Transcript CMOs, IOs, POs and Valuation

CMOs, IOs, POs and Valuation
 CMO Mechanics
 Interest Only
 Principal Only
 CMO Tranches: PACs, TACs, Floaters and Inverse Floaters
 MBS Valuation
o MBS Prices
o Static Valuation Models
 Static Yield Spread
 Spread to the Yield Curve
o Dynamic Valuation Models
 Option Adjusted Spreads
o Introduction to Interest Rate Modeling
CMOs
 Introduced by Freddie Mac in June 1983
 Created with separate classes or ‘tranches’ where
investors have distinct claims to MBS cash flows
 Typically have at least four tranches
 Prepayment risk not shared equally
 Institutional maturity intermediation
 Most have AAA ratings
CMOs
 Class A (Fast Pay) Bondholders receive
o Coupon interest
o All scheduled principal payment (amortization)
o All unscheduled principal repayment (prepayments)
o Interest that accrues to Zero Coupon Tranche is
reclassified principal and paid to Class A bondholders
 While Class A bondholders are being repaid, Class B and
C bondholders receive coupon interest only.
 Zero Coupon bonds accrue interest until other tranches
retired
CMOs
CMO Residuals
 Income from securitized mortgages exceed bond
payments and expenses because
o Overcollateralization
o Bond Coupons < Mortgage Interest Rates
 Residual typically held by issuer
 CMO residual yields typically 300-500bp above other
mortgage derivative products
Loan Amount for One Loan
Annual Interest Rate
$100,000.00
8.00%
Loan Term in Years
30
Number of Loans in Pool
1,000
Fees:
Servicing Fee (basis points)
Guarantee Fee (basis points)
Prepay Rate: % of PSA
44
6
100.00%
Computed:
Monthly Payment for One Loan
Number of Payments
Assets Pledged as Security
Debt Issued against Pool
Equity
Residual IRR
$733.76
360
100,000,000
97,000,000
3,000,000
13.56%
Tranche
Coupon
Rate
Amount
Issued
Weight
Weighted
Average
Coupon
A
6.00%
$30,000,000
0.3093
1.86%
B
6.50%
$25,000,000
0.2577
1.68%
C
7.00%
$20,000,000
0.2062
1.44%
Z
8.00%
$22,000,000
0.2268
1.81%
$97,000,000
1.0000
6.79%
Total
Prepayment
Rate
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of
Prepayments
12
36
54
52
49
47
44
41
39
36
35
32
31
28
28
25
24
23
21
20
19
18
16
16
15
14
13
12
12
11
=
Residual IRR
=
100 % of PSA
MBS Cash In
$9,967,310
$12,117,255
$13,446,880
$12,724,947
$11,933,158
$11,262,832
$10,522,280
$9,809,618
$9,213,849
$8,553,493
$8,088,003
$7,469,514
$7,042,820
$6,469,509
$6,155,155
$5,621,021
$5,260,113
$4,908,755
$4,505,546
$4,184,354
$3,874,822
$3,576,811
$3,242,892
$3,020,416
$2,758,150
$2,508,870
$2,273,357
$2,053,083
$1,857,285
$1,654,139
Fees
$496,374
$480,379
$453,813
$423,043
$393,908
$366,320
$340,148
$315,458
$292,158
$270,227
$249,284
$229,464
$210,819
$193,088
$176,357
$160,381
$145,365
$130,984
$117,402
$104,457
$92,161
$80,469
$69,316
$58,706
$48,577
$38,913
$29,675
$20,858
$12,395
$4,267
Tranche A
$5,558,313
$7,810,127
$9,287,049
$8,732,636
$3,180,383
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
MBS Cash
Out
Tranche B
$1,625,000
$1,625,000
$1,625,000
$1,625,000
$6,543,399
$9,178,955
$8,559,256
$3,917,566
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche C
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$5,444,462
$8,851,484
$8,196,380
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche Z
$0
$0
$0
$0
$0
$0
$0
$0
$0
$72,065
$7,838,719
$7,240,050
$6,832,001
$6,276,422
$5,978,797
$5,460,640
$5,114,748
$4,777,771
$4,388,144
$4,079,897
$3,782,661
$3,496,342
$3,173,576
$2,961,710
$2,684,112
$2,266,373
$1,305,268
$0
$0
$0
13.56%
Residual
$887,624
$801,750
$681,018
$544,268
$415,467
$317,558
$222,876
$132,132
$70,207
$14,820
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$25,461
$203,584
$938,414
