International Fixed Income Topic IB: Fixed Income Basics - Risk.
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Transcript of International Fixed Income Topic IB: Fixed Income Basics - Risk.
International Fixed Income
Topic IB: Fixed Income Basics - Risk
Readings
• Duration: An Introduction to the Concept and Its Uses, (Dym & 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: "How much will the value of my portfolio change if interest rates go up 10 basis points?"
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
Maturity (years)
Yield
0 3010 20
Increase in interest rates
Decrease in interest rates
Price-Yield relation: Illustration
In general, the price of a bond is given by
P
Mc
r
M
rt
tt
T
T
T
/
( / ) ( / )/
2
1 2 1 221
2
2
P
Mc
y
M
yt
t
T
T
/
( / ) ( / )
2
1 2 1 21
2
2
But, if the yield curve is flat, then each of the spot rates must equal the bond’s yield y:
Result: y provides a complete description of the term structure
Zero Prices as a Function of Yield
0
20
40
60
80
100
0 2 4 6 8 10 12 14 16
Yield (%)
Price
30-year
10-year
5-year
Consider three zeros with maturities of 5,10 and 30 years. What do their prices look like as a function of their yields?
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
Yield
P*
Y* Y Y**
P**P
Y-Y**=Y*-Y, butP-P**<P*-P
The Effect of MaturityPercent decrease in price for a 100 basis point increase
25.17
24.53
24.01
23.35
9.21
8.97
8.74
8.5
4.72
4.59
4.46
4.34
0 5 10 15 20 25 30
3%
6%
9%
12%
Leve
l of
y
% change
30-year 10-year 5-year
The Effect of the Coupon RatePercent decrease price for a 100 basis point increase
25.17
24.53
24.01
23.35
15.74
12.83
10.25
8.21
14.1
11.5
9.31
7.61
0 5 10 15 20 25 30
3%
6%
9%
12%
Level of y
% change
30-yr zero 30-yr 5% 30-yr 10%
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.
rdurP
dur
$
i.e. ,$ ratein ChangePricein Change
Example
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.2
0.4
0.6
0.8
1
1.2
30-year spot rate
Price of 30-year zero
At a rate of 5%, the price is 0.2273If rates fall to 4%,the price is 0.3048
The actual change is 0.077
dr30
3060
1
1 2
( / )
100 bp
Using a linear approximation, the change is about 0.0665
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
65.6)2/05.01(
30
)2/1()('
2273.0)2/05.01(
1
)2/1(
1)(
6112
602
tt
tt
tt
tt
r
trd
rrd
To avoid working with negative numbers,the dollar duration is quoted in positive terms,that is, 6.65.
Dollar Duration: Example
What's the dollar duration of $1 par of a 1.5-year zero if the 1.5-year discount rate is 5.47%?
actual price rise is 0.0009432
346535.142
0547.122
)1(5.1
)1(
ttrt
The Approximation
3 6 9 120
0.020.040.060.08
Bond price
change
3 6 9 12
Level of y
Decrease in bond price per 1 basis point increase
ActualChange
Delta Approx
3 6 9 1202468
Bond price
change
3 6 9 12
Level of y
Decrease in bond price per 100 basis points increase
Actualchange
Delta approx
3 6 9 120
0.20.40.60.8
Bond price
change
3 6 9 12
Level of y
Decrease in bond price per 10 basis points increase
Actualchange
Delta approx
3 6 9 1206
121824
Bond price
change
3 6 9 12
Level of y
Decrease in bond price per 300 basis points increase
Actualchange
Delta approx
DurationDuration is a measure of the interest ratesensitivity of the bond that does not depend onscale or size. It is defined as the dollar durationscaled by the value of the bond:
For a t-year zero, we have
)1(Duration
2)1(1
)1(
22
122
tttr
ttr
r
tt
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
46.12/0547.01
5.1
)2/1(duration
46.19222.0
3465.1
price
durationdollar =duration
tr
t
Duration: Example Continued...
26.292/05.01
30
)2/1(duration
2273.0
65.6
price
durationdollar =duration
tr
t
At an interest rate of 5%, the duration of a30-year zero in our example is
Macaulay DurationNote that the duration of a zero is just slightlyless than its maturity. This measure of durationis 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 allowsus to describe duration in terms of the years thecash 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
jt
t
jj
j
jr
tK
112)2/1(
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 $13.40 for every 100 basis point shift in interest rates.
