Lecture 13: Hedging with duration and
convexity and review
Read Chapter 12
problems 1-5
Finance 688: Investment Administration
Professor John Chalmers
Duration and Convexity are risk management tools

Basic ideas are applicable to all assets

Often not analytically tractable, make heroic assumptions

Primary uses

Asset liability management (managing the firm’s exposure)
 Bank
managers manage loan portfolio risk
 insurance

Portfolio selection (in which bonds do we invest)
 risk
aversion of investors
 matching

company portfolios, pension fund portfolios
particular liabilities (a retirement plan)
Security selection (how to best implement a trading strategy)
 how
 e.g.
to best play information about interest rates
if you know rates are coming down long maturities? MBS?
Convexity helps the Estimates
P 1 P
1  2P
2

 y 

(

y
)
 error
2
P P y
2 P y
80
60
40
20
0
0
0.05
0.1
0.15
0.2
-20
-40
True Price
Modified Duration (7.22)
Duration + Convexity Price
0.25
Three portfolios
120.00
115.00
Portfolio Value
• Duration increases as
coupons decrease
• Convexity increases as
coupons decrease
• Suppose your
liabilities look like the
5% bond, what can we
do to hedge with the
other two portfolios?
110.00
105.00
100.00
95.00
90.00
85.00
8.00%
9.00%
Price 10%
10.00%
1.43 5%
11.00%
12.00%
2.59 Zeros
Ten year 10%, 5% and 0% bonds
109.84
106.42
103.14
100.00
96.99
94.11
91.35
1.44 Units
5% Bond 5% bond
77.04
111.19
74.33
107.29
71.75
103.56
69.28
100.00
66.92
96.59
64.66
93.34
62.51
90.23
2.59 Units
0% Bond
Zeros
44.23
114.72
42.24
109.56
40.35
104.66
38.55
100.00
36.84
95.57
35.22
91.35
33.67
87.33
8.50%
9.00%
9.50%
10.00%
10.50%
11.00%
11.50%
dP
-3.42
-3.28
-3.14
-3.01
-2.88
-2.76
dy
0.0050
0.0050
0.0050
0.0050
0.0050
0.0050
Yield Price Chg Yield Chg
8.50%
dP
dy
9.00%
-3.91
0.0050
9.50%
-3.73
0.0050
10.00%
-3.56
0.0050
10.50%
-3.40
0.0050
11.00%
-3.25
0.0050
11.50%
-3.11
0.0050
Yield Price Chg Yield Chg
8.50%
dP
dy
9.00%
-5.15
0.0050
9.50%
-4.90
0.0050
10.00%
-4.66
0.0050
10.50%
-4.43
0.0050
11.00%
-4.22
0.0050
11.50%
-4.01
0.0050
1st Der
-684.87
-655.65
-627.88
-601.48
-576.37
-552.48
1st der/P 1st Der
6.24
6.16
29.22
6.09
27.77
6.01
26.40
5.94
25.11
5.87
23.88
dP/dy
Duration
Chg
1st Der 1st der/P 1st Der
-781.15
7.03
-745.95
6.95
35.21
-712.54
6.88
33.41
-680.84
6.81
31.71
-650.73
6.74
30.10
-622.15
6.67
28.59
2nd Der
2nd Der/P
5844.23
5554.39
5280.51
5021.63
4776.86
54.92
53.85
52.81
51.77
50.76
d2P/dy2 Convexity
2nd Der 2nd Der/P
7041.12
6681.06
6341.21
6020.36
5717.34
65.63
64.51
63.41
62.33
61.25
dP/dy
Duration
Chg
d2P/dy2 Convexity
1st Der 1st der/P 1st Der 2nd Der 2nd Der/P
-1030.99
8.99
-980.26
8.95
50.73 10146.60
92.61
-932.24
8.91
48.02
9604.35
91.77
-886.77
8.87
45.47
9093.35
90.93
-843.71
8.83
43.06
8611.68
90.11
-802.92
8.79
40.79
8157.52
89.30
Neutral hedge
 The objective of a neutral hedge is to desensitize portfolio
value from changes in interest rates.
 In general, any hedging problem solves for the amount to
buy of various instruments that you can use to hedge. The
number of assets required to hedge with will be equal to
the number of dimensions on which you wish to hedge.
 If D is zero this implies that changes in interest rates will
have no impact on the value of your portfolio. This is
portfolio immunization. Depends on parallel shift
assumption.
 Suppose liability is 10% bond. Duration hedge with zero:
 Remember the duration of portfolio is weighted average of
the duration of the assets in the portfolio
Duration Hedge
Suppose liability is 10% bond. Duration hedge with zero:
 n10%  p10%
n0%  p0%
 D (10%) 
 D (0%)  0
 n10%  p10%  n0%  p0%
 n10%  p10%  n0%  p0%
 20  100
n0%  38.55
 6.09 
 8.91  0
 20  100  n0%  38.55
 20  100  n0%  38.55
n0% 
If
20  100  6.09
 35.46
38.55  8.91
y changes from 10% to 8.5% then the actual change in portfolio value will be
 20  (109.84  100)  35.46  ( 44.23  38.55)
 196.80  201.41  4.6
Duration and Convexity Hedge
 Match the duration of your portfolio along with the
convexity of the portfolios
V  n10%  p10%  n0%  p0%  n5%  p5%
 n10%  p10%
n p
n p
 D(10%)  0% 0%  D(0%)  5% 5%  D(5%)  0
V
V
V
n
p
n p
n p
C : 10% 10%  C (10%)  0% 0%  C (0%)  5% 5%  C (5%)  0
V
V
V
D : 20  100  6.09  n0%  38.55  8.91  n5%  69.28  6.88  0
D:
C : 20  100  53.85  n0%  38.55  91.77  n5%  69.28  64.51  0
n0%  20.5
n5%  40.34
If
y  8.5% then change in value of portfolio will be
 20  (109.84  100)  20.5  (44.23  38.55)  40.34  (77.04  69.28)
 196.80  116.44  313.04  .20
Bullet versus Barbell Hedge
 Bullet effectively matches duration with assets of maturity
similar to the asset that is being hedged. For example
hedge a bond with 6 year duration with 6 year zero.
 Barbell matches duration with bonds with very different
maturities. For example, hedge a 6 year duration bond
with a 1 year zero and a 13 year zero.
 Bullet hedges will come closer to matching duration and
convexity than a barbell hedge. The barbell will have
higher convexity, which is fine if rates are a changing.
Summary
• Hedging with duration and convexity
• This is useful in many contexts, including the
corporate managers, portfolio managers and
business line people.
• Duration and PVBP are the crudest but most often
encountered measures of price sensitivity
• The topics on the exam.