Lecture 18
CHAPTER 7
Annual Premiums
Of Life Insurance policies
Net Annual Premiums
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat
A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
1
It is possible to buy either a life insurance policy or an annuity by paying a
single premium, but the amount of a single premium is too large for most
people. Instead, individuals usually pay an annual premium to the
insurance company to receive the policy benefits. Under annual premium
plan, premiums are paid in annual installments prior to the data benefits
begin.
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
2
Ordinary Whole Life (Straight Life Policy)
Let Px indicates the net annual premium for a straight life policy issued at age
x. The present value of a whole life annual due to L.E 1 N
is
therefore
the
D
N
present life value of the premiums is P D
x
x
x
x
x
This
.
value will equal the present value of the death benefits provided by the
policy, which are the same as the same as for a single premium whole life of
M x
Dx
Setting the equation of value, in which the present value of the premiums
equal to the present value of the death benefits, we obtain the form of Px as
follows
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
3
Px
Nx
Mx

Dx
Dx
and hence
Dx M x
Px 
.
N x Dx
Px
Mx

Nx
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
4
EXAMPLE page 235
Calculate the net annual premium for a L.E 10000 ordinary life policy
issued at age 45.
SOLUTION
x = 45
n=∞
S = 10000
Ordinary life policy
the net annual premium =?
Let P represents the net annual premium, we have
P  S .
M x
Nx
P  1000
P  1000
M 45
N 45
1541435.3639
 26.16
58927803.0828
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
5
EXAMPLE page235
Find the net annual premium for a LE 70000 whole life insurance policy
issued to an applicant aged 30.
SOLUTION
x = 30
n=∞
S = 70000
Whole life policy
the net annual premium ?
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
6
Let P represents the net annual premium. Then using the fundamental
equation, we obtain
x  S Ax
Pa
30  70000 A30
Pa
Solving for P, we have
70000 A30
P
a30
M 30 D30
 70000
*
D30 N 30
M 30
P  70000
N 30
1706575.67 73
 70000
1035.74
115337741. 8645
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
7
(Limited Life Policy)
Let, tPx denotes the annual premium for a limited payment life policy, with a
premium payment period of t-years. At- payment life policy is one that
provides protection for the lifetime of the insured although premiums are paid
for only t years.
Since life time protection is provided in straight and a limited payment life,
then the present value of the benefits is the same as for ordinary life policy,
The present value for a t-year temporary life annuity due of L.E 1 is
the present value of the premiums will be
t
 N x  N x t
Px 

Dx





M x
Dx
N x  N x t
Dx
Setting the equation of value, in Which the present value of the
premiums equal to the present value benefits, we have
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
8
t
 N x  N x t
Px 

Dx


Mx



Dx

and hence
t
Px
Mx

Dx
t
Px 
t
Px 
Nx
Mx
Dx
.
Dx
 N x t
Nx
Mx
 N x t
.
Nx
Dx
Mx

 N x t
N x  N x t
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
9
EXAMPLE page 237
Find the net annual premium for a L.E 90000
20 pay whole life
insurance policy issued to a man age 40.
SOLUTION
x = 40
n=∞
S = 90000
t = 20
Limited life policy
The net annual premium - Premium are payable for 20 years, but the
insurance coverage is whole life, we have
10
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
x : t  S . Ax
Pa
40 : t  90000 Ax
Pa
N 40  N 60
M 40
P.
 90000 .
D40
D40
Solving for P, we obtain
M 40
P  90000 .
N 40  N 60
1607743.1688
P  90000.
 2775.22
75194899.1714  23056044.9677
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
11
EXAMPLE page 238
Compute the net annual premium for a L.E 30000 paid -up - at age 65
policy issued a 25 year-old man.
SOLUTION
x = 25
n=∞
S = 30000
t = 40
Limited life policy
the net annual premium ?
Premium are payable for 40 years (from age 25 until age 65), but the insurance
coverage is whole life, we find
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
12
x :
P a
t
2 5 :
P a
 S . Ax
40
 30000 . Ax
N 25  N 65
M 25
P
 30000
D2 5
D2 5
Solving for p, we have
M 25
P  30000
N 25  N 65
1754288.5116
 30000
 421.83
139839496.9072  15077832.5953
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
13
Term Insurance
a) Premium payment period coincides with the term insurance period.
(t = n)
1
Let P
x : n denotes the net annual premium for term life policy for n-year issued
.
at age coincides with the term insurance period. The present value for a (t = n)
year temporary life annuity due of L.E 1 is N x  N xn , the present value
of the premiums will be Px1: n
N x  N x t
Dx
Dx
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
14
This value will equal the present value of the death benefits provided by the
policy, which are the same as for a single premium term insurance of M x  M xn
Dx
. Setting
the equation of value, in which the present value of the premiums
equal to the present value of the benefits, we obtain
 N x  N xn
Px1: n 