Portfolio Value as a Function of Interest Rate Shifts
Instead of thinking of the portfolio value as a function of all of the different discount rates, we can think of it as a function of just the change or shift s in all rates:
603 )2/)0500.0(1(
100
)2/)0547.0(1(
500)(
sssV
At s=0, V=483.85. At s=.005 (50 bp increase), V=477.41.
Dollar Duration and Its Derivative
n
jt
t
jjs
n
jt
t
j
j
j
j
j
r
tKsV
sr
KsV
1120
12
)2/1(|)('$dur
)2/)(1()(
The dollar duration of a portfolio is the sum of the dollar durations of the component zeroes.Just as with zeroes, the dollar duration of the portfolio value is related to its derivative. We take the derivative with respect to the change in interest rates, s, and evaluate it at level, s = 0:
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.
ExampleThe duration of the portfolio consisting of $500par of the 1.5-yr zero and $100 par of the 30-yrzero is
8.285.483
1340
uemarket val
durationdollar =duration
This means that the portfolio will change 2.8%for every 100 basis points change in rates.
Formula: Duration of a Portfolio
n
jtj
n
j t
jtj
j
j
j
dK
r
tdK
1
1 )2/1()(
=duration
The duration of the portfolio is the average duration of the component zeroes, weighted bytheir 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
8.273.22$12.461$
)26.2973.22($)46.112.461($
Macaulay Duration
n
jt
j
n
jjt
j
j
j
y
K
ty
K
12
12
)2/1(
)2/1(=durationMacaulay
The Macaulay duration of a portfolio is the averagematurity of each cash flow, weighted by its presentvalue at the yield on each security. [It is theModified Duration times (1+y/2)].
Coupon and Maturity Effects
5-year 10-year 30-year 15%10%
5%Zero
05
101520253035
Duration
5-year 10-year 30-year 15%10%
5%Zero
Maturity of the Bond
Macaulay Duration at current yield of 10%
15%
10%
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
Yieldyy** y*
Steeplysloped
Mildlyslope Almost
flat
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
Illustration
Yield
Actual price
y
Error in estimating pricebased only on duration
Slope at y,( i.e., dollar duration)
Price
Example
4.50% 5.00% 5.50% 6.00% 6.50% 7.00% 7.50% 8.00% 8.50%0.15
0.2
0.25
0.3
0.35
0.4
0.45
20-Year Discount Rate
20-Year Zero Price
d rr20 20
2040
1
1 2( )
( / )
approximation of price function around the point r = 6.5%
error
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
Recall from calculus that the value of a function can be approximated near a given point using its Taylor series around that point. Using only the first two derivatives, the Taylor series approximation is:
200000 )()(''
2
1)()(')()( xxxfxxxfxfxf
Derivatives!
convexitydollar )2/1(
2/)(''
durationdollar -)2/1(
)('
price)2/1(
1)(
22
2
12
2
tt
tt
tt
tt
tt
tt
r
ttrd
r
trd
rrd
Example
0043.107)2/065.01(
410
)2/1(
2/)(''
389364.5)2/065.01(
20
)2/1()('
278226.0)2/065.01(
1
)2/1(
1)(
4222
2
4112
402
tt
tt
tt
tt
tt
tt
r
ttrd
r
trd
rrd
For the 20-yr at 6.5%, we get:
The Convexity Correction
20,0,2
10,0,0, )()()()()()( tttttttttttt rrrdrrrdrdrd
Applying the Taylor series approximation, thechange in the zero price given a change in rates:
Change in price = -dollar duration x change in rates (1/2) x dollar convexity x change in rates squared
Example
How does the 20-year zero price change as its discount rate changes from 6.5% to 7.5%? The actual change is:
048888.0
278226.0229038.0
)065.0()075.0( 4040 )2/065.01(1
)2/075.01(1
2020
dd
Example continued...
