Fixed Income II
Fixed Income II
1
Essential Readings
• Z. Bodie, A. Kane, A. Marcus and R. Jain, Investments, Asia Global Edition, McGraw Hill. Chapter
16.: 16.1, 16.2 (ignore Duration and Convexity of Callable Bonds and Duration and Convexity of
Mortgage-Back Securities), 16.3
• J. Berk and P. DeMarzo, Corporate Finance, Pearson Global Edition. Chapter 30.4
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6 Mar 2020
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24 Feb 2021
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5 Aug 2022
Fixed Income II
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14Jul 2023
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Sources of Return
• There are three sources of return on a fixed-rate bond:
• Receipt of promised coupon and principal payments.
• Reinvestment of coupon payments.
• Potential capital gains/losses if the bond is sold prior to maturity.
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Sources of Return
• Consider an investor purchased a 10-year, 8% annual-pay bond at a price of $100 (per $100 of
par).
• The YTM at issuance (which is the rate at which coupon payments are reinvested) solves:
+
+ ⋯+
= 100 ⇒
= 8%
• Using financial calculator: N = 10; PMT = 8; PV = −100; FV = 100; CPT I/Y; I/Y = 8
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Sources of Return
• Suppose after the purchase the bond, but before receiving the first coupon payment, interest
rates rise to 10%. If the investor plans to sell the bond at year 4, returns from the investment are:
1) Future value for the reinvested coupons (coupon receipts plus reinvestment income) is:
8 × 1 + 10%
= 37.128 >
8 × 1 + 8%
= 36.0489
(I/Y = 10; PMT = 8; PV = 0; N = 4; CPT FV; FV = 37.128 vs. I/Y = 8; PMT = 8; PV = 0; N = 4; CPT FV; FV = 36.0489)
2) Selling price at year 4 is (N = 6; I/Y = 10; PMT = 8; FV = 100; CPT PV; PV = 91.2895):
8
1
1
1
100
1
100
−
+
=
91.2895
<
8
−
+
= 100
10% 10% 1 + 10% "
1 + 10% "
8% 8% 1 + 8% "
1 + 8% "
Fixed Income II
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0
1
2
-100
+8
+8
8
1 + 8%
0
1
2
3
4
8
8
8
8
1
8
2
3
⋯
+8
8
100 + 8
+
+ ⋯+
1 + 8% $
1 + 8%
8 × 1 + 10%
0
3
5
= 37.128 >
4
6
8 × 1 + 8%
5
6
8
8
9
10
+8
+108
= 100
⋯
9
10
9
10
8
108
= 36.0489
⋯
1
1
1
100
1
100
−
+
=
91.2895
<
8
−
+
= 100
10% 10% 1 + 10% "
1 + 10% "
8% 8% 1 + 8% "
1 + 8% "
Fixed Income II
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Sources of Return
• The realized annual yield is calculated as:
37.128 + 91.2895
100 =
⇒ y = 6.4525% < 8%
1+ %
(N = 4; FV = 128.4175; PV = -100; CPT I/Y; I/Y = 6.4525)
Fixed Income II
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Fixed Income II
https://www.ustreasuryyieldcurve.com/
12
Interest Rate Risk
• There are two types of interest rate risk, which offset each other:
• Reinvestment risk: the future value of any interim cash flows received from a bond increases
when interest rates rise and decreases when interest rates decline.
• Market price risk: the selling price of a bond (at any point during its term) decreases when
interest rates rise and increases when interest rates decline.
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Bond Performance Profile
• Assuming a flat yield curve (sport rates = YTM of bonds), face value $100, annual coupon rate '%
and ( years to maturity, price of the bond at YTM = is:
P( y) =
C
1
(1 + y )
+
C
(1 + y )
2
+
C
(1 + y )
3
+L +
• At the current yield, $1 of investment in the bond can buy 0
100 + C
T
(1 + y )
1
units of the bond.
• When yield changes by Δ , value of the investment at new yield " + Δ " is: / = 0
+Δ
×
0
1
• We can show the changes in bond investment value (y-axis) following the change in yield (x-axis)
in the bond’s Performance Profile.
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Bond Performance Profile
Performance profile of a zero-coupon bond, 20 years to maturity, current yield