Dx


M




Dx
M
.
 N xn
Px1: n

Px1: n
M x  M xn

N x  N xn
Nx
x
x
 M
Dx
 M
Dx
xn
xn
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
15
EXAMPLE page 240
Compute the net annual premium for a 5-year, L.E 150000 term policy for
a man aged 50.
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
16
SOLUTION
S = 150000
x = 50
t=n=5
term insurance
the net annual premium ?
Let P represents the net annual premium
Px1: n
M x  M xn
S .
N x  N xn
Px1: n
M 50  M 55
 150000 .
N 50  N 55
1454100 .5090  1338046 .9045
 150000 .
 1459 .72
44904190 .1550  3297858 .8638
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
17
EXAMPLE page 240
Find the net annual premium for a L.E 50000 20-year term insurance
policy issued to a man aged 35.
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
18
x = 35
t = n = 20
term insurance
the net annual premium =?
SOLUTION
S = 50000
1
35: 20  50000 A25
Pa
: 20
P
N 35  N 55
M 35  M 55
 50000
D35
D35
Solving for P, we have
M 35  M 55
P  50000
N 35  N 55
1659440.3563  1338046.9045
 50000
 263.75
93906838.6413  32978578.8638
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
19
b) Premium Payment Period Less Than The Term Insurance Period.(t < n)
Let t P1
x: n
indicate the annual premium payable for t years (t < n) to provide
on n-year term insurance issued at age x. The present value for a t-year
temporary life annuity due of LE 1 is 
N x  N x t 




, the
present value of the premiums will be
Dx
1
P
t x: n


 N x  N x t

Dx


 This value will

Equal the present value of the death benefits provided by the policy,
which are the same as for a single premium term insurance of  M  M
x

xn
Dx




Setting the equation of value, in which the present value of the premiums equal
to the present value of the benefits, we have
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
20
t
1
x: n
P
 N x  N x t


Dx


M x  M xn


Dx

and hence
t
1
x: n
P
Dx
M x  M xn
M x  M xn

.

N x  N x t
Dx
N x  N x t
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
21
EXAMPLE page 242
Find the net annual premium for a 15-year, L.E 120000 term policy issued
to a man aged 40 if
1) premiums are payable for 5 years,
2) premiums are payable to age 50.
SOLUTION
x = 40
n = 15
S = 120000
t=5
For Term insurance, the net single premium =?
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr. Raafat A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
22
Px1: n represent the net annual premium, and using the fundamental
Let t
equation, we obtain
t
Px1:
n
x:
a
5
1
P40
: 15
5
1
P40
: 15
t
40:
a
S
5
A1
x:
n
1
 120000 A40
: 15
N 40  N 45
M
 120000
D 40
 M
D 40
40
55
Solving for
1
P
5 40: 15
M 40  M 55
120000
N 40  N 45
x = 40
n = 15
S = 120000
t = 10
Term insurance the net annual premium = ?
Let P1
represent the net annual premium,
t
x: n
t
Px1:
10
n
S
M x  M xn
N x  N x t
1
P40
 120000
: 15
 120000
M 40  M 55
N 40  N 50
1607743.1688  1338046.9045
23

1068.43
Instructor: Dr. Lobna M Farid, Copy rights for Dr. Ibrahim Morgan and Dr.
Raafat
A. Ibrahim, " Life Insurance", Faculty of Commerce, Cairo University
75194899.1714  44904190.1550
Thank You
24