Description
This is a document describing bond analytics discusses Relationship Between the Yield to Maturity and Bond Prices, Mathematical Foundation for Bond Pricing, Bond Theorems, Implication about the Portfolio Management, Problems Associated with Immunization, Effective Duration and Convexity for Bonds with Embedded Options
BOND ANALYTICS
Relationship Between the Yield to Maturity and Bond Prices
1. Par Bond (Market price = par value); yield to maturity = coupon rate
2. Discount Bond (Market price < par value); yield to maturity > coupon rate
3. Premium Bond (Market price > par value); yield to maturity < coupon rate
Mathematical Foundation for Bond Pricing
Taylor series approximation of bond prices as a function of the yield to maturity:
( )
P
Error
dy
P dy
P d
dy
P dy
dP
P
dP
+ + ·
2
2
2
1
2
1 1
A change in bond prices is estimated by looking at the first two terms of the Taylor
series.
1. The first term,
y D dy
P dy
dP
m
? ? ? ?
1
, where
m
D
is known as the Modified Duration.
Differentiating the bond price formula with respect to the yield to maturity,
( )
y D y
y
D
y
y
D
P
P
m
? ? ? · ? ?
+
? ·
,
_
¸
¸
+
?
? ? ?
?
1 1
and
( )
? ?
· ·
? ? · ? ·
T
t
t
T
t
t
t t
P
C PV
D
1 1 0
The symbol D measures an “average maturity.” It is known as the Macaulay
Duration. The present value could be computed by using the spot rate, but in
practice, the yield to maturity is used. Note that for zero coupon bonds, T D · . As
compared to the symbol D,
m
D
is a bond’s Modified Duration, i.e.
( ) y
D
D
m
+
·
1
.
Consequently, y D
P
P
m
? ? ? ?
?
Q1) Consider a regular 5-year bond with an annual coupon rate equal to 8%.
Compute the Macaulay and Modified Duration, when the bond sells for $98.50.
y- y y+
Q2) Compute the percent change in the bond price if there was a 1 percent rise in the
interest rate..
Q3) Assume that a 8.5% coupon bond maturing in 6 years also has the same
Macaulay Duration. The yield to maturity is 8.75%. If the 6-year bond yield also
increases by 1%, i.e. from 8.75% to 9.38%, how would the bond price change?
2. The second term,
( )
2
2
2
1
2
1
dy
P dy
P d
, deals with the convexity of a bond’s price.
Define the dollar convexity as
2
2
$
dy
P d
C ·
. A change in a bond’s price due to the
second term of the Taylor’s expansion is now ( )
2
$
2
1
dy C dP ? · . If a bond’s
convexity is
P
C C
1
$
· , then ( )
2
2
1
dy C
P
dP
· .
The convexity is reported in years. The relationship between the convexity in years
and the convexity calculated based on the number of coupon payments in a year is:
2
m
r yea per periods m in convexity
years in Convexity · .
Now the second derivative of the bond price is
?
·
+ +
+
+
+
+
+
·
n
t
n t
y
M n n
y
C t t
dy
P d
1
2 2 2
2
) 1 (
) 1 (
) 1 (
) 1 (
, or
2 1 2 3 2
2
) 1 (
) / 100 )( 1 (
) 1 (
2
) 1 (
1
1
2
+ +
+
? +
+
+
?
1
]
1
¸
+
? ·
n n n
y
y C n n
y y
Cn
y y
C
dy
P d
.
The convexity refers to the curvature of the bond prices and is a function of:
(1) the size of the coupon payments
(2) the life of the bond
(3) its current market price.
If a bond has a call feature, the convexity will disappear at sufficiently low yields
because such a drop in yields will trigger a call, limiting the rise in the bond’s price to
its call price.
y- y y+
Therefore, to approximate the percentage change in bond prices based on both duration
and convexity,
( )
2
2
1
dy C dy D
P
dP
m
+ ? ·
Q1) Consider again the same bond with an 8 percent coupon maturing in 5 years. The
price is $98.50. Compute the regular and dollar convexity for the bond.
Q2) If there was an increase in the interest rate of 1%, how would it affect the bond price
due to the bond’s duration and convexity?
