Stat 353 Problem Set 2
1) You are given that
a) Deaths follow the constant force assumption over each year of age,
b) qx = 0.06,
c) qx+1 = 0.07,
d) A1 = 0.054
x:1
Find 2 A1 .
x:2
2) Assume that the mortality follows the DeMoivres Law where w = 100. Let Z
Stat 353 Problem Set 2 Solution
1) You are given that
a) Deaths follow the constant force assumption over each year of age,
b) qx = 0.06,
c) qx+1 = 0.07,
d) A1 = 0.054
x:1
2
Find A1 .
x:2
2) Assume that the mortality follows the DeMoivres Law where w = 10
Chapter 5: Life Annuity
5.2 Continuous Life Annuity
Here we assume the payment is a unit benefit of $1 per year.
Chapter 5: Life Annuity
In general: the APV for a continuous annuity is:
0
v t P[payments are being made at time t] [payment rate at time t]dt
Chapter 4: Life Insurance
4.1 Insurance Payable at the Moment of Death
Lets consider insurances that are payable at the moment of
Chapter 4: Life Insurance
affects when and whether the benefit is paid.
Recall: The density of T(x) is:
death (sometimes refe
Stat 353 Practice Midterm Exam
Stat 353 Practice Midterm Exam
1) The survival function S(x) = e -x/100 for x > 0. Find the
k) a(20),
instantaneous death rate (x). If K(x) is the curtate further
lifetime of a life aged x, find P(K(x) = k) and give the rang
Stat 353 Practice Problem Set #1
20000 100 x x 2
for x 0
20000
Find w (the oldest age that a life can live)
Calculate the probability of surviving to age 20.
Calculate the probability that a life aged 20 will survive to age 40.
Find the probability that a
Chapter 3:
Survival Distributions and Life Tables
Ch 3 Formula Summary
s(x) = 1 FX(x) = 1 P(X x) = P(X > x)
P(x < X z) = FX(z) - FX(x) = s(x) s(z)
CDF for T(x):
tqx
= P(T(x) t) = FT(x)(t) = P(X x t | X > x)
= P( X > x X x + t ) = P( x < X x + t )
P( X > x
Chapter 3: Survival Distributions and Life Tables
3.1 Survival Function
Chapter 3: Survival Distributions and Life Tables
Survival Function:
Let s(x) be a survival function of X:
Let X be the age-at-death of a life (ie. Newborns age at
s(x) = 1 FX (x) =
Chapter 7: Benefit Reserves
Fully Continuous Benefit Reserves
A whole life policy on a fully continuous basis with an
annual continuous benefit premium rate of P (Ax ) .
t
L = vT ( x ) t P ( Ax )aT ( x ) t
1)
Prospective Formula
V (Ax ) = Ax +t P ( Ax )a
Chapter 6: Benefit Premiums
6.1
Introduction to Benefit Premiums
Equivalent principle:
E [L ] = E [PV of Benefits ] E [PV of Premiums ] = 0
Fully Continuous Premiums
whole life insurance
L = 1 v T P aT .
P ( Ax ) =
Ax 1 ax
A
Ax
Ax
=
P ( Ax ) = x = 1 A =
Chapter 5: Life Annuity Summary
Continuous Life Annuity
Cumulative Distribution of Y
1 e T
ln(1 y )
FY ( y ) = P[Y y ] = P
y = FT
Density Function of Y
1 ln(1 y )
fY ( y ) =
fT
1 y
0< y<
Whole Life Annuity
ax = E (aT ) = at t px ( x + t )dt
Stat 353 Fall 2006
Assignment 2 Solution
Due: Friday, October 20, 2006 by 3pm
1) You are given that
a) Deaths follow the constant force assumption over each year of age,
b) qx = 0.06,
c) qx+1 = 0.07,
d) A1 = 0.054
x:1
Find 2 A1 .
x:2
2) Assume that the mo
Chapter 3:
Survival Distributions and Life Tables
Ch 3 Formula Summary
s(x) = 1 FX(x) = 1 P(X x) = P(X > x)
P(x < X z) = FX(z) - FX(x) = s(x) s(z)
CDF for T(x):
tqx
= P(T(x) t) = FT(x)(t) = P(X x t | X > x)
= P( X > x X x + t ) = P( x < X x + t )
P( X > x
Stat 353 Fall 2006
Assignment 1 Solution
Due: Friday, October 6, 2006 by 3pm
20000 100 x x 2
for x 0
20000
a) Find w (the oldest age that a life can live)
20000 100 w w 2
= 0 w = 200 or
Since s ( w) = 0, we have s ( w) =
20000
Therefore, w = 100 as w = 20
Chapter 4: Life Insurance Formula Summary
Continuous Insurance
n-year term life insurance
Ax : n | = E[Z ] = vt t px ( x + t )dt
E[ Z 2 ] = v2 t t p x ( x + t ) dt = Ax : n |
n
1
n
0
2
2
1
0
( )
Var[Z ]= Ax :n| Ax: n|
1
1
2
Whole life insurance
A x = E [Z