Stock duration of an equity stock is the average of the times until its dividends are received, weighted by their present values.
As per Dividend Discount Model: Formula for the duration of stock is as follows-
M
a
c
D
d
d
m
=
1
+
r
r
−
g
where
M
a
c
D
d
d
m
is the Macaulay duration of stock under the DDM model
r
is the discount rate
g
is the expected growth rate in perpetuity
The modified duration is the percentage change in price in response to a 1% change in the long-term return that the stock is priced to deliver. Per the relationship between Macaulay duration and Modified duration:
M
o
d
D
d
d
m
=
1
r
−
g
The other formula for the same is - D = saa
The Macaulay duration is defined as:
(
1
)
M
a
c
D
=
∑
i
t
i
P
V
i
V
where:
i
indexes the cash flows,
P
V
i
is the present value of the
i
th cash payment from an asset,
t
i
is the time in years until the
i
th payment will be received,
V
is the present value of all future cash payments from the asset.
The present value of dividends per the Dividend Discount Model is:
(
2
)
V
=
∑
t
=
1
∞
D
0
(
1
+
g
)
t
(
1
+
r
)
t
=
D
0
(
1
+
g
)
r
−
g
The numerator in the Macaulay duration formula becomes:
(
3
)
∑
i
t
i
P
V
i
=
∑
t
=
1
∞
t
D
0
(
1
+
g
)
t
(
1
+
r
)
t
=
D
0
(
1
+
g
)
(
1
+
r
)
+
2
D
0
(
1
+
g
)
2
(
1
+
r
)
2
+
3
D
0
(
1
+
g
)
3
(
1
+
r
)
3
+
.
.
.
Multiplying by
1
+
r
1
+
g
:
(
4
)
1
+
r
1
+
g
∑
i
t
i
P
V
i
=
D
0
+
2
D
0
(
1
+
g
)
(
1
+
r
)
+
3
D
0
(
1
+
g
)
2
(
1
+
r
)
2
+
.
.
.
Subtracting
(
4
)
−
(
3
)
:
1
+
r
1
+
g
∑
i
t
i
P
V
i
−
∑
i
t
i
P
V
i
=
D
0
+
D
0
(
1
+
g
)
(
1
+
r
)
+
D
0
(
1
+
g
)
2
(
1
+
r
)
2
+
.
.
.
Applying the Dividend Discount Model to the right side:
(
1
+
r
1
+
g
−
1
)
∑
i
t
i
P
V
i
=
D
0
+
D
0
(
1
+
g
)
r
−
g
=
D
0
+
V
Simplifying:
r
−
g
1
+
g
∑
i
t
i
P
V
i
=
D
0
+
V
(
5
)
∑
i
t
i
P
V
i
=
(
D
0
+
V
)
1
+
g
r
−
g
Combining (1), (2) and (5):
M
a
c
D
=
∑
i
=
1
n
t
i
P
V
i
V
=
(
D
0
+
V
)
1
+
g
r
−
g
D
0
1
+
g
r
−
g
=
D
0
+
V
D
0
=
D
0
+
D
0
1
+
g
r
−
g
D
0
=
1
+
1
+
g
r
−
g
=
1
+
r
r
−
g
For the stock market as a whole, the modified duration is the price/dividend ratio, which for the S&P 500 was about 62 in February 2004.
