Asset-Backed Securities

Chapter 11

Securitization and

Mortgage-Backed Securities

Securitization

• Securitization refers to a process in which the assets of a corporation or financial institution are pooled into a package of securities backed by the assets.

• The process starts when an originator, who owns the assets (e.g., mortgages or accounts receivable), sells them to an issuer.

• The issuer then creates a security backed by the assets called an asset-backed security or passthrough that he sells to investors.

Securitization

• The securitization process often involves a thirdparty trustee who ensures that the issuer complies with the terms underlying the asset-backed security.

• Many securitized assets are backed by credit enhancements, such as a third-party guarantee against the default on the underlying assets.

• The most common types of asset-backed securities are those secured by mortgages, automobile loans, credit card receivables, and home equity loans.

Securitization

• By far the largest type and the one in which the process of securitization has been most extensively applied is mortgages.

• Asset-backed securities formed with mortgages are called mortgage-backed securities , MBSs, or mortgage pass-throughs .

Mortgage-Backed Securities: Definition

• Mortgage-Backed Securities (MBS) or

Mortgage Pass-Throughs (PT) are claims on a portfolio of mortgages.

• The securities entitle the holder to the cash flows from a pool of mortgages.

Mortgage-Backed Securities: Creation

• Typically, the issuer of a MBS buys a portfolio or pool of mortgages of a certain type from a mortgage originator, such as a commercial bank, savings and loan, or mortgage banker.

• The issuer finances the purchase of the mortgage portfolio through the sale of the mortgage passthroughs, which have a claim on the portfolio's cash flow.

• The mortgage originator usually agrees to continue to service the loans, passing the payments on to the MBS holders.

Agency Pass-Throughs

• There are three federal agencies that buy certain types of mortgage loan portfolios (e.g., FHA- or VA-insured mortgages) and then pool them to create MBSs to sell to investors:

– Federal National Mortgage Association (FNMA)

– Government National Mortgage Association (GNMA)

– Federal Home Loan Mortgage Corporation (FHLMC)

• Collectively, the MBSs created by these agencies are referred to as agency pass-throughs .

• Agency pass-throughs are guaranteed by the agencies, and the loans they purchase must be conforming loans, meaning they meet certain standards.


Government National Mortgage Association, GNMA

• GNMA mortgage-backed securities or pass-throughs are formed with FHA- or VA-insured mortgages.

• They are put together by an originator (bank, thrift, or mortgage banker), who presents a block of FHA and VA mortgages to GNMA.

• If GNMA finds them in order, they will issue a guarantee and assign a pool number that identifies the MBS that is to be issued.

• The originator will transfer the mortgages to a trustee, and then issue the pass-throughs, often selling them to investment bankers for distribution.


Government National Mortgage Association, GNMA

• The mortgages underlying GNMA’s MBSs are very similar (e.g., single-family, 30-year maturity, and fixed rate), with the mortgage rates usually differing by no more than 50 basis point from the average mortgage rate.

• GNMA does offer programs in which the underlying mortgages are more diverse.

• Note: Since GNMA is a federal agency, its guarantee of timely interest and principal payments is backed by the full faith and credit of the U.S. government -- the only

MBS with this type of guarantee.


Federal Home Loan Mortgage Corporation, FHLMC

• The FHLMC (Freddie Mac) issues MBSs that they refer to as participation certificates (PCs).

• The FHLMC has a regular MBS (also called a cash PC), which is backed by a pool of either conventional, FHA, or

VA mortgages that the FHLMC has purchased from mortgage originators.

• They also offer a pass-though formed through their

Guarantor/Swap Program. In this program, mortgage originators can swap mortgages for a FMLMC passthrough.


Federal Home Loan Mortgage Corporation

• Unlike GNMA’s MBSs, Freddie Mac's MBSs are formed with more heterogeneous mortgages.

• Like GNMA, the Federal Home Loan Mortgage

Corporation backs the interest and principal payments of its securities, but the FHLMC's guarantee is not backed by the U.S. government.


Federal National Mortgage Association, FNMA

• FNMA (Fannie Mae) offers several types of passthroughs, referred to as FNMA mortgage-backed securities.

• Like FHLMC’s pass-throughs, FNMA’s securities are backed by the agency, but not by the government.

• Like the FHLMC, FNMA buys conventional,

FHA, and VA mortgages, and offers a SWAP program whereby mortgage loans can be swapped for FNMA-issued MBSs.


Federal National Mortgage Association

• Like the FHLMC, FNMA's mortgages are more heterogeneous than GNMA's mortgages, with mortgage rates in some pools differing by as much as 200 basis points from the portfolio's average mortgage rate.

Conventional Pass-Throughs

• Conventional pass-throughs are sold by commercial banks, savings and loans, other thrifts, and mortgage bankers.

• These nonagency pass-throughs, also called private labels , are often formed with nonconforming mortgages; that is, mortgages that fail to meet size limits and other requirements placed on agency pass-throughs.


• The larger issuers of conventional MBSs include:

– Citicorp Housing

– Countrywide

– Prudential Home

– Ryland/Saxon

– G.E. Capital Mortgage


• Conventional pass-throughs are often guaranteed against default through external credit enhancements, such as:

– Guarantee of a corporation

– Bank letter of credit

– Private insurance from such insurers as the Financial

Guarantee Insurance Corporation (FGIC), the Capital

Markets Assurance Corporation (CAPMAC), or the

Financial Security Assurance Company (FSA)


• Some conventional pass-throughs are guaranteed internally through the creation of senior and subordinate classes of bonds with different priority claims on the pool's cash flows in the case some of the mortgages in the pool default.

• Example: A conventional pass-through, known as an

A/B pass-through , consists of two types of claims on the underlying pool of mortgages - senior and subordinate.

– The senior claim is backed by the mortgages, while the subordinate claim is not.

– The more subordinate claims sold relative to senior, the more secured the senior claims.


• Conventional MBSs are rated by Moody's and Standard and Poor's.

• They must be registered with the SEC when they are issued.


• Most financial entities that issue private-labeled MBSs or derivatives of MBSs are legally set up so that they do not have to pay taxes on the interest and principal that passes through them to their MBS investors.

• The requirements that MBS issuers must meet to ensure tax-exempt status are specified in the Tax Reform Act of

1983 in the section on trusts referred to as Real Estate

Mortgage Investment Conduits, REMIC .

• Private-labeled MBS issuers who comply with these provisions are sometimes referred to as REMICs .

Market

• Primary Market: Investors buy MBSs issued by agencies or private-label investment companies either directly or through dealers. Many of the investors are institutional investors. Thus, the creation of MBS has provided a tool for having real estate financed more by institutions.

• Secondary Market:

– Existing MBS are traded by dealers on the

OTC

Cash Flows

• Cash flows from MBSs are generated from the cash flows from the underlying pool of mortgages, minus servicing and other fees.

• Typically, fees for constructing, managing, and servicing the underlying mortgages (also referred to as the mortgage collateral) and the MBSs are equal to the difference between the rates associated with the mortgage pool and the rates paid on the MBS ( pass-through (PT) rate).

Cash Flows: Terms

• Weighted Average Coupon Rate , WAC : Mortgage portfolio's (collateral’s) weighted average rate.

• Weighted Average Maturity , WAM : Mortgage portfolio's weighted average maturity.

•

Pass-Through Rate, PT Rate : Interest rate paid on the MBS; PT rate is lower than WAC -- the difference going to MBS issuer.

•

Prepayment Rate or Speed : Assumed prepayment rate.

Prepayment

• A number of prepayment models have been developed to try to predict the cash flows from a portfolio of mortgages.

• Most of these models estimate the prepayment rate, referred to as the prepayment speed or simply speed , in terms of four factors:

– Refinancing incentive

– Seasoning (the age of the mortgage)

– Monthly factors

– Prepayment burnout

Prepayment

Refinancing Incentive

• The refinancing incentive is the most important factor influencing prepayment.

• If mortgage rates decrease below the mortgage loan rate, borrowers have a strong incentive to refinance.

• This incentive increases during periods of falling interest rates, with the greatest increases occurring when borrowers determine that rates have bottomed out.

Prepayment

Refinancing Incentive

• The refinancing incentive can be measured by the difference between the mortgage portfolio's WAC and the refinancing rate, R ref .

• A study by Goldman, Sachs, and Company found that the annualized prepayment speed, referred to as the conditional prepayment rate , CPR, is greater the larger the positive difference between the WAC and R ref .

Prepayment

Seasoning

• A second factor determining prepayment is the age of the mortgage, referred to as seasoning .

• Prepayment tends to be greater during the early part of the loan, then stabilize after about three years. The figures on the next slide depicts a commonly referenced seasoning pattern known as the PSA model (Public Securities Association).

• PSA Model:

Prepayment

CPR (%)

0 1







30

150 PSA

100 PSA

50 PSA

360

Month

Prepayment

Seasoning

• In the standard PSA model, known as 100 PSA, the CPR starts at .2% for the first month and then increases at a constant rate of .2% per month to equal 6% at the 30th month; then after the 30th month the CPR stays at a constant 6%. Thus for any month t, the CPR is

CPR

CPR



.

06



.

06 , t

30





, i f t if

 t



30 ,

30

Prepayment

Seasoning

• Note that the CPR is quoted on an annual basis.

• The monthly prepayment rate, referred to as the single monthly mortality rate , SMM, can be obtained given the annual CPR by using the following formula:

SMM



1



[ 1



CPR ]

1 / 12

Prepayment

Seasoning

• The 100 PSA model is often used as a benchmark. The actual aging pattern will differ depending on where current mortgage rates are relative to the WAC.

• Analysts often refer to the applicable pattern as being a certain percentage of the PSA.

• For example:

– If the pattern is described as being 200 PSA, then the prepayment speeds are twice the 100 PSA rates.

– If the pattern is described as 50 PSA, then the CPRs are half of the 100 PSA rates (see figures on previous slide).