5% is shown
3
below:
1
1
Note: the performance profile is
ΔV / ! 1 0
Δ
!0
VALUE OF INVESTMENT
2.5
dP ×
1
P
0
1
0
1
0
0
20
Slope of Performance Profile: 0 as 2 → 0
2
0
2
1.5
Δ
!0
20
• Downward sloping: The value of
the investment goes down when
interest rates go up.
• Convex: The value goes down at a
decreasing rate.
1
0.5
• Question: What information dose the
slope of Performance Profile show?
0
0%
1%
2%
3%
4%
dy 5%
6%
7%
8%
9%
10%
YIELD
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Bond Performance Profile
• Performance profiles of two zero-coupon bonds with 5 and 20 years to maturity and current yield
5% are shown below:
3
Value (20)
Value (5)
2.5
Slope of the performance profile
increases with maturity: bonds with
longer maturity are more sensitive to
change in yield.
VALUE OF INVESTMENT
2
1.5
1
0.5
0
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
YIELD
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Bond Performance Profile
• Performance profile of two bonds with 20 years to maturity, current yield
rates are 0% and 6% are shown below:
5%, annual coupon
3
0%_20 yrs
VALUE OF INVESTMENT
2.5
6%_20 yrs
2
Slope of the performance profile
decreases as coupon rate raises:
bonds with lower coupon rates are
more sensitive to change in yield.
1.5
1
0.5
0
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
YIELD
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Tools for Measuring and Managing Interest Rate
Risk
• The main tools for measuring and managing interest rate risk are:
• Duration
• Convexity
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Macaulay Duration (effective maturity. When do I
get my $ back)
• Zero coupon bond makes no cash payment before maturity date, its effective maturity equal to
the actual term to maturity.
• For coupon bonds, each coupon payment is a zero coupon bond with maturity date equal to the
coupon payment date.
• The effective maturity of this set of zero coupon bonds is the sum of the effective maturity of
each zero coupon bond, weighted by the total value of all the bonds that it represents.
• Macaulay Duration of a coupon bond is the weighted average of the effective maturities of the
set of individual zero coupon bonds, with the weights equaling the proportion of the total value
represented by each zero coupon bond.
T
D =  wt t
t =1
CFt
t
(1
+
y
)
where wt =
P
Fixed Income II
and 0
∑G
DEF
F
19
T
1
Macaulay Duration: D =  w t where w = (1CF
+ y) P
t
t
t
t
t =1
Macaulay Duration for a 10% annual coupon rate, 10 years to maturity with face value $1,000 is:
Year
CFt
PV ( y = 10% )
Weights = PV(CFt)/P
Year
Weighted Maturity
1
2
3
4
5
6
7
8
9
10
Total
100
100
100
100
100
100
100
100
100
1100
90.9091
82.6446
75.1315
68.3013
62.0921
56.4474
51.3158
46.6507
42.4098
424.0976
1,000.0000
0.0909
0.0826
0.0751
0.0683
0.0621
0.0564
0.0513
0.0467
0.0424
0.4241
1
2
3
4
5
6
7
8
9
10
0.0909
0.1653
0.2254
0.2732
0.3105
0.3387
0.3592
0.3732
0.3817
4.2410
6.7590
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Macaulay Duration
• Macaulay Duration is therefore the weighted average of the times until each payment is
received, with the weights proportional to the present value of the payment.
• Quiz: find Macaulay Duration for a 4-year annual coupon bond with face value $100, YTM =
10%; ' 4%.
P=
4
4
4
104
+
+
+
= 80.98
1
2
3
4
(1 + 0.1) (1 + 0.1) (1 + 0.1) (1 + 0.1)
T
D =  wt t =
t =1
4
1
4
1
4
1
104
1
×
×1 +
×
×2+
×
×3+
×
× 4 = 3.7465
1
2
3
4
(1 + 0.1) 80.98
(1 + 0.1) 80.98
(1 + 0.1) 80.98
(1 + 0.1) 80.98
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What determines Macaulay Duration
• Macaulay Duration can also be understood as the year around which the bond’s discounted cash
flows are in balance, i.e., a “center of gravity” or “how soon investors will get their money back”.
1000
1 + 0.1 %
t=0
t=1
t=2
t=3
