Treasury bond
futures: pricing
and applications
for hedgers,
speculators, and
arbitrageurs
Galen Burghardt
Taifex/Taiwan
7 June 2004
Part I: Contract structure and pricing
• Futures contract specifications
• Deliverable Treasury issues
• Pricing Treasury futures relative to deliverable
cash bonds
• Components of the basis
• The short’s delivery options
• Fair value of a futures contract
Overview of Treasury futures
• Key contract specifications
• Supplies of deliverable Treasury bonds and
notes
• Shifts in trading and open interest
Key contract specifications
Long-Term U.S.
Treasury Bonds
Size
Contact Grade
Conversion
Factor
$100,000 par value
10-Year U.S. Treasury
Notes
$100,000 par value
5-Year U.S. Treasury
Notes
$100,000 par value
U.S. Treasury Notes
that have an original
maturity of not more
than 5 years and 3
months and a
remaining maturity of
U.S. Treasury bonds
U.S. Treasury Notes
not less than 4 years
with at least 15 years
maturing at least 6.5 years
and 3 months as of the
remaining to maturity if but not more than 10
first calendar day of the
not callable as of the
years from the first
delivery month. The 5first calendar day of the calendar day of the
year Treasury Note
delivery month
delivery month
issued after the last
trading day of the
contract month will not
be eligible for delivery
into that months
contract.
2-Year U.S. Treasury
Notes
$200,000 par value
U.S. Treasury Notes
that have an original
maturity of not more
than 5 years and 3
months and a remaining
maturity of not less than
1 year and 9 months
from the first day of the
calendar month but not
more than 2 years from
the last day of the
calendar
The hypothetical price per dollar of face value at which the bond would yield 6% to maturity, or to first
call if callable, as of the first day of the contract month
Price Quotes
Points and 32nds of a
point
Points and 1/2 of 32nds of
a point
Points and 1/2 of
32nds of a point
Points and 1/4 of 32nds
of a point
Tick Size & Value
1/32 of a point
($31.25)
1/2 of 1/32 of a point
($15.625)
1/2 of 1/32 of a point
($15.625)
1/4 of 1/32 of a point
($15.625)
Trading days and hours
Bonds, 10s, and 5s
2s
3 points
1 point
7:20 am to 2:00 pm (Open Outcry)
7:00 pm - 4:00 pm (Electronic)
7:20 am to 2:00 pm (Open Outcry) 7:00
pm - 4:00 pm (Electronic)
March, June, September, December
March, June, September, December
Last Trading Day
7th business day preceding the last
business day of the delivery month
The earlier of (1) the second business
day prior to the issue day of the 2-year
note auctioned in the current month, or
(2) the last business day of the
calendar month
First Delivery Day
First business day of delivery month
First business day of delivery month
Last Delivery Day
Last business day of delivery month
The third business day following the
last trading day
Daily Price Limit
Trading Hours
(Chicago Time)
Delivery Months
Deliverable Treasury bonds
Exhibit 2.2 Supplies of Deliverable Treasury Bonds
(May 24, 2004)
50
45
5.6%
Stripped
Unstripped
5.5%
CTD
35
30
25
5.4%
20
15
5.3%
10
5
0
Jul-16
5.2%
Jul-18
Jul-20
Jul-22
Jul-24
Maturity
Jul-26
Jul-28
Jul-30
Yields
Amount outstanding
(billions)
40
Deliverable Treasury 10-year notes
Supplies of Deliverable 10-year Treasury Notes
(May 24, 2004)
40
4.8
35
4.7
30
4.6
25
4.5
20
4.4
15
4.3
10
4.2
5
4.1
0
Feb-11
4.0
Feb-12
Feb-13
Maturity
Feb-14
Yield (%)
Amount outstanding
(billions)
CTD
Shifts in trading and open interest
Trading Volume, CBOT Treasury Bond and 10-Year Note Futures
12,500,000
October '97
7,500,000
5,000,000
serutuF etoN raeY-01 dna dnoB yrusaerT TOBC ,tseretnI nepO
2,500,000
000,002,1
0
Jan-01
Jan-99
Jan-97
Jan-95
Jan-93
Jan-91
Jan-89
Jan-87
Jan-85
Jan-83
Jan-81
Jan-79
89' enuJ
Jan-77
Contracts per Month
10,000,000
000,004
0
htnoM fo dnE ta tseretnI nepO
000,008
77-naJ
97-naJ
18-naJ
38-naJ
58-naJ
78-naJ
98-naJ
19-naJ
39-naJ
59-naJ
79-naJ
99-naJ
10-naJ
Pricing Treasury futures
• Identifying the cheapest to deliver
• The short’s delivery options
• Pricing and hedging tool
Treasury Pricing & Hedging Tool
Jun-04/late delivery
Key Dates(days*)
Futures
5/24/04
5/25/04
5/26/04
6/1/04
6/21/04
6/30/04
Closing
Trade
Settlement
1st Delivery
Last Trade
Last Delivery
2 Year
(-2)
(-1)
(0)
(6)
(26)
(35)
6/30/04
7/6/04
(35)
(41)
* days from settlement
Assumptions
Yield volatility
Yield distribution
Yield beta coefficient
Twist volatility coefficient
Delivery date
# B/N
2-year
5-year
104.34% 37.36%
normal
normal
0.0272
0.0272
12.3243 6.4856
7/6/04 6/30/04
10-year
19.00%
normal
0.0272
3.0143
6/30/04
Bond
17.19%
normal
0.0272
0.4517
6/30/04
(Contract Jun-04)
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Full $
Price
98.433
99.612
98.966
104.044
99.766
Term
Repo
0.800
0.500
0.821
0.850
0.500
Carry
Carry $/Day
19.1097
47.3063
31.7781
101.1133
46.1197
32-Del
2.51
6.21
4.17
13.27
6.05
0.781
-0.647
-2.814
-0.567
-1.854
Prob. Of
CTD
96.72%
0.00%
0.00%
0.00%
3.28%
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
1 T
1.5
2 T02 2.25
3 T
2
4 T
4.625
5 TWI
2.5
3/31/06
4/30/06
5/15/06
5/15/06
5/31/06
2:00 PM
Price-32nd
98-06+
99-14+
98-29
103-29
99-24+
2.5010
2.5418
2.5727
2.5783
2.6201
Jun-04
Factor
0.9263
0.9358
0.9286
0.9754
0.9375
Basis
32nd
2.575
10.372
17.278
18.638
14.609
Basis NOC
Actual
Expected
0.07
0.07
4.17
4.17
13.11
13.11
5.37
5.38
8.56
7.47
6 TBA
2.5
6/30/06
99-24
2.6183
0.9350
22.869
15.71
15.70
20.242
2.01
0.9963
100.769
0.500
54.6018
7.16
-3.784
0.00%
3.25
3.125
3.125
3.375
3.375
8/15/08
9/15/08
10/15/08
11/15/08
12/15/08
98-13
97-23
97-17
98-14
98-07
3.6599
3.7024
3.7396
3.7578
3.8043
0.8999
0.8935
0.8917
0.8994
0.8978
8.799
9.132
9.413
11.544
10.127
0.96
1.61
2.38
3.57
2.26
1.68
2.32
3.09
4.29
2.98
38.447
38.937
39.554
40.428
41.033
3.87
3.96
4.04
4.10
4.11
1.0165
1.0140
1.0118
1.0102
1.0098
99.308
98.330
97.881
98.538
99.722
0.700
0.650
0.830
0.750
0.800
69.9758
67.1644
62.8154
71.1831
70.2134
7.84
7.52
7.04
7.97
7.86
0.389
0.124
0.049
-0.415
0.065
62.69%
0.00%
0.00%
0.00%
31.95%
3.25
3
2.625
3.125
3.875
1/15/09
2/15/09
3/15/09
4/15/09
5/15/09
97-17+
96-11+
94-20
96-23+
99-30
3.8317
3.8497
3.8608
3.8640
3.8887
0.8912
0.8794
0.8622
0.8809
0.9106
11.658
14.834
19.353
21.599
20.461
4.33
7.87
13.08
13.09
10.07
5.04
8.57
13.77
13.79
10.79
41.526
41.902
42.119
43.323
44.777
4.21
4.31
4.43
4.46
4.48
1.0073
1.0045
1.0013
1.0004
1.0000
98.725
97.175
95.139
97.084
100.032
0.870
0.750
0.580
0.350
0.450
65.4271
62.1727
56.0036
75.9437
92.7949
7.33
6.96
6.27
8.51
10.39
-0.540
-1.853
-3.839
-3.985
-2.785
