# Example ```International Fixed Income
Topic IB: Fixed Income Basics Risk
• Duration: An Introduction to the Concept
and Its Uses, (Dym &amp; Garbade, Bankers
Trust (1984))
• Convexity: An Introduction, (Yawitz,
Goldman Sachs)
Outline
II. Interest rate risk
A. Interest rate sensitivity - Summary
B. Duration
C. Convexity
D. Hedging
A. Interest Rate Sensitivity
• Values of fixed income securities change as
economic conditions change.
• Even though bond prices are not perfectly
correlated, they tend to move together. People try
and relate bond prices to a single variable, “level
of interest rates”.
• They want simple answers to questions like:
&quot;How much will the value of my portfolio change
if interest rates go up 10 basis points?&quot;
Price-Yield Relation
• For zeroes, there is a very explicit formula relating
the price to its discount rate or yield.
• For coupon bonds, or portfolios with fixed cash
flows, we have a formula that gives the price as a
function of all the discount rates associated with
the cash flows. Alternatively, we have a formula
that gives price as a function of yield.
• For other instruments, there is no explicit formula
relating price to interest rates. Instead, they
require a model which incorporates both interest
rates and estimates of volatility.
Parallel Shifts
Yield
Increase in interest rates
Decrease in interest rates
0
10
20
30
Maturity (years)
Price-Yield relation: Illustration
In general, the price of a bond is given by
2T
Mc / 2
M

t
2T
1

(
r
/
2
)
1

(
r
/
2
)




t 1
t /2
T
P
But, if the yield curve is flat, then each of the spot
rates must equal the bond’s yield y:
2T
Mc / 2
M

t
1  ( y / 2)2 T
t 1 1  ( y / 2) 
P
Result: y provides a complete description of the
term structure
Zero Prices as a Function of Yield
Consider three zeros with maturities of 5,10 and 30
years. What do their prices look like as a function of
their yields?
100
80
30-year
60
10-year
Price
40
5-year
20
0
0
2
4
6
8
Yield (%)
10
12
14
16
Terminology
• Delta - measures how the price (i.e., bond)
changes as the underlying (i.e., interest rate)
changes.
• Gamma - measures how the Delta changes
as the underlying (i.e., interest rate)
changes.
Characteristics of the Price/Yield Relation
• The higher the yield, the lower the price (Delta is
negative)
• The higher the yield, the smaller the magnitude of
delta (Prices are convex in the yield, i.e., Gamma
is positive)
• The longer the maturity, the higher the magnitude
of delta (longer maturity bonds are more sensitive
to interest rate changes than shorter maturity
bonds)
The Effect of Convexity
Price
Y-Y**=Y*-Y, but
P-P**&lt;P*-P
P*
P
P**
Y*
Y
Y**
Yield
The Effect of Maturity
Percent decrease in price for a 100 basis point increase
4.34
Level of y
12%
4.46
9%
4.59
6%
4.72
3%
0
5
8.5
23.35
8.74
24.01
8.97
24.53
9.21
25.17
10
15
20
% change
30-year
10-year
5-year
25
30
The Effect of the Coupon Rate
Percent decrease price for a 100 basis point increase
7.61
8.21
12%
23.35
9.31
10.25
9%
Level of y
11.5
6%
24.01
12.83
14.1
3%
0
5
10
24.53
15.74
15
25.17
20
% change
30-yr zero
30-yr 5%
30-yr 10%
25
30
General Conclusions
• Level effect: Regardless of the coupon rate, the
magnitude of delta is decreasing in yield
(sensitivity is greater when yields are low)
• Maturity effect: Regardless of the coupon rate,
the magnitude of delta is increasing in maturity
(longer maturities are more sensitive)
• Coupon effect: The lower the coupon, the more
sensitive the price to changes in interest rates
Outline
II. Interest rate risk
A. Interest rate sensitivity - Summary
B. Duration
C. Convexity
D. Hedging
B. Duration
• Loose Definition: The duration of a bond is an
approximation of the percent change in its price given a
100 basis point change in interest rates.
• For example, a bond with a duration of 7 will gain about
7% in value if interest rates fall 100 bp.
• For zeroes, this measure is easy to define and compute
with a formula.
• For securities with fixed cash flows, we must make
assumptions about how rates shift together.
• To compute duration for other instruments requires further
assumptions and numerical estimation.
Dollar Duration of Zeroes
• Definition: The dollar duration of a zero-coupon
bond is a linear approximation of the dollar
change in its price divided by the change in its
discount rate. Because \$dur is essentially the
derivative of the bond price with respect to the
interest rate, we often call it the bond's Delta.
\$dur   
Change in Price
Change in rate
P  \$dur  r
, i.e.
Example
P
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3
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1
.
2
1
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.
8
1
d

