝑙𝑙$§− 𝑑𝑑b$§

𝑑𝑑$Dbê = 𝑙𝑙$§b𝑝𝑝$§𝑞𝑞$Dbê

Discrete Insurances

𝐴𝐴 = 𝑣𝑣bbRd𝑝𝑝$(§)

]

bcd

𝑞𝑞$DbRd(ê) 𝑏𝑏b(ê) ê Continuous Insurances

𝐴𝐴= 𝑣𝑣##𝑝𝑝$§

]

𝜇𝜇$D#ê 𝑏𝑏#ê

êcd

d𝑡𝑡

Forces of Decrement #𝑞𝑞$ê= M𝑝𝑝$(§)

null

𝜇𝜇$DM(ê) d𝑠𝑠

𝜇𝜇$D#(ê) =

1 #𝑝𝑝$§

d#𝑞𝑞$ê d𝑡𝑡

𝜇𝜇$D#(§) = 𝜇𝜇$D#(ê)

êcd

#𝑝𝑝$(§)=exp − 𝜇𝜇$DM(§)

null

d𝑠𝑠 Fractional Ages UDD in the multiple decrement table: M𝑞𝑞$(ê)= 𝑠𝑠𝑞𝑞$(ê), 0 ≤ 𝑠𝑠 ≤ 1 Constant forces of decrement:

M𝑞𝑞$ê=

𝑞𝑞$ê 𝑞𝑞$§

1− 𝑝𝑝$§

Associated Single Decrement Tables The associated single decrements are independent.

#𝑝𝑝$•(ê)=exp − 𝜇𝜇$DM(ê)

null

d𝑠𝑠

#𝑞𝑞$•(ê)= M𝑝𝑝$•(ê)𝜇𝜇$DM(ê)

null

d𝑠𝑠

#𝑝𝑝$•(ê)

êcd

= 𝑝𝑝#$(§)

𝜇𝜇$D#ê = −

1

#𝑝𝑝$•ê

d#𝑝𝑝$•ê d𝑡𝑡 = −

d d𝑡𝑡ln 𝑝𝑝$

null

UDD in Multiple-Decrement Tables (UDDMDT)

M𝑝𝑝$•ê = M𝑝𝑝$§

¶ß® ¶ß©, 0 ≤ 𝑠𝑠 ≤ 1 UDD in Associated Single Decrement Tables (UDDASDT) For 2 decrements:

#𝑞𝑞$(d)= 𝑞𝑞$•d 𝑡𝑡 −

𝑡𝑡h𝑞𝑞$•h 2

, 0 ≤ 𝑡𝑡 ≤ 1

For 3 decrements:

#𝑞𝑞$d= 𝑞𝑞$•d 𝑡𝑡 −

𝑡𝑡h𝑞𝑞$•h+𝑞𝑞$•™ 2 +

𝑡𝑡™𝑞𝑞$•h𝑞𝑞$•™ 3 , 0 ≤ 𝑡𝑡 ≤ 1

MULTIPLE LIVES

Joint Life 𝑇𝑇$ ̈= min𝑇𝑇$,𝑇𝑇 ̈

#𝑝𝑝$ ̈+ 𝑞𝑞#$ ̈= 1 #|3𝑞𝑞$ ̈ = 𝑝𝑝#$ ̈⋅ 𝑞𝑞$D#: ̈D# = 𝑝𝑝#$ ̈− 𝑝𝑝#D3$ ̈ = 𝑞𝑞#D3$ ̈− 𝑞𝑞#$ ̈ #D3𝑝𝑝$ ̈= 𝑝𝑝#$ ̈⋅ 𝑝𝑝$D#: ̈D# 𝑒𝑒

∘ $ ̈= #𝑝𝑝$ ̈

]

d𝑡𝑡

𝑒𝑒$ ̈= b𝑝𝑝$ ̈

]

bcd 𝐴𝐴$ ̈= 1−𝛿𝛿𝑎𝑎$ ̈

Independent Lives #𝑝𝑝$ ̈= 𝑝𝑝$# ⋅ 𝑝𝑝 ̈# 𝜇𝜇$D#: ̈D#= 𝜇𝜇$D#+𝜇𝜇 ̈D#

#𝑝𝑝$ ̈=exp − 𝜇𝜇$DM+𝜇𝜇 ̈DM

null

d𝑠𝑠 Last Survivor 𝑇𝑇$ ̈= max𝑇𝑇$,𝑇𝑇 ̈

#𝑝𝑝$ ̈+ 𝑞𝑞#$ ̈= 1 #|3𝑞𝑞$ ̈= 𝑝𝑝#$ ̈− 𝑝𝑝#D3$ ̈= 𝑞𝑞#D3$ ̈− 𝑞𝑞#$ ̈

𝑒𝑒

∘ $ ̈= #𝑝𝑝$ ̈

]