t=4
4
• Maturity of all cash flows of a 4-year zero occurs at t = 4 so Macaulay Duration is 4 years.
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What determines Macaulay Duration
• Adding coupon payments shifts the “center of gravity” to left. However, the bulk of the maturity
of cash flows occurs close to t = 4 because the last coupon and the par are paid at t = 4.
t=0
40
1 + 0.1
40
1 + 0.1 $
40
1 + 0.1
1040
1 + 0.1 %
t=1
t=2
t=3
t=4
3.747
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What determines Macaulay Duration
• Holding the coupon rate constant, increasing time to maturity often increases (not always for
bonds when C is low, YTM is high, and maturity is relatively long) the weighted average maturity
of the payment or shifts the “center of gravity” to right.
t=0
80
1 + 0.1
80
1 + 0.1 $
80
1 + 0.1
80
1080
1 + 0.1 %%
1 + 0.1
1080
1 + 0.1 I
t=1
t=2
t=3
t=4
t=5
3.562
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4.281
24
What determines Macaulay Duration
• Increasing coupon rate increases the weights on the early payments. It thus lowers the weighted
average maturity of the payment or shifts the “center of gravity” to left.
1080
1 + 0.1 %
t=0
JKL
JDF
F
F
80
1 + 0.1
80
1 + 0.1 $
80
1 + 0.1
40
1 + 0.1
40
1 + 0.1 $
40
1 + 0.1
t=1
t=2
t=3
→ 0 as M → ∞
1040
1 + 0.1 %
t=4
3.562 3.7465
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What determines Macaulay Duration
• Holding the coupon rate constant, a higher yield reduces the value of more-distant payments by a
greater proportional amount, thereby reducing the weighted average maturity of the payment or
shifts the “center of gravity” to left.
1080
1 + 0.2 I
80
1 + 0.2
80
1 + 0.2 $
80
1 + 0.2
80
1 + 0.1
80
1 + 0.1 $
80
1 + 0.1
t=1
t=2
t=3
t=0
JKL
J F
!M
DF
F
F
→ 0 as M → ∞
80
1 + 0.2 %
80
1 + 0.1 %
t=4
1080
1 + 0.1 I
t=5
4.1204.281
Fixed Income II
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What determines Macaulay Duration
YTM ↑, D ↓
C ↑, D ↓
From: Z. Bodie, A. Kane, A. Marcus and R. Jain, Investments
Fixed Income II
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From: Z. Bodie, A. Kane, A. Marcus and R. Jain, Investments
• Duration decreases as coupon rates increase.
• Duration increases with time to maturity (generally).
• Duration is independent of coupon rate only for perpetuities.
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What determines Duration
• Rule 1 The duration of a zero-coupon bond equals its time to maturity.
• Rule 2 Holding the coupon rate constant, a bond’s duration generally increases with its time to
maturity.
• Prices of long-term bonds tend to be more price sensitive to interest rate changes than
prices of short-term bonds.
• Rule 3 Holding maturity constant, a bond’s duration is lower when the coupon rate is higher.
• The higher the coupon rate, the higher the weights on the early payments and the lower is
the weighted average maturity of the payment.
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What determines Duration
• Rule 4 Holding other factors constant, the duration of a coupon bond is lower when the bond’s
yield to maturity is higher.
• A higher yield reduces the value of more-distant payments by a greater proportional
amount, thereby reducing effective maturity.
• Rules 5 The duration of a level perpetuity is equal to: (1 + )/
• For example, at a 10% yield, the duration of perpetuity that pays $100 once a year forever
will equal 1.10/0.10 = 11 years, but at an 8% yield it will equal 1.08/.08 = 13.5 years.
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Macaulay Duration of a Semiannual Coupon Bond
• Macaulay Duration of a semiannual coupon bond with face value 100, coupon rate '%, T years to
maturity, YTM quoted as a semiannual APR is calculated as:
C
C
100 +
2T
1
1
2
2 1
=
−
=
D =  wt t where wt =
for
t
1,
K
2
T
1
and
w
2T
t
2T
2 t =1
y P
y P