Prepayment

Seasoning

• A current mortgage pool described by a 100 PSA would have a annual prepayment rate of 2% after

10 months (or a monthly prepayment rate of

SMM = .00168), and a premium pool described as a 150 PSA would have a 3% CPR (or SMM =

.002535) after 10 months.

Prepayment

Monthly Factors

• In addition to the effect of seasoning, mortgage prepayment rates are also influenced by the month of the year, with prepayment tending to be higher during the summer months.

• Monthly factors can be taken into account by multiplying the CPR by the estimated monthly multiplier to obtain a monthly-adjusted CPR.

PSA provides estimates of the monthly multipliers.

Prepayment

Burnout Factors

• Many prepayment models also try to capture what is known as the burnout factor .

• The burnout factor refers to the tendency for premium mortgages to hit some maximum CPR and then level off.

• For example, in response to a 2% decrease in refinancing rates, a pool of premium mortgages might peak at a 40% prepayment rate after one year, then level off at approximately 25%.

Cash Flow from a Mortgage Portfolio

• The cash flow from a portfolio of mortgages consists

– Interest payments

– Scheduled principal

– Prepaid principal

Cash Flow from a Mortgage Portfolio: Example 1

• Example 1: Consider a bank that has a pool of current fixed rate mortgages that are

– worth $100 million (Par, F)

– yield a WAC of 8%, and

– have a WAM of 360 months.


• For the first month, the portfolio would generate an aggregate mortgage payment of

$733,765: p



$ 100 , 000 , 000





1



1 /( 1



(.

08

.

08 /

/

12

12 ))

360







$ 733 , 765

F

0

F

0



 t

M 



1 p

( 1



( R A / 12 )) t p





1



1 /( 1



( R

R

A

/

A /

12

12 )) M



 p



F

0





1



1 /( 1



( R

R

A

/

A /

12

12 )) M






• From the $733,765 payment, $666,667 would go towards interest and $67,098 would go towards the scheduled principal payment:

Interest







R

A

12



 F

0



Scheduled Pr incipal

.

08

12

$ 100 , 000 , 000



$ 666 , 667

Payment

 p



Interest



$ 733 , 765



$ 666 , 667



$ 67 , 098


• The projected first month prepaid principal can be estimated with a prepayment model. Using the

100% PSA model, the monthly prepayment rate for the first month (t = 1) is equal to SMM =

.0001668:

CPR

SMM



1

.

06



.

002

30



1



[ 1



.

002 ]

1 / 12 

.

0001668


• Given the prepayment rate, the projected prepaid principal in the first month is found by multiplying the balance at the beginning of the month minus the scheduled principal by the

SMM.

• Doing this yields a projected prepaid principal of

$16,671 in the first month: prepaid principal prepaid principal





SMM [ F

0



Scheduled principal ]

.

0001668 [$ 100 , 000 , 000



$ 67 , 098 ]



$ 16 , 671


• Thus, for the first month, the mortgage portfolio would generate an estimated cash flow of

$750,435, and a balance at the beginning of the next month of $99,916,231:

CF



Interest



Scheduled principal

 prepaid principal

CF



$ 666

Beginning

, 666



Balance

$ 67 for

, 098



Month

$ 16 , 671

2



F

0





$ 750 , 435

Scheduled

Beginning Balance for Month 2



$ 100 , 000 , 000

 principal

$ 67 , 098





$ prepaid

16 , 671 principal



$ 99 , 916 , 231

Cash Flow from a Mortgage Portfolio : Example 1

• In the second month (t = 2), the projected payment would be $733,642 with $666,108 going to interest and $67,534 to scheduled principal: p



$ 99 , 916 , 231







1



1 /( 1



(.

08

.

08 /

/

12

12 ))

359









$ 733 , 642

Interest



Scheduled







.

08

12 





($ 99 , 916 , 231 ) principal



$ 733 ,



642

$ 666 , 108



$ 666 , 108



$ 67 , 534


• Using the 100% PSA model, the estimated monthly prepayment rate is .000333946, yielding a projected prepaid principal in month 2 of

$33,344:

CPR



2

.

06



.

004

30

SMM prepaid



1



[ 1



.

004 ]

1 / 12 principal



.



.

000333946

000333946 [$ 99 , 916 , 231



$ 67 , 534 ]



33 , 344


• Thus, for the second month, the mortgage portfolio would generate an estimated cash flow of $766,986 and have a balance at the beginning of month three of $99,815,353:

CF



$ 666

Beginning

, 108



Balance

$ 67 for

, 534



$ 33 , 344

Month



$ 766 , 986

3





$ 99 , 916 , 231

$ 99 , 815 , 353



$ 67 , 534



$ 33 , 344


• The exhibit on the next slide summarizes the mortgage portfolio's cash flow for the first two months and other selected months.

•

In examining the exhibit, two points should be noted:

1. Starting in month 30 the SMM remains constant at

.005143; this reflects the 100% PSA model's assumption of a constant CPR of 6% starting in month 30.

2. The projected cash flows are based on a static analysis in which rates are assumed fixed over the time period.


Period Balance

100000000

90341088

89801183

54900442

54533417

54168153

53804641

53442873

53082839

496620

371778

247395

123470

100000000

99916231

99815353

99697380

99562336

99410252

99241170

94291147

93847732

93389518

92916704

92429495

91928105

91412755

90883671

Interest p Sch. Prin.

SMM

31

32

110

111

112

113

114

115

357

358

359

360

24

25

26

27

28

29

30

5

6

7

23

1

2

3

4

602274

598675

366003

363556

361121

358698

356286

353886

3311

2479

1649

823

666667

666108

665436

664649

663749

662735

661608

628608

625652

622597

619445

616197

612854

609418

605891

677943

674456

451112

448792

446484

444188

441903

439631

126231

125582

124936

124293

733765

733642

733397

733029

732539

731926

731190

703012

700259

697394

694420

691336

688146

684849

681447

75669

75781

85109

85236

85363

85490

85617

85745

122920

123103

123287

123470

67098

67534

67961

68380

68790

69191

69582

74405

74607

74798

74975

75140

75292

75430

75556

Monthly Payment

 p









1



[ 1 /( 1



Balance

(.

08 / 12 ))]

Re maining Periods

(.

08 / 12 )

Interest



(.

08 / 12 )( Balance )

Scheduled

Pr epaid Pr

Pr incipal incipal

 p



(.

08 ) Balance



SMM [ Beginning Balance









Scheduled

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.005143

0.0001668

0.0003339

0.0005014

0.0006691

0.0008372

0.0010055

0.0011742

0.0039166

0.0040908

0.0042653

0.0044402

0.0046154

0.0047909

0.0049668

0.005143

Pr incipal ]

Prepaid Prin.

464236

461459

281916

280028

278148

276278

274417

272565

1922

1279

638

0

16671

33344

50011

66664

83294

99892

116449

369010

383607

398017

412234

426250

440059

453653

467027

CF

1142179

1135915

733028

728820

724632

720466

716320

712195

128153

126861

125574

124293

750435

766986

783409

799694

815833

831817

847639

1072023

1083866

1095411

1106653

1117586

1128204

1138502

1148475

Cash Flow from a MBS: Example 2

Example 2

• The next exhibit shows the monthly cash flows for a MBS issue constructed from a

$100M mortgage pool with the following features

– Current balance = $100M

– WAC = 8%

– WAM = 355 months

– PT rate = 7.5%

– Prepayment speed equal to 150% of the standard PSA model: PSA = 150


Period Balance

100000000

100000000

99779252

99533099

99261650

98965033

98643396

91641550

90975181

90288672

89582468

88857032

88112838

87350375

86593968

85843572

85099137

84360619

83627969

82901143

82180094

44933791

44515680

44100923

43689493

16163713

15978416

148527

98569

49061

Interest p

32

33

100

101

102

103

25

26

27

28

29

30

31

200

201

353

354

355

5

6

20

21

22

23

24

1

2

3

4

625000

623620

622082

620385

618531

616521

572760

568595

564304

559890

555356

550705

545940

541212

536522

531870

527254

522675

518132

513626

280836

278223

275631

273059

101023

99865

928

616

307

M onthly Payment

 p



Balance



 



1



[ 1 /( 1



(.

08 / 12 ))] Re

(.

08 / 12 ) ma ining Pe r iods

Interest



Scheduled

(.

075 / 12 )( Balance

Pr incipal

 p



)

(.

08 ) Balance

Pr epaid P r incipal



SM M [ Beginning Balance



Scheduled



 



Pr incipal ]

660671

655499

650368

645277

640225

635213

630240

625307

620411

366433

363564

360718

357894

736268

735154

733855

732371

730702

728850

684341

679910

675324

670587

665702

166983

165676

50171

49778

49388

Scheduled SMM

Principal

69601 0.0015125

69959 0.0017671

70301 0.0020223

70627 0.0022783

70936 0.002535

71227 0.0027925

73398 0.0064757

73408 0.0067447

73399 0.0070144

73370 0.0072849

73321 0.0075563

73253 0.0078284

73164 0.0078284

73075 0.0078284

72986 0.0078284

72897 0.0078284

72809 0.0078284

72721 0.0078284

72632 0.0078284

72544 0.0078284

66874 0.0078284

66793 0.0078284

66712 0.0078284

66631 0.0078284

59225 0.0078284

59153 0.0078284

49181 0.0078284

49121 0.0078284

49061 0.0078284

689211

683243

677322

671448

665621

659840

654106

648416

642772

351237

347965

344718

341498

Prepaid Principal

Principal

151147

176194

201148

225990

250701

275262

592971

613101

632804

652066

670873

220748

246153

271449

296617

CF

845748

869773

893531

917002

321637

346489

940168

963011

666369 1239128

686510 1255105

706204 1270508

725436 1285327

744194 1299550

126073

124623

778

387

0

762463 1313169

756406 1302346

750397 1291609

744434 1280957

738519 1270388

732649 1259903

726826 1249501

721049 1239181

715317 1228942

418111

414758

411430

408129

185298

183776

49958

49508

49061

698947

692981

687061

681188

286321

283641

50887

50124

49368


Notes:

•

The first month's CPR for the MBS issue reflects a fivemonth seasoning in which t = 6, and a speed that is

150% greater than the 100 PSA. For the MBS issue, this yields a first month SMM of .0015125 and a constant

SMM of .0078284 starting in month 25.

