ST3452: Actuarial Statistics National University of Lesotho
Lecture 4: January 29
Lecturer: T. Khalema
4.1 Level annuities
Definition 4.1 An annuity is a regular series of payments. An annuity certain is an annuity payable for
a definite period of time; the payments do not depend on some factor, such as whether a person is alive or
not.
If the payments are made at the end of each time period, they are paid in arrear. If they are paid at the
beginning of the time period, they are paid in advance. An annuity paid in advance is also known as an
annuity due.
Where the first payment is made during the first time period, this is an immediate annuity. When no
payments are made during the first time period, this is a deferred annuity.
4.1.1 Payments made in arrears
Consider a series of payments of 1 unit each at times 1, 2, . . . , n. One is usually interested in the value of
such a series of payments one unit of time before the first payment is made. This is denoted a n .
The symbol a n represents the present value of an annuity of n payments of 1 unit made at the end of each
of the next n time periods. It is called an annuity paid in arrear.
An expression for this present value is
1 − vn
an = (4.1)
i
since, for i 6= 0 we have
a n =v + v 2 + v 3 + · · · + v n (4.2)
1 − vn
=v (4.3)
1−v
1 − vn
= −1 (4.4)
v −1
1 − vn
= . (4.5)
i
It is common to refer to a n as the present value of an immediate annuity-certain. Also, if a n is calculated
at i% effective then it is common to write a n i% .
Example: Calculate the present value as at 1 March 2005 of a series of payments of M 1, 000 payable on
the first day of each month from April 2005 to December 2005 inclusive, assuming a rate of interest of 6% pa
convertible monthly.
4-1
4-2 Lecture 4: January 29
Solution: An interest rate of 6% pa convertible monthly is equivalent to an effective monthly interest rate
of 0.5%. Additionally, there are 9 payments of M 1, 000 each, starting in one month’s time. So working in
terms of months the present value of the payments is:
1 − 1.005−9
1, 000a 9 0.5% =1000 × (4.6)
0.005
=M 8, 779. (4.7)
4.1.2 Payments made in advance
Consider a series of n payments of 1 unit each made at the start of each time period. The value of this series
of payments at the time the first payment is made is denoted ä n . This symbol is pronounced “A due n”.
One can show that
1 − vn
ä n = . (4.8)
d
To this end, we note from the definition that
ä n =1 + v + v 2 + · · · + v n−1 (4.9)
1 − vn
= (4.10)
1−v
1 − vn
= . (4.11)
d
It is common to refer to ä n as the present value of an annuity-due. Also, if this present value is calculated
at an effective rate of i% then it is written ä n i% .
Exercise 4.2 Calculate a 25 and ä 15 at 13.5% pa effective.
Solution: 7.095 and 7.149
4.1.3 Accumulations of annuities
The value of a series of payments of 1 unit each at the time the last payment is denoted s n . The value of
the corresponding series one unit of time after the last payment is made is denoted s̈ n .
In other words, s n considers the same series of payments as a n but it is the accumulated value at time n, as
opposed to the present value at time 0. Similarly, s̈ n is the accumulated value at time n of the same series
of payments as those in the definition of ä n .
It follows then that
s n = (1 + i)n a n
and
s̈ n = (1 + i)n ä n .
Lecture 4: January 29 4-3
It is more common to express these future values as
(1 + i)n − 1
sn = (4.12)
i
and
(1 + i)n − 1
s̈ n = . (4.13)
d
The derivations are as follows:
s n =(1 + i)n−1 + (1 + i)n−2 + (1 + i)n−3 + . . . + 1 (4.14)
n
=(1 + i) a n (4.15)
n
(1 + i) − 1
= (4.16)
i
and
s̈ n =(1 + i)n + (1 + i)n−1 + (1 + i)n−2 + . . . + (1 + i) (4.17)
n
=(1 + i) ä n (4.18)
(1 + i)n − 1
= . (4.19)
d
Exercise 4.3 Calculate s 10 and s̈ 13 at 3.5% pa effective.
Solution: 11.731 and 16.677
4.1.4 Perpetuities
An annuity that is payable forever is called a perpetuity. For example, consider an equity that pays a
dividend of M 1, 000 at the end of each year. An investor who purchases the equity pays an amount equal
to the present value of the dividends. The present value of the dividends is
1
1000v + 1000v 2 + 1000v 3 + · · · =1000v (4.20)
1−v
1000
= −1 (4.21)
v −1
1000
= (4.22)
i
Recall that
1 − vn
an i = ,
i
hence
1 − vn 1
a ∞ i = lim = .
n→∞ i i
4-4 Lecture 4: January 29
Similarly
1 − vn 1
ä ∞ i = lim = .
n→∞ d d
In summary, the present value of payments of 1 pa payable at the end of each year forever is 1i . This present
value is written
1
a∞ i = . (4.23)
i
Similarly, the present value of payments of 1 pa payable at the start of each year forever is d1 . This present
value is written
1
ä ∞ i = . (4.24)
d
Calculate the present value of payments of M 2, 000 at times 0, 1, 2, . . . using i = 7.6% pa effective.
Solution: M 28, 315.79