d𝑡𝑡

𝑒𝑒$ ̈= b𝑝𝑝$ ̈

]

bcd 𝐴𝐴$ ̈= 1−𝛿𝛿𝑎𝑎$ ̈

Independent Lives #𝑞𝑞$ ̈= 𝑞𝑞$# ⋅ 𝑞𝑞 ̈#

𝜇𝜇$ ̈𝑡𝑡 =

𝑞𝑞$∙ 𝑝𝑝 ̈𝜇𝜇 ̈D#+ 𝑞𝑞 ̈∙ 𝑝𝑝$𝜇𝜇$D#####

#𝑝𝑝$ ̈

Relationship between ** (𝒙𝒙𝒙𝒙 ) ** Status and (𝒙𝒙𝒙𝒙) ** Status** 𝑇𝑇$ ̈+𝑇𝑇$ ̈= 𝑇𝑇$+𝑇𝑇 ̈ #𝑝𝑝$ ̈+ 𝑝𝑝#$ ̈= 𝑝𝑝$# + 𝑝𝑝 ̈# 𝑒𝑒∘$ ̈+𝑒𝑒∘$ ̈= 𝑒𝑒∘$+𝑒𝑒∘ ̈ 𝑒𝑒$ ̈+𝑒𝑒$ ̈= 𝑒𝑒$+𝑒𝑒 ̈ Cov𝑇𝑇$ ̈,𝑇𝑇$ ̈ =Cov𝑇𝑇$,𝑇𝑇 ̈ +𝑒𝑒

∘ $−𝑒𝑒

∘ $ ̈ 𝑒𝑒

∘ ̈−𝑒𝑒

∘ $ ̈ Cov𝑇𝑇$,𝑇𝑇 ̈ = 0 if 𝑇𝑇$ and 𝑇𝑇 ̈ are independent 𝐴𝐴$ ̈+𝐴𝐴$ ̈= 𝐴𝐴$+𝐴𝐴 ̈ 𝑎𝑎$ ̈+𝑎𝑎$ ̈= 𝑎𝑎$+𝑎𝑎 ̈ G𝐸𝐸$ ̈+ 𝐸𝐸G$ ̈= 𝐸𝐸$G + 𝐸𝐸 ̈G Contingent Probabilities

G𝑞𝑞$ ̈d = #𝑝𝑝$ ̈∙𝜇𝜇$D#𝑑𝑑𝑡𝑡

G O G𝑞𝑞$ ̈ d= #𝑝𝑝$ ̈∙𝜇𝜇 ̈D#𝑑𝑑𝑡𝑡

G O 𝑞𝑞$ ̈d + 𝑞𝑞GG $ ̈ d=G𝑞𝑞$ ̈

G𝑞𝑞$ ̈h = 𝑝𝑝$# 1− 𝑝𝑝 ̈# ∙𝜇𝜇$D#𝑑𝑑𝑡𝑡

O G𝑞𝑞$ ̈ h= 𝑝𝑝 ̈# 1− 𝑝𝑝$# ∙𝜇𝜇 ̈D#𝑑𝑑𝑡𝑡

O 𝑞𝑞$ ̈h + 𝑞𝑞GG $ ̈ h=G𝑞𝑞$ ̈ G𝑞𝑞$ ̈d +G𝑞𝑞$ ̈h = 𝑞𝑞G$ G𝑞𝑞$ ̈ d+G𝑞𝑞$ ̈ h= 𝑞𝑞G ̈ G𝑞𝑞$ ̈d =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$| ̈= 𝑎𝑎 ̈−𝑎𝑎$ ̈

MULTIPLE DECREMENT MODELS MULTIPLE LIVES

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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 ï Salaries increase continuously 𝑠𝑠 ̈ 𝑠𝑠$=

salary rate at age 𝑦𝑦 salary rate at age 𝑥𝑥 Salary Scale Assumption ï Salaries increase at discrete intervals 𝑠𝑠 ̈ 𝑠𝑠$=

salary earned between age 𝑦𝑦 and 𝑦𝑦 + salary earned between age 𝑥𝑥 and 𝑥𝑥 + Final average salary over the last 3 years (e. retire at age 65)