1 + 
1 + 
2


 2
• E.g., YTM = 10%, coupon rate 8% with 2 years to maturity (0 964.5405), Macaulay Duration
the semiannual coupon bond is:
D=
=
1 2T
 wt t
2 t =1

1 
40
1
40
1
40
1
1, 040
1
×
×
1
+
×
×
2
+
×
×
3
+
×
×
4


2
3
4
2  (1 + 0.05)1 964.54
(1 + 0.05) 964.54
(1 + 0.05) 964.54
(1 + 0.05) 964.54 
= 1.8852 years or 3.7704 periods
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Modified Duration
• The price of a bond is the present value of its cash flows:
0
G
'R
(1 + )
'R (1 + )S +'R$ (1 + )S$ +'R (1 + )S …
• The change in price for a small change in yield gives:
20
2
!'R (1 + )S$ !2'R$ 1 +
'R
G
1+
S (!1)(1 +
'R
(!M)
(1 + ) (1 + )
S
! 3'R 1 +
S% + ⋯
)S + 'R$ (1 + )S$ (!2) 1 +
!1
(1 + )
V
S
+ 'R (1 + )S (!3) 1 +
S +⋯
M ⋅ 'R
(1 + )
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Modified Duration
20
2
!1
(1 + )
V
M ⋅ 'R
(1 + )
• Dividing both sides by price of the bond,
20
2
0
whereW ∗
!1
(1 + )
(
Y
)
∑V
M ⋅ 'R
(1 + )
0
−1
=
(1 + )
V
'R 1
−1
(
)M=
(1 + ) 0
(1 + )
G
ZM =
−W
= −W ∗
(1 + )
is the modified duration.
Fixed Income II
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Modified Duration
• We shown the slope of the performance profile at the yield
3
∆V = P ( y + ∆y )
VALUE OF INVESTMENT
2.5
dP ×
20^
0 .
is
2
1
1
− P ( y)
P ( y)
P ( y)
=  P ( y + ∆y ) − P ( y ) 
2
1
1
= dP
P ( y)
P
dP
1.5
Slope of Perf. Profile:
1
P 1
20
Note, 0
2
.
0.5
20 1
0 2
P as dy → 0
dy
20 1
2 0
20
2
0
!W
O1
P
!W ∗
0
0%
1%
2%
3%
4%
dy 5%
6%
YIELD
7%
8%
9%
10%
• Modified Duration is the slope of performance profile.
Fixed Income II
34
Modified Duration
• The slope of the performance profile at the yield
dP
dy
P
= −D * .
as 2 → 0 is Modified Duration, i.e.,
Hence, dP = − D *× P × dy
• For a small interest rate movement Δ , the change in the bond price is approximately equal to
.
∆P ≈ – D * ×P × ∆y
• In terms of percentage change,
ΔP
≈– W ∗ × a
0
Fixed Income II
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Quiz
• You predict that interest rates are about to fall, which bond will give you the highest capital gain?
a)
b)
c)
d)
Low coupon, long maturity.
High coupon, short maturity.
High coupon, long maturity.
Zero coupon, long maturity.
Fixed Income II
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Modified Duration
• Semiannual coupon bond with face value 100, coupon rate '%, T years to maturity, YTM quoted
as a semiannual APR, the Macaulay Duration is calculated as:
2T
D=
1
wt t

