Morningstar Magazine - October/November 2016 - 53
EXHIBIT 2
Maturity vs. Macaulay Duration
1%
3%
5%
Macaulay Duration
50
in E X H I B I T 2
of coupon bonds at par (so that the coupon
n = qT
,
f = n- qT
40
30
n
n = qT
-j
c
y
y
P= q
1+ q + 1+ q
j =1
f = n- qT
∑(
-n
) (
n
n
) (1 + )
y f
q
- qT
f- j - n
∑ ( ∑)( ( ) ) ( ( ) )
c
-j
y y
y
c
y
y
PP (=y; c,q T ) = 1 +q q 1++ 1q + q + 1 +1q+ q
f
j =1
j =1
20
, and
P ((y;
D
y; c,
c, TT )) ==
f- j
- qT
+ T 1+
1+
∑ ((1 + ) ) ((+ (1 +)) ) ( )
P y; c, T
j -f c
f- j
q y q
q
n
∑ j =1
c
q
y
q
y
q
- qT
y
q
j =1
10
n
P=
c
q
20
-j
-n
30
40
50
0
y
q
(
y f
q
∑(
c
q
P ( y; c, T ) =
j =1
∑
D ( y; c, T ) =
n
j =1
)
f- j
( ) (1 + )
j -f
q
c
q
y
q
(
+ T 1+
y
q
- qT
)
(
n
q
) (1 + )
y f
q
c
c q
c
q
(1 + )
y
q
P ( y;P c,
bond,
(y;Tc,) T
=
of
c
q
=
c
q
a coupon bond with a
coupon at par with
f- j
n
q
0.8
-j
(1 + )
in T
P ( y; c, )
y
q
0.4
T
c
y
q
(
y f
)
(
)
10
y
q
y f
q
0
)
2
1
2
)
1- e - T / - e - T /
D ( y; c, T )
∆ P+ 3
≈
-
D
=T / 2 y
Dm (y; c, T m) ∆y
P
1+ q
2
(
1- e - T / - e - T /
T/
(
1- e - T / - e - T /
T/ 1
1
)
)
1
2
∆ P = P D y = P Dm
DV01
≈ - D ∆y
P y (T ) =m q1 + q 1+ F (∆t)
s
q
Macaulay Duration and Maturity
f
f
q
P ((y;y; c,c,TT))- T /
D
coupon
TT=)) =
D
(y;
c,) on
=the
Dmm (y;
s (Tc,
+ 1- e
+
0 P (1y; c,y T )
1 + qT / 1
(
f
y f
q s(T )
q
e -T/
s (T ) = 0 +P 1( y;1-
c,T/T ) +
y
1
D
y ) =c, Ty ) =
Dcm ((y;
1 + q( - qf )
P y; c, T
n
q
1
0Pc ( y ) = y 1 + q
5
Yield (%)
y
Dc ( y ) = y1 1 + q - qf
n
q
(
y f
q
)
-j
1
y
1
y
c
is what
(1 + )
P ( y; c, ) (1 + )
y
q
1
y
c
in
f- j
)
y
q
s
jth 1.2
- qT
)( ( ) )
( )( )
( )
( )( )
(
1.6
P ( y; c, T )
Where
P ( y; c, T ) = P y; c,
) (
( () )
( )
(1 + )
=P ( y ) = (1 + )
y P(T()y;= c,q e) - 1
D ( y ) = (1 + ) -
P ( y ) = (1 + )
Macaulay
duration
- qT of a bond
f- j
y
y
1+ q + 1+ q
Price
) (
y
q
y f
q
n
q
(
j =1
n
y
q
f- j ln [ Z (T )]
f- j
y
y
s (T+) = -
+y f
Pc ( y;j 1c, T )q = P y;Tcc, qn 1 +1 q q
the
= q
q
y
P ( y; c, T )
P -qT n
+
ys (T ) y; c, q 1 q
f- j
Z (T ) y= f-1j + q
y
+ q y -j
+q
1
1
c
1+ q = c
q
y
=q qcP ( y; c, T )
n
1 y; c,
1+ q
-P
q
n
ys P(T )y;= c,q q Z (T ) qT - 1
∑ (1 + ) + (1 + ) (1 + )
y
q
c
q
j -f
q
-s (T)T
n = qT
0
10
f = n-toqT
Years
Maturity
f- j
( ) (1 + ) + T (1 + )
D ( y; c, T ) =
P ( y; c,( T) ) = P ( y; c, ) (1P+( y; )c, T )
Z T =e
n
∑ j =1
*
( )
1+ F 2∆t
...
q
D ∆t
DV01 = FP(T ) y T = P Dm
1+ 1 +P q( y;-c,1T )
C (y; c, T ) q=
P ( y; c, T )
*
EDXmH(y;
I B Ic,
T T1 )
=
P ( y; c, T )
P ( y; c, T )
D ( y; c, T )
Dm (y; c, T ) =
y
1+ q
∆P
≈ - Dm ∆y
P
Price
1.6
1.4
1.2
1.0
Consols
P ( y; c, T )
C (y; c, T ) =
C ( y; c, T ) = P ( y; c, T )1+ ys (T )
q
F (T ) = q
f- j-2
c
f-j
f- j-1
n
y y(T -∆t)
q ∑ j =1 q
q
1+1 + sq q
C ( y; c, T ) =
P ( y; c, T )
( )(
(
(
)(
)
f- -2
)
)
T
T-∆ t
1
∆t
-1
53
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Table of Contents for the Digital Edition of Morningstar Magazine - October/November 2016
Contents
Morningstar Magazine - October/November 2016 - Cover1
Morningstar Magazine - October/November 2016 - Cover2
Morningstar Magazine - October/November 2016 - 1
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Morningstar Magazine - October/November 2016 - Contents
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