lecture · 1999. 4. 23. · mp = mb 0 c (1 +) n (1 + c) n 1 mortgage b alanc e mb t = 0 (1 + c) n t...
TRANSCRIPT
Lecture 2
Finance Project
Somesh Jha
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Summary of Mortgages
�MBt: (mortgage balance remaining at timet)MP : (mortgage payment)c: (simple monthly interest rate (annualinterest rate/12))n: (number of months)
�Mortgage Payments
MP = MB0
c(1 + c)n
(1 + c)n � 1
�Mortgage balance
MBt = MB0
(1 + c)n� (1 + c)t
(1 + c)n� 1
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Summary of Mortgages (Contd)
� Scheduled Principal Repayments
Pt = MB0
c(1 + c)t�1
(1 + c)n � 1
� Interest for month t
It = cMt�1
MB0
c"(1 + c)n � (1 + c)t�1
#
(1 + c)n� 1
� Check that MP = It + Pt.
3
Adjustable Rate Mortgages (ARMs)
� IndexIndex at time time t denoted by x(t).The interest rate of a mortgage variesaccording to this contractually speci�edindex. The two most widely used indices arecost of funds index (COFI) and a constantmaturity (one year or �ve year) Treasuryindex.
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ARMs (Contd)
�MarginDenote by mThis is the amount the interest rate will bechanged by. Margin is also speci�ed in thecontract. The size of the margin re ects thevalue of the various options embedded in theARM, including the option to prepay, aswell as the costs of servicing a loan.
�Adjustment PeriodThis is the stipulated frequency at whichthe interest rate will be adjusted. Typicaladjustment periods are six months or oneyear.
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ARMs (Contd)
� Lifetime capDenoted by cL.This is a contractually speci�ed upperbound on the interest rate.
� Lifetime oorDenoted by cFThis is a contractually speci�ed lower boundon the interest rate.
�Periodic capDenoted by cPIn a single adjustment period cannot adjustthe interest rate by this cap.
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Formula for adjusting
�Assume that time is t and we are going toadjust the mortgage interest rate (or couponrate), which is c(t� 1).
� If x(t) +m > c(t� 1), then the formula is:
min [x(t) +m; cL; c(t� 1) + cP ]
� If x(t) � c(t� 1), then the formula is:
max [x(t) +m; cF ; c(t� 1)� cP ]
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Structure of the document
�Describe the cash- ow of the �nancialinstrument (use equations along withexplanations).
� Provide scenarios, or examples of cash- ows.
� Provide a summary of all the formulas. In areal system there will be a formal documentdescribing all the user requirements.
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Structure of the document (Contd)
�Also pay attention to extraneous factors(not directed related to the cash- ow).
�We will see examples of this later in thecontext of MBSs.
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Who should be able to understand it
�A user with basic �nance knowledge (notfamiliar with that domain) should be able tounderstand it.
�Minimize ambiguities.
� The summary is meant for developers anddesigners.
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Mortgage Backed Securities (MBSs)
� The underlying asset for MBSs is a pool ofmortgages.
� The cash- ow for MBSs consist of threecomponents
{ Interest.
{ Scheduled Principal Repayment.
{ Payments in excess of the regularlyscheduled principal repayment.
� Third component is what makes thecash- ows of MBSs tricky.
�What happens without the thirdcomponent?
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Mortgage Pass-Through Securities
� These are the simplest of MBSs. We willcall them pass-throughs from here on.
�We will illustrate the cash- ow of apass-through security through an exampleand later provide the formulas.
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Pass-Throughs (Contd)
� Suppose the pool of mortgages underlyingthe pass-through has 10 mortgages.
� Each mortgage is worth 100,000 dollars.Total value is 1 million dollars.
�Assume there are 40 pass-throughs issuedalong with this pool of mortgage.
� Each pass-through is worth 25,000 or 2.5%of the entire amount.
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Cash ow for each pass-through
� Each pass-through security gets 2.5% of thecash- ow each month.
� Basically an owner of the pass-through is avirtual loaning agency for the pool ofmortgages.
� Pass-throughs basically provide liquidity tothe mortgage market.
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Digression on pass-throughs
� There is always the risk that there might bedefaults in the pool of mortgages underlyingthe pass-through.
� Fortunately majority of pass-throughs areguaranteed by government-related entities:
{Government National MortgageAssociation (Ginnie Mae).
{ Federal National Mortgage Association(Fannie Mae).
{ Federal Home Loan MortgageCorportaion(Freddie Mac).
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Digression on pass-throughs (Contd)
�Outcome: Don't have to consider risk ofdefault.
�Notice how extraneous factors are drivingour assumptions.
� For pass-throughs backed by privateagencies this assumption is lest justi�ed.
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Collateralized Mortgage Obligations(CMOs)
� Each pass-through security is exposed to therisk of prepayment uniformly.
