accident insurance. In this example, the start of payment is not known and theamount of payment is dependent to which event.Annuity certain can be classified into two, simple annuity andgeneralannuity. In simple annuity, the payment period is the same as the interestperiod, which means that if the payment is made monthly the conversion ofmoney also occurs monthly. In general annuity, the payment period is not thesame as the interest period. There are many situations where the payment forexample is made quarterly but the money compounds in another period, saymonthly. To deal with general annuity, we can convert it to simple annuity bymaking the payment period the same as the compounding periods by theconcept of effective rates.Types of AnnuitiesIn engineering economy, annuities are classified into four categories.These are (1) ordinary annuity, (2) annuity due, (3) deferred annuity, and (4)
perpetuity.
An annuity is a series of equal payments occurring at equal periods of
time.
Symbol and Their Meaning
P = Value or sum of money at present
F = Value or sum of money at some future time
A = A series I periodic, equal amounts of money
n = Number of interest periods
I = Interest rate per interest period
Ordinary Annuity
An ordinary annuity is a series of uniform cash flows where the first
amount of the series occurs at the end of the first period and every succeeding
cash flow occurs at te end of each period.
An ordinary annuity is one where the payments are made at the end of
each period.
Characteristics of ordinary annuity:
a.) P (present equivalent value)

Occurs one interest period before the first A (uniform amount)
b.) F (future equivalent value)

Occurs at the same time as the last A and n intervals after P
c.)
A (annual equivalent vale)

Occurs at the end of each period
Finding P when A is given
P
0
1
2
3
n1
n
A
AAAA
A(P/F,i%,1)
A(P/F,i%,2)
A(P/F,i%,3)
A(P/F,i%,n1)
A(P/F,i%,n)
Cash flow diagram to find P given A
P = A (P/F, i%, 1) + A (P/F, i%, 2) + (P/F, i%, 3) +……. + A (P/F, i%, n1) + A (P/F,
1%, n)
P = A (1+i)
1
+ A (1+i)
2
+ …… + A (1+i)
(n1)
+ A (1+i)
n
Multiplying this equation by (1+i) result in
P + Pi = A + A (1+i)
1
+ A (1+i)
2
+ …… + A (1+i)
n+2
+ A (1+i)
n+1
Subtracting the first equation from the second gives
P + Pi = A + A (1+i)
1
+ A (1+i)
2
+…………… + A (1+i)
n+1
P
=
 A (1+i)
1
 A (1+i)
2
+…………… + A (1+i)
n+1
– A (1+i)
n
Pi
= A
– A (1+i)
n
Solving For P gives
P = A
[
1
−
(
1
+
i
)
−
1
i
]
= A
1
+
i
¿
n
i
¿
(
1
+
i
)
n
−
1
¿
¿
The quantity in brackets is called the “ uniform series present worth factor”
and is designated by the functional symbol P/A, i%, n, read as “P given A at I
percent in the interest periods. “ Hence Equation can be expressed as
P = A (P/A, i%, n)
Finding F when A is Given
F
0
1
2
3
n1
n
A
AAAA
A
(F/P,i%,1)
A(F/P,i%,n3)
A(F/P,i%,n2)
A(F/P,i%,n1)
Cash flow diagram to find F given A
F = A + A (F/A, i%, 1) + ……. + A (F/P, i%, n3) + A (F/P, 1%, n2) + A(F/P, i%, n
1)
F = A + A (1+i) + ……… + A (1+i)
(n3)
+ A (1+i)
n2
+ A (1+i)
n1
Multiplying this equation by (1+i) results in
F + Fi = A + A (1+i) + A (1+i)
2
+ …… + A (1+i)
n2
+ A (1+i)
n1
+ A (1+i)
n
Subtracting the First equation from the second gives
F + Fi =
A (1+i) + ………. + A (1+i)
n2
+ A (1+i)
n1