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 $100,000.00

Annual Interest Rate 8.00%

Loan Term in Years 30

Number of Loans in Pool 1,000

Fees:

Servicing Fee (basis points) 44

Guarantee Fee (basis points) 6

Prepay Rate: % of PSA 100.00%

Computed:

Monthly Payment for One Loan $733.76 Number of Payments 360

Assets Pledged as Security 100,000,000

Debt Issued against Pool 97,000,000

Equity 3,000,000

Residual IRR 13.56%

Tranche Coupon Amount Weight Weighted Average

Rate Issued 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%

Total $97,000,000 1.0000 6.79%

Prepayment Rate = 100 % of PSA

Residual IRR = 13.56%

Year Number of MBS Cash In MBS Cash

Out Prepayments Fees Tranche A Tranche B Tranche C Tranche Z Residual

1 12 $9,967,310 $496,374 $5,558,313 $1,625,000 $1,400,000 $0 $887,624 2 36 $12,117,255 $480,379 $7,810,127 $1,625,000 $1,400,000 $0 $801,750 3 54 $13,446,880 $453,813 $9,287,049 $1,625,000 $1,400,000 $0 $681,018 4 52 $12,724,947 $423,043 $8,732,636 $1,625,000 $1,400,000 $0 $544,268 5 49 $11,933,158 $393,908 $3,180,383 $6,543,399 $1,400,000 $0 $415,467 6 47 $11,262,832 $366,320 $0 $9,178,955 $1,400,000 $0 $317,558 7 44 $10,522,280 $340,148 $0 $8,559,256 $1,400,000 $0 $222,876 8 41 $9,809,618 $315,458 $0 $3,917,566 $5,444,462 $0 $132,132 9 39 $9,213,849 $292,158 $0 $0 $8,851,484 $0 $70,207

10 36 $8,553,493 $270,227 $0 $0 $8,196,380 $72,065 $14,820 11 35 $8,088,003 $249,284 $0 $0 $0 $7,838,719 $0 12 32 $7,469,514 $229,464 $0 $0 $0 $7,240,050 $0 13 31 $7,042,820 $210,819 $0 $0 $0 $6,832,001 $0 14 28 $6,469,509 $193,088 $0 $0 $0 $6,276,422 $0 15 28 $6,155,155 $176,357 $0 $0 $0 $5,978,797 $0 16 25 $5,621,021 $160,381 $0 $0 $0 $5,460,640 $0 17 24 $5,260,113 $145,365 $0 $0 $0 $5,114,748 $0 18 23 $4,908,755 $130,984 $0 $0 $0 $4,777,771 $0 19 21 $4,505,546 $117,402 $0 $0 $0 $4,388,144 $0 20 20 $4,184,354 $104,457 $0 $0 $0 $4,079,897 $0 21 19 $3,874,822 $92,161 $0 $0 $0 $3,782,661 $0 22 18 $3,576,811 $80,469 $0 $0 $0 $3,496,342 $0 23 16 $3,242,892 $69,316 $0 $0 $0 $3,173,576 $0 24 16 $3,020,416 $58,706 $0 $0 $0 $2,961,710 $0 25 15 $2,758,150 $48,577 $0 $0 $0 $2,684,112 $25,461 26 14 $2,508,870 $38,913 $0 $0 $0 $2,266,373 $203,584 27 13 $2,273,357 $29,675 $0 $0 $0 $1,305,268 $938,414 28 12 $2,053,083 $20,858 $0 $0 $0 $0 $2,032,224 29 12 $1,857,285 $12,395 $0 $0 $0 $0 $1,844,890 30 11 $1,654,139 $4,267 $0 $0 $0 $0 $1,649,873