�We might want di�ering risk-levels.
�Again take the same pool of mortgages asbefore.
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CMOs (contd)
� Suppose we have three classes A, B, and C.
� Class A has par value 400,000.
� Class B has par value 300,000.
� Class C has par value 300,000.
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CMOs (Distribution of cash- ows)
� Remember the cash ow of the pool ofmortgages has two components: interestand principal.
� Interest is distributed among the threeclasses proportional to the par value, e.g,class A receives 40% of the interestpayments.
�All principal payments go to class A untilthe mortgage balance reaches 600,000.
�While the mortgage balance is between600,000 and 300,000, the principal paymentsgo to class B.
� The rest of the time the principal paymentsgo to class C.
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CMOs (Contd)
� CMOs described earlier have a very simplerule for prioritizing the distributionprincipal.
� There are much more complicated schemessuch as:
{ Planned amortization class (PAC) bonds.
{ Targeted amortization class (TAC) bonds.
{Very accurately determined maturity(VADM) bonds.
� These di�erent kind of instruments providedi�erent kind of prepayment risks.
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Stripped MBS
� CMOs specify a set of rules for prioritizingthe distribution of the principal paymentsamong the various classes (also calledtranches).
�A stripped MBS has only two classes:principal-only or PO class, andinterest-only or IO class.
� PO class only receives the principal.
� IO class only receives the interest.
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Cash ow characteristics of CMOs
�MPt mortgage payment for month t.
�MBt�1 mortgage balance at the start ofmonth t.
� It (monthly interest form month t).
�NIt (net interest for month t).
� St (servicing fee for month t).
� s (rate of servicing fee).
� Pt (principal payment form month t).
� PPt (prepayment for month t).
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Cash ow (contd)
We have following relationships between thequantities:
MPt = MBt�1
c(1 + c)n�t+1
(1 + c)n�t+1 � 1It = cMBt�1
NIt = (c� s)MBt�1
St = sMBt�1
MBt = Mt�1 � Pt � PPt
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Pass-Throughs
�Assume that we have n pass-throughs andthe underlying mortgage pool has thecash- ow at time t of
CFt = NIt + Pt + PPt
� Cash ow for each pass-through is simplyCt
n.
� Cash- ow divided evenly between eachpass-through security.
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CMOs
� The interest component of the cash- ow isNIt.
�Assume we have k classes or tranchesT1; � � � ; Tk and wi is the weight of tranchTi.
�Weight wi is the par value of that tranchdivided by the total value of the mortgagepool.
� Tranch Ti receives wiNIt from the interestrate component.
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CMOs
�Once a tranch receives principal paymentsequal to its par value it is said to retire.
� The principal component of the mortgagepayment is PPt + Pt.
� Let Ti be the tranch with the highestpriority that has not retired.
� Pay the principal payment to that tranch.
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Modeling Prepayment
� Prepayment is a�ected by several factors.We will list some of them here.
�Re�nancing opportunitiesPrevailing re�nancing rates. Lowerre�nancing rates relative to the mortgagecoupon/interest rate leads to higherlikelihood of prepayment.
�Age or seasoningThe age of a mortgage a�ects prepaymentdecisions. Newly originated mortgages areunlikely to be prepaid, while agedmortgages, for example, held by retirees, aremore likely to be prepaid.
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Modeling Prepayment (Contd)
� SeasonailityThe time of year a�ects prepayments.Spring and summer months tend toexperience greater prepayment activity.
�BurnoutBorrowers in a pool are heterogenous intheir response to re�nancing opportunities.Once the eager borrowers have re�nancedthe rest of the borrowers (the laggards) willnot prepay for a while. This phenomenon iscalled burnout.
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Simple Model of prepayment
� Forecasting prepayments is crucial tocomputing the cash- ow of MBSs.
� Current practice is to use Public SecuritiesAssociation (PSA) prepayment benchmark.
� In this simple model a fraction of theprincipal is assumed to be prepayed eachmonth.
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SMM and CPR
�Conditional Prepayment Rate (CPR) isthe annual prepayment rate.
� CPR is based on the characteristics of thepool and the current expected economicenvironment.
� Single Monthly Mortality rate (SMM) isthe fraction of the principal payed o� eachmonth, and is related to CPR in thefollowing way:
SMM = 1 � (1� CPR)1
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� Check that the following formula is true:
P (1 + SMM)12 = P (1� CPR)
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Prepayment and SMM
� Let MBt�1 be the mortgage balance attime t� 1 and SPt the servicing fee formonth t.
� The prepayment for month t is given by thefollowing formula:
PPt = SMMt(MBt�1 � St)
� Recall that St is the servicing fee for thatmonth.