1 3 𝑠𝑠≤h+𝑠𝑠≤™+𝑠𝑠≤≥ 𝑠𝑠$ ⋅Salary between age 𝑥𝑥 and 𝑥𝑥 + Salary rate to salary scale : 𝑠𝑠$= 𝑠𝑠$D#

d𝑡𝑡

Salary scale to salary rate : 𝑠𝑠 $= 𝑠𝑠$RO.μ Normal Contribution 𝐶𝐶#= 𝑣𝑣 𝑝𝑝d$OO#Dd𝑉𝑉− 𝑉𝑉# + EPV(mid-year exits benefits) ï TUC if the actuarial liability is calculated with the traditional unit method ï PUC if the actuarial liability is calculated with the projected unit method. Under constant and independent of salary accrual rate with no exit benefits: ï TUC: 𝑉𝑉∂ß∑∏∂ ß

GDd O G −1 PUC: 𝑉𝑉

d O G

INTEREST RATE RISK

Replicating Cash Flows Spot rate, 𝑦𝑦#: effective interest rate paid by a zero- coupon bond maturing at time 𝑡𝑡 𝑣𝑣𝑡𝑡: Present value of 1 paid at time 𝑡𝑡 𝑣𝑣𝑡𝑡=

1 1+𝑦𝑦##

Forward rate, 𝑓𝑓𝑡𝑡,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+𝑓𝑓𝑡𝑡,𝑡𝑡 +𝑘𝑘 b=

𝑣𝑣𝑡𝑡

𝑣𝑣𝑡𝑡 +𝑘𝑘 =

1+𝑦𝑦#Db#Db 1+𝑦𝑦## Variance of loss per policy

Var

𝐿𝐿ã 𝑛𝑛 =Var𝐸𝐸𝐿𝐿

v𝐼𝐼 +𝐸𝐸Var𝐿𝐿

d𝐼𝐼 𝑛𝑛

PROFIT TESTS

Asset Shares b𝐴𝐴𝐴𝐴= 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⋅ 𝑝𝑝bRd$ , 𝑘𝑘 ≥ 1 Πb= Prb , 𝑘𝑘 = 0 Change in reserve Δb𝑉𝑉 = 1+𝑖𝑖bRd𝑉𝑉−𝑝𝑝$DbRd(§) b𝑉𝑉

IRR: GbcOΠb𝑣𝑣b= 0 NPV = ]bcOΠb𝑣𝑣øb, where 𝑟𝑟 = discount/hurdle rate Partial NPV

NPV𝑡𝑡 = Πb𝑣𝑣øb

null

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

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+𝑖𝑖 −𝑞𝑞$D#RdFA 1−𝑞𝑞$D#Rd Type B (Death Benefit = Face Amount + AV√) AV#= AV#Rd+𝑃𝑃#−𝑒𝑒# 1+𝑖𝑖 −𝑞𝑞$D#RdFA

Corridor Factor, γ AV#=

AV#Rd+𝑃𝑃#−𝑒𝑒# 1+𝑖𝑖 1+𝑞𝑞$D#Rd𝛾𝛾 − _If _ 𝛾𝛾 ⋅ AV #> _death benefit, set death benefit _ = 𝛾𝛾 ⋅ AV #.

Note : For all types, replace 𝑞𝑞$D#Rd with 𝑞𝑞$D#Rd1+𝑖𝑖𝑣𝑣u 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∗ bRd𝑉𝑉+𝐺𝐺b−𝑒𝑒b Expense : 𝑒𝑒b∗−𝑒𝑒b• 1+𝑖𝑖 +𝑞𝑞$DbRd𝐸𝐸b∗−𝐸𝐸b• Mortality : 𝑞𝑞$DbRd∗ −𝑞𝑞$DbRd• 𝑏𝑏b+𝐸𝐸b− 𝑉𝑉b Lapse : 𝑞𝑞$DbRd∫∗ −𝑞𝑞$DbRd∫

...

kCV+𝐸𝐸b∫− 𝑉𝑉b

PEN S IO N MATH EMATIC S INTEREST RATE RISK

PROFIT TESTS

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Formula MLC by Coaching Actuaties