3
0
6
0
(/
1

r
2
)
3
0
0
.
6
0
.
4
0
.
2
Using a linear
approximation,
the change is
A
t
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5
%
,
t
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0
.
2
2
7
3
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%
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4
8
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0
7
7
100 bp
0
00
.
0
1
0
.
0
2
0
.
0
3
0
.
0
4
0
.
0
5
0
.
0
6
0
.
0
7
0
.
0
8
0
.
0
9
0
.
1
3
0
y
e
a
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s
p
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a
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e
Example: Dollar Duration
• The dollar duration of \$1 par of a 30-year zero at
an interest rate of 5% is 6.65, as illustrated in the
last slide.
• -0.0665/(-0.01)=0.0665/0.01=6.65.
• The illustration shows that the dollar duration is
related to the slope of the price-rate function.
• We can use calculus to get an explicit formula for
the dollar duration of any zero.
Dollar Duration: Formula
1
1
d t (rt ) 

 0.2273
2t
60
(1  rt / 2)
(1  0.05 / 2)
t
 30
d t ' (rt ) 

 6.65
2 t 1
61
(1  rt / 2)
(1  0.05 / 2)
To avoid working with negative numbers,
the dollar duration is quoted in positive terms,
that is, 6.65.
Dollar Duration: Example
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4
7
%
?
t
rt 2 t 1
(1 2 )
 (1 .0547)4  1.346535
1.5
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2
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actual price rise is 0.0009432
The Approximation
Decrease in bond price per 1 basis point
increase
Bond
price
change
0.08
0.06
0.04
0.02
0
Act ual
Change
3
6
9
12
Decrease in bond price per 10 basis point s
increase
Bond
price
change
Delt a Approx
0.8
0.6
0.4
0.2
0
Act ual
change
3
Decrease in bond price per 100 basis point s
increase
Bond
price
change
Act ual
change
Delt a approx
3
6
9
Level of y
12
9
12
Delt a approx
Level of y
Level of y
8
6
4
2
0
6
Decrease in bond price per 300 basis point s
increase
Bond
price
change
24
18
12
6
0
Act ual
change
3
6
9
Level of y
12
Delt a approx
Duration
Duration is a measure of the interest rate
sensitivity of the bond that does not depend on
scale or size. It is defined as the dollar duration
scaled by the value of the bond:
For a t-year zero, we have
t
Duration 
(1 2t ) 2 t 1
r
1
r
(1 2t ) 2 t
t

rt
(1  2 )
Duration: Example
• Duration approximates the percent change
in price for a 100 basis point change in rates
• For example, at an interest rate of 5.47%,
the duration of the 1.5-year zero is
dollar duration 1.3465
duration =

 1.46
price
0.9222
t
1.5
 duration 

 1.46
(1  rt / 2) 1  0.0547 / 2
Duration: Example Continued...
At an interest rate of 5%, the duration of a
30-year zero in our example is
dollar duration
6.65
duration =

