Corporate Finance

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