$2,032,224
$1,844,890
$1,649,873
Prepayment
Rate
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of
Prepayments
25
70
99
91
82
71
64
57
50
44
40
35
31
27
24
22
19
17
15
13
12
10
10
8
7
7
5
5
5
4
=
Residual IRR
=
200 % of PSA
MBS Cash In
$11,225,966
$15,235,852
$17,249,869
$15,523,777
$13,798,570
$11,984,479
$10,641,439
$9,377,650
$8,186,882
$7,155,438
$6,359,801
$5,528,949
$4,833,837
$4,184,363
$3,656,075
$3,239,362
$2,777,331
$2,420,671
$2,089,736
$1,785,891
$1,566,621
$1,307,697
$1,174,218
$956,240
$806,390
$703,875
$552,349
$473,166
$392,010
$307,743
Fees
$494,468
$467,224
$420,777
$369,537
$323,824
$283,442
$247,836
$216,419
$188,617
$164,133
$142,598
$123,580
$106,874
$92,173
$79,216
$67,879
$57,825
$49,095
$41,398
$34,690
$28,890
$23,661
$19,225
$15,315
$11,920
$8,998
$6,452
$4,276
$2,393
$781
Tranche A
$6,824,589
$10,981,344
$13,222,180
$2,484,055
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
MBS Cash
Out
Tranche B
$1,625,000
$1,625,000
$1,625,000
$10,869,218
$11,803,090
$4,475,946
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche C
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$7,066,065
$10,304,047
$4,703,978
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche Z
$0
$0
$0
$0
$0
$0
$0
$4,435,811
$7,998,265
$6,991,305
$6,217,203
$5,405,369
$4,726,963
$4,092,190
$3,576,860
$3,171,483
$2,719,507
$2,370,701
$1,946,794
$1,503,449
$1,027,616
$0
$0
$0
$0
$0
$0
$0
$0
$0
11.38%
Residual
$881,908
$762,285
$581,911
$400,966
$271,656
$159,026
$89,556
$21,441
$0
$0
$0
$0
$0
$0
$0
$0
$0
$875
$101,544
$247,751
$510,114
$1,284,036
$1,154,992
$940,926
$794,470
$694,877
$545,896
$468,890
$389,618
$306,962
Prepayment
Rate
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of
Prepayments
50
132
169
140
110
86
67
53
42
32
26
20
16
12
10
7
6
5
4
2
3
2
1
1
1
1
0
1
0
0
=
Residual IRR
=
400 % of PSA
MBS Cash In
$13,642,784
$20,900,811
$23,032,404
$18,666,718
$14,539,056
$11,274,860
$8,714,605
$6,804,439
$5,312,897
$4,029,561
$3,193,882
$2,432,957
$1,901,934
$1,418,596
$1,135,543
$806,816
$655,390
$518,838
$398,054
$230,486
$258,976
$171,759
$98,951
$84,139
$70,154
$56,268
$17,610
$33,835
$8,805
$8,806
Fees
$490,575
$441,940
$361,912
$281,712
$218,527
$169,331
$131,046
$101,338
$78,215
$60,240
$46,287
$35,560
$27,261
$20,772
$15,809
$12,000
$9,091
$6,789
$5,080
$3,758
$2,789
$2,018
$1,432
$1,009
$707
$489
$271
$202
$64
$23
Tranche A
$9,256,982
$16,747,438
$6,657,899
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
MBS Cash
Out
Tranche B
$1,625,000
$1,625,000
$14,188,266
$12,501,125
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche C
$1,400,000
$1,400,000
$1,400,000
$5,658,137
$14,205,995
$2,176,753
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche Z
$0
$0
$0
$0
$0
$8,917,308
$8,583,559
$6,703,101
$5,234,682
$3,969,321
$3,147,594
$2,333,606
$1,647,412
$304,582
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
10.89%
Residual
$870,227
$686,433
$424,326
$225,744
$114,533
$11,468
$0
$0
$0
$0
$0
$63,791
$227,261
$1,093,242
$1,119,734
$794,816
$646,299
$512,049
$392,973
$226,728
$256,188
$169,741
$97,519
$83,130
$69,447
$55,780
$17,340
$33,633
$8,742
$8,782
Prepayment
Rate
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of
Prepayments
6
18
28
28
27
27
25
25
24
24
22
22
22
21
20
19
19
19
18
17
17
16
16
15
15
15
14
14
13
13
=
Residual IRR
=
50 % of PSA
MBS Cash In
$9,388,229
$10,463,189
$11,219,397
$10,942,251
$10,572,151
$10,303,908
$9,851,699
$9,594,515
$9,252,289
$9,001,329
$8,580,569
$8,342,000
$8,104,413
$7,786,262
$7,476,005
$7,173,584
$6,949,536
$6,724,347
$6,433,378
$6,149,600
$5,932,416
$5,659,181
$5,443,159
$5,181,342
$4,967,210
$4,747,252