Course: Actuarial Science (CS242)

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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

= 𝐴𝐴$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$−#D3𝑝𝑝$

=#D3 𝑞𝑞$−#𝑞𝑞$

LifeTableFunctions

𝑑𝑑

G $ = 𝑙𝑙$− 𝑙𝑙$DG

𝑝𝑝

# $ =𝑙𝑙$D#

𝑙𝑙$

𝑞𝑞

# $ =𝑑𝑑

# $

𝑙𝑙$

=  𝑙𝑙$− 𝑙𝑙$D#

𝑙𝑙$

𝑞𝑞

#|3 $=𝑑𝑑

3 $D#

𝑙𝑙$

=  𝑙𝑙$D# − 𝑙𝑙$D#D3

𝑙𝑙$

ForceofMortality

𝜇𝜇$D# =𝑓𝑓

$𝑡𝑡

𝐴𝐴$𝑡𝑡

𝜇𝜇$D# = − 𝑑𝑑

d𝑡𝑡ln 𝐴𝐴$𝑡𝑡

𝜇𝜇$D# = − 𝑑𝑑

d𝑡𝑡ln 𝑝𝑝

# $

𝑓𝑓

$𝑡𝑡 =  𝑝𝑝

# $ ⋅ 𝜇𝜇$D# 

𝑝𝑝

# $ =exp− 𝜇𝜇$DMd𝑠𝑠

𝑞𝑞

# $ =𝑝𝑝

.$⋅ 𝜇𝜇$DMd𝑠𝑠



𝑞𝑞

#|3 $=𝑝𝑝

.$⋅ 𝜇𝜇$DMd𝑠𝑠

#D3



MortalityLaws

ConstantForceofMortality

𝜇𝜇$= 𝜇𝜇

𝑝𝑝

# $ = 𝑒𝑒RS#

UniformDistribution

𝜇𝜇$=1

𝜔𝜔 − 𝑥𝑥 , 0 ≤ 𝑥𝑥 < 𝜔𝜔

𝑝𝑝

# $ =𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡

𝜔𝜔 − 𝑥𝑥 , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥

𝑞𝑞

#|3 $=𝑢𝑢

𝜔𝜔 − 𝑥𝑥 , 0 ≤ 𝑡𝑡 + 𝑢𝑢 ≤ 𝜔𝜔 − 𝑥𝑥

BetaDistribution

𝜇𝜇$=𝛼𝛼

𝜔𝜔 − 𝑥𝑥 , 0 ≤ 𝑥𝑥 < 𝜔𝜔

𝑝𝑝

# $ =𝜔𝜔 − 𝑥𝑥 − 𝑡𝑡

𝜔𝜔 − 𝑥𝑥

, 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 − 𝑥𝑥

Gompertz’sLaw

𝜇𝜇$= 𝐵𝐵𝑐𝑐$, 𝑐𝑐 > 1

𝑝𝑝

# $ =exp−𝐵𝐵𝑐𝑐$𝑐𝑐#− 1

ln 𝑐𝑐

Makeham’sLaw

𝜇𝜇$= 𝐴𝐴 + 𝐵𝐵𝑐𝑐$, 𝑐𝑐 > 1

𝑝𝑝

# $ =exp −𝐴𝐴𝑡𝑡 −𝐵𝐵𝑐𝑐$𝑐𝑐#− 1

ln 𝑐𝑐

Moments

CompleteFutureLifetime

General

𝑒𝑒

∘

$= 𝑝𝑝

# $

]

d𝑡𝑡

ConstantForceofMortality

𝑒𝑒

∘

$=1

𝜇𝜇

UniformDistribution

𝑒𝑒

∘

$=𝜔𝜔 − 𝑥𝑥

BetaDistribution

𝑒𝑒

∘

$=𝜔𝜔 − 𝑥𝑥

𝛼𝛼 + 1

n-yearTemporaryCompleteFutureLifetime

𝑒𝑒

∘

$:G| = 𝑝𝑝

# $

d𝑡𝑡

•UniformDistribution

𝑒𝑒

∘

$:G| = 𝑝𝑝

G $ 𝑛𝑛 + 𝑞𝑞

G $

𝑛𝑛

CurtateFutureLifetime

𝑒𝑒$= 𝑘𝑘 ⋅

]

bcd

𝑞𝑞

b| $=𝑝𝑝

]