price
0.2273
t
30
 duration 

 29.26
(1  rt / 2) 1  0.05 / 2
Macaulay Duration
Note that the duration of a zero is just slightly
less than its maturity. This measure of duration
is known as MODIFIED duration.
This is to distinguish itself from another measure
of duration, MACAULAY duration, which equals:
MODIFIED(1+r/2)=t years.
Macaulay duration is popular because it allows
us to describe duration in terms of the years the
cash flows of the bond will be around.
Duration of a Portfolio of Cash Flows
• Definition: The dollar duration of a portfolio approximates
the dollar change in portfolio value divided by the change
in interest rates, assuming all rates change by the same
amount.
• It follows that the dollar duration of a portfolio is the sum
of the dollar durations of each of the cash flows in the
portfolio.
• Why? The change in the portfolio value is the sum of the
changes in the value of each cash flow.
– The dollar duration of each cash flow describes its value change.
– The sum of all the dollar durations describes the total change.
Formula
• Suppose the portfolio has cash flows K1,
K2, K3,... at times t1, t2, t3,.... Then its
dollar duration would be
n
K j t j
 (1  r
j 1
t
j
/ 2)
2 t j 1
Example
• What is the dollar duration of a portfolio of
consisting of \$500 par of the 1.5-year zero
and \$100 par of the 30-year zero?
– (500 x 1.35) + (100 x 6.65) = 1340
– This means the portfolio value will change
about \$13.40 for every 100 basis point shift in
interest rates.
Portfolio Value as a Function of Interest Rate
Shifts
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:
500
100
V ( s) 

3
(1  (0.0547  s) / 2) (1  (0.0500  s) / 2) 60
At s=0, V=483.85. At s=.005 (50 bp
increase), V=477.41.
Dollar Duration and Its Derivative
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s
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:
n
V ( s)  
j 1
Kj
(1  (rt j  s) / 2)
n
\$dur  V ' ( s) |s 0  
j 1
2t j
K j t j
(1  rt j / 2)
2 t j 1
Duration of a Portfolio
• Just as with a zero, the duration of a
portfolio is its dollar duration divided by its
market value.
• The duration gives the percent change in
value for each 100 basis point change in all
rates.
Example
The duration of the portfolio consisting of \$500
par of the 1.5-yr zero and \$100 par of the 30-yr
zero is
dollar duration
1340
duration =

 2.8
market val ue
483.85
This means that the portfolio will change 2.8%
for every 100 basis points change in rates.
Formula: Duration of a Portfolio
n
 (K
duration =
j 1
j
 dt j ) 
n
K
j 1
j
tj
(1  rt j / 2)
 dt j
The duration of the portfolio is the average
duration of the component zeroes, weighted by
their market values.
Example
• Recall the portfolio consisting of \$500 par of the
1.5-year zero and \$100 par of the 30-year zero.
– The market value of the 1.5-year zero is 500 x 0.92224
= \$461.12. Its duration is 1.46.
– The market value of the 30-year zero is 100 x 0.2273 =
\$22.73. Its duration is 29.26.
• The duration of the portfolio is
(\$461.12 1.46)  (\$22.73  29.26)
 2.8
\$461.12  \$22.73
Macaulay Duration
Kj
n
Macaulay duration =
 (1  y / 2)
j 1
n
2t j
Kj
 (1  y / 2)
j 1
t j
2t j
The Macaulay duration of a portfolio is the average
maturity of each cash flow, weighted by its present
value at the yield on each security. [It is the
Modified Duration times (1+y/2)].
Coupon and Maturity Effects
Macaulay Duration at current yield of 10%
35
30
25
20
Duration
15
10
5
0
15%
10%
5-year
10-year
30-year
Maturity of the Bond
10%
15%
Zero
5%
5%
Zero
Problems with Duration
• Accuracy: duration is accurate only for
small yield changes.
• Applicability: duration begins to break
down for nonparallel shifts in the yield
curve.
• Generality: duration is only valid for
option-free bonds.
Outline
II. Interest rate risk
A. Interest rate sensitivity - Summary
B. Duration
C. Convexity
D. Hedging
C. Convexity
• Convexity is a measure of the curvature of the
value of a security or portfolio as a function of
interest rates. It tells you how the duration changes
as interest rates change.
• As its name suggests, convexity is related to the
second derivative if the price function. As such, it
is often called a bond's Gamma.
• Using convexity together with duration gives a
better approximation of the change in value given
a change in interest rates than using duration
alone.
Illustration
As y changes to y** (y*), the slope of
the bond pricing function increases
(decreases). This slope is simply the
dollar duration of the bond.
Price
Steeply
sloped
Mildly
slope
y**
Almost
flat
y
y* Yield
Illustration
Price
Error in estimating price
based only on duration
Actual
price
Slope at y,( i.e.,
dollar duration)
y
Yield
Example
2
0
Y
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4
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0
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(
r
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
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2
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1
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/
2
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Correcting the Duration Error
• The price-rate function is not linear.
• Duration and dollar duration use a linear
approximation to the price rate function to
measure the change in price given a change
in rates.
• The error in the approximation can be
substantially reduced by making a
convexity correction.
Taylor Series
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:
1
2
f ( x)  f ( x0 )  f ' ( x0 )  ( x  x0 )  f ' ' ( x0 )  ( x  x0 )
2
Derivatives!
1
d t (rt ) 
 price
2t
(1  rt / 2)
t
d t ' (rt ) 
 - dollar duration
2 t 1
(1  rt / 2)
t2  t / 2
d t ' ' (rt ) 
 dollar convexity
2t  2
(1  rt / 2)
Example
For the 20-yr at 6.5%, we get:
1
1
d t (rt ) 

