# Time Value of Money formula

**Topics:**Time value of money, Annual percentage rate, Compound interest

**Pages:**9 (385 words)

**Published:**October 31, 2013

#

TVM Formula For:

1

Future Value of a

Lump Sum. (FVIFi,n)

Compounded/Payments

(m) Times per Year

Annual Compounding

FVn = PV( 1 + i )n

2

FV

1 i

PV =

Present Value of a

Lump Sum. (PVIFi,n)

-n

Future Value of an

Annuity. (FVIFAi,n)

FVAn = CF

4

Present Value of an

Annuity. (PVIFAi,n)

1 - ( 1 + i )-n

PVAn = CF

i

5

Present Value of

Perpetuity. (PVA )

6

Effective Annual Rate

given the APR.

7

The length of time

required for a PV to

grow to a FV.

8

The APR required for

a PV to grow to a FV.

9

Present Value of

a Growing Annuity.

10

Present Value of

a Growing

Perpetuity.

11

The length of time

required for a series

of PMT’s to grow to a

future amount (FVAn).

12

EAR = APR

n=

(-m n)

m

-1

ln ( FV/PV)

m ln (1 i/m)

FV

PV

i= m

EAR = ei - 1

n=

ln (FV/PV)

i

i=

ln (FV/PV)

n

[1/(m n)]

-1

n

(FVA)(i)

+1

CF

ln (1 + i)

ln 1

n

n=

(1/n)

1 g

1 i

(-i n)

CF0 1 g

i g

PVA

ln

i

EAR = 1

m

-1

PV = FV e

CF

[(1 i/m) m 1]

PVA

i

CF0 1 g

1

i g

PVAn

1 - 1 + i/m

PVAn = CF

i/m

ln (FV/PV)

ln (1 + i )

FV

i=

PV

(-m n)

( 1 + i/m )(m n) - 1

i/m

FVAn = CF

CF

i

FV

(i n)

e

or

PV = FV 1 i/m

( 1 + i )n - 1

i

3

n=

PV =

(m n)

(i n)

FVn = PV e

or

PV = FV( 1 + i )

The length of time

required for a series

of PMT’s to exhaust a

specific present

amount (PVAn).

FV

1 i/m

PV =

n

or

PVA

(m n)

FVn = PV 1 i/m

Continuous

Compounding

i

m

ln

n=

(PVA)(i)

CF

,

ln (1 i)

m

ln 1

n

for PVA(i) < CF

m

FVA m

+

CF

i

ln (1 i/m)

(PVA)(i/m)

CF

,

ln(1 i/m)

for PVA(i/m) < CF

Legend

i = APR, the nominal or Annual Percentage Rate

m = the number of compounding periods per year

ln = the natural logarithm, the logarithm to the base e

CF = PMT = the periodic payment or cash flow

n = the number of periods

EAR = the Effective Annual Rate

e = the base of the natural logarithm 2.71828…

(Calculator: 1

ex )

Perpetuity = an infinite annuity

Prepared by Jim Keys

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