bcd



•UniformDistribution

𝑒𝑒$= 𝑒𝑒

∘

$− 0.5

n-yearTemporaryCurtateFutureLifetime

𝑒𝑒$:G| = 𝑘𝑘 ⋅

GRd

bcd

𝑞𝑞

b| $+ 𝑛𝑛 ⋅ 𝑝𝑝

G $ =𝑝𝑝

bcd



•UniformDistribution

𝑒𝑒$:G| = 𝑒𝑒

∘

$:G| − 0.5 𝑞𝑞

RecursiveFormulas

𝑒𝑒

∘

$= 𝑒𝑒

∘

$:G| + 𝑝𝑝

G $ ⋅ 𝑒𝑒

∘

$DG

𝑒𝑒

∘

$:G| = 𝑒𝑒

∘

$:f| + 𝑝𝑝

f $ ⋅ 𝑒𝑒

∘

$Df:GRf|, 𝑚𝑚 < 𝑛𝑛

𝑒𝑒$= 𝑒𝑒$:G| + 𝑝𝑝

G $ ⋅ 𝑒𝑒$DG = 𝑒𝑒$:GRd| + 𝑝𝑝

G $ 1 + 𝑒𝑒$DG

𝑒𝑒$= 𝑝𝑝$1 + 𝑒𝑒$Dd 

𝑒𝑒$:G| = 𝑒𝑒$:f| + 𝑝𝑝

f $ ⋅ 𝑒𝑒$Df:GRf|, 𝑚𝑚 < 𝑛𝑛

𝑒𝑒$:G| = 𝑒𝑒$:fRd| + 𝑝𝑝

f $ 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 $D# =𝑠𝑠 ⋅ 𝑞𝑞$

1 − 𝑡𝑡 ⋅ 𝑞𝑞$

𝜇𝜇$DM =𝑞𝑞$

1 − 𝑠𝑠 ⋅ 𝑞𝑞$

𝑞𝑞$=  𝑝𝑝

M $ ⋅ 𝜇𝜇$DM

ConstantForceofMortality 0 ≤ 𝑠𝑠 + 𝑡𝑡 ≤ 1

𝑙𝑙$DM = 𝑙𝑙$dRM ⋅𝑙𝑙$Dd M

𝑝𝑝

M $ = 𝑝𝑝

M $D# =𝑝𝑝$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.

After a certain number of years of‘select period’,

mortalityiscalledthe‘ultimate’mortality.

𝑞𝑞$ D# = 𝑞𝑞$D#.

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#⋅

]