 0.278226
2t
40
(1  rt / 2)
(1  0.065 / 2)
t
 20
d t ' (rt ) 

 5.389364
2 t 1
41
(1  rt / 2)
(1  0.065 / 2)
t2  t / 2
410
d t ' ' (rt ) 

 107.0043
2t  2
42
(1  rt / 2)
(1  0.065 / 2)
The Convexity Correction
Applying the Taylor series approximation, the
change in the zero price given a change in rates:
dt (rt )  dt (rt ,0 )  dt(rt ,0 )  (rt  rt ,0 )  dt(rt ,0 )  (rt  rt ,0 )
1
2
Change in price = -dollar duration x change in rates
(1/2) x dollar convexity x change in rates squared
2
Example
H
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6
.
5
%
t
o
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5
%
?
T
h
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a
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c
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e
i
s
:
d 20 (0.075)  d 20 (0.065) 
1
(1 0.075/ 2 ) 40
1
 (1 0.065
/ 2 ) 40
 0.229038  0.278226
 0.048888
Example continued...
Change in price = -dollar duration x change in rates
-5.38964 x 0.01 = -0.538964
Duration approximation is far off
Change in price = -dollar duration x change in rates
(1/2) x dollar convexity x change in rates squared
-0.538964 + [(1/2) x 107.0043 x 0.0001]=-0.048543
Duration/Convexity approximation does much
better
Summary
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8
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.
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3
Duration/Convexity Approximations for 10-year zero
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6
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Estimating Price Movements
Decrease in bond price per 100 basis points increase in yield
2.73
2.87
2.74
Level of y
12%
3.68
3.88
3.69
9%
4.99
5.27
5.00
6%
6.84
7.22
6.85
3%
0
2
4
6
8
Bond price change
Actual change
Delta approx
Gamma approx
10
Estimating Price Movements
Decrease in bond price per 300 basis points increase in yield
7.35
12%
8.62
7.48
9.87
11.63
10.04
13.34
9%
Level of y
6%
13.60
15.80
18.20
3%
0
18.57
5
10
15
20
Bond price change
Actual change
Delta approx
Gamma approx
21.67
25
Convexity
To get a scale-free measure of curvature, convexity
is defined as
dollar convexity
convexity =
value
t2  t / 2
(1  rt / 2) 2t  2
t2  t / 2
convexity =