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
Change in price = -dollar duration x change in rates -5.38964 x 0.01 = -0.538964
Duration approximation is far off
Duration/Convexity approximation does muchbetter
Summary
Rate (%) 20-Year Price Actual Change
Duration Approximation
Duration and Convexity
5.50 0.337852 0.059626 0.053894 0.0592446.50 0.2782267.50 0.229338 -0.048888 -0.053894 -0.048543
Duration/Convexity Approximations for 10-year zero
Decrease in bond price per 1 basis point increase in y
3% 6% 9% 12%0
0.02
0.04
0.06
0.08
Level of y
Bond price change
Actual ChangeGamma ApproxDelta Approx
Decrease in bond price per 100 basis points increase in y
3% 6% 9% 12%0
2
4
6
8
Level of y
Bond price change
Actual changeGamma approxDelta approx
Decrease in bond price per 10 basis points increase in y
3% 6% 9% 12%0
0.2
0.4
0.6
0.8
Level of y
Bond price change
Actual changeGamma approxDelta approx
Decrease in bond price per 300 basis points increase in y
3% 6% 9% 12%0
5
10
15
20
25
Level of y
Bond price change
Actual changeGamma approxDelta approx
Estimating Price Movements
3%
6%
9%
12%
6.85 7.22
6.84 5.00
5.27 4.99
3.69 3.88
3.68 2.74 2.87
2.73
0 2 4 6 8 10
Bond price change
3%
6%
9%
12%
Leve
l of
y
Decrease in bond price per 100 basis points increase in yield
Actual change Delta approx Gamma approx
Estimating Price Movements
3%
6%
9%
12%
18.57 21.67
18.20 13.60
15.80 13.34
10.04 11.63
9.87 7.48
8.62 7.35
0 5 10 15 20 25
Bond price change
3%
6%
9%
12%
Level of y
Decrease in bond price per 300 basis points increase in yield
Actual change Delta approx Gamma approx
Convexity
2
2
2
22
2
)2/1(
2/
)2/1(1
)2/1(2/
=convexity
value
convexitydollar =convexity
tt
t
tt
r
tt
r
rtt
To get a scale-free measure of curvature, convexityis defined as
Convexity of a zero is its maturity squared.
Example
Maturity Rate Price Dollar Duration
Duration Dollar Convexity
Convexity
10 6.00% 0.553676 5.375493 9.70874 54.7987 98.972620 6.50% 0.278226 5.389364 19.37046 107.0043 384.595130 6.40% 0.151084 4.391974 29.06977 129.8015 859.1356
10-, 20-, 30-yr zero:
Dollar Convexity of a Portfolio
n
jt
t
jjj j
jr
ttK
122
2
)2/1(
2/
Suppose the portfolio has cash flows K1,K2,K3,…at times t1,t2,t3,… then the dollar convexity is
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 convexitydivided by its value
n
jt
t
j
n
jt
t
jjj
j
j
j
j
r
K
r
ttK
12
122
2
)2/1(
)2/1(
2/
=convexity
Convexity of a Portfolio
Alternatively, market-weighted averageof convexities of zeroes
n
jt
t
j
n
j t
jjt
t
j
j
j
j
j
j
r
K
r
tt
r
K
12
12
2
2
)2/1(
)2/1(
2/
)2/1(=convexity
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:
32.478884,13938,13
)1356.859884,13()9726.98938,13(
Effects of Maturity, Coupons, and Yields
5yr 10yr 30yr0
200
400
600
800
1000
Convexity
5yr 10yr 30yr
Maturity of the Bond
Convexity at current yield of 10%
15%10%5%Zero
5yr 10yr 30yr0
200
400
600
800
1000
Convexity
5yr 10yr 30yr
Maturity of the Bond
Convexity at current yield of 5%
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
37.19884,13938,13
)06977.29884,13()70874.9938,13(=duration
Example
• The convexity of the bullet is 385.
• The convexity of the barbell is 478.
Value of Barbell and Bullet
-0.03 -0.02 -0.01 0 0.01 0.02 0.0310,000
20,000
30,000
40,000
50,000
60,000
Shift in Rates
Portfolio Value
Bullet
Barbell
bullet:
barbell:
V ss
V ss s
1 40
2 20 60
100 000
1 0 065 2
25174
1 0 06 2
91898
1 0 064 2
( ),
( ( . ) / )
( ),
( ( . ) / )
,
( ( . ) / )
Does the Barbell Always Do Better?
If there is an immediate parallel shift in interest rates, either up or down, then the barbell will outperform the bullet.If the shift is not parallel, anything could happen.If the rates on the bonds stay exactly the same, then as time passes the bullet will actually outperform the barbell:
the bullet will return 6.5%the barbell will return about 6.2%, a blend of the 6% and 6.4% on the 10- and 30-year zeroes.
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 $1,000,000 par value of a 7.5%-coupon 29-year bond.
• This liability has a duration of 12.58.
Mkt. Val. Of Liabilities
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
50,000
100,000
150,000
200,000
Maturity
Market Value
Example
• Construct an asset portfolio that has the same market value and duration as the liabilities using– a 12-year zero and– a 15-year zero.