𝑝𝑝

# $ ⋅ 𝜇𝜇$D# d𝑡𝑡

TermLife

Discrete

𝐴𝐴$:G|

d= 𝐴𝐴$− 𝐸𝐸

G $ ⋅ 𝐴𝐴$DG

Continuous

𝐴𝐴$∶G|

d = 𝐴𝐴$− 𝐸𝐸

G $ ⋅ 𝐴𝐴$DG

DeferredLife

Discrete

𝐴𝐴

G| $= 𝐴𝐴$− 𝐴𝐴$:G|

d= 𝐸𝐸

G $ ⋅ 𝐴𝐴$DG

Continuous

𝐴𝐴

G| $= 𝐴𝐴$− 𝐴𝐴$∶G|

d = 𝐸𝐸

G $ ⋅ 𝐴𝐴$DG

Pure

Endowment

Discrete

𝐴𝐴$:G|

d = 𝐸𝐸

G $ = 𝑣𝑣G𝑝𝑝

G $

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

𝑞𝑞 + 𝑖𝑖

𝐴𝐴$=𝜇𝜇

𝜇𝜇 + 𝛿𝛿

𝐴𝐴$:G|

d=𝑞𝑞

𝑞𝑞 + 𝑖𝑖 1 − 𝐸𝐸

G $

𝐴𝐴$:G|

d =𝜇𝜇

𝜇𝜇 + 𝛿𝛿 1 − 𝐸𝐸

G $

𝐴𝐴

G| $=𝑞𝑞

𝑞𝑞 + 𝑖𝑖 ⋅ 𝐸𝐸

G $

𝐴𝐴

G| $=𝜇𝜇

𝜇𝜇 + 𝛿𝛿 ⋅ 𝐸𝐸

G $

𝐸𝐸

G $ = 𝑣𝑣G𝑝𝑝G

𝐸𝐸

G $ = 𝑒𝑒R(SDo)G

EPVunderUniformDistribution

Discrete

Continuous

𝐴𝐴$=𝑎𝑎rR$|

𝜔𝜔 − 𝑥𝑥

𝐴𝐴$=𝑎𝑎rR$|

𝜔𝜔 − 𝑥𝑥

𝐴𝐴$:G|

d=𝑎𝑎G|

𝜔𝜔 − 𝑥𝑥

𝐴𝐴$:G|

d =𝑎𝑎G|

𝜔𝜔 − 𝑥𝑥

𝐸𝐸

G $ = 𝑣𝑣G⋅𝜔𝜔 − 𝑥𝑥 − 𝑛𝑛

𝜔𝜔 − 𝑥𝑥

𝐸𝐸

G $ = 𝑣𝑣G⋅𝜔𝜔 − 𝑥𝑥 − 𝑛𝑛

𝜔𝜔 − 𝑥𝑥

SURVIVAL DISTRIBUTIONS

INSURANCE

Formula MLC by Coaching Actuaties

Actuarial Science (CS242)

Universiti Teknologi MARA

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Raise Your Odds® with Adapt

Exam MLC

SURVIVAL DISTRIBUTIONS

𝑙𝑙$D#

𝑙𝑙$

# $𝑞𝑞 =

# $𝑑𝑑

𝑙𝑙$

=

𝑙𝑙$−𝑙𝑙$D#

𝑙𝑙$

#|3𝑞𝑞$=

3 $D#𝑑𝑑

𝑙𝑙$ =

𝑙𝑙$D#−𝑙𝑙$D#D

𝑙𝑙$

𝑓𝑓$𝑡𝑡

𝐴𝐴$𝑡𝑡

𝜇𝜇$D#= −

𝑑𝑑

𝑑𝑑

1

𝜔𝜔 −𝑥𝑥

, 0 ≤ 𝑥𝑥 < 𝜔𝜔

# $𝑝𝑝 =

𝜔𝜔 −𝑥𝑥 −𝑡𝑡

𝜔𝜔 −𝑥𝑥 , 0 ≤ 𝑡𝑡 ≤ 𝜔𝜔 −𝑥𝑥

#|3𝑞𝑞$=

𝑢𝑢

𝜔𝜔 −𝑥𝑥, 0 ≤ 𝑡𝑡 +𝑢𝑢 ≤ 𝜔𝜔 −𝑥𝑥

𝛼𝛼

𝜔𝜔 −𝑥𝑥, 0 ≤ 𝑥𝑥 < 𝜔𝜔

# $𝑝𝑝 =

𝜔𝜔 −𝑥𝑥 −𝑡𝑡

𝜔𝜔 −𝑥𝑥

𝐵𝐵𝑐𝑐$𝑐𝑐#−

𝐵𝐵𝑐𝑐$𝑐𝑐#−

1

𝜇𝜇

𝜔𝜔 −𝑥𝑥

2

𝜔𝜔 −𝑥𝑥

𝛼𝛼 +

𝑛𝑛

2

𝑠𝑠 ⋅𝑞𝑞$

1−𝑡𝑡 ⋅𝑞𝑞$

𝜇𝜇$DM=

𝑞𝑞$

1−𝑠𝑠 ⋅𝑞𝑞$

𝑞𝑞$= 𝑝𝑝M $⋅𝜇𝜇$DM

INSURANCE

𝑞𝑞

𝑞𝑞 +𝑖𝑖 𝐴𝐴$=

𝜇𝜇

𝜇𝜇 +𝛿𝛿

𝑞𝑞

𝑞𝑞 +𝑖𝑖

𝜇𝜇

𝜇𝜇 +𝛿𝛿 1− 𝐸𝐸G $

G|𝐴𝐴$=

𝑞𝑞

𝑞𝑞 +𝑖𝑖⋅ 𝐸𝐸G $ G|𝐴𝐴$=

𝜇𝜇

𝜇𝜇 +𝛿𝛿⋅ 𝐸𝐸G $

𝑎𝑎G|

𝜔𝜔 −𝑥𝑥 𝐴𝐴 $:G|

𝜔𝜔 −𝑥𝑥 −𝑛𝑛

𝜔𝜔 −𝑥𝑥

G $𝐸𝐸= 𝑣𝑣G⋅

𝜔𝜔 −𝑥𝑥 −𝑛𝑛

𝜔𝜔 −𝑥𝑥