1
(1  rt / 2) 2
2t
(1  rt / 2)
Convexity of a zero is its maturity squared.
Example
10-, 20-, 30-yr zero:
M
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1
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.
1
3
5
6
Dollar Convexity of a Portfolio
Suppose the portfolio has cash flows K1,K2,K3,…
at times t1,t2,t3,… then the dollar convexity is
n
K

 j
j 1
tj tj /2
2
(1  rt j / 2)
2t j  2
Example
• Consider a portfolio consisting of
– \$25,174 par value of the 10-year zero
– \$91,898 par value of the 30-year zero.
• The dollar convexity of the portfolio is
– (25,174 x 54.7987) + (91,898 x 129.8015) =
13,307,997
Convexity of a Portfolio
The convexity of a portfolio is dollar convexity
divided by its value
K
convexity =
j 1
tj tj /2
2
n
j
n

(1  rt j / 2)
Kj
 (1  r
j 1
2t j  2
tj
/ 2)
2t j
Convexity of a Portfolio
Alternatively, market-weighted average
of convexities of zeroes
n
convexity =
Kj
 (1  r
j 1
tj
n
/ 2)
2t j

(1  rt j / 2)
Kj
 (1  r
j 1
tj tj /2
2
tj
/ 2)
2t j
2
Example
• Consider the portfolio of 10- and 30-year
zeroes.
– The 10-year zeroes have market value
• \$25,174 x 0.553676 = \$13,938.
– The 30-year zeroes have market value
• \$91,898 x 0.151084 = \$13,884.
– The market value of the portfolio is \$27,822.
• The convexity of the portfolio is
– 13,307,997/27,822 = 478.32.
Example continued...
• Alternatively, the convexity of the portfolio
is the average convexity of each zero
weighted by market value:
(13,938  98.9726)  (13,884  859.1356)
 478.32
13,938  13,884
Effects of Maturity, Coupons, and Yields
Convexity at current yield of 5%
Convexity
Convexity at current yield of 10%
1000
1000
800
800
600
400
200
0
5yr
10yr 30yr
Maturity of
the Bond
15%
10%
5%
Zero
Convexity
600
400
200
0
5yr
10yr 30yr
Maturity of
the Bond
15%
10%
5%
Zero
Barbells and Bullets
• We can construct a portfolio of a long-term
and short-term zero (a barbell) that has the
same market value and duration as an
intermediate-term zero (a bullet).
• The barbell will have more convexity.
Example
• Bullet portfolio: \$100,000 par of 20-year zeroes
– market value = \$100,000 x 0.27822 = 27,822
– duration = 19.37
• Barbell portfolio: from previous example
– \$25,174 par value of the 10-year zero
– \$91,898 par value of the 30-year zero.
– market value = 27,822
(13,938  9.70874)  (13,884  29.06977)
duration =
 19.37
13,938  13,884
Example
• The convexity of the bullet is 385.
• The convexity of the barbell is 478.
Value of Barbell and Bullet
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Does the Barbell Always Do Better?
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Outline
II. Interest rate risk
A. Interest rate sensitivity - Summary
B. Duration
C. Convexity
D. Hedging
D. Hedging Interest Rate Risk
• Suppose you have liabilities or obligations
consisting of a stream of fixed cash flows
you must pay in the future.
• How can you structure an asset portfolio to
fund these liabilities?
Dedication
• The only completely riskless approach is to
construct an asset portfolio with cash flows
that exactly match the liability cash flows.
• This funding method is called dedication.
• This approach may be infeasible or
excessively costly.
• In some situations, risk managers may want
more flexibility.
Immunization
• Consider a more flexible but more risky
approach, called immunization.
– The liabilities have a certain market value.
– That market value changes over time as interest
rates change.
– Construct an asset portfolio with the same
market value and the same interest rate
sensitivity as the liabilities so that the asset
value tracks the liability value over time.
Immunization continued...
• If the assets and liabilities have the same
market value and interest rate sensitivity,
the net position is said to be hedged or
immunized against interest rate risk.
• The approach can be extended to settings
with debt instruments that do not have fixed
cash flows.
Duration Matching
• The most common form of immunization
matches the duration and market value of
the assets and liabilities.
• This hedges the net position against small
parallel shifts in the yield curve.
Example
• Suppose the liabilities consist of \$1,000,000
par value of a 7.5%-coupon 29-year bond.
• This liability has a duration of 12.58.
Mkt. Val. Of Liabilities
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2
3
2
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2
9
M
a
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u
r
i
t
y
Example
• Construct an asset portfolio that has the
same market value and duration as the
liabilities using
– a 12-year zero and
– a 15-year zero.
Example