$4,497,197
$4,277,007
$4,040,401
$3,820,206
Fees
$497,368
$487,203
$471,206
$452,637
$434,251
$416,202
$398,281
$380,716
$363,488
$346,248
$329,211
$312,436
$295,798
$279,180
$262,707
$246,326
$229,956
$213,614
$197,308
$180,908
$164,528
$148,018
$131,408
$114,693
$97,824
$80,742
$63,442
$45,903
$28,074
$9,950
Tranche A
$4,975,255
$6,128,765
$6,989,994
$6,831,564
$6,583,133
$4,212,187
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
MBS Cash
Out
Tranche B
$1,625,000
$1,625,000
$1,625,000
$1,625,000
$1,625,000
$3,850,583
$7,714,274
$7,555,638
$7,314,752
$2,786,676
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche C
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$1,400,000
$5,776,702
$8,218,776
$8,029,443
$471,454
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Tranche Z
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$7,337,161
$7,507,082
$7,213,299
$6,927,258
$6,719,579
$6,510,733
$6,236,070
$5,968,691
$5,767,888
$5,511,163
$5,311,751
$5,066,649
$4,869,386
$4,666,510
$4,433,755
$4,228,829
$3,150,093
$0
16.73%
Residual
$890,606
$822,222
$733,197
$633,051
$529,766
$424,935
$339,143
$258,161
$174,049
$91,703
$32,581
$120
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$2,275
$862,235
$3,810,256
CMOs
Interest Only
 IO Strip: the interest only portion of a CMO (tranche)
 Investor benefits from slowing prepayments--interest payments
made for longer than expected period when prepayments are
slower than expected
 Bearish security: prices tend to rise when interest rates rise
CMOs
Principal Only
 PO Strip: principal only strip purchased at a discount
 If prepayments are faster than expected (e.g. in a declining
interest rate environment), then investors receive CFs sooner
than expected increasing the investors' yield
Prepayment
Rate
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of
Prepayments
12
36
54
52
49
47
44
41
39
36
35
32
31
28
28
25
24
23
21
20
19
18
16
16
15
14
13
12
12
11
=
100 %
MBS Cash In
$9,967,310
$12,117,255
$13,446,880
$12,724,947
$11,933,158
$11,262,832
$10,522,280
$9,809,618
$9,213,849
$8,553,493
$8,088,003
$7,469,514
$7,042,820
$6,469,509
$6,155,155
$5,621,021
$5,260,113
$4,908,755
$4,505,546
$4,184,354
$3,874,822
$3,576,811
$3,242,892
$3,020,416
$2,758,150
$2,508,870
$2,273,357
$2,053,083
$1,857,285
$1,654,139
MBS Cash Out
Principal
Interest
$2,025,331
$7,445,606
$4,431,197
$7,205,680
$6,185,875
$6,807,192
$5,956,264
$6,345,640
$5,630,627
$5,908,623
$5,401,709
$5,494,803
$5,079,914
$5,102,218
$4,762,286
$4,731,874
$4,539,326
$4,382,365
$4,229,857
$4,053,409
$4,099,456
$3,739,263
$3,798,097
$3,441,954
$3,669,715
$3,162,285
$3,380,103
$2,896,319
$3,333,437
$2,645,360
$3,054,927
$2,405,713
$2,934,273
$2,180,475
$2,813,014
$1,964,757
$2,627,119
$1,761,025
$2,513,043
$1,566,855
$2,400,243
$1,382,417
$2,289,302
$1,207,040
$2,133,831
$1,039,745
$2,081,128
$880,583
$1,980,916
$728,657
$1,886,268
$583,689
$1,798,551
$445,131
$1,719,351
$312,874
$1,658,967
$185,923
$1,585,874
$63,999
Fees
$496,374
$480,379
$453,813
$423,043
$393,908
$366,320
$340,148
$315,458
$292,158
$270,227
$249,284
$229,464
$210,819
$193,088
$176,357
$160,381
$145,365
$130,984
$117,402
$104,457
$92,161
$80,469
$69,316
$58,706
$48,577
$38,913
$29,675
$20,858
$12,395
$4,267
Principal Only
Interest Only
Discount Rate
8.25%
7.75%
Price @ 100 PSA
$44,922,302.05
$51,558,714.96
Yields:
@ 50 PSA
@ 100 PSA
@ 200 PSA
@ 400 PSA
6.08%
8.25%
12.64%
20.12%
10.20%
7.75%
2.88%
-6.72%
CMO Tranches
Introduction
In addition to the sequential pay tranches that include a zero-coupon
(or accretion) bond, collateralized mortgage obligations typically
have one (or more) of the following tranches:

