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Stock duration of an equity stock is the average of the times until its dividends are received, weighted by their present values.[citation needed
Duration
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
{\displaystyle MacD_{ddm}={\frac {1+r}{r-g}}}
where
M
a
c
D
d
d
m
{\displaystyle MacD_{ddm}}
Macaulay duration of stock under the DDM model
r
{\displaystyle r}
g
{\displaystyle g}
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
{\displaystyle ModD_{ddm}={\frac {1}{r-g}}}
The other formula for the same is - D = saa[citation needed
Derivation
The Macaulay duration is defined as:
(
1
)
M
a
c
D
=
∑
i
t
i
P
V
i
V
{\displaystyle (1)\ \ \ \ MacD={\frac {\sum _{i}{t_{i}PV_{i}}}{V}}}
where:
i
{\displaystyle i}
P
V
i
{\displaystyle PV_{i}}
present value of the
i
{\displaystyle i}
asset ,
t
i
{\displaystyle t_{i}}
i
{\displaystyle i}
V
{\displaystyle V}
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
{\displaystyle (2)\ \ \ \ V=\sum _{t=1}^{\infty }{D_{0}}{\frac {(1+g)^{t}}{(1+r)^{t}}}={\frac {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
+
.
.
.
{\displaystyle (3)\ \ \ \ \sum _{i}t_{i}PV_{i}=\sum _{t=1}^{\infty }t{D_{0}}{\frac {(1+g)^{t}}{(1+r)^{t}}}=D_{0}{\frac {(1+g)}{(1+r)}}+2{D_{0}}{\frac {(1+g)^{2}}{(1+r)^{2}}}+3{D_{0}}{\frac {(1+g)^{3}}{(1+r)^{3}}}+...}
Multiplying by
1
+
r
1
+
g
{\displaystyle {\frac {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
+
.
.
.
{\displaystyle (4)\ \ \ \ {\frac {1+r}{1+g}}\sum _{i}t_{i}PV_{i}=D_{0}+2{D_{0}}{\frac {(1+g)}{(1+r)}}+3{D_{0}}{\frac {(1+g)^{2}}{(1+r)^{2}}}+...}
Subtracting
(
4
)
−
(
3
)
{\displaystyle (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
+
.
.
.
{\displaystyle {\frac {1+r}{1+g}}\sum _{i}t_{i}PV_{i}-\sum _{i}t_{i}PV_{i}=D_{0}+D_{0}{\frac {(1+g)}{(1+r)}}+D_{0}{\frac {(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
{\displaystyle \left({\frac {1+r}{1+g}}-1\right)\sum _{i}t_{i}PV_{i}=D_{0}+{\frac {D_{0}(1+g)}{r-g}}=D_{0}+V}
Simplifying:
r
−
g
1
+
g
∑
i
t
i
P
V
i
=
D
0
+
V
{\displaystyle {\frac {r-g}{1+g}}\sum _{i}t_{i}PV_{i}=D_{0}+V}
(
5
)
∑
i
t
i
P
V
i
=
(
D
0
+
V
)
1
+
g
r
−
g
{\displaystyle (5)\ \ \ \ \sum _{i}t_{i}PV_{i}=(D_{0}+V){\frac {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
{\displaystyle MacD={\frac {\sum _{i=1}^{n}{t_{i}PV_{i}}}{V}}={\frac {(D_{0}+V){\frac {1+g}{r-g}}}{D_{0}{\frac {1+g}{r-g}}}}={\frac {D_{0}+V}{D_{0}}}={\frac {D_{0}+D_{0}{\frac {1+g}{r-g}}}{D_{0}}}=1+{\frac {1+g}{r-g}}={\frac {1+r}{r-g}}}
Modified duration
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.[citation needed
See also
External links