The table gives relevant information on the market
value and duration of the securities (using class’
discount rates):
M
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\$
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3
8
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1
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6
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2
1
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5
3
Example
Note that if the assets have the same market value
and dollar duration as the liability, then they hage
the same duration as the liability:
To construct the hedge portfolio, solve two
equations:
Asset mkt. Val. = Liability mkt. Val
Asset \$ duration = Liability \$ duration
Example
With N12 and N15 representing the par amounts
of the 12- and 15-year zero, we have
0.4784 N12  0.3881N15  1,151,802
5.5668 N12  5.6412 N15  14,486,304
Example Solution
SOLUTION: N12=1,626,424; N15=962,969
• In other words, the immunizing asset
portfolio consists of \$1,626,424 face value
of 12-year zeroes and \$962,969 face value
of 15-year zeroes. By construction it has
– the same market value (\$1,151,802) and
– the same dollar duration (14,486,304), and
therefore
– the same duration (12.58), as the liability.
Mkt. Val. Of Duration-Matched Portfolio
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Performance of Hedge
1
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8
Duration/Convexity Hedge
• The duration match performed well for small
parallel shifts in the yield curve, but not for large
shifts.
• Also the durations and dollar durations of the
assets changed with interest rates by different
amounts.
• For large interest rate changes, the durationmatched hedge has to be rebalanced.
• A way to mitigate this problem is to match the
convexity of assets and liabilities as well as
duration and market value.
Example
• Consider structuring an asset portfolio that
matches the convexity of the liabilities as
well as their duration and market value.
• Use the following instruments for the asset
portfolio.
– a 2-year zero
– a 15-year zero
– a 25-year zero
Example
Note that if the assets have the same market value
\$ duration and \$convexity as the liability, then
they have the same duration and convexity as the liability:
To construct the hedge portfolio, solve three
equations:
Asset mkt. Val. = Liability mkt. Val
Asset \$ duration = Liability \$ duration
Asset \$ convexity = Liability \$ convexity
Example
Numbers are from class discount rates:
C
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Example
For our example, the three equations become:
0.8972 N 2  0.3881N15  0.1977 N 25  1,151,802
1.7463N 2  5.6412 N15  4.7852 N 25  14,486,304
4.2489 N 2  84.7226 N15  118.129 N 25  288,068,417
The solution is:
N2=497,576; N15=920,680; N25=1,760,379
Mkt. Val. Of Duraation/Convexity Matched
Portfolios
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Duration/Convexity Performance
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Yield Curve Shift One Day Later
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M
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\$ Duration Liabilities
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\$ Duration of Duration Matched
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\$ Duration of Dur/Conv. Matched
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M
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Effect of Yield Curve Shift
• The average change in rates was +1 bp.
• If the interest rate shift had been parallel, dollar
duration of 14,486,304 would have predicted a
change of -14,486,304 x 0.0001 = -\$1449 in the
value of the liability and each asset portfolio.
• The actual change in the liability was -\$2126
• The dollar duration of the liability is concentrated
on year 29. The 29-year discount rate increased 2
bp.
Effect of Actual Yield Curve Shift
• The value of the duration-matched portfolio
changed by only \$-889.
– The 12-year discount rate did not change at all.
– The 15-year discount rate rose 2 bp.
• Net equity under this immunization would
have increased to \$1237.
Effect of Actual Yield Curve Shift
• The value of the duration-convexitymatched portfolio changed by \$-3365.
– Most of its dollar duration was on year 25.
The 25-year discount rate rose 3 bp.
• Net equity under this immunization would
have fallen to -\$1239.
Lesson
• Duration or duration-convexity matching
hedges against parallel shifts of the yield
curve.
• To hedge against other shifts, the cash flows
of the assets and liabilities must have
similar exposure to different parts of the
yield curve.
```