Formulas
1. Present and future values of a cash flow stream (
C
1
, C
2
, ..., C
T
)
PV
=
C
1
1 +
r
+
C
2
(1 +
r
)
2
+
...
C
T
(1 +
r
)
T
=
T
X
t
=1
C
t
(1 +
r
)
t
FV
=
PV
⇥
(1 +
r
)
T
2. Geometric series: (
a
⇥
p, a
⇥
p
2
, a
⇥
p
3
, ...
): sum of first
N
terms:
S
N
=
a
⇥
p
⇥
1

p
N
1

p
. If
p <
1
then sum of the whole series is
S
1
=
a
⇥
p
1

p
3. Annuity present value:
PV
=
C
⇥
1

1
(1+
r
)
T
r
4. Annuity due present value:
PV
=
C
⇥
1

1
(1+
r
)
T
r
⇥
(1 +
r
)
5. Perpetuity present value:
PV
=
C
r
6. Growing annuity present value:
PV
=
C
⇥
1

(
1+
g
1+
r
)
T
r

g
7. Growing perpetuity present value (
r > g
):
PV
=
C
r

g
8. EAR and APR:
EAR
=
✓
1 +
APR
m
◆
m

1
m
!1
!
exp(
APR
)

1
9. Bond value:
P
=
C
⇥
1

1
(1+
r
)
T
r
+
Par
(1 +
r
)
T
10. Stock value after current dividends:
P
0
=
1
X
t
=1
D
t
(1 +
r
)
t
(a) Constant dividends:
P
0
=
D
1
r
(b) Growing dividends:
P
0
=
D
1
r

g
11. Nominal rate, real rate and inflation: (1 +
r
nom.
) = (1 +
r
real
)(1 +
⇡
)
12. NPV=PV of benefits

PV of costs
13. Internal rate of return: discount rate that makes NPV = 0
14. (Discounted) payback period = Number of years that pass before the sum of (discounted) invest
ment's cash flows equals the cost of the investment
15. Profitability index: PV of future cash flows divided by initial investment
1