Bond Theorems
1. If a bond’s market price increases, then its yield must decrease; and conversely.
2. If a bond’s yield does not change over its life, then the size of its discount or premium
will decrease as its life get shorter.
3. If a bond’s yield does not change over its life, then the size of its discount or premium
will decrease at an increasing rate as its life gets shorter.
P+
P
P-
y- y y+
4. A decrease in a bond’s yield will raise the bond’s price by an amount that is greater in
size than the corresponding fall in the bond’s price that would occur if there were an
equal sized increase in the bond’s yield.
5. The percentage change in a bond’s price owing to a change in its yield will be smaller
if its coupon rate is higher.
Q1) Show a numerical example for each theorem.
Theorems 1 and 4
?
the convexity, that is, bond prices and yields are inversely related
but the relationship is not linear. The convexity creates values in that
1. The greater the convexity, the capital loss will be less for bonds with greater convexity
when the required yield rises, and the capital appreciation will also be greater when
the required yield falls. In times of rising interest rate volatility, the market will price
convexity by offering a lower yield on bonds with greater convexity.
2. As the required yield increases (decreases), the convexity of a bond decreases
(increases). This is known as positive convexity. Note a change in the slope or
duration.
3. For a given yield and maturity, the lower the coupon, the greater the convexity of a
bond.
4. For a given yield and modified duration, the lower the coupon, the smaller the
convexity.
Par
Price
Maturity
Implication about the Portfolio Management
The interest rate risk is comprised of the reinvestment rate risk and the market value risk.
The immunization technique is used to meet a given promised stream of cash outflows
regardless of what happens to the interest rate later.
If the fund obligation occurs in Year 2, buy a 2-year zero with D = 2. Alternatively, the
portfolio manager can run a portfolio whose weighted average duration is 2. Assume that
there are two bonds in the portfolio, one zero coupon bonds maturing in 1 year and one
three year regular coupon bond with D = 2.78. To find the portfolio weight, notice that
( ) ( ) 2 78 . 2 1
3 1 3 3 1 1
· ? + ? · ? + ? D D
where
1
3 1
· ? + ?
. The solution is 4382 . 0
1
· ? and
5618 . 0
3
· ?
.
Now suppose that the Bond 1 has the cash flow at Year 1 equal to $1,070, and sells for
1
P = $972.73 today, i.e. the yield to maturity is 10%.
If the fund manager buys the Bond 1, he must face the reinvestment rate risk.
Now suppose that the Bond 3 pays an 8% coupon and matures in 3 years. Then, at the
yield to maturity of 10%, the bond’s duration equals 2.78 years. It sells for
3
P
= $950.25.
If the interest rate rises at the end of 2 years, however, we suffer a capital loss.
The following is the immunization strategy when the fund obligation is $1,000,000 in
two years.
1. The investment required is $826,446, i.e.
( )
2
10 . 1
000 , 000 , 1
.
2. Spend $362,149 ( ) 446 , 826 $ 4382 . 0 × · on Bond 1, and spend $464,297
( ) 446 , 826 $ 5618 . 0 × · on Bond 3. That is, buy 372 one-year bonds
,
_
¸
¸
·
73 . 972
149 , 362
and
489 three-year bonds
,
_
¸
¸
·
25 . 950
297 , 464
.
Q1) Construct a cash flow table to show that at the end of two years, the fund manager
should be able to meet the fund obligation regardless of what happens to the interest rate.
Problems Associated with Immunization
1. Default and call risk
2. Multiple nonparallel shifts in a nonhorizontal yield curve
3. Rebalancing
4. Many candidates
Effective Duration and Convexity for Bonds with Embedded Options
A bond’s duration and convexity for bonds including derivative securities or optionable
bonds can be approximated as follows.
1. Decrease the yield on the bond by a percent to calculate
+
P .
2. Increase the yield by the same percent to calculate
?
P .
3. Then,
y P
P P
Duration e Approximat
?
?
·
? +
0
2
, where y is in decimal form.
2
0
0
) (
2
y P
P P P
Convexity e Approximat
?
