TIME VALUE OF MONEY FORMULA SHEET
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TVM Formula For:
Annual Compounding
Compounded (m) Times
per Year
Continuous
Compounding
1
Future Value of a
Lump Sum. (FVIFk,n)
FV = PV( 1 + k )n
FV = PV  1  k/m  nm
FV = PV(e )kn
2
Present Value of a
Lump Sum. (PVIFk,n)
PV = FV( 1 + k )-n
PV = FV( 1 + k/m )-nm
PV = FV(e )-kn
3
Future Value of an
Annuity. (FVIFAk,n)
4
Present Value of an
Annuity. (PVIFAk,n)
 ( 1 + k )n - 1
FVA = PMT 

k


-n
1 - ( 1 + k ) 
PVA = PMT 

k


 ( 1 + k/m ) nm - 1
FVA = PMT 

k/m


-nm
1 -  1 + k/m  
PVA = PMT 

k/m


5
Present Value of
Perpetuity. (PVp)
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.
 FV 
k=
 -1
 PV 
9
Present Value of a
Growing Annuity.
n
CF0 1  g    1  g  
PV 
 
1  
k  g    1  k  
10
The length of time
required for a series
of PMT’s to grow to
a future amount
(FVAn).
 (FVA)(k) 
ln 
+ 1
PMT


n=
ln (1 + k)
 k  FVA m 
ln  
+ 
 m  PMT k 

n=
m * ln (1  k/m) 
11
The length of time
required for a series
of PMT’s to exhaust
a specific present
amount (PVAn).
 (PVA)(k) 
ln 1 
PMT 
,
n 
ln (1  k)
 (PVA)(k/m) 
ln 1 
PMT 
,
n 
m * ln(1  k/m) 
for PVA(k) < PMT
for PVA(k/m) < PMT
PVperpetuity 
PMT
k
EAR = APR
n=
ln (FV/PV)
ln (1 + k )
1/n
Legend
k = the nominal or Annual Percentage Rate
m = the number of compounding periods per year
ln = the natural logarithm, the logarithm to the base e
PMT = the periodic payment or cash flow
PVperpetuity 
PMT
[(1  k)1/m  1]
EAR = (1  k/m) m - 1
n=
ln ( FV/PV)
m * ln (1  k/m)
 FV 1/(nm) 
k = m * 
- 1

PV




EAR = ek - 1
n=
ln (FV/PV)
k
k=
ln (FV/PV)
n
n = the number of periods
EAR = the Effective Annual Rate
e = the base of the natural logarithm ≈ 2.71828
Perpetuity = an infinite annuity
Prepared by Jim Keys