Chapter 8
LIFE ANNUITIES
• Basic Concepts
• Commutation Functions
8.1 Basic Concepts
• We know how to compute present value of
contingent payments
• Life tables are sources of probabilities of
surviving
• We can use data from life tables to compute
present values of payments which are
contingent on either survival or death
Example (pure endowment), p. 155
• Yuanlin is 38 years old. If he reaches
age 65, he will receive a single payment
of 50,000. If i = .12, find an expression
for the value of this payment to Yuanlin
today. Use the following entries in the
life table: l38 = 8327, l65 = 5411
Pure Endowment
• Pure endowment: 1 is paid t years from
now to an individual currently aged x if
the individual survives
• Probability of surviving is t px
• Therefore the present value of this
payment is the net single premium for
the pure endowment =
= t Ex = (t px ) (1 + t) – t = v t t px
Example (life annuity), p. 156
• Aretha is 27 years old. Beginning one year
from today, she will receive 10,000 annually
for as long as she is alive. Find an expression
for the present value of this series of
payments assuming i = .09
• Find numerical value of this expression if
px = .95 for each x
Life annuity
Series of payments of 1 unit
as long as individual is alive
present value
(net single premium)
of annuity
ax
1
age
probability
x
1
x+1 x+2
px
2px
1
…..
x+n
…..
npx

a x  vpx  v 2 2 p x  v 3 3 p x    v n n p x    v t t p x
t 1
Temporary life annuity
Series of n payments of 1 unit
(contingent on survival)
present value
last payment
ax:n|
1
age
probability
x
1
x+1 x+2
px
2px
1
…..
x+n
npx
n
a x:n|  vpx  v 2 2 p x  v 3 3 p x    v n n p x  v t t p x
t 1
n - years deferred life annuity
Series of payments of 1 unit as long as individual is alive
in which the first payment is at x + n + 1
present value
first payment
n|ax
1
age
x
x+1 x+2
…
1
x + n x + n +1 x + n + 2
probability
n+1px
…
n+2px
n2
n 3
n t
n1
|
a

v
p

v
p

v
p



v
n
x
n 1 x
n2 x
n 3 x
n t p x   

  v n t n t p x 
t 1
Note:
n
| ax  ax  ax:n|

s
v
 s px
s  n 1
äx
Life annuities-due
1
x
äx:n|
1
x
1
…..
x+1 x+2
px
1
1
1
px
…
x+n
2px
ax  1  a x  1   v t t p x
t 1
npx
1
x+1 x+2

1
n 1
….. x + n-1
ax:n|  1  a x:n 1|  1   v t t p x
t 1
x+n
n-1px
2px
n|äx
1
x
x+1 x+2
…
1
1
x + n x + n +1 x + n + 2
npx
n+1px
n+2px
…
n
| ax  n1 | ax
Note
ax:n|  1  ax:n1|
but
ax:n|  (1  i)ax:n|
ax1:n|  vpx1ax:n|
8.2 Commutation Functions
• Recall: present value of a pure endowment
of 1 to be paid n years hence to a life
currently aged x
xn


v lxn
n
n lxn
n Ex  v n px  v 
 x
v lx
 lx 
= v xlx
• Then nEx = Dx+n / Dx
• Denote Dx
Life annuity and commutation function


a x  v t p x   t E x
t
t 1
t 1
Since nEx =
we have
Dx+n / Dx

Dx t
1
Dx1  Dx2  Dx3  
ax  

Dx
t 1 Dx
Define commutation
Nx 
function Nx as follows:
Then:


t 0
t 0
x t
D

v
 x t  l x t
N x 1
ax 
Dx
Note:
Dx  N x 1  N x
Identities for other types of life annuities
temporary life annuity
Dx t N x 1  N x  n1


Dx
t 1 Dx
n
ax:n|
n-years delayed life annuity
N x  n 1
n | ax 
Dx
temporary life annuity-due
ax:n|
N x  N xn

Dx
Accumulated values of life annuities
temporary life annuity ax:n| n Ex  sx:n|
since a  N x 1  N x  n 1
x:n|
Dx
we have
s x:n|
Dx  n
and n E x 
Dx
N x 1  N x  n 1

Dx  n
similarly for temporary life annuity-due:
ax:n| n Ex  sx:n|
and
sx:n|
N x  N xn

Dx  n
Examples (p. 162 – p. 164)
• (life annuities and commutation functions) Marvin, aged
38, purchases a life annuity of 1000 per year. From
tables, we learn that N38 = 5600 and N39 = 5350. Find
the net single premium Marvin should pay for this
annuity
– if the first 1000 payment occurs in one year
– if the first 1000 payment occurs now
• Stay verbally the meaning of (N35 – N55) / D20
• (unknown rate of interest) Given Nx = 5000, Nx+1=4900,
Nx+2 = 4810 and qx = .005, find i
Select group
• Select group of population is a group with the
probability of survival different from the
probability given in the standard life tables
• Such groups can have higher than average
probability of survival (e.g. due to excellent
health) or, conversely, higher mortality rate
(e.g. due to dangerous working conditions)
Notations
• Suppose that a person aged x is
in the first year of being in the select group
• Then p[x] denotes the probability of survival for 1 year
and q[x] = 1 – p[x] denotes the probability of dying during
1 year for such a person
• If the person stays within this group for subsequent
years, the corresponding probabilities of survival for 1
more year are denoted by p[x]+1, p[x]+2, and so on
• Similar notations are used for life annuities:
a[x] denotes the net single premium for a life annuity of
1 (with the first payment in one year) to a person aged
x in his first year as a member of the select group
• A life table which involves a select group is called a
select-and-ultimate table
Examples (p. 165 – p. 166)
• (select group) Margaret, aged 65, purchases a life annuity
which will provide annual payments of 1000 commencing at
age 66. For the next year only, Margaret’s probability of
survival is higher than that predicted by the life tables and, in
fact, is equal to p65 + .05, where p65 is taken from the
standard life table. Based on that standard life table, we have
the values D65 = 300, D66 = 260 and N67 = 1450. If i = .09,
find the net single premium for this annuity
• (select-and-ultimate table) A select-and-ultimate table has a
select period of two years. Select probabilities are related to
ultimate probabilities by the relationships p[x] = (11/10) px and
p[x]+1 = (21/20) px+1. An ultimate table shows D60 = 1900,
D61 = 1500, and ä 60:20| = 11, when i = .08. Find the select
temporary life annuity ä[60]:20|
• The following values are based on a unisex life table:
N38 = 5600, N39 = 5350, N40 = 5105, N41 = 4865,
N42 = 4625.
It is assumed that this table needs to be set forward
one year for males and set back two years for
females. If Michael and Brenda are both age 40, find
the net single premium that each should pay for a life
annuity of 1000 per year, if the first payment occurs
immediately.