• The WAC of 8% is used to determine the mortgage payment and scheduled principal, while the PT rate of

7.5% is used to determine the interest.

• The monthly fees implied on the MBS issue are equal to

.04167% = (8% - 7.5%)/12 of the monthly balance.


• First Month’s Payment: p



$ 100 M





1



1 /( 1



(.

08

.

08 /

/

12

12 ))

355







$ 736 , 268

WAC


• From the $736,268 payment, $625,000 would go towards interest and $69,601 would go towards the scheduled principal payment :

PT Rate WAC

Interest



Scheduled





R

12

A



 F

0

Pr incipal



.

075

12





$ 100 , 000 , 000

Payment

 p



Interest





$ 625 , 000

$ 736 , 268



[(.

08 / 12 )($ 100 , 000 , 000 )]



$ 69 , 601


• Using 150% PSA model and seasoning of 5 months the first month SMM = .0015125:

CPR

SMM



1 .

50

6

.

06

30



1



[ 1



.

018 ]

1 / 12



.

018



.

0015125


• Given the prepayment rate, the projected prepaid principal in the first month is $151,147 prepaid principal prepaid principal





SMM [ F

0



Scheduled principal ]

.

0015125 [$ 100 , 000 , 000



$ 69 , 601 ]



$ 151 , 147

Allow for slight rounding difference s


• Thus, for the first month, the MBS would generate an estimated cash flow of $845,748 and a balance at the beginning of the next month of

$99,779,252:

CF



Interest



Scheduled principal

 prepaid principal

CF



$ 625

Beginning

, 000



Balance

$ 69 for

, 601



Month

$ 151 , 147

2



F

0





$ 845 , 748

Scheduled

Beginning Balance for Month 2



$ 100 , 000 , 000

 principal

$ 69 , 601



 prepaid

$ 151 , 147 principal



$ 99 , 779 , 252



• Second Month: Payment, Interest, Scheduled Principal,

Prepaid Principal, and Cash flow: p



$ 99 , 779 , 252





1



1 /( 1



(.

08

.

08 /

/

12

12 ))

354







$ 735 , 154

Interest







R A

12



 F

0



Scheduled Pr incipal

.

075

$ 99 , 779 , 252



$ 623 , 620

12

Payment

 p



Interest



$ 735 , 154



[(.

08 / 12 )($ 99 , 779 , 252 )]



$ 69 , 959

CPR

SMM





1 .

50







7

30







.

06

1



[ 1



.

021 ]

1 / 12





.

021

.

0017671 prepaid principal prepaid principal





SMM [ F

0



Scheduled principal

.

0017671 [$ 99 , 779 , 252



$ 69 ,

]

959 ]



$ 176 , 194

CF



CF



Interest



$ 623 , 620

Scheduled



$ 69 , 959 principal





$ 176 , 194 prepaid



$ 869 principal

, 773


Market

• As noted, investors can acquire newly issued mortgage-backed securities from the agencies, originators, or dealers specializing in specific pass-throughs.

• There is also a secondary market consisting of dealers who operate in the OTC market as part of the Mortgage-Backed Security Dealers

Association .

• These dealers form the core of the secondary market for the trading of existing pass-throughs.

Market

• Mortgage pass-throughs are normally sold in denominations ranging from $25,000 to

$250,000, although some privately-placed issues are sold with denominations as high as $1 million.

Price Quotes

• The prices of MBSs are quoted as a percentage of the underlying MBS issue’s balance.

• The mortgage balance at time t, F t

, is usually calculated by the servicing institution and is quoted as a proportion of the original balance,

F

0

.

•

This proportion is referred to as the pool factor , pf: pf

 t

F t

F

0

Price Quotes

• Example: A MBS backed by a mortgage pool originally worth $100M, a current pf of .92, and quoted at 95 - 16 (Note: 16 is 16/32) would have a market value of $87.86M:

F t





( pf t

) F

0

(.

92 ) ($ 100 M )



$ 92 M

Market Value



(.

9550 ) ($ 92 M )



$ 87 .

86 M

Price Quotes

• The market value is the clean price; it does not take into account accrued interest, ai.

• For MBS, accrued interest is based on the time period from the settlement date (two days after the trade) and the first day of the next month.

• Example: If the time period is 20 days, the month is

30 days, and the WAC = 9%, then ai is $.46M: ai



20



30  

.

09

12

$ 92 M



$ 0 .

46 M

Price Quotes

• The full market value would be $88.32M:

Full Mkt Value





$ 87 .

86 M

$ 88 .

32 M



$ 0 .

46 M

Price Quotes

• The market price per share is the full market value divided by the number of shares.

• If the number of shares is 400, then the price of the MBS based on a 95 - 16 quote would be

$220,800:

MBS price per share



$ 88 .

32 M

400



$ 220 , 800

Extension Risk

• Like other fixed-income securities, the value of a MBS is determined by the MBS's future cash flow (CF), maturity, default risk, and other features germane to fixed-income securities.

V

MBS

V

MBS

 t

M 



1

( 1

CF t



R ) t

 f ( CF t

, R )

Extension Risk

• In contrast to other bonds, MBSs are also subject to prepayment risk.

• Prepayment affects the MBS’s CF.

• Prepayment, in turn, is affected by interest rates.

• Thus, interest rates affects the MBS’s CFs.

V

MBS

 f ( CF , R )

CF

 f ( R )

Extension Risk

• With the CF a function of rates, the value of a MBS is more sensitive to interest rate changes than those bonds whose CFs are not.

• This sensitivity is known as extension risk .

Extension Risk

• If interest rates decrease, then the prices of

MBSs, like the prices of most bonds, increase as a result of the lower discount rates.

•

However, the decrease in rates will also augment prepayment speed, causing the earlier cash flow of the mortgages to be larger which, depending on the level of rates and the maturity remaining, could also contribute to increasing the MBS’s price.

Extension Risk

• Rate Decrease like most bonds if R

  lower discount rate



V

M

 if R

 

Increases prepayment



Earlier CFs

 

V

M

 or



Extension Risk

• If interest rates increase, then the prices of

MBSs will decrease as a result of higher discount rates and possibly the smaller earlier cash flow resulting from lower prepayment speeds.

Extension Risk

• Rate Increase like most bonds if R

  greater discount rate



V

M

 if R

 

Decreases prepayment



Earlier CFs

 

V

M

 or



Average Life

• The average life of a MBS or mortgage portfolio is the weighted average of the security’s time periods, with the weights being the periodic principal payments (scheduled and prepaid principal) divided by the total principal:

Average Life



1

12 t

T 



1 t



 principal total received principal at t





Average Life

• The average life for the MBS issue with WAC = 8%,

WAM = 355, PT Rate = 7.5%, and PSA = 150 is

9.18 years

Average Life





1

12 





1 ($ 220 , 748 )

9 .

18 years



2 ($ 246 , 153 )

$ 100 , 000 ,

 

000

 

355 ($ 49 , 061







Average Life

• The average life of a MBS depends on prepayment speed:

– If the PSA speed of the $100M MBS issue were to increase from 150 to 200, the MBS’s average life would decrease from 9.18 to 7.55, reflecting greater principal payments in the earlier years.

– If the PSA speed were to decrease from 150 to

100, then the average life of the MBS would increase to 11.51.

Average Life and Prepayment Risk

•

For MBSs and mortgage portfolios, prepayment risk can be evaluated in terms of how responsive a MBS's or mortgage portfolio’s average life is to changes in prepayment speeds: prepayment risk





Average



PSA

Life

•

A MBS with an average life that did not change with PSA speeds, in turn, would have stable principal payments over time and would be absent of prepayment risk.



Av life



PSA



0



Zero prepayment risk

MBS Derivatives

• One of the more creative developments in the security market industry over the last two decades has been the creation of derivative securities formed from MBSs and mortgage portfolios that have different prepayment risk characteristics, including some that are formed that have average lives that are invariant to changes in prepayment rates.

•

The most popular of these derivatives are

– Collateralized Mortgage Obligations, CMOs

– Stripped MBS

Collateralized Mortgage Obligations

• Collateralized mortgage obligations, CMOs , are formed by dividing the cash flow of an underlying pool of mortgages or a MBS issue into several classes, with each class having a different claim on the mortgage collateral and with each sold separately to different types of investors.

Collateralized Mortgage Obligations

• The different classes making up a CMO are called tranches or bond classes.

• There are two general types of CMO tranches:

– Sequential-Pay Tranches

– Planned Amortization Class Tranches, PAC

Sequential-Pay Tranches

• A CMO with sequential-pay tranches, called a sequential-pay CMO , is divided into classes with different priority claims on the collateral's principal.

•

The tranche with the first priority claim has its principal paid entirely before the next priority class, which has its principal paid before the third class, and so on.

•

Interest payments on most CMO tranches are made until the tranche's principal is retired.


• Example: A sequential-pay CMO is shown in the next exhibit.

•

This CMO consist of three tranches, A, B, and C, formed from the collateral making up the $100M

MBS in the previous example: F = $100M, WAM

= 355, WAC = 8%, PT Rate = 7.5%, PSA = 150.

– Tranche A = $50M

– Tranche B = $30M

– Tranche C = $20M


•

In terms of the priority disbursement rules:

– Tranche A receives all principal payment from the collateral until its principal of $50M is retired. No other tranche's principal payments are disbursed until the principal on A is paid.

– After tranche A's principal is retired, all principal payments from the collateral are then made to tranche B until its principal of $30M is retired.

– Finally, tranche C receives the remaining principal that is equal to its par value of $20M.

– Note: while the principal is paid sequentially, each tranche does receive interest each period equal to its stated PT rate

(7.5%) times its outstanding balance at the beginning of each month.