Prepayment Rate = 200 % of PSA

Residual IRR = 11.38%

Year Number of MBS Cash In MBS Cash

Out Prepayments Fees Tranche A Tranche B Tranche C Tranche Z Residual

1 25 $11,225,966 $494,468 $6,824,589 $1,625,000 $1,400,000 $0 $881,908 2 70 $15,235,852 $467,224 $10,981,344 $1,625,000 $1,400,000 $0 $762,285 3 99 $17,249,869 $420,777 $13,222,180 $1,625,000 $1,400,000 $0 $581,911 4 91 $15,523,777 $369,537 $2,484,055 $10,869,218 $1,400,000 $0 $400,966 5 82 $13,798,570 $323,824 $0 $11,803,090 $1,400,000 $0 $271,656 6 71 $11,984,479 $283,442 $0 $4,475,946 $7,066,065 $0 $159,026 7 64 $10,641,439 $247,836 $0 $0 $10,304,047 $0 $89,556 8 57 $9,377,650 $216,419 $0 $0 $4,703,978 $4,435,811 $21,441 9 50 $8,186,882 $188,617 $0 $0 $0 $7,998,265 $0

10 44 $7,155,438 $164,133 $0 $0 $0 $6,991,305 $0 11 40 $6,359,801 $142,598 $0 $0 $0 $6,217,203 $0 12 35 $5,528,949 $123,580 $0 $0 $0 $5,405,369 $0 13 31 $4,833,837 $106,874 $0 $0 $0 $4,726,963 $0 14 27 $4,184,363 $92,173 $0 $0 $0 $4,092,190 $0 15 24 $3,656,075 $79,216 $0 $0 $0 $3,576,860 $0 16 22 $3,239,362 $67,879 $0 $0 $0 $3,171,483 $0 17 19 $2,777,331 $57,825 $0 $0 $0 $2,719,507 $0 18 17 $2,420,671 $49,095 $0 $0 $0 $2,370,701 $875 19 15 $2,089,736 $41,398 $0 $0 $0 $1,946,794 $101,544 20 13 $1,785,891 $34,690 $0 $0 $0 $1,503,449 $247,751 21 12 $1,566,621 $28,890 $0 $0 $0 $1,027,616 $510,114 22 10 $1,307,697 $23,661 $0 $0 $0 $0 $1,284,036 23 10 $1,174,218 $19,225 $0 $0 $0 $0 $1,154,992 24 8 $956,240 $15,315 $0 $0 $0 $0 $940,926 25 7 $806,390 $11,920 $0 $0 $0 $0 $794,470 26 7 $703,875 $8,998 $0 $0 $0 $0 $694,877 27 5 $552,349 $6,452 $0 $0 $0 $0 $545,896 28 5 $473,166 $4,276 $0 $0 $0 $0 $468,890 29 5 $392,010 $2,393 $0 $0 $0 $0 $389,618 30 4 $307,743 $781 $0 $0 $0 $0 $306,962

Prepayment Rate = 400 % of PSA

Residual IRR = 10.89%

Year Number of MBS Cash In MBS Cash

Out Prepayments Fees Tranche A Tranche B Tranche C Tranche Z Residual

1 50 $13,642,784 $490,575 $9,256,982 $1,625,000 $1,400,000 $0 $870,227 2 132 $20,900,811 $441,940 $16,747,438 $1,625,000 $1,400,000 $0 $686,433 3 169 $23,032,404 $361,912 $6,657,899 $14,188,266 $1,400,000 $0 $424,326 4 140 $18,666,718 $281,712 $0 $12,501,125 $5,658,137 $0 $225,744 5 110 $14,539,056 $218,527 $0 $0 $14,205,995 $0 $114,533 6 86 $11,274,860 $169,331 $0 $0 $2,176,753 $8,917,308 $11,468 7 67 $8,714,605 $131,046 $0 $0 $0 $8,583,559 $0 8 53 $6,804,439 $101,338 $0 $0 $0 $6,703,101 $0 9 42 $5,312,897 $78,215 $0 $0 $0 $5,234,682 $0

