University : Université Française d’Égypte
Faculty : Management
Course title : Financial Mathematics
Year / level : 2nd year
Lecture 7
The Annuities
Dr. Awad Talal
Definition: An annuity is any finite sequence of payments made at fixed periods of time
over a given interval (which is called term of the annuity).
The fixed periods of time that we consider will always be equal length (which call the
payment period).
- The payments are always equal values.
Ordinary annuity
- Suppose that n payments (each of amount R) occur at time 1,2,…,n , we can
consider that an ordinary annuity’s payments are at the end of each payment period
(like a worker’s salary ).
Annuity due
- Suppose that n payments (each of amount R) occur at time
0,1, 2,…n-1. , we can consider that an annuity due’s payments are at the beginning
of each payment period. (like a rent)
Dr. Awad Talal
Future value of an annuity
Definition
If R represent the deposit made at each payment period for an annuity at i interest
per payment period. The amount A of the annuity after N payment periods (the
future value) is
(𝟏 + 𝒊)𝑵 − 𝟏
𝑨 = 𝑹.
𝒊
Example
Find the amount of an annuity if a deposit of 100$ per year is made for 5 years at 10 %
compounded annually. How much interest is earned?
Solution
R=100 number of payment periods N=5 years i=0.1
(𝟏 + 𝒊)𝑵 − 𝟏 (𝟏 + 𝟎. 𝟏)𝟓 − 𝟏
𝑨 = 𝑹. = 𝟏𝟎𝟎 = 𝟔𝟏𝟎. 𝟓𝟏
𝒊 𝟎. 𝟏
The interest after 5 years is I=610.51-(5)(100)=110.51 $
Example
Find the future value of an annuity consisting of a payment of 50$ every 3 months for 3
years at the rate of 6 % compounded quarterly. How much interest is earned?
(𝟏 + 𝒊)𝑵 − 𝟏 (𝟏 + 𝟎. 𝟎𝟔/𝟒)(𝟒)(𝟑) − 𝟏
𝑨 = 𝑹. = 𝟓𝟎 = 𝟔𝟓𝟐. 𝟎𝟔
𝒊 𝟎. 𝟎𝟔/𝟒
The interest after 3 years is I=652.06-[(𝟑)(𝟒)](50)=52.06 $
Remark
(𝟏+𝒊)𝑵 −𝟏
The expression in business mathematics, we replace it by the formula
𝒊
(𝟏+𝒊)𝑵 −𝟏
𝒂 𝑵 ⌉𝒊 = and 𝒂𝑵 ⌉𝒊 𝒊𝒔 𝒔𝒐𝒎𝒕𝒊𝒎𝒆𝒔 𝒓𝒆𝒂𝒅 𝒂 𝒂𝒏𝒈𝒍𝒆 𝑵 𝒂𝒕 𝒓 and we can find its
𝒊
value directly from the table of annuity
Then 𝑨 = 𝑹 𝒂 𝑵 ⌉𝒊
Dr. Awad Talal
Present value of an annuity
Definition
The Present value of an annuity is the sum of the present value of all n payments.
It represents the amount that must be invested now to purchase all n of them
We consider the case of an ordinary annuity ant let P the present value
The formula of the present value of an annuity
𝟏 − (𝟏 + 𝒊)−𝑵
𝑷 = 𝑹.
𝒊
Where: R is the annuity per payment period for N period at the interest rate of i per
period.
Remark:
𝟏−(𝟏+𝒊)−𝑵
The expression in business mathematics, we replace it by the formula
𝒊
𝟏−(𝟏+𝒊)−𝑵
𝒑 𝑵 ⌉𝒊 = and 𝒑𝑵 ⌉𝒊 𝒊𝒔 𝒔𝒐𝒎𝒕𝒊𝒎𝒆𝒔 𝒓𝒆𝒂𝒅 𝒂 𝒂𝒏𝒈𝒍𝒆 𝑵 𝒂𝒕 𝒓 and we can find its
𝒊
value directly from the table of annuity
Then 𝑷 = 𝑹 𝒑 𝑵 ⌉𝒊
Example
Find the present value of an annuity of 100 $ per month for 3.5 years at an interest
rate of 6% compounded monthly.
Solution
R=100 i=r/n =0.06/12=0.005 N=(3.5)(12)=42
𝟏 − (𝟏 + 𝒊)−𝑵 𝟏 − (𝟏 + 𝟎. 𝟎𝟎𝟓)−𝟒𝟐
𝑷 = 𝑹. = 𝟏𝟎𝟎. = 𝟑𝟕𝟕𝟗. 𝟖𝟑 $
𝒊 𝟎. 𝟎𝟎𝟓
Example
Ahmed agrees to pay 300$ per month for 48 months to pay off a car loan if interest
of 12% per year is charged monthly. How interest was paid?
Solution
R=300 i=r/n =0.12/12=0.01 N =48
𝟏 − (𝟏 + 𝒊)−𝑵 𝟏 − (𝟏 + 𝟎. 𝟎𝟏)−𝟒𝟖
𝑷 = 𝑹. = 𝟑𝟎𝟎. = 𝟏𝟏, 𝟑𝟗𝟐. 𝟏𝟗 $
𝒊 𝟎. 𝟎𝟏
I = (300)(48)-(11392.19)=3007.81 $
Dr. Awad Talal