? +
·
? +
doc_297109302.doc
This is a document describing bond analytics discusses Relationship Between the Yield to Maturity and Bond Prices, Mathematical Foundation for Bond Pricing, Bond Theorems, Implication about the Portfolio Management, Problems Associated with Immunization, Effective Duration and Convexity for Bonds with Embedded Options
BOND ANALYTICS
Relationship Between the Yield to Maturity and Bond Prices
1. Par Bond (Market price = par value); yield to maturity = coupon rate
2. Discount Bond (Market price < par value); yield to maturity > coupon rate
3. Premium Bond (Market price > par value); yield to maturity < coupon rate
Mathematical Foundation for Bond Pricing
Taylor series approximation of bond prices as a function of the yield to maturity:
( )
P
Error
dy
P dy
P d
dy
P dy
dP
P
dP
+ + ·
2
2
2
1
2
1 1
A change in bond prices is estimated by looking at the first two terms of the Taylor
series.
1. The first term,
y D dy
P dy
dP
m
? ? ? ?
1
, where
m
D
is known as the Modified Duration.
Differentiating the bond price formula with respect to the yield to maturity,
( )
y D y
y
D
y
y
D
P
P
m
? ? ? · ? ?
+
? ·
,
_
¸
¸
+
?
? ? ?
?
1 1
and
( )
? ?
· ·
? ? · ? ·
T
t
t
T
t
t
t t
P
C PV
D
1 1 0
The symbol D measures an “average maturity.” It is known as the Macaulay
Duration. The present value could be computed by using the spot rate, but in
practice, the yield to maturity is used. Note that for zero coupon bonds, T D · . As
compared to the symbol D,
m
D
is a bond’s Modified Duration, i.e.
( ) y
D
D
m
+
·
1
.
Consequently, y D
P
P
m
? ? ? ?
?
Q1) Consider a regular 5-year bond with an annual coupon rate equal to 8%.
Compute the Macaulay and Modified Duration, when the bond sells for $98.50.
y- y y+
Q2) Compute the percent change in the bond price if there was a 1 percent rise in the
interest rate..
Q3) Assume that a 8.5% coupon bond maturing in 6 years also has the same
Macaulay Duration. The yield to maturity is 8.75%. If the 6-year bond yield also
increases by 1%, i.e. from 8.75% to 9.38%, how would the bond price change?
2. The second term,
( )
2
2
2
1
2
1
dy
P dy
P d
, deals with the convexity of a bond’s price.
Define the dollar convexity as
2
2
$
dy
P d
C ·
. A change in a bond’s price due to the
second term of the Taylor’s expansion is now ( )
2
$
2
1
dy C dP ? · . If a bond’s
convexity is
P
C C
1
$
· , then ( )
2
2
1
dy C
P
dP
· .
The convexity is reported in years. The relationship between the convexity in years
and the convexity calculated based on the number of coupon payments in a year is:
2
m
r yea per periods m in convexity
years in Convexity · .
Now the second derivative of the bond price is
?
·
+ +
+
+
+
+
+
·
n
t
n t
y
M n n
y
C t t
dy
P d
1
2 2 2
2
) 1 (
) 1 (
) 1 (
) 1 (
, or
2 1 2 3 2
2
) 1 (
) / 100 )( 1 (
) 1 (
2
) 1 (
1
1
2
+ +
+
? +
+
+
?
1
]
1
¸
+
? ·
n n n
y
y C n n
y y
Cn
y y
C
dy
P d
.
The convexity refers to the curvature of the bond prices and is a function of:
(1) the size of the coupon payments
(2) the life of the bond
(3) its current market price.
If a bond has a call feature, the convexity will disappear at sufficiently low yields
because such a drop in yields will trigger a call, limiting the rise in the bond’s price to
its call price.
y- y y+
Therefore, to approximate the percentage change in bond prices based on both duration
and convexity,
( )
2
2
1
dy C dy D
P
dP
m
+ ? ·
Q1) Consider again the same bond with an 8 percent coupon maturing in 5 years. The
price is $98.50. Compute the regular and dollar convexity for the bond.
Q2) If there was an increase in the interest rate of 1%, how would it affect the bond price
due to the bond’s duration and convexity?