0.00%
0.00%
0.00%
0.00%
5.36%
17 TBA 3.875
6/15/09
99-30+
3.8824
0.9094
25.512
15.07
15.79
45.493
4.47
1.0001
101.690
0.450
93.2668
10.45
-4.351
0.00%
18
19
20
21
22
N
N
N
N
N
5
5
4.875
4.375
4
2/15/11
8/15/11
2/15/12
8/15/12
11/15/12
103-31+
103-19
102-13
98-19+
95-26+
4.3099
4.4124
4.5016
4.5790
4.6001
0.9468
0.9435
0.9328
0.8979
0.8713
21.274
20.298
19.662
20.033
23.920
8.68
8.22
7.57
9.21
13.84
11.48
11.02
10.33
11.88
16.42
59.191
62.585
65.558
67.338
67.783
5.62
5.96
6.32
6.75
7.06
1.0610
1.0516
1.0419
1.0303
1.0216
105.372
104.981
103.759
99.823
95.948
0.850
1.013
0.900
0.850
0.700
112.4832
107.8124
107.9888
96.6229
90.0392
12.60
12.07
12.09
10.82
10.08
-1.797
-1.504
-1.444
-2.116
-3.935
41.48%
2.61%
2.29%
0.00%
0.00%
23
24
25
26
27
N
N
N
N
N
3.875
3.625
4.25
4.25
4
2/15/13
5/15/13
8/15/13
11/15/13
2/15/14
94-18+
92-22+
96-17+
96-08
94-14
4.6371
4.6281
4.7152
4.7460
4.7192
0.8601
0.8401
0.8797
0.8771
0.8568
23.031
32.871
17.588
17.167
30.054
13.36
23.86
7.09
6.78
18.34
15.91
26.35
9.70
9.38
20.88
68.799
69.579
72.581
73.877
74.677
7.19
7.50
7.43
7.67
7.82
1.0181
1.0099
1.0118
1.0053
1.0011
95.653
92.811
97.726
96.377
95.525
0.756
0.700
0.850
0.850
0.200
86.3688
80.4588
93.6840
92.7334
104.5831
9.67
9.01
10.49
10.39
11.71
-3.733
-7.563
-1.484
-1.411
-5.971
0.00%
0.00%
2.43%
50.97%
0.00%
28 N10
4.75
5/15/14
100-02
4.7419
0.9086
29.169
16.31
19.00
78.733
7.86
1.0000
100.179
0.512
114.8285
12.86
-4.721
0.21%
29
30
31
32
33
B
B
B
B
B
8.125
8.5
8.75
8.75
7.875
8/15/19
2/15/20
5/15/20
8/15/20
2/15/21
129-09
133-19+
136-19+
136-25+
127-12+
5.2974
5.3176
5.3189
5.3315
5.3711
1.2083
1.2500
1.2775
1.2803
1.1947
62.612
60.500
63.770
60.328
47.972
41.76
38.63
41.45
37.79
27.83
42.90
39.81
42.66
39.00
28.95
123.214
128.748
132.154
133.604
129.198
9.37
9.47
9.66
9.60
9.97
1.1215
1.1187
1.1137
1.1153
1.1051
131.536
135.968
136.871
139.225
129.576
1.013
1.013
1.013
1.013
1.013
186.1894
195.2440
199.2451
201.1954
179.8730
20.85
21.87
22.32
22.53
20.15
-9.191
-8.119
-8.722
-7.712
-5.889
0.00%
0.21%
0.00%
0.00%
0.00%
34
35
36
37
38
B
B
B
B
B
8.125
8.125
8
7.25
7.625
5/15/21
8/15/21
11/15/21
8/15/22
11/15/22
130-13
130-17
129-09
120-23+
125-09+
5.3716
5.3821
5.3913
5.4411
5.4371
1.2224
1.2245
1.2130
1.1365
1.1786
51.067
47.986
46.764
31.222
35.261
30.46
27.17
26.50
12.78
16.01
31.61
28.33
27.64
13.85
17.12
132.600
133.894
134.082
130.667
135.311
10.15
10.08
10.35
10.65
10.78
1.1002
1.1020
1.0947
1.0867
1.0831
130.649
132.786
129.520
122.746
125.525
1.013
1.013
1.013
1.013
1.013
184.0127
185.8376
180.9337
164.6251
171.8682
20.61
20.81
20.26
18.44
19.25
-6.480
-5.564
-5.563
-2.334
-3.087
0.00%
0.00%
0.00%
4.84%
0.00%
39
40
41
42
43
B
B
B
B
B
7.125
6.25
7.5
7.625
6.875
2/15/23
8/15/23
11/15/24
2/15/25
8/15/25
119-13
108-31
124-22
126-11
117-00
5.4565
5.4881
5.4788
5.4813
5.5061
1.1247
1.0281
1.1743
1.1902
1.1037
28.512
20.247
30.260
29.646
22.324
10.42
4.51
11.37
10.23
4.31
11.47
5.47
12.48
11.35
5.35
131.646
125.062
142.915
145.331
138.969
10.85
11.30
11.44
11.31
11.69
1.0813
1.0690
1.0651
1.0686
1.0584
121.383
110.703
124.912
128.459
118.908
1.013
1.013
1.013
1.013
0.850
161.5746
140.5425
168.6440
173.3190
160.7982
18.10
15.74
18.89
19.41
18.01
-1.745
-0.295
-1.913
-1.547
-0.316
0.00%
38.04%
0.00%
0.00%
19.42%
44
45
46
47
48
B
B
B
B
B
6
6.75
6.5
6.625
6.375
2/15/26
8/15/26
11/15/26
2/15/27
8/15/27
106-00
115-22+
112-16
114-06
111-00+
5.5216
5.5146
5.5220
5.5208
5.5263
1.0000
1.0910
1.0608
1.0766
1.0465
20.000
23.648
22.982
23.705
23.702
4.93
6.59
6.75
6.98
7.64
5.87
7.61
7.75
7.99
8.63
130.926
141.256
139.099
141.491
140.114
12.16
12.01
12.34
12.19
12.42
1.0455
1.0495
1.0406
1.0446
1.0384
107.665
117.576
112.694
116.026
112.785
1.013
1.013
1.013
1.013
1.013
134.5295
152.3441
144.9091
149.3464
143.3906
15.07
17.06
16.23
16.73
16.06
-0.459
-0.787
-0.913
-0.920
-1.165
16.70%
0.00%
0.00%
0.00%
17.10%
49
50
51
52
53
B
B
B
B
B
6.125
5.5
5.25
5.25
6.125
11/15/27
8/15/28
11/15/28
2/15/29
8/15/29
107-24
99-18
96-08
96-10
108-04+
5.5312
5.5323
5.5313
5.5246
5.5223
1.0154
0.9368
0.9047
0.9044
1.0161
24.071
27.110
29.352
32.363
34.211
8.83
13.37
16.41
19.29
18.27
9.79
14.26
17.26
20.14
19.23
137.760
132.169
129.633
130.422
143.250
12.76
13.07
13.45
13.34
13.04
1.0291
1.0207
1.0105
1.0134
1.0216
107.933
101.089
96.407
97.769
109.840
1.013
1.013
1.013
1.013
0.850
136.0590
122.6443
115.5263
116.7105
142.3348
15.24
13.74
12.94
13.07
15.94
-1.617
-3.239
-4.459
-5.329
-4.496
0.00%
3.00%
0.00%
0.00%
0.67%
54 B
6.25
55 B30 5.375
5/15/30
2/15/31
110-01+
99-04
5.5173
5.4368
1.0325
0.9176
39.910
77.853
23.80
63.98
24.78
64.84
147.202
139.195
13.35
13.83
1.0131
1.0000
110.234
100.616
0.850
0.850
143.8096
123.9082
16.11
13.88
-6.091
-19.587
0.00%
0.00%
7
8
9
10
11
Coupon Maturity
F
F
F
F
F
12 F
13 F
14 F
15 F
16 F05
Yield
DV01/
$100,000
17.737
18.651
18.960
19.566
19.228
Modified Assumed
Duration Yield Beta
1.80
1.0019
1.87
1.0000
1.92
0.9988
1.88
0.9998
1.93
0.9985
Imp-RP
Calculating an implied RP (repo) rate (1)
Buy the 8.125s of 8/15/19 in the cash market (settle 5/26/04)
Spot price = 129-09 ( or 129.2813)
Accrued interest at settlement = (8.125/2)x(101/182) = 2.2545
Full price = 131.5357 [ = 129.2813 + 2.2545 ]
Deliver the bond on 6/30/04 at today’s futures price
Futures price = 105-12 (105.3750)
June ‘04 conversion factor = 1.2083
Converted futures price = 127.3246 [ = 1.2083 x 105.3750 ]
Accrued interest at futures delivery = (8.125/2)x(136/182) = 3.0357
Futures invoice price = 130.3603 [ = 127.3246 + 3.0357 ]
Calculating an implied RP (repo) rate (2)
Calculate the rate of return
Implied repo rate
= ((130.3603/131.5357)-1)x(360/35)
= 0.09191 (or -9.191%)
Note: This implied return is expressed as a money market rate
using an actual/360 money market day-count convention.
Identifying the cheapest to deliver
• As a rule, the issue with the highest implied
RP (repo) rate is identified as the cheapest to
deliver
• As a better rule, the issue with the smallest
difference between its own term repo rate and
its implied repo is the cheapest to deliver
Basis concepts
•
•
•
•
Definition
What drives the basis
Changes in the cheapest to deliver
Fair value of a futures contract
Basis defined
Basis = Spot price - (Conversion factor x Futures price)
Example of 6.875s of 8/15/25
Spot price = 117-00 (or 117.0000)
June ‘04 Factor = 1.1037
June ‘04 futures price = 105-12 (or 105.3750)
Basis
= 117.0000 - 1.1037 x 105.3750
= 117.0000 - 116.3024
= 0.6976 or 23.32/32nds [ = 0.6976 x 32 ]
What drives the basis?