10 32 $4,029,561 $60,240 $0 $0 $0 $3,969,321 $0 11 26 $3,193,882 $46,287 $0 $0 $0 $3,147,594 $0 12 20 $2,432,957 $35,560 $0 $0 $0 $2,333,606 $63,791 13 16 $1,901,934 $27,261 $0 $0 $0 $1,647,412 $227,261 14 12 $1,418,596 $20,772 $0 $0 $0 $304,582 $1,093,242 15 10 $1,135,543 $15,809 $0 $0 $0 $0 $1,119,734 16 7 $806,816 $12,000 $0 $0 $0 $0 $794,816 17 6 $655,390 $9,091 $0 $0 $0 $0 $646,299 18 5 $518,838 $6,789 $0 $0 $0 $0 $512,049 19 4 $398,054 $5,080 $0 $0 $0 $0 $392,973 20 2 $230,486 $3,758 $0 $0 $0 $0 $226,728 21 3 $258,976 $2,789 $0 $0 $0 $0 $256,188 22 2 $171,759 $2,018 $0 $0 $0 $0 $169,741 23 1 $98,951 $1,432 $0 $0 $0 $0 $97,519 24 1 $84,139 $1,009 $0 $0 $0 $0 $83,130 25 1 $70,154 $707 $0 $0 $0 $0 $69,447 26 1 $56,268 $489 $0 $0 $0 $0 $55,780 27 0 $17,610 $271 $0 $0 $0 $0 $17,340 28 1 $33,835 $202 $0 $0 $0 $0 $33,633 29 0 $8,805 $64 $0 $0 $0 $0 $8,742 30 0 $8,806 $23 $0 $0 $0 $0 $8,782

Prepayment Rate = 50 % of PSA

Residual IRR = 16.73%

Year Number of MBS Cash In MBS Cash

Out Prepayments Fees Tranche A Tranche B Tranche C Tranche Z Residual

1 6 $9,388,229 $497,368 $4,975,255 $1,625,000 $1,400,000 $0 $890,606 2 18 $10,463,189 $487,203 $6,128,765 $1,625,000 $1,400,000 $0 $822,222 3 28 $11,219,397 $471,206 $6,989,994 $1,625,000 $1,400,000 $0 $733,197 4 28 $10,942,251 $452,637 $6,831,564 $1,625,000 $1,400,000 $0 $633,051 5 27 $10,572,151 $434,251 $6,583,133 $1,625,000 $1,400,000 $0 $529,766 6 27 $10,303,908 $416,202 $4,212,187 $3,850,583 $1,400,000 $0 $424,935 7 25 $9,851,699 $398,281 $0 $7,714,274 $1,400,000 $0 $339,143 8 25 $9,594,515 $380,716 $0 $7,555,638 $1,400,000 $0 $258,161 9 24 $9,252,289 $363,488 $0 $7,314,752 $1,400,000 $0 $174,049

10 24 $9,001,329 $346,248 $0 $2,786,676 $5,776,702 $0 $91,703 11 22 $8,580,569 $329,211 $0 $0 $8,218,776 $0 $32,581 12 22 $8,342,000 $312,436 $0 $0 $8,029,443 $0 $120 13 22 $8,104,413 $295,798 $0 $0 $471,454 $7,337,161 $0 14 21 $7,786,262 $279,180 $0 $0 $0 $7,507,082 $0 15 20 $7,476,005 $262,707 $0 $0 $0 $7,213,299 $0 16 19 $7,173,584 $246,326 $0 $0 $0 $6,927,258 $0 17 19 $6,949,536 $229,956 $0 $0 $0 $6,719,579 $0 18 19 $6,724,347 $213,614 $0 $0 $0 $6,510,733 $0 19 18 $6,433,378 $197,308 $0 $0 $0 $6,236,070 $0 20 17 $6,149,600 $180,908 $0 $0 $0 $5,968,691 $0 21 17 $5,932,416 $164,528 $0 $0 $0 $5,767,888 $0 22 16 $5,659,181 $148,018 $0 $0 $0 $5,511,163 $0 23 16 $5,443,159 $131,408 $0 $0 $0 $5,311,751 $0 24 15 $5,181,342 $114,693 $0 $0 $0 $5,066,649 $0 25 15 $4,967,210 $97,824 $0 $0 $0 $4,869,386 $0 26 15 $4,747,252 $80,742 $0 $0 $0 $4,666,510 $0 27 14 $4,497,197 $63,442 $0 $0 $0 $4,433,755 $0 28 14 $4,277,007 $45,903 $0 $0 $0 $4,228,829 $2,275 29 13 $4,040,401 $28,074 $0 $0 $0 $3,150,093 $862,235 30 13 $3,820,206 $9,950 $0 $0 $0 $0 $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