2 t =1
C
1
2 1
=
K
−
=
where wt =
for
t
1,
2
T
1
and
w
2
T
t
2T
y P
y P


1 + 
1 + 
 2
 2
• The modified duration is: D* =
C
2
100 +
D
1+
y
2
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Using Exel
A
B
C
D
E
F
Settlement date
1/1/2010
1/1/2010
1/1/2010
1/1/2010
1/1/2010
Maturity date
1/1/2014
1/1/2014
1/1/2014
1/1/2015
1/1/2015
Coupon rate
0%
4%
8%
8%
8%
Yield to maturity
10%
10%
10%
10%
20%
Coupons per year
Redumption value
1
1
1
1
1
1
1
1
1
1
Macaulay duration
4
3.747
3.562
4.281
4.120
Modified duration
3.636
3.406
3.238
3.892
3.433
Macaulay duration: =DURATION(B2,B3,B4,B5,B6)
Modified duration: =MDURATION(B2,B3,B4,B5,B6)
Fixed Income II
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Application
• 2-year maturity 8% coupon bond with semiannual coupon payments selling at a price of $964.54
when YTM = 10%. Macaulay duration is 1.8852 years and modified duration is
D* =
1.8852
= 1.7954
10%
1+
2
• Suppose the interest rate increases by 0.02%. Bond price changes by:
∆P ≈ − D* P∆y = −1.7954 × 964.54 × 0.02% = −0.3463
• In percentage term, the price of the bond changes by:
∆P
≈ − D*∆y = −1.7954 × −0.02% = −0.0359%
P
• Suppose there is a $1,000 par value zero coupon bond also with maturity of 1.8852 years, the
impact on the change in interest rate will be approximately the same so long as the two bonds
Fixed Income II
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have the same duration D.
Application
• As the bond makes payment semiannually, the duration is 1.8852 × 2 = 3.7704 periods (each
period is a half-year) with a per period interest of 5%. The modified duration of the bond is
3.7704/(1 + 0.05) = 3.591 periods. The semiannual interest rate increases by 0.01%. Bond price
changes by:
∆P ≈ − D* P∆y = −3.591× 964.54 × 0.01% = −0.3463
• In percentage term, the price of the bond changes by:
∆P
≈ − D*∆y = −3.591× 0.01% = −0.0359%
P
Fixed Income II
40
Convexity
• Annual coupon rate = 8%, 30-year bond
selling at its initial YTM = 8%. The
selling price is $1,000.
∆P
= − D*∆y = −11.26 × 0.02 = −22.52%
P
• Modified Duration D* = 11.26.
• If the interest rate increases from 8% to
10%, the actual bond price falls to
$811.462.
• The price change will be: -18.85%.
• Duration rule gives fall in bond price of:
!11.26 × 0.02 = -22.52%
From: Z. Bodie, A. Kane, A. Marcus and R. Jain, Investments
Fixed Income II
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Convexity
• Modified Duration assumes that the price changes are linear with respect to changes in the yield
to maturity. Duration rule is a good approximation for only small changes in bond yields.
∆P
≈ − D*∆y
P
• Duration rule is less accurate for larger changes in bond yields because the relationship between
bond prices and yields is convex.
• Duration model always understates the capital gain if bond yields go down and overstates the
capital loss if bond yields go up.
• These approximation errors depend on the curvature of the performance profile.
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Taylor Series Approximation
• A Taylor series expansion is a representation of a function by an infinite series of polynomials around
a point, i.e., it uses the series to represent a number as a polynomial that has a very similar value to
the number in a neighbourhood around a specified b value:
c(b)
c′(d)(b ! d) c′′(d)(b ! d)$ c
c(d) +
+
+
2!
1!
(d)(b ! d)
+ ⋯.
3!
• Read this for a short introduction: https://brilliant.org/wiki/taylor-series-approximation/
• A short video: https://www.youtube.com/watch?v=-CCJNbXzcKU
• A longer one: https://www.youtube.com/watch?v=3d6DsjIBzJ4
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43
Convexity
• The initial yield is , following Taylor Series Approximation, the price of the bond for nearby yields
can be approximated as:
0
≈0
+
0′
1!
−
+
0′′
2!
−
$
• Approximated change of bond price is:
Δ0
⇒
0
!0
≈ 0′
Δ0
0′
≈
0
0
!
where W ∗ = −
0′
0
!
+
1 0′′
2! 0
and Γ =
+
1
0′′
2!
!
!
$
$
1
≈ !W ∗ Δ + Γ Δ
2
$
0′′
0
Fixed Income II
44
20
2
Convexity
'R 1 +
S
(!1)(1 + )S + 'R$ (1 + )S$ (!2) 1 +
S
+ 'R (1 + )S (!3) 1 +
• The convexity of a bond with annual coupon rate C% and T years to maturity is:
Γ=
T
1
1
CFt
w t ( t + 1) where w =