Components of the basis
spot price
carry
(18.01/32nds)
basis
(22.32/32nds)
forward price
factor x futures price
value of strategic
delivery options, or
basis net of carry, or
BNOC
(4.31/32nds)
delivery
Basis = Spot price - Conversion factor x Futures price
Basis = Carry + Value of the short's strategic delivery options
Futures price = (Spot price - Basis) / Conversion faactor
Futures price = (Spot price - Carry - Delivery option value) / Conversion factor
How carry is calculated
Example of 6.875s of 8/15/25 on 5/25/04 (to settle 5/26/04)
Spot price = 117-00 (117.000)
Accrued interest = (6.875/2) x (101/182) = 1.9076
where 101 is actual days from 2/15/04 to 5/26/04)
and 182 is actual days from 2/15/04 to 8/15/04
Full price = 118.9076 [ = 117.0000 + 1.9076 ]
Coupon income = (6.875/2) x ( 35/182) = 0.6611
Financing expense = 118.9076 x .00850 x (35/360) = 0.0983
Carry = Coupon income - Financing expense = 0.6611 - 0.0983 = 0.5628
Carry = 18.01/32nds [ = 0.5628 x 32 ]
Shifts in the cheapest to deliver
Relative duration and cheapest to deliver
price/factor
high duration bond
100
low duration bond
6%
yield
Cash/futures price relationships
(with crossover points on April 5, 2001)
price/factor
111-26**
103-30
futures
price
8 7/8's of 8/15/17
97-20+**
7 5/8's of 11/15/27
5 1/2's of 8/15/28
4.98*
5.64 6.13*
*Approximate crossover yields as of 4/5/01
**Approximate crossover futures prices as of 4/5/01
Issue
5 1/2's of 8/15/28
7 5/8's of 11/15/22
8 7/8's of 8/15/17
Yield
5.65
5.64
5.53
Yield of 7 5/8's
Basis of 5-1/2s is like a call option on bond futures
basis*
price/factor
5 1/2's
(high duration)
YA
YB
futures
price
YA
YB
*basis = price - factor x futures
YC
Yield of 7 5/8's
YC
Yield of 7 5/8's
Basis of 8-7/8s is like a put option on bond futures
basis*
price/factor
YA
YB
8 7/8's
(low duration)
futures
price
YA
YB
* basis = price - factor x futures
YC
Yield of 7 5/8's
YC
Yield of 7 5/8's
Basis of 7-5/8s is like a straddle on bond futures
basis*
price/factor
YA
YB
futures
price
7 5/8's
(medium duration)
YA
* basis = factor x futures
YB
YC
Yield of 7 5/8's
YC
Yield of 7 5/8's
Price
ISSUE 6.875 08/15/25
(05/26/04, Based on 05/25/04 closing prices)
3.165%
2.640%
Forward yield Shift probability
-0.472
-0.404
at expiry
Yield shift (%)
Twist probability
6.6807%
117-09
Basis
BNOC
Jun-04/late delivery
21.39
3.9032
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
3.165%
0.472
Curve Flattening
Yield twist (%)
0.0502
5.0910
5.1584
5.2258
5.2932
5.3606
5.4280
5.4955
5.5629
5.6303
5.6977
5.7651
5.8325
5.8999
5.9673
6.0347
0.0251
5.0659
5.1333
5.2007
5.2681
5.3356
5.4030
5.4704
5.5378
5.6052
5.6726
5.7400
5.8074
5.8748
5.9422
6.0096
38.2925%
0.0000
5.0408
5.1082
5.1757
5.2431
5.3105
5.3779
5.4453
5.5127
5.5801
5.6475
5.7149
5.7823
5.8497
5.9171
5.9845
24.1730%
-0.0251
5.0158
5.0832
5.1506
5.2180
5.2854
5.3528
5.4202
5.4876
5.5550
5.6224
5.6898
5.7572
5.8246
5.8920
5.9594
6.6807%
-0.0502
4.9907
5.0581
5.1255
5.1929
5.2603
5.3277
5.3951
5.4625
5.5299
5.5973
5.6647
5.7321
5.7995
5.8669
5.9343
3.165%
2.640%
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
3.165%
24.1730%
Curve Steepening
BNOC
at expiry
Shift probability
Yield shift (%)
-0.472
-0.404
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
0.472
Twist probability
6.6807%
Yield twist (%)
0.0502
18.4708
15.7699
13.4789
11.6441
10.1315
8.9580
7.9176
7.0454
6.3452
5.8252
5.4379
5.0776
4.7541
4.5909
4.7052
8.0277
24.1730%
0.0251
13.4377
11.2010
9.3552
7.8571
6.7962
5.9239
5.2245
4.7814
4.3953
4.0219
3.7003
3.8343
4.2036
4.7360
5.3840
5.6507
38.2925%
0.0000
9.0309
7.0786
6.1085
5.1919
4.4967
3.8612
3.3060
3.1502
3.0154
3.4045
3.9247
4.5435
5.3430
6.1262
6.9912
4.3193
24.1730%
-0.0251
5.4604
4.6203
3.7900
3.3773
3.0302
2.8398
2.7978
3.3865
4.1291
4.9340
5.8363
6.7920
7.8415
9.1394
10.4447
4.5405
6.6807%
-0.0502
3.7313
3.3102
2.9451
2.6197
3.0686
3.8962
4.8444
5.8824
6.9838
8.1077
9.2658
10.6430
12.0107
13.3540
14.6635
6.5170
Average over
shifts
Net P/L
at expiry*
Shift probability
3.165%
2.640%
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
3.165%
Yield shift (%)
-0.472
-0.404
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
0.472
Twist probability
6.6807%
Yield twist (%)
0.0502
14.5677
11.8667
9.5758
7.7410
6.2284
5.0548
4.0144
3.1423
2.4420
1.9220
1.5347
1.1744
0.8509
0.6877
0.8020
4.1245
0.0251
9.5346
7.2979
5.4520
3.9539
2.8930
2.0207
1.3214
0.8783
0.4922
0.1187
-0.2029
-0.0689
0.3005
0.8328
1.4808
1.7475
38.2925%
0.0000
5.1278
3.1754
2.2053
1.2887
0.5936
-0.0419
-0.5972
-0.7530
-0.8877
-0.4987
0.0216
0.6403
1.4398
2.2230
3.0880
0.4161
24.1730%
-0.0251
1.5573
0.7171
-0.1132
-0.5258
-0.8730
-1.0634
-1.1053
-0.5167
0.2259
1.0308
1.9331
2.8888
3.9383
5.2362
6.5415
0.6373
6.6807%
-0.0502
-0.1719
-0.5929
-0.9581
-1.2835
-0.8345
-0.0070
0.9412
1.9792
3.0806
4.2045
5.3626
6.7399
8.1075
9.4508
10.7604
2.6138
24.1730%
* BNOC at expiry - current BNOC
Net P/L
(32nds)
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
-2.00
-4.00
-0.472
-0.337
-0.202
-0.067
0.067
0.202
0.337
0.472
Average over
shifts
Treasury Pricing & Hedging Tool
Jun-04/late delivery
Key Dates(days*)
Futures
5/24/04
5/25/04
5/26/04
6/1/04
6/21/04
6/30/04
Closing
Trade
Settlement
1st Delivery
Last Trade
Last Delivery
2 Year
(-2)
(-1)
(0)
(6)
(26)
(35)
6/30/04
7/6/04
(35)
(41)
* days from settlement
Assumptions
Yield volatility
Yield distribution
Yield beta coefficient
Twist volatility coefficient
Delivery date
# B/N
2-year
5-year
104.34% 37.36%
normal
normal
0.0272
0.0272
12.3243 6.4856
7/6/04 6/30/04
10-year
19.00%
normal
0.0272
3.0143
6/30/04
Bond
17.19%
normal
0.0272
0.4517
6/30/04
(Contract Jun-04)
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Full $
Price
98.433
99.612
98.966
104.044
99.766
Term
Repo
0.800
0.500
0.821
0.850
0.500
Carry
Carry $/Day
19.1097
47.3063
31.7781
101.1133
46.1197
32-Del
2.51
6.21
4.17
13.27
6.05
0.781
-0.647
-2.814
-0.567
-1.854
Prob. Of
CTD
96.72%
0.00%
0.00%
0.00%
3.28%
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
1 T
1.5
2 T02 2.25
3 T
2
4 T
4.625
5 TWI
2.5
3/31/06
4/30/06
5/15/06
5/15/06
5/31/06
2:00 PM
Price-32nd
98-06+
99-14+
98-29
103-29
99-24+
2.5010
2.5418
2.5727
2.5783
2.6201
Jun-04
Factor
0.9263
0.9358
0.9286
0.9754
0.9375
Basis
32nd
2.575
10.372
17.278
18.638
14.609
Basis NOC
Actual
Expected
0.07
0.07
4.17
4.17
13.11
13.11
5.37
5.38
8.56
7.47
6 TBA
2.5
6/30/06
99-24
2.6183
0.9350
22.869
15.71
15.70
20.242
2.01
0.9963
100.769
0.500
54.6018
7.16
-3.784
0.00%
3.25
3.125
3.125
3.375
3.375
8/15/08
9/15/08
10/15/08
11/15/08
12/15/08
98-13
97-23
97-17
98-14
98-07
3.6599
3.7024
3.7396
3.7578
3.8043
0.8999
0.8935
0.8917
0.8994
0.8978
8.799
9.132
9.413
11.544
10.127
0.96
1.61
2.38
3.57
2.26
1.68
2.32
3.09
4.29
2.98
38.447
38.937
39.554
40.428
41.033
3.87
3.96
4.04
4.10
4.11
1.0165
1.0140
1.0118
1.0102
1.0098
99.308
98.330
97.881
98.538
99.722
0.700
0.650
0.830
0.750
0.800
69.9758
67.1644
62.8154
71.1831
70.2134
7.84
7.52
7.04
7.97
7.86
0.389
0.124
0.049
-0.415
0.065
62.69%
0.00%
0.00%
0.00%
31.95%
3.25
3
2.625
3.125
3.875
1/15/09
2/15/09
3/15/09
4/15/09
5/15/09
97-17+
96-11+
94-20
96-23+
99-30
3.8317
3.8497
3.8608
3.8640
3.8887
0.8912
0.8794
0.8622
0.8809
0.9106
11.658
14.834
19.353
21.599
20.461
4.33
7.87
13.08
13.09
10.07
5.04
8.57
13.77
13.79
10.79
41.526
41.902
42.119
43.323
44.777
4.21
4.31
4.43
4.46