PAC--planned amortization class

TAC--targeted amortization class

Floating rate tranche

Inverse floating rate tranche
CMO Tranches
Planned Amortization Classes
 Introduced in 1986
 Prepayment "protected" CMO tranche
 Bond pays scheduled principal payments (irrespective of CFs into
the CMO) for a range of actual prepayment speeds
 Range of prepay rates: 75% PSA to 240% PSA
 A companion PAC bond absorbs any cash flow uncertainty
o PAC and companion PAC receive coupon interest, but
o All CMO principal in excess of PAC scheduled go to companion
o All CMO principal below PAC scheduled paid from companion
CMO Tranches
Targeted Amortization Classes
CMO pricing speed = prepayment rate that was assumed when the
issue was priced
 A TAC is a PAC with
o A lower band = CMO pricing speed
o An upper band similar to a PAC upper band
 Like PACs, TACs repay principal according to a schedule as long
as actual prepayments are within a specified range
 Unlike PACs, TACs DO NOT protect against WAL extensions
when prepayments are slower than pricing speed
 Investors tradeoff extension risk for higher yields
CMO Tranches
Floaters and Inverse Floaters
 Introduced by Shearson Lehman Brothers in 1986
 Divide a fixed rate CMO tranche into two tranches with variable
coupons
o Floater rate moves with some index rate
o Inverse floater varies inversely with the index
 Collateral weighted average coupon = fixed rate coupon
 Principal payments are the same as they are for the underlying
fixed rate tranche
CMO Tranches
Floating Rate Tranche
 Floating rate = Index + spread (in basis points)
 Also have maximums, or caps, on floating rate
 Typical index rates include
o LIBOR: London Interbank Overnight Rate
o COFI: FHLB 11th District Cost of Funds Index
o CMT: Constant Maturity Treasuries
 Example:
Floating rate = LIBOR + 65bp with cap of 12%
CMO Tranches
Inverse Floater
C
Inverse Floater Coupon 
 (Index  Spread)  Multiplier ,
IFF
where C = the coupon on the fixed rate collateral
IFF  Inverse floater factor 
Multiplier 
Inverse floater face value
Collateral face value
Floater face value
Inverse floater face value
Inverse floater tranches have interest rate floors.
CMO Tranches
Inverse Floater
Example: Suppose a $30M 6% fixed rate coupon CMO tranche was
divided into a $20M floating rate bond and a $10M inverse floater.
If the coupon on the floating rate bond is LIBOR + 50bp, what is the
coupon on the inverse floater?
Multiplier = $20M/$10M = 2;
Inverse Floater Coupon 
IFF = 10/30 = 1/3
6
 (LIBOR  0.50)  2
1
3
 18 - 2  LIBOR - 1.00
 17.0 - 2  LIBOR
CMO Tranches
Inverse Floater
Note that the (collateral) weighted coupon is:
$20 M
$10M
 ( LIBOR  0.5) 
 (17.0  2  LIBOR)
$30 M
$30M

