54 Corporate Finance
Derivation of Future Value of an Annuity (Cash flows occur at the beginning of the period):
FVAn = A(1 + r)n + A(1 + r)n–1 + ··· + A(1 + r) (2E)Divide equation (2E) by (1 + r)(1 )rFVAn
+= A (1 + r)n–1 + A(1 + r)n–2 + ··· + A (2F)Subtracting equation (2F) from (2E), we getFVAn[1 – {1 – (1 + r)}] = A(1 + r)n – A
i.e., r/(1 + r). FVAn= A[(1 + r)n – 1]
FVAn= )1(1)1(
r
rr
An
+
⎥⎥
⎦
⎤
⎢
⎢
⎣
⎡ −+
Note that the FVA of this series is FV of regular annuity multiplied by (1 + r).
Derivation of Present Value of an Annuity (Cash flows occur at the beginning of the period):
PVAn= 2 –1
(1 )···
(1 )
(1 )
(^) n
r
A
rA
rA
A
+
++
+
+
+
+ (2G)
Multiply both sides by (1 + r):(1 )
PVA
rn
+ = rnA
rA
rA
)(1) (1
)(1^2 +
++
+
+
+
··· (2H)
Subtract equation (2G) from (2H):PVAn [1 – {1/(1 + r)}] = n
rA
A
(1 )
–
+
PVAn [r/(1 + r)] =
⎥⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
+
nnrr
A
) 1(1–) 1(
PVAn=^ (1 )
) 1(1–) 1(
r
rrr
A nn
+
⎥⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
+
= Present value of a regular annuity Y (1 + r).