4.48
1.0073
1.0045
1.0013
1.0004
1.0000
98.725
97.175
95.139
97.084
100.032
0.870
0.750
0.580
0.350
0.450
65.4271
62.1727
56.0036
75.9437
92.7949
7.33
6.96
6.27
8.51
10.39
-0.540
-1.853
-3.839
-3.985
-2.785
0.00%
0.00%
0.00%
0.00%
5.36%
17 TBA 3.875
6/15/09
99-30+
3.8824
0.9094
25.512
15.07
15.79
45.493
4.47
1.0001
101.690
0.450
93.2668
10.45
-4.351
0.00%
18
19
20
21
22
N
N
N
N
N
5
5
4.875
4.375
4
2/15/11
8/15/11
2/15/12
8/15/12
11/15/12
103-31+
103-19
102-13
98-19+
95-26+
4.3099
4.4124
4.5016
4.5790
4.6001
0.9468
0.9435
0.9328
0.8979
0.8713
21.274
20.298
19.662
20.033
23.920
8.68
8.22
7.57
9.21
13.84
11.48
11.02
10.33
11.88
16.42
59.191
62.585
65.558
67.338
67.783
5.62
5.96
6.32
6.75
7.06
1.0610
1.0516
1.0419
1.0303
1.0216
105.372
104.981
103.759
99.823
95.948
0.850
1.013
0.900
0.850
0.700
112.4832
107.8124
107.9888
96.6229
90.0392
12.60
12.07
12.09
10.82
10.08
-1.797
-1.504
-1.444
-2.116
-3.935
41.48%
2.61%
2.29%
0.00%
0.00%
23
24
25
26
27
N
N
N
N
N
3.875
3.625
4.25
4.25
4
2/15/13
5/15/13
8/15/13
11/15/13
2/15/14
94-18+
92-22+
96-17+
96-08
94-14
4.6371
4.6281
4.7152
4.7460
4.7192
0.8601
0.8401
0.8797
0.8771
0.8568
23.031
32.871
17.588
17.167
30.054
13.36
23.86
7.09
6.78
18.34
15.91
26.35
9.70
9.38
20.88
68.799
69.579
72.581
73.877
74.677
7.19
7.50
7.43
7.67
7.82
1.0181
1.0099
1.0118
1.0053
1.0011
95.653
92.811
97.726
96.377
95.525
0.756
0.700
0.850
0.850
0.200
86.3688
80.4588
93.6840
92.7334
104.5831
9.67
9.01
10.49
10.39
11.71
-3.733
-7.563
-1.484
-1.411
-5.971
0.00%
0.00%
2.43%
50.97%
0.00%
28 N10
4.75
5/15/14
100-02
4.7419
0.9086
29.169
16.31
19.00
78.733
7.86
1.0000
100.179
0.512
114.8285
12.86
-4.721
0.21%
29
30
31
32
33
B
B
B
B
B
8.125
8.5
8.75
8.75
7.875
8/15/19
2/15/20
5/15/20
8/15/20
2/15/21
129-09
133-19+
136-19+
136-25+
127-12+
5.2974
5.3176
5.3189
5.3315
5.3711
1.2083
1.2500
1.2775
1.2803
1.1947
62.612
60.500
63.770
60.328
47.972
41.76
38.63
41.45
37.79
27.83
42.90
39.81
42.66
39.00
28.95
123.214
128.748
132.154
133.604
129.198
9.37
9.47
9.66
9.60
9.97
1.1215
1.1187
1.1137
1.1153
1.1051
131.536
135.968
136.871
139.225
129.576
1.013
1.013
1.013
1.013
1.013
186.1894
195.2440
199.2451
201.1954
179.8730
20.85
21.87
22.32
22.53
20.15
-9.191
-8.119
-8.722
-7.712
-5.889
0.00%
0.21%
0.00%
0.00%
0.00%
34
35
36
37
38
B
B
B
B
B
8.125
8.125
8
7.25
7.625
5/15/21
8/15/21
11/15/21
8/15/22
11/15/22
130-13
130-17
129-09
120-23+
125-09+
5.3716
5.3821
5.3913
5.4411
5.4371
1.2224
1.2245
1.2130
1.1365
1.1786
51.067
47.986
46.764
31.222
35.261
30.46
27.17
26.50
12.78
16.01
31.61
28.33
27.64
13.85
17.12
132.600
133.894
134.082
130.667
135.311
10.15
10.08
10.35
10.65
10.78
1.1002
1.1020
1.0947
1.0867
1.0831
130.649
132.786
129.520
122.746
125.525
1.013
1.013
1.013
1.013
1.013
184.0127
185.8376
180.9337
164.6251
171.8682
20.61
20.81
20.26
18.44
19.25
-6.480
-5.564
-5.563
-2.334
-3.087
0.00%
0.00%
0.00%
4.84%
0.00%
39
40
41
42
43
B
B
B
B
B
7.125
6.25
7.5
7.625
6.875
2/15/23
8/15/23
11/15/24
2/15/25
8/15/25
119-13
108-31
124-22
126-11
117-00
5.4565
5.4881
5.4788
5.4813
5.5061
1.1247
1.0281
1.1743
1.1902
1.1037
28.512
20.247
30.260
29.646
22.324
10.42
4.51
11.37
10.23
4.31
11.47
5.47
12.48
11.35
5.35
131.646
125.062
142.915
145.331
138.969
10.85
11.30
11.44
11.31
11.69
1.0813
1.0690
1.0651
1.0686
1.0584
121.383
110.703
124.912
128.459
118.908
1.013
1.013
1.013
1.013
0.850
161.5746
140.5425
168.6440
173.3190
160.7982
18.10
15.74
18.89
19.41
18.01
-1.745
-0.295
-1.913
-1.547
-0.316
0.00%
38.04%
0.00%
0.00%
19.42%
44
45
46
47
48
B
B
B
B
B
6
6.75
6.5
6.625
6.375
2/15/26
8/15/26
11/15/26
2/15/27
8/15/27
106-00
115-22+
112-16
114-06
111-00+
5.5216
5.5146
5.5220
5.5208
5.5263
1.0000
1.0910
1.0608
1.0766
1.0465
20.000
23.648
22.982
23.705
23.702
4.93
6.59
6.75
6.98
7.64
5.87
7.61
7.75
7.99
8.63
130.926
141.256
139.099
141.491
140.114
12.16
12.01
12.34
12.19
12.42
1.0455
1.0495
1.0406
1.0446
1.0384
107.665
117.576
112.694
116.026
112.785
1.013
1.013
1.013
1.013
1.013
134.5295
152.3441
144.9091
149.3464
143.3906
15.07
17.06
16.23
16.73
16.06
-0.459
-0.787
-0.913
-0.920
-1.165
16.70%
0.00%
0.00%
0.00%
17.10%
49
50
51
52
53
B
B
B
B
B
6.125
5.5
5.25
5.25
6.125
11/15/27
8/15/28
11/15/28
2/15/29
8/15/29
107-24
99-18
96-08
96-10
108-04+
5.5312
5.5323
5.5313
5.5246
5.5223
1.0154
0.9368
0.9047
0.9044
1.0161
24.071
27.110
29.352
32.363
34.211
8.83
13.37
16.41
19.29
18.27
9.79
14.26
17.26
20.14
19.23
137.760
132.169
129.633
130.422
143.250
12.76
13.07
13.45
13.34
13.04
1.0291
1.0207
1.0105
1.0134
1.0216
107.933
101.089
96.407
97.769
109.840
1.013
1.013
1.013
1.013
0.850
136.0590
122.6443
115.5263
116.7105
142.3348
15.24
13.74
12.94
13.07
15.94
-1.617
-3.239
-4.459
-5.329
-4.496
0.00%
3.00%
0.00%
0.00%
0.67%
54 B
6.25
55 B30 5.375
5/15/30
2/15/31
110-01+
99-04
5.5173
5.4368
1.0325
0.9176
39.910
77.853
23.80
63.98
24.78
64.84
147.202
139.195
13.35
13.83
1.0131
1.0000
110.234
100.616
0.850
0.850
143.8096
123.9082
16.11
13.88
-6.091
-19.587
0.00%
0.00%
7
8
9
10
11
Coupon Maturity
F
F
F
F
F
12 F
13 F
14 F
15 F
16 F05
Yield
DV01/
$100,000
17.737
18.651
18.960
19.566
19.228
Modified Assumed
Duration Yield Beta
1.80
1.0019
1.87
1.0000
1.92
0.9988
1.88
0.9998
1.93
0.9985
Imp-RP
Futures are fairly priced
if expected basis net of
carry equals actual basis
net of carry
Futures are rich if
expected basis net of
carry exceeds actual
basis net of carry (that is,
if the basis is cheap)
Futures are cheap if
expected basis net of
carry is less than actual
basis net of carry (and
the basis is rich)
Part II: Applications
•
•
•
•
•
•
Hedging
Basis trading
Speculation (duration management)
Curve trading
Synthetic asset construction
Volatility arbitrage
Hedging
• Finding the option-adjusted value of a basis
point
• Measuring “stub” risk (repo exposure)
• Yield betas
Cash/futures price relationships
(with crossover points on April 5, 2001)
price/factor
111-26**
103-30
futures
price
8 7/8's of 8/15/17
97-20+**
7 5/8's of 11/15/27
5 1/2's of 8/15/28
4.98*
5.64 6.13*
*Approximate crossover yields as of 4/5/01
**Approximate crossover futures prices as of 4/5/01
Issue
5 1/2's of 8/15/28
7 5/8's of 11/15/22
8 7/8's of 8/15/17
Yield
5.65
5.64
5.53
Yield of 7 5/8's
Option-adjusted DV01s
Futures
(Contract Jun-04)
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Hedging example
Hedge $100 million of the 5.375s of 2/15/31 (on-the-run)
DV01/$100,000 = $139.195
Position DV01
= $100,000,000 x ($139.195/$100,000)
= $139,195
Option-adjusted futures DV01 = $126.078
For parallel shifts in the yield curve,
Hedge ratio = $139,195 / $126.078 = 1,104 contracts
Treasury Pricing & Hedging Tool
Jun-04/late delivery
Key Dates(days*)
Futures
5/24/04
5/25/04
5/26/04
6/1/04
6/21/04
6/30/04
Closing
Trade
Settlement
1st Delivery
Last Trade
Last Delivery
2 Year
(-2)
(-1)
(0)
(6)
(26)
(35)
6/30/04
7/6/04
(35)
(41)
* days from settlement
Assumptions
Yield volatility
Yield distribution
Yield beta coefficient
Twist volatility coefficient
Delivery date
# B/N
2-year
5-year
104.34% 37.36%
normal
normal
0.0272
0.0272
12.3243 6.4856
7/6/04 6/30/04
10-year
19.00%
normal
0.0272
3.0143
6/30/04
Bond
17.19%
normal
0.0272
0.4517
6/30/04
(Contract Jun-04)
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Full $
Price
98.433
99.612
98.966
104.044
99.766
Term
Repo
0.800
0.500
0.821
0.850
0.500
Carry
Carry $/Day
19.1097
47.3063
31.7781
101.1133
46.1197
32-Del
2.51
