# TVM Formula

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```					N
u
Time Value of
m                                                                 Compounded (m) Times per                         Continuous
Money Formula           Annual Compounding
b                                                                         Year                                    Compounding
For:
e
r
nm
Future Value of a                                                               i 
1                                 F V = P V (1+ i )    n
FV = PV 1 +                               FV = PV( e )in
Lump Sum. ( FVIFi,n )
 m
- nm
Present Value of a                                                                     i 
2                                 PV = FV ( 1 + i )-n                 PV = FV 1 +                               PV = FV( e )-in
Lump Sum. ( PVIFi,n )
 m
Future Value of an                      ( 1 + i )n - 1                      1  (i / m) nm  1
3                            FVA = PMT                          FVA  PMT                                
Annuity. ( FVIFAi,n )
        i                                    i/m             
Present Value of an                    1 - ( 1 + i ) 
-n
1 -  1 + (i / m)  - nm

4                            PVA = PMT                         PVA = PMT                                  
Annuity. ( PVIFAi,n )
       i                                     i /m             
Present Value of a                         PMT                                       PMT
5                               PVperpetuity                       PV perpetuity 
Perpetuity.                                   i                                 [(1  i )1/ m  1]
m
Effective Annual                                                              i 
6
Rate given the APR.             EAR = APR                           EAR =  1 +  - 1                            EAR = e i - 1
    m

The length of time                                                         ln ( FV/PV)
ln (FV/PV)                         n=                                           1
7    required for a PV to          n=
m * ln  1               
i                     n=     * ln ( FV/PV)
grow to a FV.                     ln (1 + i )                                                                    i
m

The APR required for
 FV 
1/ n                            FV 1/( nm )                       1
8    a PV to grow to a               i=     -1                       i = m *             - 1               i=     * ln (FV/PV)
FV.                                PV                                    PV 
                 
                    n

The length of time                                                     i  FVA m 
required for a series             (FVA)( i )                     ln         + 
ln               + 1                   m  PMT i 
9    of PMT’s to grow to               PMT                         n=
a future amount
n=                                                   i 
ln (1 + i )                       m * ln 1 + 
(FVA).                                                                       m 

 (PVA )(i / m) 
 (PVA )(i )                      ln 1             
The length of time           ln 1                             n                  ,
PMT
required for a series                   PMT 
10   of PMT’s to exhaust       n                 ,                              i 
m * ln 1  
ln (1  i )
a specific present                                                           m 
amount (PVA).
for PVA(i) < PMT
for PVA(i/m) < PMT

Legend
i = the nominal or Annual Percentage Rate                                   n = the number of periods
m = the number of compounding periods per year                                EAR = the Effective Annual Rate
ln = the natural logarithm, the logarithm to the base e                 e = the base of the natural logarithm ≈ 2.71828
PMT = the periodic payment or cash flow                                 Perpetuity = an infinite annuity

Prepared by Jim Keys

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