Samenvatting Life Insurance Mathematics
Lecture 1
Tx = future lifetime of (x)
x + Tx = age at death random variable for (x)
Fx = P[Tx < t] = cumulative dist. Function of Tx
Sx = survival function = 1 - Fx
Assumptions Fx
- Fx must be a non decreasing function
- lim(t-> inf) Fx (t) = 1
- lim(t-> - inf) Fx (t) = 0
Assumptions survival function
- Sx (0) = 1
- Survival function must be a non-increasing function of t
- Sx (t) is di erentiable, t>0
- lim (t->inf) Sx (t) = 0
- lim(t->inf) t Sx (t) = 0
- lim(t->inf) t^2 Sx (t) = 0
μx = force of mortality = Hazard rate
For small dx: μx dx = probability that (x) dies before attaining age x + dx
Lecture 2
E[Tx] = ėx E[Kx] = ex
Var (Tx ) = E[Tx ^2] - (ėx)^2
Curtate future lifetime = Kx = Expected # of whole years lived by (x)
ėx is almost always a decreasing function of x
t = representation of # of survivors to age x+t
E[ _t] = x+t
d_x = represents # of deaths at age x
Multiplication rule
As the outstanding loan ↓ => Term insurance sum insured ↓
Lecture 3
Period life tables ( drawback: generations are mixed )
Cohort life tables ( preferred, but drawback: di cult to predict )
Infant mortality, accident hump and aging process
Fractional age assumptions : about the prob. dist. of the future lifetime r.v. between integer ages
Force of mortality is constant between integer values
Underwriting : sign and accept the liability under an insurance policy
==> guaranteeing payment in case loss or damage occurs
= is a way of reducing the impact of adverse selection for life insurance
𝓁𝓁
𝓁 ff ffi
Lecture 1
Tx = future lifetime of (x)
x + Tx = age at death random variable for (x)
Fx = P[Tx < t] = cumulative dist. Function of Tx
Sx = survival function = 1 - Fx
Assumptions Fx
- Fx must be a non decreasing function
- lim(t-> inf) Fx (t) = 1
- lim(t-> - inf) Fx (t) = 0
Assumptions survival function
- Sx (0) = 1
- Survival function must be a non-increasing function of t
- Sx (t) is di erentiable, t>0
- lim (t->inf) Sx (t) = 0
- lim(t->inf) t Sx (t) = 0
- lim(t->inf) t^2 Sx (t) = 0
μx = force of mortality = Hazard rate
For small dx: μx dx = probability that (x) dies before attaining age x + dx
Lecture 2
E[Tx] = ėx E[Kx] = ex
Var (Tx ) = E[Tx ^2] - (ėx)^2
Curtate future lifetime = Kx = Expected # of whole years lived by (x)
ėx is almost always a decreasing function of x
t = representation of # of survivors to age x+t
E[ _t] = x+t
d_x = represents # of deaths at age x
Multiplication rule
As the outstanding loan ↓ => Term insurance sum insured ↓
Lecture 3
Period life tables ( drawback: generations are mixed )
Cohort life tables ( preferred, but drawback: di cult to predict )
Infant mortality, accident hump and aging process
Fractional age assumptions : about the prob. dist. of the future lifetime r.v. between integer ages
Force of mortality is constant between integer values
Underwriting : sign and accept the liability under an insurance policy
==> guaranteeing payment in case loss or damage occurs
= is a way of reducing the impact of adverse selection for life insurance
𝓁𝓁
𝓁 ff ffi