6.21
4.17
13.27
6.05
0.781
-0.647
-2.814
-0.567
-1.854
Prob. Of
CTD
96.72%
0.00%
0.00%
0.00%
3.28%
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
1 T
1.5
2 T02 2.25
3 T
2
4 T
4.625
5 TWI
2.5
3/31/06
4/30/06
5/15/06
5/15/06
5/31/06
2:00 PM
Price-32nd
98-06+
99-14+
98-29
103-29
99-24+
2.5010
2.5418
2.5727
2.5783
2.6201
Jun-04
Factor
0.9263
0.9358
0.9286
0.9754
0.9375
Basis
32nd
2.575
10.372
17.278
18.638
14.609
Basis NOC
Actual
Expected
0.07
0.07
4.17
4.17
13.11
13.11
5.37
5.38
8.56
7.47
6 TBA
2.5
6/30/06
99-24
2.6183
0.9350
22.869
15.71
15.70
20.242
2.01
0.9963
100.769
0.500
54.6018
7.16
-3.784
0.00%
3.25
3.125
3.125
3.375
3.375
8/15/08
9/15/08
10/15/08
11/15/08
12/15/08
98-13
97-23
97-17
98-14
98-07
3.6599
3.7024
3.7396
3.7578
3.8043
0.8999
0.8935
0.8917
0.8994
0.8978
8.799
9.132
9.413
11.544
10.127
0.96
1.61
2.38
3.57
2.26
1.68
2.32
3.09
4.29
2.98
38.447
38.937
39.554
40.428
41.033
3.87
3.96
4.04
4.10
4.11
1.0165
1.0140
1.0118
1.0102
1.0098
99.308
98.330
97.881
98.538
99.722
0.700
0.650
0.830
0.750
0.800
69.9758
67.1644
62.8154
71.1831
70.2134
7.84
7.52
7.04
7.97
7.86
0.389
0.124
0.049
-0.415
0.065
62.69%
0.00%
0.00%
0.00%
31.95%
3.25
3
2.625
3.125
3.875
1/15/09
2/15/09
3/15/09
4/15/09
5/15/09
97-17+
96-11+
94-20
96-23+
99-30
3.8317
3.8497
3.8608
3.8640
3.8887
0.8912
0.8794
0.8622
0.8809
0.9106
11.658
14.834
19.353
21.599
20.461
4.33
7.87
13.08
13.09
10.07
5.04
8.57
13.77
13.79
10.79
41.526
41.902
42.119
43.323
44.777
4.21
4.31
4.43
4.46
4.48
1.0073
1.0045
1.0013
1.0004
1.0000
98.725
97.175
95.139
97.084
100.032
0.870
0.750
0.580
0.350
0.450
65.4271
62.1727
56.0036
75.9437
92.7949
7.33
6.96
6.27
8.51
10.39
-0.540
-1.853
-3.839
-3.985
-2.785
0.00%
0.00%
0.00%
0.00%
5.36%
17 TBA 3.875
6/15/09
99-30+
3.8824
0.9094
25.512
15.07
15.79
45.493
4.47
1.0001
101.690
0.450
93.2668
10.45
-4.351
0.00%
18
19
20
21
22
N
N
N
N
N
5
5
4.875
4.375
4
2/15/11
8/15/11
2/15/12
8/15/12
11/15/12
103-31+
103-19
102-13
98-19+
95-26+
4.3099
4.4124
4.5016
4.5790
4.6001
0.9468
0.9435
0.9328
0.8979
0.8713
21.274
20.298
19.662
20.033
23.920
8.68
8.22
7.57
9.21
13.84
11.48
11.02
10.33
11.88
16.42
59.191
62.585
65.558
67.338
67.783
5.62
5.96
6.32
6.75
7.06
1.0610
1.0516
1.0419
1.0303
1.0216
105.372
104.981
103.759
99.823
95.948
0.850
1.013
0.900
0.850
0.700
112.4832
107.8124
107.9888
96.6229
90.0392
12.60
12.07
12.09
10.82
10.08
-1.797
-1.504
-1.444
-2.116
-3.935
41.48%
2.61%
2.29%
0.00%
0.00%
23
24
25
26
27
N
N
N
N
N
3.875
3.625
4.25
4.25
4
2/15/13
5/15/13
8/15/13
11/15/13
2/15/14
94-18+
92-22+
96-17+
96-08
94-14
4.6371
4.6281
4.7152
4.7460
4.7192
0.8601
0.8401
0.8797
0.8771
0.8568
23.031
32.871
17.588
17.167
30.054
13.36
23.86
7.09
6.78
18.34
15.91
26.35
9.70
9.38
20.88
68.799
69.579
72.581
73.877
74.677
7.19
7.50
7.43
7.67
7.82
1.0181
1.0099
1.0118
1.0053
1.0011
95.653
92.811
97.726
96.377
95.525
0.756
0.700
0.850
0.850
0.200
86.3688
80.4588
93.6840
92.7334
104.5831
9.67
9.01
10.49
10.39
11.71
-3.733
-7.563
-1.484
-1.411
-5.971
0.00%
0.00%
2.43%
50.97%
0.00%
28 N10
4.75
5/15/14
100-02
4.7419
0.9086
29.169
16.31
19.00
78.733
7.86
1.0000
100.179
0.512
114.8285
12.86
-4.721
0.21%
29
30
31
32
33
B
B
B
B
B
8.125
8.5
8.75
8.75
7.875
8/15/19
2/15/20
5/15/20
8/15/20
2/15/21
129-09
133-19+
136-19+
136-25+
127-12+
5.2974
5.3176
5.3189
5.3315
5.3711
1.2083
1.2500
1.2775
1.2803
1.1947
62.612
60.500
63.770
60.328
47.972
41.76
38.63
41.45
37.79
27.83
42.90
39.81
42.66
39.00
28.95
123.214
128.748
132.154
133.604
129.198
9.37
9.47
9.66
9.60
9.97
1.1215
1.1187
1.1137
1.1153
1.1051
131.536
135.968
136.871
139.225
129.576
1.013
1.013
1.013
1.013
1.013
186.1894
195.2440
199.2451
201.1954
179.8730
20.85
21.87
22.32
22.53
20.15
-9.191
-8.119
-8.722
-7.712
-5.889
0.00%
0.21%
0.00%
0.00%
0.00%
34
35
36
37
38
B
B
B
B
B
8.125
8.125
8
7.25
7.625
5/15/21
8/15/21
11/15/21
8/15/22
11/15/22
130-13
130-17
129-09
120-23+
125-09+
5.3716
5.3821
5.3913
5.4411
5.4371
1.2224
1.2245
1.2130
1.1365
1.1786
51.067
47.986
46.764
31.222
35.261
30.46
27.17
26.50
12.78
16.01
31.61
28.33
27.64
13.85
17.12
132.600
133.894
134.082
130.667
135.311
10.15
10.08
10.35
10.65
10.78
1.1002
1.1020
1.0947
1.0867
1.0831
130.649
132.786
129.520
122.746
125.525
1.013
1.013
1.013
1.013
1.013
184.0127
185.8376
180.9337
164.6251
171.8682
20.61
20.81
20.26
18.44
19.25
-6.480
-5.564
-5.563
-2.334
-3.087
0.00%
0.00%
0.00%
4.84%
0.00%
39
40
41
42
43
B
B
B
B
B
7.125
6.25
7.5
7.625
6.875
2/15/23
8/15/23
11/15/24
2/15/25
8/15/25
119-13
108-31
124-22
126-11
117-00
5.4565
5.4881
5.4788
5.4813
5.5061
1.1247
1.0281
1.1743
1.1902
1.1037
28.512
20.247
30.260
29.646
22.324
10.42
4.51
11.37
10.23
4.31
11.47
5.47
12.48
11.35
5.35
131.646
125.062
142.915
145.331
138.969
10.85
11.30
11.44
11.31
11.69
1.0813
1.0690
1.0651
1.0686
1.0584
121.383
110.703
124.912
128.459
118.908
1.013
1.013
1.013
1.013
0.850
161.5746
140.5425
168.6440
173.3190
160.7982
18.10
15.74
18.89
19.41
18.01
-1.745
-0.295
-1.913
-1.547
-0.316
0.00%
38.04%
0.00%
0.00%
19.42%
44
45
46
47
48
B
B
B
B
B
6
6.75
6.5
6.625
6.375
2/15/26
8/15/26
11/15/26
2/15/27
8/15/27
106-00
115-22+
112-16
114-06
111-00+
5.5216
5.5146
5.5220
5.5208
5.5263
1.0000
1.0910
1.0608
1.0766
1.0465
20.000
23.648
22.982
23.705
23.702
4.93
6.59
6.75
6.98
7.64
5.87
7.61
7.75
7.99
8.63
130.926
141.256
139.099
141.491
140.114
12.16
12.01
12.34
12.19
12.42
1.0455
1.0495
1.0406
1.0446
1.0384
107.665
117.576
112.694
116.026
112.785
1.013
1.013
1.013
1.013
1.013
134.5295
152.3441
144.9091
149.3464
143.3906
15.07
17.06
16.23
16.73
16.06
-0.459
-0.787
-0.913
-0.920
-1.165
16.70%
0.00%
0.00%
0.00%
17.10%
49
50
51
52
53
B
B
B
B
B
6.125
5.5
5.25
5.25
6.125
11/15/27
8/15/28
11/15/28
2/15/29
8/15/29
107-24
99-18
96-08
96-10
108-04+
5.5312
5.5323
5.5313
5.5246
5.5223
1.0154
0.9368
0.9047
0.9044
1.0161
24.071
27.110
29.352
32.363
34.211
8.83
13.37
16.41
19.29
18.27
9.79
14.26
17.26
20.14
19.23
137.760
132.169
129.633
130.422
143.250
12.76
13.07
13.45
13.34
13.04
1.0291
1.0207
1.0105
1.0134
1.0216
107.933
101.089
96.407
97.769
109.840
1.013
1.013
1.013
1.013
0.850
136.0590
122.6443
115.5263
116.7105
142.3348
15.24
13.74
12.94
13.07
15.94
-1.617
-3.239
-4.459
-5.329
-4.496
0.00%
3.00%
0.00%
0.00%
0.67%
54 B
6.25
55 B30 5.375
5/15/30
2/15/31
110-01+
99-04
5.5173
5.4368
1.0325
0.9176
39.910
77.853
23.80
63.98
24.78
64.84
147.202
139.195
13.35
13.83
1.0131
1.0000
110.234
100.616
0.850
0.850
143.8096
123.9082
16.11
13.88
-6.091
-19.587
0.00%
0.00%
7
8
9
10
11
Coupon Maturity
F
F
F
F
F
12 F
13 F
14 F
15 F
16 F05
Yield
DV01/
$100,000
17.737
18.651
18.960
19.566
19.228
Modified Assumed
Duration Yield Beta
1.80
1.0019
1.87
1.0000
1.92
0.9988
1.88
0.9998
1.93
0.9985
Imp-RP
Hedging example using yield betas
Hedge $100 million of the 5.375s of 2/15/31 (on-the-run)
DV01/$100,000 = $139.195
Position DV01
= $100,000,000 x ($139.195/$100,000)
= $139,195
Option-adjusted futures DV01 = $126.078
Yield betas
1.0000 (for 5.375s)
1.0584 (for the cheapest to deliver)
Hedge ratio
= ($139,195 / $126.078) x (1.0000/1.0584)
= 1,043 contracts
Other ways to think about hedging
• Duration adjustment
• Interest rate immunization
• Asset allocation
Duration adjustment
• Hedging, as usually described, is simply a
way to reduce net duration to zero
• Speculating, on the other hand, is a matter of
choosing a target duration to take advantage
of a view on the level of interest rates.
