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
2
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
2
