Time Value of Money 53
Future Value, FVAn= A(1 + r)n–1 + A(1 + r)n–2 + ··· + A (2A)
Multiply both sides of the equation by (1 + r):
FVAn(1 + r)= A(1 + r)n + A(1 + r)n–1 + ··· + A(1 + r) (2B)
Subtract equation (2A) from (2B):
FVAn. r= A(1 + r)n – A
FVAn. r= A[(1 + r)n – 1]
FVAn. r= ⎥
⎦
⎤
⎢
⎣
⎡ −+
r
rn 1)1(
A
Derivation of Present Value of an Annuity (Cash flows occur at the end of the period):
PVAn = n
r
A
r
A
r
A
(^2) + )1()1()1(
++
+
+
+
··· (2C)
Multiply both sides of the equation by (1 + r):
PVAn(1 + r) = 1
)1()1(
(^) −
++
+
+ n
r
A
r
A
A ··· (2D)
Subtract equation (2C) from (2D):
PVAn. r= n
r
A
A
)1(
+
−
PVAn. r= ⎥
⎦
⎤
⎢
⎣
⎡
+
− n
r)1(
1
A 1
PVAn. r= ⎥
⎦
⎤
⎢
⎣
⎡
+
−+
n
n
r
r
A
)1(
1)1(
PVAn. r= ⎥
⎦
⎤
⎢
⎣
⎡
+
−+
r.)r1(
1)r1(
A n
n