178

181

182

183

184

353

88

89

90

91

92

354

355

4

5

85

86

87

1

2

3

Period

Month

Par = $100M Rate = 7.5%

Collateral Collateral Collateral

Interest Principal Balance

100000000

100000000

99779252

99533099

99261650

98965033

51626473

51154749

50686799

50222595

49762107

49305305

48852161

48402646

20650839

19990210

19773585

19558729

19345627

148527

98569

49061

625000

623620

622082

620385

618531

322665

319717

316792

313891

311013

308158

305326

302517

129068

124939

123585

122242

120910

928

616

307

220748

246153

271449

296617

321637

471724

467949

464204

460488

456802

453144

449515

445915

222016

216625

214856

213101

211360

49958

49508

49061

A: Par = $50M Rate = 7.5%

Tranch A

Balance

50000000

A A

Interest Principal

50000000

49779252

49533099

49261650

48965033

1626473

1154749

686799

312500

311120

309582

307885

306031

10165

7217

4292

220748

246153

271449

296617

321637

471724

467949

464204

222595

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1391

0

0

0

0

0

0

222595

0

0

0

0

0

0

0

0

0

0

0

0

B: Par =$30M Rate = 7.5%

Tranche B B

Principal Balance

30000000

30000000

30000000

30000000

30000000

30000000

30000000

30000000

30000000

30000000

29762107

29305305

28852161

28402646

0

0

0

0

0

0

0

0

650839

0

0

0

0

0

0

0

237893

456802

453144

449515

445915

222016

0

0

0

0

0

0

0

187500

187500

187500

187500

187500

187500

187500

187500

187500

186013

183158

180326

177517

4068

0

0

0

0

0

0

0

B

Interest

C: Par = $20M Rate = 7.5%

Tranche C

Balance

20000000

C C

Principal Interest

0

20000000

20000000

20000000

20000000

20000000

20000000

20000000

20000000

0

0

0

0

0

0

0

0

125000

125000

125000

125000

125000

125000

125000

125000

20000000

20000000

20000000

20000000

20000000

20000000

19990210

19773585

19558729

19345627

148527

98569

49061

0

0

0

0

0

125000

125000

125000

125000

125000

0 125000

216625 124939

214856 123585

213101 122242

211360 120910

49958 928

49508

49061

616

307


• Given the assumed PSA of 150, the first month cash flow for tranche A consist of a principal payment (scheduled and prepaid) of $220,748 and an interest payment of $312,500:

[(.075/12)($50M) = $312,500]

•

In month 2, tranche A receives an interest payment of $311,120 based on the balance of

$49.779252M and a principal payment of

$246,153.


•

Based on the assumption of a 150% PSA speed, it takes

88 months before A's principal of $50M is retired.

•

During the first 88 months, the cash flows for tranches B and C consist of just the interest on their balances, with no principal payments made to them.

•

Starting in month 88, tranche B begins to receive the principal payment.

•

Tranche B is paid off in month 180, at which time principal payments begin to be paid to tranche C.

• Finally, in month 355 tranche C's principal is retired.


Features of Sequential-Pay CMOs

• By creating sequential-pay tranches, issuers of CMOs are able to offer investors maturities, principal payment periods, and average lives different from those defined by the underlying mortgage collateral.


Features of Sequential-Pay CMOs

•

Maturity:

–

Collateral's maturity = 355 months (29.58 years)

– Tranche A’s maturity = 88 months (7.33 years)

–

Tranche B's maturity = 180 months (15 years)

– Tranche C’s maturity = 355 months (29.58 years)

• Window: The period between the beginning and ending principal payment is referred to as the principal pay-down window :

– Collateral’s window = 355 months

– Tranche A’s window = 87 months

– Tranche B's window = 92 months

–

Tranche C's window =176 months

• Average Life:

– Collateral's average = 9.18 years

– Tranche A’s average life = 3.69 years

– Tranche B’s average life = 10.71 years

– Tranche C’s Average life = 20.59 years


Tranche Maturity Window Average Life

A

B

C

88 Months

179 Months

87 Months

92 Months

355 Months 176 Months

3.69 years

10.71 years

20.59 years

Collateral 355 Months 355 Months 9.18 years


•

Note: A CMO tranche with a lower average life is still susceptible to prepayment risk.

•

The average lives for the collateral and the three tranches are shown below for different PSA models

•

Note that the average life of each of the tranches still varies as prepayment speed changes.

PSA

50

100

150

200

300

Collateral

14.95

11.51

9.18

7.55

5.5

Tranche A

7.53

4.92

3.69

3.01

2.26

Tranche B Tranche C

19.4

14.18

10.71

8.51

6.03

26.81

23.99

9.18

17.46

12.82


• Note: Issuers of CMOs are able to offer a number of CMO tranches with different maturities and windows by simply creating more tranches.

Different Types of Sequential-Pay Tranches

• Sequential-pay CMOs often include traches with special features. These include:

– Accrual Bond Trache

– Floating-Rate Tranche

– Notional Interest-Only Tranche

Accrual Tranche

• Many sequential-pay CMOs have an accrual bond class .

•

Such a tranche, also referred to as the Z bond , does not receive current interest but instead has it deferred.

•

The Z bond's current interest is used to pay down the principal on the other tranches, increasing their speed and reducing their average life.

Accrual Tranche

• Example: suppose in our preceding equential-pay

CMO example we make tranche C an accrual tranche in which its interest of 7.5% is to paid to the earlier tranches and its principal of $20M and accrued interest is to be paid after tranche B's principal has been retired

• The next exhibit shows the principal and interest payments from the collateral and Tranches A, B, and Z.

Accrual Tranche

Par = $100M Rate = 7.5%

Period Collateral Collateral Collateral

Month Balance

100000000

Interest Principal

A: Par = $50M Rate = 7.5%

A

Balance

50000000

A

Interest

A

Principal

70

71

72

122

123

125

126

354

355

4

5

68

69

1

2

3

100000000

99779252

99533099

99261650

98965033

60253239

59712571

59176219

58644150

58116329

36470935

36120824

35429038

35087319

98569

49061

625000

623620

622082

620385

618531

376583

373204

369851

366526

363227

227943

225755

221431

219296

616

307

220748

246153

271449

296617

321637

540668

536352

532069

527821

523605

350111

347292

341719

338966

49508

49061

50000000

49654252

49283099

48886650

48465033

1878239

1212571

551219

0

0

0

0

0

0

0

0

312500

310339

308019

305542

302906

11739

7579

3445

0

0

0

0

0

0

0

0

345748

371153

396449

421617

446637

665668

661352

551219

0

0

0

0

0

0

0

0

B: Par = $30M Rate = 7.5%

B B

Balance

30000000

Principal

B

Interest

Z: Par = $20M Rate = 7.5%

Z

20000000

Z

Bal.+Cum Int Principal

0

Z

Interest

30000000

30000000

30000000

30000000

30000000

30000000

30000000

30000000

29894150

29241329

1345935

870824

0

0

0

0

105850

652821

648605

475111

472292

0

0

0

0

0

0

0

0

0

0

0

187500

187500

187500

187500

187500

187500

187500

187500

186838

182758

8412

5443

0

0

0

0

20000000

20125000

20250000

20375000

20500000

28375000

28500000

28625000

28750000

28875000

35125000

35250000

35429038

35087319

98569

49061

0

0

0

0

0

341719

338966

49508

49061

0

0

0

0

0

0

0

0

0

0

0

0

221431

219296

616

307

0

0

0

0

0

0

0

Accrual Tranche

• Since the accrual tranche's current interest of $125,000 is now used to pay down the other classes' principals, the other tranches now have lower maturities and average lives.

•

For example, the principal payment on tranche A is

$345,748 in the first month ($220,748 of scheduled and projected prepaid principal and $125,000 of Z's interest); in contrast, the principal is only $220,748 when there is no Z bond.

•

As a result of the Z bond, trache A's window is reduced from 87 months to 69 months and its average life from

3.69 years to 3.06 years.

Accrual Tranche

Tranche

A

B

Window Average Life

69 Months 3.06 Years

54 Months 8.23 Years

Floating-Rate Tranche

• In order to attract investors who prefer variable rate securities, CMO issuers often create floatingrate and inverse floating-rate tranches .

•

The monthly coupon rate on the floating-rate tranche is usually set equal to a reference rate such as the London Interbank Offer Rate, LIBOR, while the rate on the inverse floating-rate tranche is determined by a formula that is inversely related to the reference rate.


• Example: Sequential-pay CMO with a floating and inverse floating tranches

Tranche

A

FR

IFR

Z

Total

Par Value

$50M

$22.5M

$7.5M

$20M

$100

PT Rate

7.5%

LIBOR +50BP

28.3 – 3 LIBOR

7.5%

7.5%

• Note: The CMO is identical to our preceding CMO, except that tranche B has been replaced with a floating-rate tranche, FR, and an inverse floating-rate tranche, IFR.


• The rate on the FR tranche, R

FR

, is set to the

LIBOR plus 50 basis points, with the maximum rate permitted being 9.5%.

•

The rate on the IFR tranche, R

IFR by the following formula:

, is determined

R

IFR

= 28.5 - 3 LIBOR

•

This formula ensures that the weighted average coupon rate (WAC) of the two tranches will be equal to the coupon rate on tranche B of 7.5%, provided the LIBOR is less than 9.5%.


•

For example, if the LIBOR is 8%, then the rate on the FR tranche is 8.5%, the IFR tranche's rate is 4.5%, and the

WAC of the two tranches is 7.5%:

LIBOR

R

FR





8 %

LIBOR

R

IFR



28 .

5





50 BP

3 LIBOR





8 .

5 %

4 .

5 %

WAC



.

75 R

FR



.

25 R

IFR



7 .

5 %

Notional Interest-Only Class

• Each of the fixed-rate tranches in the previous CMOs have the same coupon rate as the collateral rate of 7.5%.

