TIME VALUE OF MONEY FORMULA SHEET
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TVM Formula For:
Annual Compounding
Compounded/Payments
(m) Times per Year
Continuous
Compounding
1
Future Value of a
Lump Sum. (FVIFi,n)
FV = PV( 1 + i )n
FV = PV ( 1 + i/m ) nm
FV = PV(e )in
2
Present Value of a
Lump Sum. (PVIFi,n)
PV = FV( 1 + i )-n
PV = FV( 1 + i/m )-nm
PV = FV(e )-in
3
Future Value of an
Annuity. (FVIFAi,n)
4
Present Value of an
Annuity. (PVIFAi,n)
5
Present Value of
Perpetuity. (PVp)
⎡ ( 1 + i )n - 1⎤
FVA = PMT ⎢
⎥
i
⎣
⎦
-n
⎡1 - ( 1 + i ) ⎤
PVA = PMT ⎢
⎥
i
⎣
⎦
PMT
PVperpetuity =
i
⎡ ( 1 + i/m )nm - 1⎤
FVA = PMT ⎢
⎥
i/m
⎣
⎦
-nm
⎡1 - ( 1 + i/m ) ⎤
PVA = PMT ⎢
⎥
i/m
⎣
⎦
PMT
PVperpetuity =
[(1 + i)1/m − 1]
6
Effective Annual
Rate given the APR.
EAR = APR
EAR = (1 + i/m)m - 1
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
The length of time
required for a series
of PMT’s to grow to
a future amount
(FVAn).
11
The length of time
required for a series
of PMT’s to exhaust
a specific present
amount (PVAn).
n=
ln (FV/PV)
ln (1 + i )
1/n
⎛ FV ⎞
i=⎜
⎟ -1
⎝ PV ⎠
PV =
n=
ln ( FV/PV)
m * ln (1 + i/m)
⎡⎛ FV ⎞1/(nm) ⎤
i = m * ⎢⎜
- 1⎥
⎟
⎢⎣⎝ PV ⎠
⎥⎦
EAR = ei - 1
n=
ln (FV/PV)
i
i=
ln (FV/PV)
n
n
CF0 (1 + g ) ⎡ ⎛ 1 + g ⎞ ⎤
−
1
⎜
⎟
(i − g ) ⎢⎣⎢ ⎝ 1 + i ⎠ ⎥⎦⎥
⎡ (FVA)(i) ⎤
ln ⎢
+ 1⎥
PMT
⎣
⎦
n=
ln (1 + i)
⎡⎛ i ⎞⎛ FVA m ⎞⎤
ln ⎢⎜ ⎟⎜
+ ⎟⎥
⎝ m ⎠⎝ PMT i ⎠⎦
⎣
n=
m * [ln (1 + i/m)]
⎡ (PVA)(i) ⎤
ln ⎢1 −
PMT ⎥⎦
n=− ⎣
,
ln (1 + i)
⎡ (PVA)(i/m) ⎤
ln ⎢1 −
PMT ⎥⎦
n=− ⎣
,
m * [ln(1 + i/m)]
for PVA(i) < PMT
for PVA(i/m) < PMT
Legend
i = 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
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