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PSA benchmark
� PSA standard benchmark assumes thefollowing prepayment rates for 30-yearmortgages:
1. A CPR of 0.2% for the �rst month,increased by 0.2% per month for the next30 months, when it reaches 6% year.
2. A 6% CPR for the remaining years.
� The standard PSA benchmark (written as100% PSA) can be expressed using thefollowing formula:
CPR =6%t30
if t � 306% if t > 30
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PSA (contd)
�A x% PSA varies the speed of theprepayment by x
100.
�A 50% will assume one-half the CPR of thestandard PSA.
�A 150% will assume 1.5 times the CPR ofthe standard PSA.
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References
� F.J. Fabozzi, Fixed Income Mathematics(Analytical and Statistical Techniques),Third Edition, Irwin ProfessionalPublishing, 1997.
�W.N. Torus, Mortgage Backed Securities,Handbooks in Operations Research andManagement Science (Volume 9),Editors: R.A. Jarrow, V. Maksimovic,W.T. Ziemba, Informs, North-Holland,1995.
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Scenarios
� Pass-through security with originalmortgage balance of 100 million dollars.
�Mortgage rate is 9.5%.
� Servicing fee is 0.5%.
� 360 months to maturity.
� Prepayment according to 100% PSA.
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Parameters
�MB0 = 100; 000; 000.
� n = 360.
� c = :0079167.
� s = :0004167.
�CPRt is given by the following formula:
6%t30
if t � 306% if t > 30
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Example cash- ows (Check)
� t = 1MPt = 841, SPt = 49, It = 792, PPt = 17,St = 42, CFt = 816
� t = 100MPt = 544, SPt = 25, It = 475,PPt = 308, St = 25, CFt = 827
� t = 358MPt = 144, SPt = 141, It = 2, PPt = 1,St = 0, CFt = 142
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150% PSA
�CPRt given by the following formula:
9%t30
if t � 309% if t > 30
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50% PSA
�CPRt given by the following formula:
3%t30
if t � 303% if t > 30
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Drawbacks of PSA
� The PSA model is very simple. It is abenchmark and is simply a marketconvention.
�Does not model seasonal or burnout e�ects.
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Vector of PSAs
�One could vary the speed of the prepaymentfor di�erent time.
� For example,
{Months 1-43 (100% PSA)
{Months 44-204 (150% PSA)
{Months 205-360 (50% PSA)
� Speed of prepayment could also depend onmortgage balance.
� Probably can handle seasonal e�ects in thismanner.
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More complicated prepayment mod-els
�Assume that the number of prepayments inmonth t (denoted by pnt) follows a Poissondistribution with the parameter �(Xt; �).
� Let Nt be the number of mortgages in thepool.
� In this case expected number ofprepayments is given by:
Nt�(Xt; �)
�Xt set of variables to be de�ned.
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� � parameters to be de�ned.
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Form of �
�Re�nancing opportunitiesrf7The ratio of 7 year Treasury bond to themortgage pool's weighted-average-coupon(WAC).
�Age or Seasoning�0(t)The ratio of old mortgages to the newmortgages at time t.
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Form of � (Contd)
� Seasonal a�ectsseason
Discrete variable indicating which season weare in. For example, season = 1 for monthsof April through September and zero everywhere else.
�Burnoutburnout
Ratio of the pools' current principal and theprincipal if the mortgage pool had beenpayed o� according to a standardizedbenchmark (say 100% PSA).
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Form of �
� �(Xt; �) is given by the following equation:
�0(t) exp(�1 � rf7+�2 ln(burnout) + �3 � season)
�How does one estimate the parameters�1; �2; �3.
�Use Maximum-Likelihood-Estimation. Willbe covered in a statistics course.
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Requirements Document
�Due by: Wednesday, Jan 27, 1999.
�Describe the �nancial instruments and theunderlying assets in great detail. Equationsfor cash- ows.
� Example of cash- ows (scenarios).
� Summary of formulas.
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Forming sub-teams
�Divide your team into the followingsub-teams. Everyone should be involved insome role.
�ResearchersStudents in this sub-team will actually dothe research and write down the equationsdescribing the cash- ows of the instrumentsand the assets. Mention your resources veryclearly.
� Scenario buildersThese students will look at the documentprovided the researchers and provideexamples of cash- ows.
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Forming sub-teams (Contd)
�ReviewersThe document produced by the researchersand the scenario-builders will be reviewedand a summary will be written by thissub-team. This sub-team will also makesure that everything �ts together.
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Document
�Mention the composition of the sub-teamson the top page.
�Use a standard format to compose thedocument.
� Researchers should mention their sourcesvery clearly.
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Penalty
� If you are late by more than 2 days, 10%deducted for every 2 days.
� Be on time. We will provide feedback onyour document in a week or so.
�Mention all the background for the assetsand the �nancial instruments.
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