Option-adjusted DV01s
Futures
(Contract Jun-04)
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Duration adjustment (example)
Problem: You have a $200 million portfolio of 10-year notes
with a duration of 7 and want to reduce the portfolio’s duration
to 5.
Solution: Find the number of futures, N, that sets
($200,000,000 x .07 + N x $109,031.25 x .07)/$200,000,000
equal to .05. Solving, we find N = -524 (rounded).
Shorting 524 futures, each with a portfolio equivalent value of
$109,031.25 and an option-adjusted duration of .07 would be
like shorting, fully leveraged, a note portfolio of $57,132,375
with a duration of .07.
Immunization
• Duration is a concept that is defined for
parallel shifts in the yield curve.
• Immunization is an approach to hedging that
provides protection against non-parallel shifts
in the yield curve
Treasury Pricing & Hedging Tool
Jun-04/late delivery
Key Dates(days*)
Futures
5/24/04
5/25/04
5/26/04
6/1/04
6/21/04
6/30/04
Closing
Trade
Settlement
1st Delivery
Last Trade
Last Delivery
2 Year
(-2)
(-1)
(0)
(6)
(26)
(35)
6/30/04
7/6/04
(35)
(41)
* days from settlement
Assumptions
Yield volatility
Yield distribution
Yield beta coefficient
Twist volatility coefficient
Delivery date
# B/N
2-year
5-year
104.34% 37.36%
normal
normal
0.0272
0.0272
12.3243 6.4856
7/6/04 6/30/04
10-year
19.00%
normal
0.0272
3.0143
6/30/04
Bond
17.19%
normal
0.0272
0.4517
6/30/04
(Contract Jun-04)
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Full $
Price
98.433
99.612
98.966
104.044
99.766
Term
Repo
0.800
0.500
0.821
0.850
0.500
Carry
Carry $/Day
19.1097
47.3063
31.7781
101.1133
46.1197
32-Del
2.51
6.21
4.17
13.27
6.05
0.781
-0.647
-2.814
-0.567
-1.854
Prob. Of
CTD
96.72%
0.00%
0.00%
0.00%
3.28%
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
1 T
1.5
2 T02 2.25
3 T
2
4 T
4.625
5 TWI
2.5
3/31/06
4/30/06
5/15/06
5/15/06
5/31/06
2:00 PM
Price-32nd
98-06+
99-14+
98-29
103-29
99-24+
2.5010
2.5418
2.5727
2.5783
2.6201
Jun-04
Factor
0.9263
0.9358
0.9286
0.9754
0.9375
Basis
32nd
2.575
10.372
17.278
18.638
14.609
Basis NOC
Actual
Expected
0.07
0.07
4.17
4.17
13.11
13.11
5.37
5.38
8.56
7.47
6 TBA
2.5
6/30/06
99-24
2.6183
0.9350
22.869
15.71
15.70
20.242
2.01
0.9963
100.769
0.500
54.6018
7.16
-3.784
0.00%
3.25
3.125
3.125
3.375
3.375
8/15/08
9/15/08
10/15/08
11/15/08
12/15/08
98-13
97-23
97-17
98-14
98-07
3.6599
3.7024
3.7396
3.7578
3.8043
0.8999
0.8935
0.8917
0.8994
0.8978
8.799
9.132
9.413
11.544
10.127
0.96
1.61
2.38
3.57
2.26
1.68
2.32
3.09
4.29
2.98
38.447
38.937
39.554
40.428
41.033
3.87
3.96
4.04
4.10
4.11
1.0165
1.0140
1.0118
1.0102
1.0098
99.308
98.330
97.881
98.538
99.722
0.700
0.650
0.830
0.750
0.800
69.9758
67.1644
62.8154
71.1831
70.2134
7.84
7.52
7.04
7.97
7.86
0.389
0.124
0.049
-0.415
0.065
62.69%
0.00%
0.00%
0.00%
31.95%
3.25
3
2.625
3.125
3.875
1/15/09
2/15/09
3/15/09
4/15/09
5/15/09
97-17+
96-11+
94-20
96-23+
99-30
3.8317
3.8497
3.8608
3.8640
3.8887
0.8912
0.8794
0.8622
0.8809
0.9106
11.658
14.834
19.353
21.599
20.461
4.33
7.87
13.08
13.09
10.07
5.04
8.57
13.77
13.79
10.79
41.526
41.902
42.119
43.323
44.777
4.21
4.31
4.43
4.46
4.48
1.0073
1.0045
1.0013
1.0004
1.0000
98.725
97.175
95.139
97.084
100.032
0.870
0.750
0.580
0.350
0.450
65.4271
62.1727
56.0036
75.9437
92.7949
7.33
6.96
6.27
8.51
10.39
-0.540
-1.853
-3.839
-3.985
-2.785
0.00%
0.00%
0.00%
0.00%
5.36%
17 TBA 3.875
6/15/09
99-30+
3.8824
0.9094
25.512
15.07
15.79
45.493
4.47
1.0001
101.690
0.450
93.2668
10.45
-4.351
0.00%
18
19
20
21
22
N
N
N
N
N
5
5
4.875
4.375
4
2/15/11
8/15/11
2/15/12
8/15/12
11/15/12
103-31+
103-19
102-13
98-19+
95-26+
4.3099
4.4124
4.5016
4.5790
4.6001
0.9468
0.9435
0.9328
0.8979
0.8713
21.274
20.298
19.662
20.033
23.920
8.68
8.22
7.57
9.21
13.84
11.48
11.02
10.33
11.88
16.42
59.191
62.585
65.558
67.338
67.783
5.62
5.96
6.32
6.75
7.06
1.0610
1.0516
1.0419
1.0303
1.0216
105.372
104.981
103.759
99.823
95.948
0.850
1.013
0.900
0.850
0.700
112.4832
107.8124
107.9888
96.6229
90.0392
12.60
12.07
12.09
10.82
10.08
-1.797
-1.504
-1.444
-2.116
-3.935
41.48%
2.61%
2.29%
0.00%
0.00%
23
24
25
26
27
N
N
N
N
N
3.875
3.625
4.25
4.25
4
2/15/13
5/15/13
8/15/13
11/15/13
2/15/14
94-18+
92-22+
96-17+
96-08
94-14
4.6371
4.6281
4.7152
4.7460
4.7192
0.8601
0.8401
0.8797
0.8771
0.8568
23.031
32.871
17.588
17.167
30.054
13.36
23.86
7.09
6.78
18.34
15.91
26.35
9.70
9.38
20.88
68.799
69.579
72.581
73.877
74.677
7.19
7.50
7.43
7.67
7.82
1.0181
1.0099
1.0118
1.0053
1.0011
95.653
92.811
97.726
96.377
95.525
0.756
0.700
0.850
0.850
0.200
86.3688
80.4588
93.6840
92.7334
104.5831
9.67
9.01
10.49
10.39
11.71
-3.733
-7.563
-1.484
-1.411
-5.971
0.00%
0.00%
2.43%
50.97%
0.00%
28 N10
4.75
5/15/14
100-02
4.7419
0.9086
29.169
16.31
19.00
78.733
7.86
1.0000
100.179
0.512
114.8285
12.86
-4.721
0.21%
29
30
31
32
33
B
B
B
B
B
8.125
8.5
8.75
8.75
7.875
8/15/19
2/15/20
5/15/20
8/15/20
2/15/21
129-09
133-19+
136-19+
136-25+
127-12+
5.2974
5.3176
5.3189
5.3315
5.3711
1.2083
1.2500
1.2775
1.2803
1.1947
62.612
60.500
63.770
60.328
47.972
41.76
38.63
41.45
37.79
27.83
42.90
39.81
42.66
39.00
28.95
123.214
128.748
132.154
133.604
129.198
9.37
9.47
9.66
9.60
9.97
1.1215
1.1187
1.1137
1.1153
1.1051
131.536
135.968
136.871
139.225
129.576
1.013
1.013
1.013
1.013
1.013
186.1894
195.2440
199.2451
201.1954
179.8730
20.85
21.87
22.32
22.53
20.15
-9.191
-8.119
-8.722
-7.712
-5.889
0.00%
0.21%
0.00%
0.00%
0.00%
34
35
36
37
38
B
B
B
B
B
8.125
8.125
8
7.25
7.625
5/15/21
8/15/21
11/15/21
8/15/22
11/15/22
130-13
130-17
129-09
120-23+
125-09+
5.3716
5.3821
5.3913
5.4411
5.4371
1.2224
1.2245
1.2130
1.1365
1.1786
51.067
47.986
46.764
31.222
35.261
30.46
27.17
26.50
12.78
16.01
31.61
28.33
27.64
13.85
17.12
132.600
133.894
134.082
130.667
135.311
10.15
10.08
10.35
10.65
10.78
1.1002
1.1020
1.0947
1.0867
1.0831
130.649
132.786
129.520
122.746
125.525
1.013
1.013
1.013
1.013
1.013
184.0127
185.8376
180.9337
164.6251
171.8682
20.61
20.81
20.26
18.44
19.25
-6.480
-5.564
-5.563
-2.334
-3.087
0.00%
0.00%
0.00%
4.84%
0.00%
39
40
41
42
43
B
B
B
B
B
7.125
6.25
7.5
7.625
6.875
2/15/23
8/15/23
11/15/24
2/15/25
8/15/25
119-13
108-31
124-22
126-11
117-00
5.4565
5.4881
5.4788
5.4813
5.5061