Bond Theorems
1. If a bond’s market price increases, then its yield must decrease; and conversely.
2. If a bond’s yield does not change over its life, then the size of its discount or premium
will decrease as its life get shorter.
3. If a bond’s yield does not change over its life, then the size of its discount or premium
will decrease at an increasing rate as its life gets shorter.
P+
P
P-
y- y y+
4. A decrease in a bond’s yield will raise the bond’s price by an amount that is greater in
size than the corresponding fall in the bond’s price that would occur if there were an
equal sized increase in the bond’s yield.
5. The percentage change in a bond’s price owing to a change in its yield will be smaller
if its coupon rate is higher.
Q1) Show a numerical example for each theorem.
Theorems 1 and 4
?
the convexity, that is, bond prices and yields are inversely related
but the relationship is not linear. The convexity creates values in that
1. The greater the convexity, the capital loss will be less for bonds with greater convexity
when the required yield rises, and the capital appreciation will also be greater when
the required yield falls. In times of rising interest rate volatility, the market will price
convexity by offering a lower yield on bonds with greater convexity.
2. As the required yield increases (decreases), the convexity of a bond decreases
(increases). This is known as positive convexity. Note a change in the slope or
duration.
3. For a given yield and maturity, the lower the coupon, the greater the convexity of a
bond.
4. For a given yield and modified duration, the lower the coupon, the smaller the
convexity.
Par
Price
Maturity
Implication about the Portfolio Management
The interest rate risk is comprised of the reinvestment rate risk and the market value risk.
The immunization technique is used to meet a given promised stream of cash outflows
regardless of what happens to the interest rate later.
If the fund obligation occurs in Year 2, buy a 2-year zero with D = 2. Alternatively, the
portfolio manager can run a portfolio whose weighted average duration is 2. Assume that
there are two bonds in the portfolio, one zero coupon bonds maturing in 1 year and one
three year regular coupon bond with D = 2.78. To find the portfolio weight, notice that
( ) ( ) 2 78 . 2 1
3 1 3 3 1 1
· ? + ? · ? + ? D D
where
1
3 1
· ? + ?
. The solution is 4382 . 0
1
· ? and
5618 . 0
3
· ?
.
Now suppose that the Bond 1 has the cash flow at Year 1 equal to $1,070, and sells for
1
P = $972.73 today, i.e. the yield to maturity is 10%.
If the fund manager buys the Bond 1, he must face the reinvestment rate risk.
Now suppose that the Bond 3 pays an 8% coupon and matures in 3 years. Then, at the
yield to maturity of 10%, the bond’s duration equals 2.78 years. It sells for
3
P
= $950.25.
If the interest rate rises at the end of 2 years, however, we suffer a capital loss.
The following is the immunization strategy when the fund obligation is $1,000,000 in
two years.
1. The investment required is $826,446, i.e.
( )
2
10 . 1
000 , 000 , 1
.
2. Spend $362,149 ( ) 446 , 826 $ 4382 . 0 × · on Bond 1, and spend $464,297
( ) 446 , 826 $ 5618 . 0 × · on Bond 3. That is, buy 372 one-year bonds
,
_
¸
¸
·
73 . 972
149 , 362
and
489 three-year bonds
,
_
¸
¸
·
25 . 950
297 , 464
.
Q1) Construct a cash flow table to show that at the end of two years, the fund manager
should be able to meet the fund obligation regardless of what happens to the interest rate.
Problems Associated with Immunization
1. Default and call risk
2. Multiple nonparallel shifts in a nonhorizontal yield curve
3. Rebalancing
4. Many candidates
Effective Duration and Convexity for Bonds with Embedded Options
A bond’s duration and convexity for bonds including derivative securities or optionable
bonds can be approximated as follows.
1. Decrease the yield on the bond by a percent to calculate
+
P .
2. Increase the yield by the same percent to calculate
?
P .
3. Then,
y P
P P
Duration e Approximat
?
?
·
? +
0
2
, where y is in decimal form.
2
0
0
) (
2
y P
P P P
Convexity e Approximat
?
? +
·
? +
doc_297109302.doc