• Many CMOs, though, are structured with tranches that have different rates. When CMOs are formed this way, an additional tranche, known as a notional interest-only (IO) class, is often created.

•

This tranche receives the excess interest on the other tranches’ principals, with the excess rate being equal to the difference in the collateral’s PT rate minus the tranches’ PT rates.


• Example: A sequential-pay CMO with a Z bond and notional IO tranche is shown in the next exhibit.

•

This CMO is identical to our previous CMO with a Z bond, except that each of the tranches has a coupon rate lower than the collateral rate of 7.5% and there is a notional IO class.


• The notional IO class receives the excess interest on each tranche's remaining balance, with the excess rate based on the collateral rate of 7.5%.

•

In the first month, for example, the IO class would receive interest of $87,500:

Interest

Interest







.

075



12

.

06

$ 62 , 500



$ 50 , 000

$ 25 , 000



, 000

$ 87



, 500



.

075



12

.

065

$ 30 , 000 , 000


Period

Month

124

125

126

127

353

354

355

5

70

71

72

122

123

3

4

1

2

Collateral:

Collateral

Balance

100000000

100000000

99779252

99533099

99261650

98965033

59176219

58644150

58116329

36470935

36120824

35773533

35429038

35087319

34748353

148527

98569

49061

Par = $100M Rate = 7.5% Tranche A: Par = $50M Rate = 6%

Collateral Collateral

Interest

625000

623620

622082

620385

618531

369851

366526

363227

227943

225755

223585

221431

219296

217177

928

616

307

Principal

220748

246153

271449

296617

321637

532069

527821

523605

350111

347292

344494

341719

338966

336235

49958

49508

49061

A

Balance

50000000

50000000

49654252

49283099

48886650

48465033

551219

0

0

0

0

0

0

0

0

0

0

0

A A Notional

0

0

0

0

0

0

0

0

Interest Principal Interest

0.06

0.015

250000

248271

345748

371153

62500

62068

246415

244433

242325

2756

396449

421617

446637

551219

61604

61108

60581

689

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 B: Par = $30M Rate = 6.5%

B B B

Balance

30000000

30000000

30000000

30000000

30000000

30000000

30000000

29894150

29241329

1345935

870824

398533

0

0

0

0

0

0

Principal

0

0

0

0

0

105850

652821

648605

475111

472292

398533

0

0

0

0

0

0

Interest

0.065

162500

162500

162500

162500

162500

162500

161927

158391

7290

4717

2159

0

0

0

0

0

0

Notional

Tranche Z: Par = $20M Rate = 7%

Z

Interest Balance

0.01

20000000

25000

25000

20000000

20125000

25000

25000

25000

25000

20250000

20375000

20500000

28625000

24912

24368

1122

726

332

0

0

0

0

0

0

28750000

28875000

35125000

35250000

35375000

35429038

35087319

34748353

148527

98569

49061

Z

Principal

0

0

0

0

0

0

0

0

0

0

0

-54038

341719

338966

336235

49958

49508

49061

Z Notional

Interest Interest

0.07

0.005

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

206354

206669

204676

202699

866

575

286

0

0

0

0

14740

14762

14620

14478

62

41

20

Notional Par = $15.333M

Notional

Total CF

87500

87068

86604

86108

85581

25689

24912

24368

1122

726

15072

14762

14620

14478

62

41

20


• The IO class is described as paying 7.5% interest on a notional principal of $15,333,333.

•

This notional principal is determined by summing each tranche's notional principal.

•

A tranche's notional principal is the number of dollars that makes the return on the tranche's principal equal to 7.5%.


• The notional principal for tranche A is

$10,000,000, for B, $4,000,000, and for Z,

$1,333,333, yielding a total notional principal of

$15,333,333:

A ' s

B

Z

'

' s s

Notional

Notional

Notional principal principal principal







($ 50 , 000 , 000 )(.

075



.

06

.

075

$ 30 , 000 , 000 )(.

075



.

065 )

)

.

075

($ 20 , 000 , 000 )(.

075



.

07 )

.

075



$ 10 , 000 , 000





$ 4

$ 1

, 000 , 000

, 333 , 333

Total Notional Pr incipal



$ 15 , 333 , 333

Planned Amortization Class, PAC

• A CMO with a planned amortization class , PAC , is structured such that there is virtually no prepayment risk.

•

In a PAC-structured CMO, the underlying mortgages or MBS (i.e., the collateral) is divided into two general tranches:

– The PAC (also called the PAC bond)

– The support class (also called the support bond or the companion bond )


• The two tranches are formed by generating two monthly principal payment schedules from the collateral:

– One schedule is based on assuming a relatively low

PSA speed – lower collar .

– The other schedule is based on assuming a relatively high PSA speed – upper collar .

•

The PAC bond is then set up so that it will receive a monthly principal payment schedule based on the minimum principal from the two principal payments.


• The PAC bond is designed to have no prepayment risk provided the actual prepayment falls within the minimum and maximum assumed PSA speeds.

• The support bond, on the other hand, receives the remaining principal balance and is therefore subject to prepayment risk.


• To illustrate, suppose we form PAC and support bonds from the $100M collateral that we used to construct our sequential-pay tranches:

– Underlying MBS = $100M, WAC = 8%, WAM = 355 months, and PT rate = 7.5%

• To generate the minimum monthly principal payments for the PAC, assume:

– Minimum speed of 100% PSA; lower collar = 100

PSA

– Maximum speed of 300% PSA; upper collar = 300

PSA


• The next exhibit shows the cash flows for the

PAC, collateral, and support bond. The exhibit shows:

– In columns 2 and 3 the principal payments (scheduled and prepaid) for selected months at both collars.

– In the fourth column the minimum of the two payments.

•

For example, in the first month the principal payment is $170,085 for the 100% PSA and

$374,456 for the 300% PSA; thus, the principal payment for the PAC would be $170,085.


Period

Month

101

102

201

202

203

204

205

206

354

355

5

98

99

100

3

4

1

2

Pac Pac Pac

Low PSA Pr high PSA Pr Min. Principal

100 300

170085

187135

204125

221048

237895

381871

380032

378204

376384

374575

235460

234395

233336

232283

231235

230193

124660

124203

374456

425190

475588

525572

575064

386139

379499

372970

366552

360242

61932

60806

59699

58611

57542

56492

2559

2493

170085

187135

204125

221048

237895

381871

379499

372970

366552

360242

61932

60806

59699

58611

57542

56492

2559

2493

Par = 63777030

Pac

Int

0.075

398606

397543

396374

395098

393716

135237

132851

130479

128148

125857

19312

18925

18545

18172

17806

17446

32

16

Pac

CF

Collateral Collateral Collateral Collateral

Balance

100000000

Interest Prncipal CF

568692 100000000 625000

584678 99779252 623620

600499 99533099 622082

616147 99261650 620385

631612 98965033 618531

517108 45780181 286126

512349 45355283 283471

503449 44933791 280836

494700 44515680 278223

486099 44100923 275631

81245

79731

15978416

15794640

99865

98716

78244

76783

75348

73938

2591

2509

15612374

15431606

15252325

15074517

98569

49061

97577

96448

95327

94216

616

307

220748

246153

271449

296617

321637

424898

421491

418111

414758

411430

183776

182266

180768

179282

177807

176344

49508

49061

845748

869773

893531

917002

940168

711025

704962

698947

692981

687061

283641

280982

278345

275729

273134

270560

50124

49368

Support

Principal

Col Pr - PAC Pr

50662

59018

67324

75568

83742

43028

41993

45141

48205

51188

121844

121460

121069

120671

120265

119852

46948

46568

Par = 36222970

Support

Balance

36222970

36172308

36113290

36045966

35970398

24142190

24099163

24057170

24012029

23963824

12888435

12766592

12645131

12524062

12403392

12283127

93517

46568

Support Support

Interest

0.075

CF

226394 277056

226077 285095

225708 293032

225287 300856

224815 308557

150889 193916

150620 192613

150357 195498

150075 198281

149774 200962

80553

79791

202396

201251

79032

78275

77521

76770

584

291

200101

198946

197786

196622

47533

46859


• Note: For the first 98 months, the minimum principal payment comes from the 100%

PSA model, and from month 99 on the minimum principal payment comes from the 300% PSA model.


• Based on the 100-300 PSA range, a PAC bond can be formed that would promise to pay the principal based on the minimum principal payment schedule shown in the exhibit.

• The support bond would receive any excess monthly principal payment.


• The sum of the PAC's principal payments is

$63.777M. Thus, the PAC can be described as having:

– Par value of $63.777M

– Coupon rate of 7.5%

– Lower collar of 100% PSA

– Upper collar of 300% PSA

•

The support bond, in turn, would have a par value of $36.223M ($100M - $63.777M) and pay a coupon of 7.5%.


• The PAC bond has no prepayment risk as long as the actual prepayment speed is between 100 and

300.

•

This can be seen by calculating the PAC's average life given different prepayment rates.

•

The next exhibit shows the average lives for the collateral, PAC bond, and support bond for various prepayment speeds ranging from 50%

PSA to 350% PSA.

PSA

50

100

150

200

250

300

350


Collateral

14.95

11.51

9.18

7.55

6.37

5.50

4.84

Average Life

PAC

7.90

6.98

6.98

6.98

6.98

6.98

6.34

Support

21.50

19.49

13.05

8.55

5.31

2.91

2.71


Note:

•

The PAC bond has an average life of 6.98 years between 100% PSA and 300% PSA; its average life does change, though, when prepayment speeds are outside the 100-300 PSA range.

•

In contrast, the support bond's average life changes as prepayment speed changes.

•

Changes in the support bond's average life due to changes in speed are greater than the underlying collateral's responsiveness.

Other PAC-Structured CMOs

• The PAC and support bond underlying a CMO can be divided into different classes. Often the

PAC bond is divided into several sequential-pay tranches, with each PAC having a different priority in principal payments over the other.

• Each sequential-pay PAC, in turn, will have a constant average life if the prepayment speed is within the lower and upper collars.