Example
Coupon (%) Maturity (Years)
Par Value ($)
Market Value ($)
Dollar Duration
Duration
7.5 29 1000000 1151802 14486304 12.580 12 1 0.4784 5.5668 11.640 15 1 0.3881 5.6412 14.53
The table gives relevant information on the marketvalue and duration of the securities (using class’discount rates):
Example
Note that if the assets have the same market valueand dollar duration as the liability, then they hagethe same duration as the liability:
To construct the hedge portfolio, solve twoequations:Asset mkt. Val. = Liability mkt. ValAsset $ duration = Liability $ duration
Example
304,486,146412.55668.5
802,151,13881.04784.0
1512
1512
NN
NN
With N12 and N15 representing the par amountsof the 12- and 15-year zero, we have
Example Solution
• 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 ($1,151,802) and – the same dollar duration (14,486,304), and
therefore – the same duration (12.58), as the liability.
SOLUTION: N12=1,626,424; N15=962,969
Mkt. Val. Of Duration-Matched Portfolio
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
200,000
400,000
600,000
800,000
Maturity
Market Value
Performance of Hedge-100 bp -10 bp 0 +10 bp +100 bp
MARKET VALUEAssets 1306689 1166384 1151802 1137411 1016080Liabilities 1312293 1166433 1151802 1137458 1020267Net Equity -5604 -49 0 -47 -4188
DOLLAR DURATIONAssets 16538729 14678971 14486304 14296286 12699204Liabilities 17742932 14777840 14486304 14201643 11921307Net Equity -1204203 -98869 0 96462 777898
DURATIONAssets 12.66 12.59 12.58 12.57 12.50Liabilities 13.52 12.67 12.58 12.49 11.68
Duration/Convexity Hedge• The duration match performed well for small parallel
shifts in the yield curve, but not for large shifts.
• Also the durations and dollar durations of the assets changed with interest rates by different amounts.
• For large interest rate changes, the duration-matched hedge has to be rebalanced.
• A way to mitigate this problem is to match the convexity of assets and liabilities as well as duration and market value.
Example
• Consider structuring an asset portfolio that matches the convexity of the liabilities as well as their duration and market value.
• Use the following instruments for the asset portfolio.– a 2-year zero– a 15-year zero– a 25-year zero
Example
Note that if the assets have the same market value$ duration and $convexity as the liability, then they have the same duration and convexity as the liability:
To construct the hedge portfolio, solve threeequations:Asset mkt. Val. = Liability mkt. ValAsset $ duration = Liability $ durationAsset $ convexity = Liability $ convexity
Example
Coupon (%)
Maturity (Years)
Par Value ($)
Market Value ($)
Dollar Duration
Dollar Convexity
Duration Convexity
7.5 29 1000000 1151802 14486304 288068417 12.58 250.10
0 2 1 0.8972 1.7463 4.2489 1.95 4.74
0 15 1 0.3881 5.6412 84.7226 14.53 218.28
0 25 1 0.1977 4.7852 118.1290 24.20 597.48
Numbers are from class discount rates:
Example
For our example, the three equations become:
417,068,288129.1187226.842489.4
304,486,147852.46412.57463.1
802,151,11977.03881.08972.0
25152
25152
25152
NNN
NNN
NNN
The solution is:N2=497,576; N15=920,680; N25=1,760,379
Mkt. Val. Of Duraation/Convexity Matched Portfolios
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
100,000
200,000
300,000
400,000
500,000
Maturity
Market Value
Duration/Convexity Performance-100 bp -10 bp 0 +10 bp +100 bp
MARKET VALUEAssets 1312210 1166433 1151802 1137458 1020328Liabilities 1312293 1166433 1151802 1137458 1020267Net Equity -82 -0.07 0 0.07 60
DOLLAR DURATIONAssets 17716763 14777623 14486304 14201435 11904016Liabilities 17742932 14777840 14486304 14201643 11921307Net Equity -26169 -217 0 -208 -17290
DURATIONAssets 13.50 12.67 12.58 12.49 11.67Liabilities 13.52 12.67 12.58 12.49 11.68
Yield Curve Shift One Day Later
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Maturity
Change in Discount Rate
$ Duration Liabilities
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
Maturity
Dollar Duration
$ Duration of Duration Matched
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
Maturity
Dollar Duration
$ Duration of Dur/Conv. Matched
0.52.0
3.55.0
6.58
9.511
12.514
15.517
18.520
21.523
24.526
27.529
0
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
Maturity
Dollar Duration
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-convexity-matched 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.