CMOsPrincipal 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 = 100 % Year Number of MBS Cash In MBS Cash Out Prepayments Principal Interest Fees

1 12 $9,967,310 $2,025,331 $7,445,606 $496,374 2 36 $12,117,255 $4,431,197 $7,205,680 $480,379 3 54 $13,446,880 $6,185,875 $6,807,192 $453,813 4 52 $12,724,947 $5,956,264 $6,345,640 $423,043 5 49 $11,933,158 $5,630,627 $5,908,623 $393,908 6 47 $11,262,832 $5,401,709 $5,494,803 $366,320 7 44 $10,522,280 $5,079,914 $5,102,218 $340,148 8 41 $9,809,618 $4,762,286 $4,731,874 $315,458 9 39 $9,213,849 $4,539,326 $4,382,365 $292,158

10 36 $8,553,493 $4,229,857 $4,053,409 $270,227 11 35 $8,088,003 $4,099,456 $3,739,263 $249,284 12 32 $7,469,514 $3,798,097 $3,441,954 $229,464 13 31 $7,042,820 $3,669,715 $3,162,285 $210,819 14 28 $6,469,509 $3,380,103 $2,896,319 $193,088 15 28 $6,155,155 $3,333,437 $2,645,360 $176,357 16 25 $5,621,021 $3,054,927 $2,405,713 $160,381 17 24 $5,260,113 $2,934,273 $2,180,475 $145,365 18 23 $4,908,755 $2,813,014 $1,964,757 $130,984 19 21 $4,505,546 $2,627,119 $1,761,025 $117,402 20 20 $4,184,354 $2,513,043 $1,566,855 $104,457 21 19 $3,874,822 $2,400,243 $1,382,417 $92,161 22 18 $3,576,811 $2,289,302 $1,207,040 $80,469 23 16 $3,242,892 $2,133,831 $1,039,745 $69,316 24 16 $3,020,416 $2,081,128 $880,583 $58,706 25 15 $2,758,150 $1,980,916 $728,657 $48,577 26 14 $2,508,870 $1,886,268 $583,689 $38,913 27 13 $2,273,357 $1,798,551 $445,131 $29,675 28 12 $2,053,083 $1,719,351 $312,874 $20,858 29 12 $1,857,285 $1,658,967 $185,923 $12,395 30 11 $1,654,139 $1,585,874 $63,999 $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 6.08% 10.20% @ 100 PSA 8.25% 7.75% @ 200 PSA 12.64% 2.88% @ 400 PSA 20.12% -6.72%

CMO TranchesIntroduction

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 TranchesPlanned 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 TranchesTargeted 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 TranchesFloaters 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 TranchesFloating 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 TranchesInverse Floater

Multiplier Spread) Index ( IFF

C Coupon Floater Inverse , w h e r e C = t h e c o u p o n o n t h e f i x e d r a t e c o l l a t e r a l

valueface Collateral valuefacefloater Inverse factor floater Inverse IFF

valuefacefloater Inverse valuefaceFloater Multiplier

I n v e r s e f l o a t e r t r a n c h e s h a v e i n t e r e s t r a t e f l o o r s .

CMO TranchesInverse Floater

E x a m p l e : S u p p o s e a $ 3 0 M 6 % f i x e d r a t e c o u p o n C M O t r a n c h e w a s d i v i d e d i n t o a $ 2 0 M f l o a t i n g r a t e b o n d a n d a $ 1 0 M i n v e r s e f l o a t e r . I f t h e c o u p o n o n th e f l o a t i n g r a t e b o n d i s L I B O R + 5 0 b p , w h a t i s t h e c o u p o n o n t h e i n v e r s e f l o a t e r ? M u l t i p l i e r = $ 2 0 M / $ 1 0 M = 2 ; I F F = 1 0 / 3 0 = 1 / 3

LIBOR2 - 17.0

1.00 - LIBOR2 - 18

2 0.50) LIBOR( 3

16 Coupon Floater Inverse

CMO TranchesInverse Floater

Note that the (collateral) weighted coupon is:

trancherate fixed on thecoupon the%,63

18

LIBOR32 0.17

31

31 LIBOR

32

LIBOR)20.17($30M$10M 0.5) LIBOR(

M30$M20$

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 125 PSA Yield at Projected Speed 6.42% WAL @ Projected Speed 9.4 Years Yield of WAL (9.4yr) Treasury 4.90% Spread over Treasury 152bp