2
1
1
2
LIBOR    17.0  LIBOR
3
3
3
3
18

 6%, the coupon on the fixed rate tranche
3
Valuation
MBS Prices
 Stated as a percentage of the face amount (e.g. 102)
 Fractions of one percent are expressed in thirty-seconds
(1/32)
 Sixty fourths (1/64) are represented with a ‘+’
Valuation
Ginnie Mae 6.5; WAC 7.0%
Price 100-24, Assumed WAM 29-08
Prepayment Speed
Yield at Projected Speed
WAL @ Projected Speed
Yield of WAL (9.4yr) Treasury
Spread over Treasury
125 PSA
6.42%
9.4 Years
4.90%
152bp
Valuation
Scenario Analysis for Ginnie Mae 6.5% Pass Through
(Short-Term ) Interest Rate Moves (bp)
-100
Prepay rate (PSA)
WAL (years)
Yield
WAL Treasury Yield
Spread/WAL (bp)
Yield Curve
1 Yr
4.63
0
425
3.2
6.19%
3.78%
241
2 Yr
4.77
Source: Salomon Smith Barney
3 Yr
4.78
125
9.4
6.43%
4.90%
142
5 Yr
4.79
+100
112
10.4
10.4%
5.91%
53
10 Yr
4.91
30 Yr
5.35
Valuation
Things Influencing MBS Cash Flows
 Aging: newer MBSs tend to have slower prepay speeds than
seasoned MBSs because households that just moved are
unlikely to change homes soon after moving
 Burnout: when market interest rates fall below coupon, prepay
speeds accelerate rapidly. After the most financially
sophisticated borrowers exit the pool, prepay speeds slow down
significantly (burnout)
 Seasoning: home buying (selling) has a seasonal component
with more transactions (and prepayments) in the summer
months and fewer transactions (and prepayments) in the winter
months.
Valuation
Valuation Models
1. Static yield spread (Spread/WAL): the spread between the
MBS bond yield and the yield on the benchmark Treasury.
2. Yield curve spread: discount each monthly cash flow from
the MBS using the yield on the same maturity Treasury plus a
constant spread
a. Spread to Treasury zero
b. Spread to forward rate
3. Option Adjusted Spread: use Monte Carlo methods to
simulate alternative interest rate paths; associate expected
prepayments; take average of resulting PVs
Valuation
Yield Curve Spread
For a given yield curve (or series of zero Treasury rates), rt, the PV
of the MBS cash flows is:
PV(s) 
CF3
CF1
CF2
CFn



....