•

In addition, it is possible that some PACs will have ranges of stability that will increase beyond the actual collar range, expanding their effective collars.

Other PAC-Structured CMOs

• A PAC-structured CMO can also be formed with

PAC classes having different collars.

•

Some PACs are formed with just one PSA rate.

These PACs are referred to as targeted amortization class (TAC) bonds .

•

Different types of tranches can also be formed out of the support bond class. These include sequential-pay, floating and inverse-floating rate, and accrual bond classes.

Stripped MBS

•

Stripped MBSs consist of two classes:

1. Principal-only (PO) class that receives only the principal from the underlying mortgages.

2. Interest-only (IO) class that receives just the interest.

Principal-Only Stripped MBS

• The return on a PO MBS is greater with greater prepayment speed.

•

For example, a PO class formed with $100M of mortgages (principal) and priced at $75M would yield an immediate return of $25M if the mortgage borrowers prepaid immediately. Since investors can reinvest the $25M, this early return will have a greater return per period than a $25M return that is spread out over a longer period.


• Because of prepayment, the price of a PO MBS tends to be more responsive to interest rate changes than an option-free bond.

• That is, if interest rates are decreasing, then like the price of most bonds, the price of a PO MBS will increase. In addition, the price of a PO MBS is also likely to increase further because of the expectation of greater earlier principal payments as a result of an increase in prepayment caused by the lower rates.

R



( 1 )

 prepayment

  return

 

V

P O



( 2 )

 lower discount rate



V

P O




• In contrast, if rates are increasing, the price of a

PO MBS will decrease as a result of both lower discount rates and lower returns from slower principal payments.

R



( 1 )

 prepayment

  return

 

V

P O



( 2 )

 greater discount rate



V

P O




•

Thus, like most bonds, the prices of PO

MBSs are inversely related to interest rates, and, like other MBSs with embedded principal prepayment options, their prices tend to be more responsive to interest rate changes.



V

P O



R



0

Interest-Only Stripped MBS

• Cash flows from an I0 MBS come from the interest paid on the mortgages portfolio’s principal balance.

•

In contrast to a PO MBS, the cash flows and the returns on an IO MBS will be greater, the slower the prepayment rate.


•

If the mortgages underlying a $100M, 7.5% MBS with PO and IO classes were paid off in the first year, then the IO

MBS holders would receive a one-time cash flow of $7.5M:

$7.5M = (.075)($100M)

•

If $50M of the mortgages were prepaid in the first year and the remaining $50M in the second year, then the IO MBS investors would receive an annualized cash flow over two years totaling $11.25M:

$11.25M = (.075) ($100M) + (.075)($100M-$50M)

• If the mortgage principal is paid down $25M per year, then the cash flow over four years would total $18.75M:

$18.75M = (.075)($100M) + (.075)($100M-$25M)

+ (.075)($75M-$25M) + (.075)($50M-$25M))


•

Thus, IO MBSs are characterized by an inverse relationship between prepayment speed and returns: the slower the prepayment rate, the greater the total cash flow on an IO MBS.

•

Interestingly, if this relationship dominates the price and discount rate relation, then the price of an IO MBS will vary directly with interest rates.

( 1 )

 prepayment

  return

 

V

IO



R



If 1 st

( 2 )

 effect

 greater

2 nd discount effect , then rate





V

IO



R

V



0

IO



IO and PO Stripped MBS

•

An example of a PO MBS and an IO MBS are shown in next exhibit.

•

The stripped MBSs are formed from collateral with

– Mortgage Balance = $100M

– WAC = 8%

– PT Rate = 8%

– WAM = 360

– PSA = 100


Period Collateral

Month Balance

100000000

100

101

200

201

358

359

360

1

2

3

4

5

100000000

99916231

99815353

99697380

99562336

58669646

58284486

27947479

27706937

371778

247395

123470

Collateral Collateral

Interest Scheduled

Principal

666667

666108

665436

664649

663749

391131

388563

186317

184713

2479

1649

823

67098

67534

67961

68380

68790

83852

83977

97308

97453

123103

123287

123470

Collateral Collateral Collateral Stripped Stripped

Prepaid

Principal

Total

Principal

CF PO IO

16671

33344

50011

66664

83294

301307

299326

143234

141996

1279

638

0

83769

100878

117973

135044

152084

385159

383303

240542

239449

124382

123925

123470

750435

766986

783409

799694

815833

776290

771866

426858

424162

126861

125574

124293

83769

100878

117973

135044

152084

385159

383303

240542

239449

124382

123925

123470

666667

666108

665436

664649

663749

391131

388563

186317

184713

2479

1649

823


• The table shows the values of the collateral, PO

MBS, and IO MBS for different discount rate and PSA combinations of 8% and 150, 8.5% and

125, and 9% and 100.

•

Note: The IO MBS is characterized by a direct relation between its value and rate of return.

PSA Discount

Rate

8%

8.50%

9.00%

150

125

100

Value of

PO

Value of

IO

Value of

Collateral

$54,228,764 $47,426,196 $101,654,960

$49,336,738 $49,513,363 $98,850,101

$44,044,300 $51,795,188 $95,799,488

Other Asset-Backed Securities

•

MBSs represent the largest and most extensively developed asset-backed security. A number of other asset-backed securities have been developed.

•

The three most common types are those backed by

–

Automobile Loans

– Credit Card Receivables

– Home Equity Loans

• These asset-backed securities are structured as passthroughs and many have tranches.


Automobile Loan-Backed Securities

• Automobile loan-backed securities are often referred to as CARS ( certificates for automobile receivables ).

• The automobile loans underlying these securities are similar to mortgages in that borrowers make regular monthly payments that include interest and a scheduled principal.

• Like mortgages, automobile loans are characterized by prepayment. For such loans, prepayment can occur as a result of car sales, trade-ins, repossessions and subsequent resales, wrecks, and refinancing when rates are low.

• Some CARS are structured as PACS.


Automobile Loan-Backed Securities

•

CARS differ from MBSs in that

– they have much shorter lives

– their prepayment rates are less influenced by interest rates than mortgage prepayment rates

– they are subject to greater default risk.


Credit-Card Receivable-Backed Securities

•

Credit-card receivable-backed securities are commonly referred to as CARDS ( certificates for amortizing revolving debts ).

•

In contrast to MBSs and CARS, CARDS investors do not receive an amortorized principal payment as part of their monthly cash flow.


Credit-Card Receivable-Backed Securities

•

CARDS are often structured with two periods:

– In one period, known as the lockout period , all principal payments made on the receivables are retained and either reinvested in other receivables or invested in other securities.

– In the other period, known as the principalamortization period , all current and accumulated principal payments are paid.


Home Equity Loan-Backed Securities

• Home-equity loan-backed securities are referred to as HELS .

•

They are similar to MBSs in that they pay a monthly cash flow consisting of interest, scheduled principal, and prepaid principal.

•

In contrast to mortgages, the home equity loans securing HELS tend to have a shorter maturity and different factors influencing their prepayment rates.

Websites

• For more information on the mortgage industry, statistics, trends, and rates go to www.mbaa.org

•

Mortgage rates in different geographical areas can be found by going to www.interest.com

.

•

For historical mortgage rates go to http://research.stlouisfed.org/fred2 and click on

“Interest Rates.”

Websites

• Agency information: www.fanniemae.com

, www.ginniemae.gov

, www.freddiemac.com

• For general information on MBS go to www.ficc.com

• For information on the market for mortgage-backed securities go to www.bondmarkets.com

and click on

“Research Statistics and “Mortgage-Backed Securities.”

For information on links to other sites click on “Gateway to Related Links.”

•

For information on the market for asset-backed securities go to www.bondmarkets.com

and click on “Research

Statistics and “Asset-Backed Securities.”

Evaluating Mortgage-Backed

Securities

Monte Carlo Simulation

• Objective: Determine the MBS’s theoretical value

• Steps:

1. Simulation of interest rates: Use a binomial interest rate tree to generate different paths for spot rates and refinancing rates.

2. Estimate the cash flows of a mortgage portfolio, MBS, or tranche for each path given a specified prepayment model based on the spot rates

3. Determine the present values of each path.

4. Calculate the average value – theoretical value.

Step 1: Simulation of Interest Rate

• Determination of interest rate paths from a binomial interest rate tree.

• Example:

– Assume three-period binomial tree of one-year spot rates (S) and refinancing rates, (R ref ) where:

S

0 u





6

1 .

1

% d



.

9091



1 / 1 .

1

R ref

0 u

 d





1 .

1

8 %

.

9091



1 / 1 .

1

– With three periods, there are four possible rates after three periods (years) and there are eight possible paths.

u = 1.10, d = .9091

Step 1: Binomial Tree for Spot and Refinancing Rates

S uuu



7 .

9860

R ref uuu



10 .

648

S uu



7 .

260

R ref uu



9 .

680

S u



6 .

600

R ref u



8 .

800

S uud



R ref uud



6 .

600

8 .

800

S

0



6 .

00 %

R ref

0



8 .

00 %

S ud



6 .

00

R ref ud



8 .

00

S d



5 .

4546

R ref d



7 .

2728

S udd



5 .

4546

R ref udd



7 .

2728

S dd



4 .

9588 ref

R dd



6 .

6117

S ddd



4 .

5080

R ref ddd



6 .

0107

Path 1

Step 1: Interest Rate Paths

Path 2 Path 3 Path 4

6.0000%

5.4546

4.9588

4.5080

Path 5

6.0000%

5.4546

6.0000

6.6000

6.0000%

5.4546

4.9588

5.4546

Path 6

6.0000%

6.6000

6.0000

6.6000

6.0000%

5.4546

6.0000

5.4546

Path 7

6.0000%

6.6000

7.2600

6.6000

6.0000%

6.6000

6.0000

5.4546

Path 8

6.0000%

6.6000

7.2600

7.9860

Step 2: Estimating Cash Flows

• The second step is to estimate the cash flow for each interest rate path.

• The cash flow depends on the prepayment rates assumed.