Valuation

Scenario Analysis for Ginnie Mae 6.5% Pass Through

(Short-Term ) Interest Rate Moves (bp)

-100 0 +100

Prepay rate (PSA) 425 125 112 WAL (years) 3.2 9.4 10.4 Yield 6.19% 6.43% 10.4% WAL Treasury Yield 3.78% 4.90% 5.91% Spread/WAL (bp) 241 142 53 Yield Curve 1 Yr 2 Yr 3 Yr 5 Yr 10 Yr 30 Yr 4.63 4.77 4.78 4.79 4.91 5.35 Source: Salomon Smith Barney

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.

ValuationValuation 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

ValuationYield Curve Spread

For a given yield curve (or series of zero Treasury rates), rt, the PV

of the MBS cash flows is:

nn

n3

3

32

2

2

1

1

s)r (1CF....

s)r (1CF

s)r (1CF

sr 1CFPV(s)

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% 29-10 98-17 85 6.24 11.4yrs 130bp 110bp 70bp 6.5% 29-08 100-23 125 6.42 9.5 153 134 75 7.0% 28-04 102-10 210 6.51 6.5 166 153 78 7.5% 28-02 103-10 340 6.42 4.2 160 161 85 8.0% 28-01 104-10 390 6.41 3.6 160 167 105 8.5% 28-00 106-02 385 6.33 3.7 152 158 108 Source: Salomon Smith Barney

ValuationIntroduction 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 1-mo 3-mo 6-mo 1-yr 2-yr 3-yr 5-yr 7-yr 10-yr 20-yr 06/01/2004 0.97 1.17 1.44 1.89 2.60 3.14 3.86 4.31 4.71 5.45 06/02/2004 0.97 1.17 1.45 1.92 2.65 3.19 3.91 4.35 4.74 5.47 06/03/2004 0.96 1.17 1.45 1.91 2.63 3.17 3.89 4.34 4.71 5.46 06/04/2004 0.96 1.21 1.51 1.97 2.70 3.25 3.97 4.41 4.78 5.51 06/07/2004 0.97 1.24 1.54 1.96 2.67 3.22 3.95 4.39 4.78 5.51 06/08/2004 1.03 1.27 1.56 2.02 2.73 3.24 3.96 4.40 4.78 5.50 06/09/2004 1.03 1.27 1.61 2.14 2.79 3.31 4.01 4.44 4.82 5.54 06/10/2004 1.02 1.30 1.65 2.14 2.81 3.32 4.00 4.43 4.81 5.52 06/14/2004 1.07 1.41 1.77 2.25 2.97 3.45 4.10 4.51 4.89 5.59 06/15/2004 1.09 1.34 1.68 2.16 2.77 3.26 3.90 4.31 4.69 5.41 06/16/2004 1.05 1.30 1.69 2.24 2.84 3.31 3.96 4.36 4.74 5.46 06/17/2004 1.02 1.28 1.66 2.21 2.81 3.28 3.93 4.33 4.71 5.42 06/18/2004 1.03 1.29 1.68 2.24 2.81 3.28 3.94 4.34 4.72 5.43 06/21/2004 1.04 1.33 1.70 2.17 2.80 3.26 3.91 4.33 4.70 5.42 06/22/2004 1.08 1.32 1.69 2.21 2.80 3.28 3.92 4.34 4.72 5.43 The 30-yr has been discontinued. See www.treas.gov/domfin/statistics.htm

ValuationRelationship Between Short and Long Interest Rates

I. Expectations Hypothesis (with certainty)

( 1 + tRN) = [(1 + tR1)(1 + t+1r1)(1 + t+2r1)…(1 + t+N-1r1)]1/N 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 FV of $1 Implied 1-Year Yield Rate change (bp) 1 1.02210 2.2100 2 1.05678 3.3934 ↑ 118bp 3 1.10166 4.2467 ↑ 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/interest-rate/yield.html

ValuationA 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….