(1 + y ) P
1+ y
(
2
) t =1
t
t
t
• Convexity improves the duration approximation for bond price changes. The change in bond price
resulting from a change in bond yield can be more accurately approximated as:
∆P
1
2
≈ − D*∆y + Γ ( ∆y )
P
2
Fixed Income II
45
S
+⋯
Example
• A 30-year annual coupon bond with a coupon rate of 12% has Macaulay duration of 11.54 years
and convexity of 192.4. The bond currently sells at a yield to maturity of 8%. Its current price is
$1,450.31. Suppose its yield to maturity falls to 7%, the actual price of the bond becomes
$1,620.45.
• Actual ΔP = $1,620.45 - $1,450.31 = $170.14
• Using the Duration Rule, the predicted approximate price change is:
 D 
 11.54 
∆P ≈  −
 × ∆y × P0 =  −
 × (−0.01) × $1, 450.31 = $155.06
1
+
y
1.08




• Therefore, predicted new price ≈ $1,450.31 + $155.06 = $1,605.37
Fixed Income II
46
Example
• Using Duration-with-Convexity Rule, the predicted approximate price change is:

 1
  11.54 
 1
  D 
2 
2 
∆P ≈    −
×
∆
y
+
×
Γ
×
(
∆
y
)
×
P
==
−
×
(
−
0.01)
+
×
192.4
×
(
−
0.01)
 
 


 2
  0
  × $1, 450.31 = $168.99
1
+
y
2
1.08








 




• Predicted new price ≈ $1,450.31 + $168.99 = $1,619.30 and is closer to actual new price.
• The duration-with-convexity rule provides more accurate approximations to the true change in
price.
Fixed Income II
47
Convexity
• Semiannual coupon rate C%, T years to maturity, YTM quoted as a semiannual APR, Macaulay
Duration is:
2T
D=
1
 wt t
2 t =1
C
1
2 1
where wt =
for t = 1,K 2T − 1 and w2T =
t
2T
y P
y P


1 + 
1 + 
 2
 2
• The modified duration is:
C
2
100 +
D
D* =
1+
• The convexity of the bond is: Γ =
y
2
2T
1
y