• Most analysts use a prepayment model in which the conditional prepayment rate (CPR) is determined by the seasonality of the mortgages, and by a refinancing incentive that ties the interest rate paths to the proportion of the mortgage collateral prepaid.


• To illustrate, consider a MBS formed from a mortgage pool with a par value of $1M, WAC =

8%, and WAM = 10 years.

• To fit this example to the three-year binomial tree assume that

– the mortgages in the pool all make annual cash flows

(instead of monthly)

– all have a balloon payment at the end of year 4

– the pass-through rate on the MBS is equal to the WAC of 8


• The mortgage pool can be viewed as a four-year asset with a principal payment made at the end of year four that is equal to the original principal less the amount paid down.

• As shown in the next exhibit, if there were no prepayments, then the pool would generate cash flows of $149,029M each year and a balloon payment of $688,946 at the end of year 4.


•

Mortgage Portfolio:

– Par Value = $1M, WAC = 8%, WAM = 10 Yrs,

– PT Rate = 8%, Balloon at the end of the 4th Year

Year Balance P

1

2

$1,000,000

$930,971

$149,029

$149,029

3 $856,419 $149,029

Balloon

4



$775,903

Balance ( yr 4 )



$149,029

Sch .

prin ( yr 4 )

CF

4





$ 775 , 903



$ 86 , 957



$ 688 , 946

Balloon

 p



$ 688 , 946



$ 149 , 029



$ 837 , 975

CF

4



Balance ( yr 4 )



Interest



$ 775 , 903



$ 62 , 072



$ 837 , 975

Interest Scheduled

Principal

$80,000

$74,478

$68,513

$62,072

$69,029

$74,552

$80,516

$86,957

Cash

Flow

$149,029

$149,029

$149,029

$837,975


• Such a cash flow is, of course, unlikely given prepayment. A simple prepayment model to apply to this mortgage pool is shown in next exhibit.


• The prepayment model assumes:

1. The annual CPR is equal to 5% if the mortgage pool rate is at a par or discount (that is, if the current refinancing rate is equal to the WAC of 8% or greater).

2. The CPR will exceed 5% if the rate on the mortgage pool is at a premium.

3. The CPR will increase within certain ranges as the premium increases.

4. The relationship between the CPRs and the range of rates is the same in each period; that is, there is no seasoning factor.


CPR Range

X = WAC – R ref

X



0

0.0% < X



0.5%

0.5% < X



1.0%

1.0% < X



1.5%

1.5% < X



2.0%

2.0% < X



2.5%

2.5% < X



3.0%

X > 3.0%

5%

10%

20%

30%

40%

50%

60%

70%


• With this prepayment model, cash flows can be generated for the eight interest rate paths.

• These cash flows are shown in Exhibit A

(next two slides).

Path 1

Year

3

4

1

2

R

1 ref

2

Balance WAC

0.072728

1000000

0.066117

744776

0.060107

479594

260703

0.08

0.08

0.08

0.08

Path 2

Year

3

4

1

2

R

1 ref

2

Balance WAC

0.072728

1000000

0.066117

744776

0.072728

479594

347604

0.08

0.08

0.08

0.08

Path 3 1.000000

Year R ref

3

4

1

2

0.072728

0.080000

0.072728

2

Balance

1000000

744776

650878

471749

WAC

0.08

0.08

0.08

0.08

Path 4

Year

3

4

1

2

R

1 ref

2

Balance WAC

0.088000

1000000

0.080000

884422

0.072728

772918

560202

0.08

0.08

0.08

0.08

Exhibit A

3 4 5 6

Interest Sch. Prin.

CPR Prepaid Prin.

80000

59582

38368

20856

69029

59641

45089

0.20

0.30

0.40

186194

205540

173802

7

CF

8

Z

1,t-1

9

Z t0

10

Value

11

Prob.

335224 0.080000 0.080000

310392

324764 0.074546 0.077270

279846

257259 0.069588 0.074703

207255

281560 0.065080 0.072289

212972

0.5

0.5

0.5

Value = 1010465 0.125

3 4 5 6



80000

59582

38368

27808

69029

59641

45089

0.20

0.30

0.20

186194

205540

86901

7

CF

8

Z

1,t-1

9

Z t0

10

Value

11

Prob.

335224 0.080000 0.080000

310392

324764 0.074546 0.077270

279846

170358 0.069588 0.074703

137245

375413 0.074546 0.074664

281461

0.5

0.5

0.5

Value = 1008945 0.125

3 4 5 6



80000

59582

52070

37740

69029

59641

61192

0.20

0.05

0.20

186194

34257

117937

7

CF Z

8

1,t-1

Z

9 t0

10

Value

11

Prob.

335224 0.080000 0.080000

310392

153480 0.074546 0.077270

132253

231200 0.080000 0.078179

184465

509489 0.074546 0.077270

378301

0.5

0.5

0.5

Value = 1005411 0.125

3 4 5 6



80000

70754

61833

44816

69029

70824

72666

0.05

0.05

0.20

46549

40680

140050

7

CF Z

8

1,t-1

Z

9 t0

10

Value

195578 0.080000 0.080000

181091

182258 0.086000 0.082996

155393

274550 0.080000 0.081996

216742

605018 0.074546 0.080129

444494

Value = 997720

11

Prob.

0.5

0.5

0.5

0.125

Exhibit A

Path 5

Year

3

4

1

2

R

1 ref

2

0.08

0.08

0.08

0.08

3

80000

59582

52070

44816

4 5 6

0.072728

0.080000

0.088000

Balance WAC Interest Sch. Prin.


1000000

744776

650878

560202

69029

59641

61192

0.20

0.05

0.05

186194

34257

29484

7

CF Z

8

1,t-1

Z

9 t0

10

Value

11

Prob.

335224 0.080000 0.080000

310392

153480 0.074546 0.077270

132253

142747 0.080000 0.078179

113892

605018 0.086000 0.080129

444494

0.5

0.5

0.5

Value = 1001031 0.125

Path 6

Year

3

4

1

2

R

1 ref

2 3 4 5 6



0.088000

1000000

0.080000

884422

0.088000

772918

665240

0.08

0.08

0.08

0.08

80000

70754

61833

53219

69029

70824

72666

0.05

0.05

0.05

46549

40680

35013

7

CF

8

Z

1,t-1

9

Z t0

10

Value

195578 0.080000 0.080000

181091

182258 0.086000 0.082996

155393

169512 0.080000 0.081996

133820

718459 0.086000 0.082996

522269

Value = 992574

11

Prob.

0.5

0.5

0.5

0.125

Path 7

Year

3

4

1

2

R

1 ref

2 3 4 5 6



0.088000

1000000

0.096000

884422

0.088000

772918

665240

0.08

0.08

0.08

0.08

80000

70754

61833

53219

69029

70824

72666

0.05

0.05

0.05

46549

40680

35013

7

CF

8

Z

1,t-1

Z

9 t0

10

Value

195578 0.080000 0.080000

181091

182258 0.086000 0.082996

155393

169512 0.092600 0.086188

132277

718459 0.086000 0.086141

516247

Value = 985008

11

Prob.

0.5

0.5

0.5

0.125

Path 8

Year

1

2

3

4

R

1 ref

2 3 4 5 6



0.088000

1000000

0.096000

884422

0.106480

772918

665240

0.08

0.08

0.08

0.08

80000

70754

61833

53219

69029

70824

72666

0.05

0.05

0.05

46549

40680

35013

7

CF

8

Z

1,t-1

9

Z t0

10

Value

195578 0.080000 0.080000

181091

182258 0.086000 0.082996

155393

169512 0.092600 0.086188

132277

718459 0.099860 0.089590

509741

Value = 978502

11

Prob.

0.5

0.5

0.5

0.125

Wt. Value $997,457


Path 1

• The cash flows for path 1 (the path with three consecutive decreases in rates) consist of

– $335,224 in year 1 (interest = $80,000, scheduled principal = $69,029.49, and prepaid principal =

$186,194.10, reflecting a CPR of .20)

– $324,764 in year 2, with $205,540 being prepaid principal (CPR = .30)

– $257,259 in year 3, with $173,802 being prepaid principal (CPR = .40)

– $251,560 in year 4

• The year 4 cash flow with the balloon payment is equal to the principal balance at the beginning of the year and the 8% interest on that balance.


Calculations for CF for Path 1:

•

$335,224 in year 1:

•

Interest = $80,000

• scheduled principal = $69,029.49

• prepaid principal = $186,194.10, reflecting a CPR of .20 p



$ 1 , 000 , 000

1



( 1 /( 1 .

08 )

10



$ 149 , 029 int erest scheduled prepaid

CF

1





.

08

.

08 ($ 1 , 000 , 000 ) principal

$ principal

80 , 000





$ 149



$ 80

, 029

, 000



$ 80 , 000



$ 69 , 029



$ 69

.

20 ($ 1 , 000 , 000



$ 69 , 029 )

, 029



$ 186 , 194



$ 335



, 224

$ 186 , 194




•

$324,764 in year 2:

•

Interest = $59,582

• scheduled principal = $59,641

• prepaid principal = $205,540, reflecting a CPR of .30

Balance



$ 1 , 000 , 000



($ 69 , 029



$ 186 , 194 )



$ 744 , 776 p



$ 744 , 776

1



( 1 /( 1 .

08 )

9



$ 119 , 223 int erest





.

08

.

08 ($ 744 , 776 )

$ 59 , 582





$ 59 , 582 scheduled prepaid principal principal





$ 119 , 223



$ 59 , 582



.

30 ($ 744 , 776



$ 59 , 641 )

CF

2

$ 59 , 641



$ 205 , 540



$ 324

$ 59 , 641



$ 205 , 540

, 764




•

$257,259 in year 3:

•

Interest = $38,368

• scheduled principal = $45,089

• prepaid principal = $173,802, reflecting a CPR of .40

Balance



$ 744 , 776



($ 59 , 641



$ 205 , 540 )



$ 479 , 594 p



$ 479 , 594

1



( 1 /( 1 .