1.1247
1.0281
1.1743
1.1902
1.1037
28.512
20.247
30.260
29.646
22.324
10.42
4.51
11.37
10.23
4.31
11.47
5.47
12.48
11.35
5.35
131.646
125.062
142.915
145.331
138.969
10.85
11.30
11.44
11.31
11.69
1.0813
1.0690
1.0651
1.0686
1.0584
121.383
110.703
124.912
128.459
118.908
1.013
1.013
1.013
1.013
0.850
161.5746
140.5425
168.6440
173.3190
160.7982
18.10
15.74
18.89
19.41
18.01
-1.745
-0.295
-1.913
-1.547
-0.316
0.00%
38.04%
0.00%
0.00%
19.42%
44
45
46
47
48
B
B
B
B
B
6
6.75
6.5
6.625
6.375
2/15/26
8/15/26
11/15/26
2/15/27
8/15/27
106-00
115-22+
112-16
114-06
111-00+
5.5216
5.5146
5.5220
5.5208
5.5263
1.0000
1.0910
1.0608
1.0766
1.0465
20.000
23.648
22.982
23.705
23.702
4.93
6.59
6.75
6.98
7.64
5.87
7.61
7.75
7.99
8.63
130.926
141.256
139.099
141.491
140.114
12.16
12.01
12.34
12.19
12.42
1.0455
1.0495
1.0406
1.0446
1.0384
107.665
117.576
112.694
116.026
112.785
1.013
1.013
1.013
1.013
1.013
134.5295
152.3441
144.9091
149.3464
143.3906
15.07
17.06
16.23
16.73
16.06
-0.459
-0.787
-0.913
-0.920
-1.165
16.70%
0.00%
0.00%
0.00%
17.10%
49
50
51
52
53
B
B
B
B
B
6.125
5.5
5.25
5.25
6.125
11/15/27
8/15/28
11/15/28
2/15/29
8/15/29
107-24
99-18
96-08
96-10
108-04+
5.5312
5.5323
5.5313
5.5246
5.5223
1.0154
0.9368
0.9047
0.9044
1.0161
24.071
27.110
29.352
32.363
34.211
8.83
13.37
16.41
19.29
18.27
9.79
14.26
17.26
20.14
19.23
137.760
132.169
129.633
130.422
143.250
12.76
13.07
13.45
13.34
13.04
1.0291
1.0207
1.0105
1.0134
1.0216
107.933
101.089
96.407
97.769
109.840
1.013
1.013
1.013
1.013
0.850
136.0590
122.6443
115.5263
116.7105
142.3348
15.24
13.74
12.94
13.07
15.94
-1.617
-3.239
-4.459
-5.329
-4.496
0.00%
3.00%
0.00%
0.00%
0.67%
54 B
6.25
55 B30 5.375
5/15/30
2/15/31
110-01+
99-04
5.5173
5.4368
1.0325
0.9176
39.910
77.853
23.80
63.98
24.78
64.84
147.202
139.195
13.35
13.83
1.0131
1.0000
110.234
100.616
0.850
0.850
143.8096
123.9082
16.11
13.88
-6.091
-19.587
0.00%
0.00%
7
8
9
10
11
Coupon Maturity
F
F
F
F
F
12 F
13 F
14 F
15 F
16 F05
Yield
DV01/
$100,000
17.737
18.651
18.960
19.566
19.228
Modified Assumed
Duration Yield Beta
1.80
1.0019
1.87
1.0000
1.92
0.9988
1.88
0.9998
1.93
0.9985
Imp-RP
Immunization with futures
On 5/26/04, CTD Treasury issues had maturities of
3/31/06
(2-year)
8/15/08
(5-year)
11/15/13
(10-year)
8/15/25
(bond)
To “immunize” the portfolio from a complex change in
yields, determine the partial sensitivity of your portfolio to
changes in each of these yields and hedge accordingly.
(Other terms are buckets, key rates, and factors.)
Asset allocation
• Futures are excellent tools for changing your
macro asset mix without disturbing the real
underlying portfolio.
• For example, one could buy bond futures and
sell equity futures as a way of increasing a
portfolio’s exposure to bonds and decreasing
its net exposure to equity prices.
• Advantages?
– Lower transactions costs
– Preserving or transporting alpha
Futures hedges are imperfect
Futures hedges will make or lose money because of
unexpected changes :
the short’s delivery option value (due to changes in
cheapest to deliver or changes in yield volatility)
term repo rates
the slope of the deliverable yield curve
Basis trades
Definition of a long basis trade
You are long the basis if, for every $100,000 par amount of a bond (or
note) that you are long you are short a number of futures equal to the
bond’s conversion factor
Example of $100 million of the 6.875s of 8/15/25
June ‘04 factor = 1.1037
Buy $100 million of the 6.875s at 117-00
Sell 1,104 [ = $100,000,000 x (1.1037/$100,000) ] June ‘04 futures
You are long the basis at 22.32/32nds
P/L of a basis trade
A basis trade has four components
Change in the price of the cash bond
Change in the futures price
Coupon income
Financing (repo) expense
By construction, the combined effect of price changes will
equal the change in the basis (times the value of a 32nd).
For example, each 32nd on a $100 million basis position is worth
$31,250.
Why trade the basis
•
•
•
•
•
•
Convergence (sale of delivery options)
Yield volatility
Changes in the slope of the yield curve
Superior financing position
Mispriced basis
Changes in term repo rates
Changes in yield levels and spreads
Changes in the cheapest to deliver
Net basis of the 5s of 2/11
Speculating with Treasury futures
The P/L of a cash bond trade has three components
Change in the price of the bond
Coupon income
Financing (repo) expense
The P/L of a futures trade has only one component
Change in the futures price
Note of caution: An appropriate comparison of the two
strategies requires a complete accounting for all sources of
gain or loss
Curve trading
• You can trade the slope of the curve using the
basis as your trading vehicle
• You can trade the slope of the curve using
combinations of 2s, 5s, 10s, and bonds
Unweighted and weighted NOB spreads
An unweighted long NOB spread (10-year notes versus
Bonds) would be long one 10-year note contract and short
one bond contract
Chief advantage is a simple P/L structure. As the price
spread widens, you make money. As the price spread
narrows, you lose money.
Chief drawback is its exposure to a change in the slope
of the yield curve. With a long unweighted NOB
spread, you would profit from a parallel downward
shift in the yield curve, but lose from a parallel upward
shift in yields.
Option-adjusted DV01s
Futures
(Contract Jun-04)
Market Price
Theoretical Price
Rule-of-Thumb DV01
Option-adjusted DV01 wrt
CTD yield
OTR yield
Option-adjusted Duration wrt
CTD yield
OTR yield
Option-adjusted Convexity
Repo DV01
Vegas wrt
yield volatility
yield beta
twist volatility
2-year
105-29.75
105-29+
38.2973
5-year
109-01+
109-00+
42.7231
10-year
109-04
109-01
84.2285
Bond
105-12
105-11
125.9123
38.4872
38.5608
43.9634
44.6866
75.9411
76.3415
126.0783
133.4414
1.8166
1.8201
4.0325
4.0989
6.9650
7.0017
11.9681
12.6670
-2.4243
-1.0737
-1.0747
-1.0467
-0.00007
0.00012
-0.00001
-0.00092
0.00149
-0.00048
-0.00599
0.00757
-0.00491
-0.00434
0.01159
-0.01203
Constructing a weighted NOB spread
Futures DV01s
10-year note futures -- $75.94
Bond futures -- $126.08
Trade ratio
Buy 1.66 [ = $126.08 / $75.94 ] 10-year futures for each bond
contract you sell.
