Summary Formularium- Insurance: financial and actuarial methods (2113TEWFIN)

FORMULARIUM

CHAPTER 1: Basic concepts in life insurance

1. Survival Models
1.1 Future lifetime and survival function

 Lifetime distribution : F x (t )=P (T x ≤ t )

 Survival distribution : S x ( t )=P ( T x >t ) =1−F x ( t )

 F x (t ) + S x ( t )=1

F 0 ( x +t )−F 0 ( x)
 F x (t )=
S 0 ( x)

 S0 ( x+ t ) =S 0 ( x )∗S x (t )  S x ( t + k )=S x ( t )∗S x+ t (k )


1.2 Basic actuarial probabilities
 probability that a life aged 𝑥 survives at least 𝑡 years :
𝑡𝑝𝑥 = P ( T x >t ) =¿ S x ( t )


 probability that a life aged 𝑥 does not survive beyond age 𝑥 + t:
𝑞
𝑡 𝑥= P ( T x ≤ t )=1−S x ( t ) =F x ( t )





 Mortality rate: q x =P ( T x ≤1 )



1.3 Force of mortality μ x
−1 d
 μ x= S (x)
S 0 ( x ) dx 0


−d
 μ x+t = ln S x (t )
dt




1

, t t


 Probability of life: ∫ μ x+ s ds=−¿ ln( 𝑡𝑝𝑥 )  : 𝑡𝑝𝑥 = −∫ μx+ s ds

0
e 0




1.4 Life expectancy
 Complete life expectancy :


∞ ∞
 Curtate life expectancy = e x =∑ ❑ or e x =∑ ❑
t =1 t =1









1.5 Laws of mortality
 Law of Makeham: μ x =A + B∗C x met 0 < A < 1, 0 < B < 1, and c > 1




2. Life tables
2.1 Actuarial probabilities
 l x +n=l x∗¿ n𝑝𝑥

 d x =l x∗q x


 n- year probabilities
o Probability that a life aged x survives at least n years

l x+n
n𝑝𝑥 =
lx


o Probability that a life aged x dies within a period of n years

l x −l x+n
nq𝑥 =
lx


 Deferred probabilities
o Probability that a life aged x dies within one year at the age of x + m

m|q𝑥 = m𝑝𝑥 * q𝑥+m

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