08 )

8



$ 83 , 456 int erest



.

08

.

08 ($ 479 , 594 )



$ 38 , 368 scheduled prepaid

CF

3



$ 38 , principal principal

368



$



45

.

,



40

$ 83

($

089

,



456

479

$

,



$ 38 , 368

594

173 ,



$ 45 , 089

802





)

$ 45



,

$

089

173

$ 257 , 259

, 802




• $281,560 in year 4:

• The year 4 cash flow with the balloon payment is equal to the principal balance at the beginning of the year and the

8% interest on that balance.

Balance



$ 479 , 594



($ 45 , 089



$ 173 , 802 ) int erest

CF

4



$



.

260

08 ($

, 703

260



, 703 )



$ 20 , 856

$ 20 , 856



$ 281 , 560



$ 260 , 703



Path 8

• In contrast, the cash flows for path 8 (the path with three consecutive interest rate increases) are smaller in the first three years and larger in year 4, reflecting the low CPR of 5% in each period.

Step 3: Valuing Each Path

• Like any bond, a MBS or CMO tranche should be valued by discounting the cash flows by the appropriate riskadjusted spot rates.

• For a MBS or CMO tranche, the risk-adjusted spot rate, z t is equal to the riskless spot rate, S t

, plus a risk premium.

,

• If the underlying mortgages are insured against default, then the risk premium would only reflect the additional return needed to compensate investors for the prepayment risk they are assuming.

•

As noted in Chapter 4, this premium is referred to as the option-adjusted spread (OAS).


• If we assume no default risk, then the riskadjusted spot rate can be defined as z t



S t

 k t where: k = OAS


• The value of each path can be defined as

V i

Path 

M

T 



1

( 1



CF

M z

M

)

M



1

CF

1

 z

1



CF

2

( 1

 z

2

)

2



( 1

CF

3

 z

3

)

3

   

( 1

CF

T

 z

T

)

T where:

• i = ith path

• z

M

= spot rate on bond with M-year maturity

• T = maturity of the MBS


• For this example, assume the optionadjusted spread is 2% greater than the oneyear, risk-free spot rates.


Binomial Tree for

Discount Rates

8 .

6 %

8%

8%

Z

10

Z

11

Z

12

Z

13


• From these current and future one-year spot rates, the current 1-year, 2-year, 3-year, and 4-year equilibrium spot rates can be obtained for each path by using the geometric mean: z

M





( 1

 z

10

) ( 1

 z

11

)

  

( 1

 z

1 , M



1

)



1 / M 

1


• The set of spot rates z

1

, z

2

, z

3

, and z

4 needed to discount the cash flows for path 1 would be: z

1 z

2

 z

3

 z

4











.

08



( 1

 z

10

) ( 1

 z

11

)



( 1 .

08 ) ( 1 .

074546



1

)

/



1

2

/



1

2 

1



.

07727



( 1

 z

10

) ( 1

 z

11

)( 1



( 1 .

08 ) ( 1 .

074546

 z

12

)



1 / 3 

1

)( 1 .

069588 )



1 / 3 

1



.

074703



( 1

 z

10

) ( 1

 z

11

)( 1

 z

12

)( 1

 z

13

)



1 / 4 

1



( 1 .

08 ) ( 1 .

074546 )( 1 .

069588 )( 1 .

06508 )



1 / 4 

1



.

072289


• Using these rates, the value of the MBS following path 1 is $1,010,465:

V

1

Path 

$ 335 , 224



1 .

08

$ 324 , 764

( 1 .

07727 )

2



$ 257 , 259

( 1 .

074703 )

3



$ 281 , 560

( 1 .

072289 )

4



$ 1 , 010 , 465

The spot rates and values of each of the eight paths are shown in columns 9 and 10 of Exhibit A.

Step 4: Theoretical Path

• The theoretical value of the MBS is defined as the average of the values of all the interest rate paths:

V



1

N i

N 



1

V i path

• In this example, the theoretical value of the MBS issue is $997,457 or 99.7457% of its par value (see bottom of Exhibit A).

Step 4: Theoretical Path

• The theoretical value along with the standard deviation of the path values are useful measures in evaluating a MBS or CMO tranche relative to other securities.

• A MBS's theoretical value can also be compared to its actual price to determine if the MBS is over or underpriced.

• For example, if the theoretical value is 98% of par and the actual price is at 96%, then the mortgage security is underpriced, '$2 cheap', and if it is priced at par, then it is considered overpriced, '$2 rich.'

Option-Adjusted Spread

• Instead of determining the theoretical value of the

MBS or tranche given a path of spot rates and option-adjusted spreads, analysts can use a Monte

Carlo simulation to estimate the mortgage security's rate of return given its market price.


• Since the security's rate of return is equal to a riskless spot rate plus the OAS (assuming no default risk), many analysts use the simulation to estimate just the OAS.

• From the simulation, the OAS is determined by finding that OAS that makes the theoretical value of the MBS equal to its market price.


• This spread can be found by iteratively solving for the k that satisfies the following equation:

Market Pr ice



1

N











M

T 



1

CF

( 1 ) M

( 1



S

( 1 ) M

 k )

M















M

T 



1

( 1



CF

( 2 ) M

S

( 2 ) M

 k )

M







   







M

T 



1

CF

( N ) M

( 1



S

( N ) M

 k )

M









 where: N = number of paths

Effective Duration and Convexity

• Effective duration and convexity can be used with a binomial tree to measure the duration of a MBS.

Duration



P



2 ( P

0



P



)

 y

; Convexity



P





( P

0

P





) (

 y

2 ( P

0

)

2

) where :

P



 price

P



 price associated associated with a small decrease with a small increase in rates in rates

Effective Duration and Convexity

•

Steps for using the binomial tree to estimate duration and convexity:

1.

Take yield curve estimated with bootstrapping and value the

MBS (theoretical value), P

0

, using the calibration approach.

2.

Let the yield curve estimated with bootstrapping decrease by a small amount and then estimate the price of the MBS using the calibration approach -- P

-

.

3.

Let the yield curve estimated with bootstrapping increase by a small amount and then estimate the price of the MBS using the calibration approach -- P

+

.

4.

Calculate effective duration and convexity.

D



P



2 ( P

0



P



)

 y

; Convexity



P





( P

0

P





2 ( P

0

) (

 y )

2

)

Yield Analysis

• Yield analysis involves calculating the yields on

MBSs or CMO tranches given different prices and prepayment speed assumptions or alternatively calculating the values on MBSs or tranches given different rates and speeds.

Yield Analysis

• For example, suppose an institutional investor is interested in buying a MBS issue that has a par value of $100M, WAC = 8, WAM = 355 months, and a PT rate of 7.5%.

• The value, as well as average life, maturity, duration, and other characteristics of this security would depend on the rate the investor requires on the MBS and the prepayment speed she estimates.

Yield Analysis

• If the investor’s required return on the MBS is 9% and her estimate of the PSA speed is 150, then she would value the

MBS issue at $93,702,142.

• At that rate and speed, the MBS would have an average life of 9.18 years. Whether a purchase of the MBS issue at

$93,702,142 to yield 9% represents a good investment depends, in part, on rates for other securities with similar maturities, durations, and risk, and in part, on how good the prepayment rate assumption is.

•

For example, if the investor felt that the prepayment rate should be 100% PSA and her required rate with that level of prepayment is 9%, then she would price the MBS issue at $92,732,145 and the average life would be 11.51 years.

Yield Analysis

• In general, for many institutional investors the decision on whether or not to invest in a particular

MBS or tranche depends on the price the institution can command.

• For example, based on an expectation of a 100%

PSA, our investor might conclude that a yield of

9% on the MBS would make it a good investment.

In this case, the investor would be willing to offer no more than $92,732,145 for the MBS issue.

Yield Analysis

• One common approach used in conducting a yield analysis is to generate a matrix of different yields by varying the prices and prepayment speeds.

• The next exhibit shows the different values for our illustrative MBS given different required rates and different prepayment speeds.

•

Using this matrix, an investor could determine, for a given price and assumed speed, the estimated yield, or determine, for a given speed and yield, the price. Using this approach, an investor can also evaluate for each price the average yield and standard deviation over a range of PSA speeds.

Yield and Vector Analysis

Mortgage Portfolio = $100M, WAC = 8%, WAM = 355 Months, PT Rate = 7.5

Rate/PSA

7%

8%

9%

10%

Average Life

Rate

7%

8%

9%

10%

50

Value

$106,039,631

$98,251,269

$91,442,890

$85,457,483

14.95

Vector

Month Range: PSA

1-50: 200

51-150: 250

151-250: 150

251-355: 200

Value

$103,729,227

$98,893,974

$94,465,328

$90,395,704

100

Value

$105,043,489

$98,526,830

$92,732,145

$87,554,145

11.51

Vector


1-50: 200

51-150: 300

151-250: 350

251-355: 400

Value

$103,473,139

$98,964,637

$94,794,856

$90,929,474

150

Value

$104,309,207

$98,732,083

$93,702,142

$89,146,871

9.18

Vector


1-50: 200

51-150: 150

151-250: 100

251-355: 50

Value

$104,229,758

$98,756,370

93,826,053

89,,364,229

Yield Analysis

• One of the limitations of the above yield analysis is the assumption that the PSA speed used to estimate the yield is constant during the life of the

MBS.

• In fact, such an analysis is sometimes referred to as static yield analysis .

• In practice, prepayment speeds change over the life of a MBS as interest rates change in the market.

Vector Analysis

• A more dynamic yield analysis, known as vector analysis , can be used.

• In applying vector analysis, PSA speeds are assumed to change over time.

• In the above case, a matrix of values for different rates can be obtained for different PSA vectors formed by dividing the total period into a number of periods with different PSA speeds assumed for each period.

• A vector analysis example is also shown at the bottom of the last exhibit.

Asset-Backed Securities

Chapter 11

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Securitization

Cash Flows

Extension Risk

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