1  r1  s (1  r2  s) 2 (1  r3  s) 3
(1  rn  s) n
where CFi is the ith period cash flow, ri is the ith period zero rate,
and s is the spread to the yield curve. The value of s that equates
PV(s) to the market price of the MBS is the Yield Curve Spread.
Valuation
Option Adjusted Spread
 The Yield Curve Spread (and the Spread/WAL) ignore
uncertainty in future interest rates (e.g. they use the current
yield on Treasuries to value MBSs)
 Future prepayments are going to depend on future interest
rates—these are likely to be different from current rates
 In addition, investors’ reinvestment rates will depend on
future interest rates, not on the current yield curve.
Valuation
Option Adjusted Spread
1. Use some interest rate model to generate an interest rate series
2. Attach prepayment behavior to the generated interest rates
3. Compute the expected MBS cash flows for the assumed interest
rate/prepayment behavior.
4. Use the yield curve spread to compute PV(s) for the assumed
interest rate series.
5. Repeat 1-4 one thousand times.
6. The value of s that equates the average PV(s) to the current
MBS price is the Option Adjusted Spread.
Valuation
OASs for Ginnie Mae Pass-Throughs
Coupon WAM Price PSA Yield WAL Spread/ YCS OAS
WAL
6.0%
6.5%
7.0%
7.5%
8.0%
8.5%
29-10
29-08
28-04
28-02
28-01
28-00
98-17
100-23
102-10
103-10
104-10
106-02
85
125
210
340
390
385
6.24 11.4yrs 130bp
6.42 9.5
153
6.51 6.5
166
6.42 4.2
160
6.41 3.6
160
6.33 3.7
152
Source: Salomon Smith Barney
110bp
134
153
161
167
158
70bp
75
78
85
105
108
Valuation
Introduction to Interest Rate Modeling
Interest Rate Models:
1. Must be consistent with the current yield curve—benchmark
Treasuries must be fairly priced
2. Cannot lead to arbitrage opportunities
3. Must be consistent with historical experience
a. No negative rates
b. Cannot allow rates to increase without bound
c. Must incorporate relative volatility in the TERM
STRUCTURE: historically, yields on 1-Yr Treasuries
are more volatile then either 1-Mo rates or 10-Yr rates.
Valuation
TREASURY YIELD CURVE RATES FOR JUN 2004
Date
06/01/2004
06/02/2004
06/03/2004
06/04/2004
06/07/2004
06/08/2004
06/09/2004
06/10/2004
06/14/2004
06/15/2004
06/16/2004
06/17/2004
06/18/2004
06/21/2004
06/22/2004
1-mo
0.97
0.97
0.96
0.96
0.97
1.03
1.03
1.02
1.07
1.09
1.05
1.02
1.03
1.04
1.08
3-mo
1.17
1.17
1.17
1.21
1.24
1.27
1.27
1.30
1.41
1.34
1.30
1.28
1.29
1.33
1.32
6-mo
1.44
1.45
1.45
1.51
1.54
1.56
1.61
1.65
1.77
1.68
1.69
1.66
1.68
1.70
1.69
1-yr
1.89
1.92
1.91
1.97
1.96
2.02
2.14
2.14
2.25
2.16
2.24
2.21
2.24
2.17
2.21
The 30-yr has been discontinued.
2-yr
2.60
2.65
2.63
2.70
2.67
2.73
2.79
2.81
2.97
2.77
2.84
2.81
2.81
2.80
2.80
3-yr
3.14
3.19
3.17
3.25
3.22
3.24
3.31
3.32
3.45
3.26
3.31
3.28
3.28
3.26
3.28
5-yr
3.86
3.91
3.89
3.97
3.95
3.96
4.01
4.00
4.10
3.90
3.96
3.93
3.94
3.91
3.92
7-yr 10-yr 20-yr
4.31 4.71 5.45
4.35 4.74 5.47
4.34 4.71 5.46
4.41 4.78 5.51
4.39 4.78 5.51
4.40 4.78 5.50
4.44 4.82 5.54
4.43 4.81 5.52
4.51 4.89 5.59
4.31 4.69 5.41
4.36 4.74 5.46
4.33 4.71 5.42
4.34 4.72 5.43
4.33 4.70 5.42
4.34 4.72 5.43
See www.treas.gov/domfin/statistics.htm
Valuation
Relationship Between Short and Long Interest Rates
I. Expectations Hypothesis (with certainty)
( 1 + tRN) = [(1 + tR1)(1 +
t+1r1)(1
+
t+2r1)…(1
+
1/N
t+N-1r1)]
Where R = actual market interest rate
r = expected interest rate
pre-subscript is the time period for which rates
are applicable
post-subscript is the maturity of the bond
so tR1 is today’s (period t) one year actual yield (e.g. 2.21%)
and t+1r1 is the one-year expected yield one year from today.
Maturity
1
2
3
FV of $1
1.02210
1.05678
1.10166
Implied 1-Year Yield
2.2100
3.3934
4.2467
Rate change (bp)
↑ 118bp
↑ 85bp
Daily Treasury Yield Curve Rates
Historical Data
November 2004
Date
1 mo
3 mo
6 mo
1 yr
2 yr
3 yr
5 yr
7 yr
10 yr
20 yr
11/01/04
1.79
1.99
2.20
2.34
2.61
2.89
3.36
3.76
4.11
4.84
11/02/04
1.86
1.97
2.19
2.33
2.60
2.86
3.34
3.75
4.10
4.84
11/03/04
1.83
1.96
2.18
2.32
2.60
2.85
3.35
3.74
4.09
4.83
11/04/04
1.85
1.98
2.19
2.34
2.63
2.89
3.37
3.76
4.10
4.82
11/05/04
1.86
2.03
2.27
2.44
2.80
3.04
3.51
3.88
4.21
4.92
11/08/04
1.88
2.07
2.30
2.47
2.80
3.08
3.51
3.88
4.22
4.95
* 30-year Treasury constant maturity series was discontinued as of 2/18/02. See Long-Term Average Rate for more
information.
Source: http://www.treas.gov/offices/domestic-finance/debt-management/interestrate/yield.html
Valuation
A Generic Interest Rate Model
Change in interest rate = drift parameter * small time interval
+ volatility parameter * random shock
Or, in the language of Calculus, dr
= drift * dt + volatility * dz
where
dr = (very short term) change in interest rates
drift parameter = a (rLT - rt) where 0 < a < 1 (mean reversion)
dt = change in time period
volatility parameter, σ = standard deviation of interest rates
random shock = Brownian motion
The drift and volatility parameters need not be constants—they can depend
on time period and/or interest rates.
Valuation
Brownian Motion

Normally distributed (e.g. bell shaped)

The distribution is centered at zero

Variance scaled to one

Draws are serially independent
Valuation
Some Interest Rate Models
Vasicek:
dr = a(rLT – r)dt + σdz
Hull-White:
dr = a(t)(θ(t) – r)dt + σ(t)dz
Cox-Ingersoll-Ross: dr = a(t)(θ(t) – r)dt + σ(t)√rdz
There are others….