The resulting trade is duration or DV01 neutral
1/1/03
5/19/04
5/5/04
4/21/04
4/7/04
3/24/04
3/10/04
2/25/04
2/11/04
1/28/04
1/14/04
12/31/03
12/17/03
12/3/03
11/19/03
11/5/03
10/22/03
10/8/03
9/24/03
9/10/03
8/27/03
8/13/03
7/30/03
7/16/03
7/2/03
6/18/03
6/4/03
5/21/03
5/7/03
4/23/03
4/9/03
3/26/03
3/12/03
2/26/03
2/12/03
1/29/03
1/15/03
5 and 10-year yields and yield spreads
6
GT5 Govt
GT10 Govt
Spread
5
4
3
2
1
0
1/
1/
1/ 03
15
/
1/ 03
29
/
2/ 03
12
/
2/ 03
26
/
3/ 03
12
/
3/ 03
26
/0
4/ 3
9/
4/ 03
23
/0
5/ 3
7/
5/ 03
21
/0
6/ 3
4/
6/ 03
18
/0
7/ 3
2/
7/ 03
16
/
7/ 03
30
/
8/ 03
13
/
8/ 03
27
/
9/ 03
10
/
9/ 03
24
/
10 03
/8
10 /03
/2
2/
11 03
/5
11 /03
/1
9/
12 03
/3
12 /03
/1
7
12 /03
/3
1/
1/ 03
14
/
1/ 04
28
/
2/ 04
11
/
2/ 04
25
/
3/ 04
10
/
3/ 04
24
/0
4/ 4
7/
4/ 04
21
/0
5/ 4
5/
5/ 04
19
/0
4
10-year less 5-year yield spread
OTR10 less OTR5
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
1/
2/
0
1/ 3
16
/0
1/ 3
30
/0
2/ 3
13
/0
2/ 3
27
/0
3/ 3
13
/0
3/ 3
27
/0
4/ 3
10
/0
4/ 3
24
/0
3
5/
8/
03
5/
22
/0
3
6/
5/
03
6/
19
/0
3
7/
3/
03
7/
17
/0
7/ 3
31
/0
8/ 3
14
/0
8/ 3
28
/0
9/ 3
11
/0
9/ 3
25
/0
10 3
/9
10 /03
/2
3/
0
11 3
/6
11 /03
/2
0/
0
12 3
/4
/
12 03
/1
8/
03
1/
1/
0
1/ 4
15
/0
1/ 4
29
/0
2/ 4
12
/0
2/ 4
26
/0
3/ 4
11
/0
3/ 4
25
/0
4
4/
8/
04
4/
22
/0
4
5/
6/
04
5/
20
/0
4
Cumulative P/Ls
3
2
-1:+1 -FV:+TY
-1.5:+1 -FV:+TY
1
0
-1
-2
-3
Characteristics of real bonds
•
•
•
•
Wealth
Yield exposure
Credit spread exposure
Other (covenants, taxes, safekeeping, etc.)
Constructing synthetic bonds
To create a synthetic Treasury, combine:
– Term cash
– Duration equivalent futures position
To create a synthetic corporate, add:
– Credit derivative
Experience with selling the 10-year basis
Coupon
Maturity
4.75
11/15/2008
4.75
11/15/2008
5.5
2/15/2008
5.5
2/15/2008
5.5
2/15/2008
5.5
2/15/2008
6
8/15/2009
6.5
2/15/2010
6.5
2/15/2010
5.75
8/15/2010
Average
Std Dev
*at futures expiration
Contract
Mar-00
Jun-00
Sep-00
Dec-00
Mar-01
Jun-01
Sep-01
Dec-01
Mar-02
Jun-02
Initial
BNOC
12.19
13.20
15.28
11.25
8.25
8.36
15.77
16.59
16.40
12.81
Ending
BNOC
0.83
4.56
1.15
10.41
0.78
8.43
2.95
1.08
1.44
1.98
Difference
11.36
8.64
14.13
0.84
7.47
-0.07
12.81
15.51
14.97
10.83
9.65
5.53
Worst
Drawdown
0.000
11.525
0.264
0.087
0.795
18.607
0.713
6.176
0.000
0.293
3.85
6.42
Most profitable example
TYZ1 (6.5 02/15/10)
2.70
20
BNOC(32nd)
2.50
Repo
2.30
Repo
BNOC (32nds)
30
10
2.10
0
1.90
-10
-20
9/6/01
1.70
9/26/01 10/16/01
11/5/01
11/25/01 12/15/01
1.50
1/4/02
Least profitable example
TYM1 (5.5 2/15/08)
5.10
4.90
4.70
20
4.50
10
4.30
4.10
0
-10
-20
3/10/01
3/30/01
BNOC(32nd)
3.90
Repo
3.70
4/19/01
3.50
7/8/01
5/9/01
5/29/01
6/18/01
Repo
BNOC (32nds)
30
Yield enhancement trades come and go
• Mispricings in futures can last several years
• Opportunities to do yield enhancement
migrate from market to market
– Bond futures in the late 80s
– Treasury note (both 5s and 10s) and bund futures
in the early 90s
– 10-year futures in the early 00s
Volatility arbitrage
If implied yield volatility in the basis market is low relative
to implied volatility in the outright options market, buy the
basis and sell a collection of real options
Price
ISSUE 6.875 08/15/25
(05/26/04, Based on 05/25/04 closing prices)
3.165%
2.640%
Forward yield Shift probability
-0.472
-0.404
at expiry
Yield shift (%)
Twist probability
6.6807%
117-09
Basis
BNOC
Jun-04/late delivery
21.39
3.9032
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
3.165%
0.472
Curve Flattening
Yield twist (%)
0.0502
5.0910
5.1584
5.2258
5.2932
5.3606
5.4280
5.4955
5.5629
5.6303
5.6977
5.7651
5.8325
5.8999
5.9673
6.0347
0.0251
5.0659
5.1333
5.2007
5.2681
5.3356
5.4030
5.4704
5.5378
5.6052
5.6726
5.7400
5.8074
5.8748
5.9422
6.0096
38.2925%
0.0000
5.0408
5.1082
5.1757
5.2431
5.3105
5.3779
5.4453
5.5127
5.5801
5.6475
5.7149
5.7823
5.8497
5.9171
5.9845
24.1730%
-0.0251
5.0158
5.0832
5.1506
5.2180
5.2854
5.3528
5.4202
5.4876
5.5550
5.6224
5.6898
5.7572
5.8246
5.8920
5.9594
6.6807%
-0.0502
4.9907
5.0581
5.1255
5.1929
5.2603
5.3277
5.3951
5.4625
5.5299
5.5973
5.6647
5.7321
5.7995
5.8669
5.9343
3.165%
2.640%
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
3.165%
24.1730%
Curve Steepening
BNOC
at expiry
Shift probability
Yield shift (%)
-0.472
-0.404
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
0.472
Twist probability
6.6807%
Yield twist (%)
0.0502
18.4708
15.7699
13.4789
11.6441
10.1315
8.9580
7.9176
7.0454
6.3452
5.8252
5.4379
5.0776
4.7541
4.5909
4.7052
8.0277
24.1730%
0.0251
13.4377
11.2010
9.3552
7.8571
6.7962
5.9239
5.2245
4.7814
4.3953
4.0219
3.7003
3.8343
4.2036
4.7360
5.3840
5.6507
38.2925%
0.0000
9.0309
7.0786
6.1085
5.1919
4.4967
3.8612
3.3060
3.1502
3.0154
3.4045
3.9247
4.5435
5.3430
6.1262
6.9912
4.3193
24.1730%
-0.0251
5.4604
4.6203
3.7900
3.3773
3.0302
2.8398
2.7978
3.3865
4.1291
4.9340
5.8363
6.7920
7.8415
9.1394
10.4447
4.5405
6.6807%
-0.0502
3.7313
3.3102
2.9451
2.6197
3.0686
3.8962
4.8444
5.8824
6.9838
8.1077
9.2658
10.6430
12.0107
13.3540
14.6635
6.5170
Average over
shifts
Net P/L
at expiry*
Shift probability
3.165%
2.640%
4.123%
5.938%
7.887%
9.659%
10.908%
11.360%
10.908%
9.659%
7.887%
5.938%
4.123%
2.640%
3.165%
Yield shift (%)
-0.472
-0.404
-0.337
-0.270
-0.202
-0.135
-0.067
0.000
0.067
0.135
0.202
0.270
0.337
0.404
0.472
Twist probability
6.6807%
Yield twist (%)
0.0502
14.5677
11.8667
9.5758
7.7410
6.2284
5.0548
4.0144
3.1423
2.4420
1.9220
1.5347
1.1744
0.8509
0.6877
0.8020
4.1245
0.0251
9.5346
7.2979
5.4520
3.9539
2.8930
2.0207
1.3214
0.8783
0.4922
0.1187
-0.2029
-0.0689
0.3005
0.8328
1.4808
1.7475
38.2925%
0.0000
5.1278
3.1754
2.2053
1.2887
0.5936
-0.0419
-0.5972
-0.7530
-0.8877
-0.4987
0.0216
0.6403
1.4398
2.2230
3.0880
0.4161
24.1730%
-0.0251
1.5573
0.7171
-0.1132
-0.5258
-0.8730
-1.0634
-1.1053
-0.5167
0.2259
1.0308
1.9331
2.8888
3.9383
5.2362
6.5415
0.6373
6.6807%
-0.0502
-0.1719
-0.5929
-0.9581
-1.2835
-0.8345
-0.0070
0.9412
1.9792
3.0806
4.2045
5.3626
6.7399
8.1075
9.4508
10.7604
2.6138
24.1730%
* BNOC at expiry - current BNOC
Sell a straddle against this position
choosing a strike price that
corresponds to a forward yield
approximately 6.7 basis points
higher than the current level of
forward yields
Net P/L
(32nds)
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
-2.00
-4.00
-0.472
-0.337
-0.202
-0.067
Average over
shifts
0.067
0.202
0.337
0.472
Roll of money market futures
• Term repo rates play a large role in the
success of hedges and basis trades
• Short-term money market futures can be very
useful in reducing term financing risk
• In the U.S., Fed funds futures traded at the
CBOT provide an excellent hedge for term
repo risk