4 1 + 
 2
2
 w t ( t + 1)
t
t =1
Fixed Income II
48
Convexity
• Bonds with greater curvature gain more
in price when yields fall than they lose
when yields rise.
• The more volatile interest rates, the
more attractive this asymmetry.
• Bonds with greater convexity tend to
have higher prices and/or lower yields,
all else equal.
From: Z. Bodie, A. Kane, A. Marcus and R. Jain, Investments
Fixed Income II
49
11/03/2023
Fixed Income II
50
Managing Interest Rate Risk
• Fluctuation of interest rate affects net worth (Assets!Liabilities) of many market participants. For
example,
1) Banks:
• Assets: long-term loans or mortgages, more sensitive to interest rate fluctuation.
• Liabilities: short-tem demand deposits with low duration.
2) Pension fund:
• Assets: bonds, stocks, properties, etc.
• Liabilities: promise future payments/obligations to retirees.
3) Insurance companies:
• Assets: bonds, stocks, properties, etc.
• Liabilities: promise future payments to policy holders.
Fixed Income II
51
Immunization
• Strategies to shield the overall financial status from exposure to interest rate fluctuations.
• For example:
o Banks liabilities are primarily the deposits with very short-term in nature and, hence, of low
duration.
o Banks assets such as mortgage or loans are largely long-term in nature and, hence, of longer
duration.
o A sudden rise in interest rates result in large fall in assets values than liability values, i.e.,
reduces banks’ net worth.
• To shield the risk, the bank can construct a portfolio that matches the interest rate exposure of
assets and liabilities such that price risk and reinvestment rate risk exactly cancel out and value
of assets will track the value of liabilities whether rates rise or fall.
Fixed Income II
52
Example
• The yield curve is flat at 8%. GIC issues a guaranteed investment contract for $10,000 with 5-year
maturity and guaranteed interest payment of 8% (a zero coupon bond selling at $10,000 with FV =
$14,693.28).
• Liability (obligation): GIC will need to pay 10,000 × 1.085 = $14,693.28 in year 5.
• Asset: to fund the its obligation, GIC invests $10,000 in 8% annual coupon bonds, selling at par, with 6
years to maturity. Macaulay duration of this 6-year maturity bonds is 5 years (4.9927 years).
• Investment in the coupon bonds faces 2 types of risks. Suppose interest rate rises:
o Market price risk: the coupon bond price falls
o Reinvestment risk: higher interest rate increases the reinvestment value of coupon payment
• So long as the duration of the coupon bonds equals the liability duration, price risk and reinvestment
risk will cancel out.
Fixed Income II
53
Example
• Duration matching balances the difference between
the accumulated value of the coupon payments and
the sale value of the bond.
• When the interest rates fall, the coupons grow
less but the higher value of the bond offsets this.
• When the interest rates rise, price of the bond
falls, but the coupons can be reinvested at the
higher rate which make up for the loss in bond
price.
Fixed Income II
Payment
Years Remaining
Accumulated
Number
until Obligation
Reinvestment Value
1
4
800 × 1.084 1088.39
2
3
800 × 1.083 1007.77
3
2
800 × 1.082 933.12
4
1
800 × 1.081 864.00
5
0
800 × 1.080 800.00
Sale of bond
0
10,800/1.08 10000.00
Value of coupon bond when rate remains at 8% 14693.28
1
4
800 × 1.074 1048.64
2
3
800 × 1.073 980.03
3
2
800 × 1.072 915.92
800 × 1.071 856.00
4
1
800 × 1.070 800.00
5
0
Sale of bond
0
10,800/1.07 10093.46
Value of coupon bond when rate falls to 7%
14694.05
1
4
800 × 1.094 1129.27
800 × 1.093 1036.02
2
3
3
2
800 × 1.092 950.48
4
1
800 × 1.091 872.00
5
0
800 × 1.090 800.00
Sale of bond
0
10,800/1.09 9908.26
Value of coupon bond when rate rises to 9%
14696.03
54
Example
• Duration matching balances the difference between
the accumulated value of the coupon payments and
the sale value of the bond.
• When the interest rates fall, the coupons grow
less but the higher value of the bond offsets this.
• When the interest rates rise, price of the bond
falls, but the coupons can be reinvested at the
higher rate which make up for the loss in bond
price.
Fixed Income II
Years Remaining
Accumulated Reinvestment
Payment Number until Obligation
Value
4
1
4
800 × 1.08
1088.39
3
2
3
800 × 1.08
1007.77
2
3
2
800 × 1.08
933.12
1
4
1
800 × 1.08
864.00
0
5
0
800 × 1.08
800.00
6
0
10,800/1.08 10000.00
Value of coupon bond when rate remains at 8%
14693.28
4
972.41
926.10
882.00
840.00
972.41
10285.71
14878.62
1
4
800 × 1.1
2
3
800 × 1.13
2
3
2
800 × 1.1
1
4
1
800 × 1.1
5
0
800 × 1.10
6
0
10,800/1.1
Value of coupon bond when rate rises to 10%
1171.28
1064.80
968.00
880.00
800.00
9818.18
14702.26
1
4
800 × 1.05
2
3
800 × 1.053
2
800 × 1.05
3
2
1
4
1
800 × 1.05
5
0
800 × 1.050
6
0
10,800/1.05
Value of coupon bond when rate falls to 5%
4
55
Macaulay Duration
• Macaulay Duration is therefore the weighted average of the times until each payment is received,
with the weights proportional to the present value of the payment.
• Quiz: find Macaulay Duration for a 4-year annual coupon bond with face value $100, YTM =
10%; ' = 4%.
P=
4
4
4
104
+
+
+
= 80.98
1
2
3
4
(1 + 0.1) (1 + 0.1) (1 + 0.1) (1 + 0.1)
T
D =  wt t =
t =1
4
1
4
1
4
1
104
1
×
×1 +
×
×2+
×
×3+
×
× 4 = 3.7465
1
2
3
4
(1 + 0.1) 80.98
(1 + 0.1) 80.98
(1 + 0.1) 80.98
(1 + 0.1) 80.98
Fixed Income II
56