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A Variable-Rate Loan-Prepayment Model
for Australian Mortgages
by
John Daniel
Abstract:
This paper is an investigation of Australian mortgage-loan prepayment from a
modelling perspective. A prepayment model for loans of mortgage-backed securitiesis developed specifically for the Australian mortgage market, and then empiricallytested using (Reuters) Australian mortgage-backed security data. The model hasorigins in the variable-rate loan-prepayment models of the U.S., but is designed anddeveloped to take into account the Australian mortgage-market structure. The model
proves very successful when tested empirically, and is able to explain the partial-prepayment features of the Australian market as well as full prepayments.
Keywords:
PREPAYMENT MODELLING; AUSTRALIAN AND U.S. MORTGAGE MARKETS;
VARIABLE-RATE LOANS; FIXED-RATE LOANS; MORTGAGE-BACKED SECURITIES;MORTGAGE RATES; PARTIAL PREPAYMENT; INVESTMENTS.
School of Finance and Applied Statistics, Australian National University. Email:John.Daniel@anu.edu.au
The author would like to acknowledge that the mortgage-backed security prepayment data, used forempirical tests of the models, was provided by SIRCASecurities Industry Research Centre of Asia-Pacific. Also comments and suggestions by Tom Smith are acknowledged with thanks.
Australian Journal of Management, Vol. 33, No. 2 December 2008, The University of New South Wales
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1. Introduction
hepr
of th
development and empirical testing of a mortgage-backed security, (MBS),epayment model appropriate for the Australian mortgage market is the focus
is paper.T
The main reason for creating prepayment models is to inform MBS investorsabout what the determinants of prepayments are, and how changes in thesedeterminants will affect prepayments. Prepayment estimation (achieved through
prepayment modelling) is a necessary intermediate step in the process of MBSvaluation. Prepayment models are also more broadly relevant to all those associatedwith mortgage finance.
Previous Australian prepayment modelling research has been restricted (byresearcher choice) to fixed-rate loan prepayment, following the United States(U.S.) fixed-rate prepayment modelling paradigms. This paper expands Australian
prepayment modelling research to include variable-rate loans and partialprepayment.1 Australian mortgage-holder partial prepayment is recognized in suchpapers as Westpac (2002) and Deutsche Morgan Grenfell (1998); both these papersoffer various explanations for partial prepayment. Adding a partial-prepaymentcomponent to prepayment modelling is one of the main research contributions ofthis paper.
Most of the research in the field of MBSs (and in particular MBS prepayment) has occurred in the U.S.; hence throughout the paper the U.S.mortgage market is used as a recurring source for comparisons and modelling
prototypes. The prepayment model which is investigated here is anadaptation/augmentation of the variable-rate mortgage models of McConnell andSingh (1991), and Sanyal (1994). Although based on the formerly mentioned
models, the model developed and empirically tested here, is new, and incorporatesunique Australian features.
The data used in the empirical testing of the model (which will be referred toas: the Australian variable-rate mortgage model, or just: the variable-ratemortgage model, when appropriate) revealed that Australian MBS pools arecharacterized by a wide variation in the percentage of variable-rate mortgages and
percentage of fixed-rate mortgages, and are invariably not composed of purelyvariable-rate mortgages, (VRMs), or purely fixed-rate mortgages,(FRMs). TheAustralian variable-rate mortgage prepayment model is intended as a general-
purpose prepayment model, and hence does also include independent variableswhich account for FRM prepayment. Given the statement of the last sentence, the
prepayment model of this paper might be more accurately referred to as acombined prepayment model (that is, a prepayment model for variable-rate andfixed-rate prepayment); the variable-rate label can be understood more as
1. For the reader unfamiliar with the terms partial prepayment and full prepayment: Whereasrefinancing prepayment results in the mortgage holder completely paying out the loan, partial
prepayments are payments which exceed the mortgage holders required scheduled monthly paymentswithout paying out the loan completely. Prepayment where the loan is paid out completely is referred to
as full prepayment. Hence the division of prepayment into these components can be described by theequation:
(total)prepayment = full prepayment + partial prepayment
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describing the origins of the model, rather than the purpose or intendedapplication.2
1.1 The Main Findings of the Research were:
(i) For variable-rate mortgage holders:Both partial prepayment and full prepayment are strongly affected by theamount by which the mortgage holders highest interest-rate level attainedover the prior course of the loan exceeds the current market mortgagevariable-rate.
(ii) The ratio of average after-tax share market returns to mortgage rates wasintroduced as a variable following from the idea of partial prepayment as analternative investment to shares. This ratio was found to be highly negativelyassociated with full prepayments.
(iii) The prepayment data revealed that in Australia partial prepayment is on
average approximately one third of full prepayments for variable-rate loans,and slightly less (closer to a quarter) of full prepayments for fixed-rate loans.
(iv) Partial prepayment is not confined to variable-rate mortgage holders;Australian fixed-rate mortgage holders also choose to partially prepay, as isshown by the next finding.
(v) For fixed-rate mortgage holders:There is weak evidence to suggest that the amount by which the fixed-ratemortgage holders interest-rate exceeds the current market mortgage fixed-rate, determines their propensity to partially prepay.
One finding unrelated to interest-rates was:(vi) (As is the case in the U.S.) the age of the pool (or the average number ofmonths since origination of the mortgages in the pool) is a strong influence onfull prepayments.
An overview of the paper by sections is as follows:There are, as is expected given the similarities of the U.S. and Australian
economies, many similarities between the U.S. and Australian mortgage markets.There are also some important differences; these are discussed in section 2. Section3 is a description of how the two leading (U.S.) articles on variable-rate loan MBS
prepayment modelling: McConnell and Singh (1991), and Sanyal (1994),
contribute to the creation of an Australian variable-rate prepayment model. Section4 presents an empirical test of the variable-rate McConnell and Singh/Sanyal modeladapted and augmented as necessary for the Australian mortgage market. Section 5
presents concluding remarks.
2. There was no evidence that the US practice of forming MBS mortgage pools containing exclusivelyvariable-rate mortgages, occurs in Australia; however, as can be seen from table 2, two of the twelve
pools used for empirical tests contained over 98% variable-rate mortgages, so 100% variable-ratemortgage pools possibly do occur in Australian MBS pools.
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2. Australian Mortgage MarketHow Different from U.S.
The essential differences between the Australian and U.S. mortgage markets arediscussed in this section as preparation for the adaptation/expansion of U.S.-basedvariable-rate prepayment models.
2.1 Tax
To clarify the differences between Australian and U.S. mortgage holder taxregimes the terms owner/occupier mortgage holder and investment mortgageholder are introduced. An investment mortgage holder is defined as a mortgageholder who is purchasing for rental purposes; the term owner/occupier mortgageholder is self-explanatory. Australian owner/occupier mortgage holders cannotclaim interest paid on their mortgage loans as a tax deduction, whereas Australianinvestment mortgage holders can. All U.S. mortgage holders, both owner/occupierand investment mortgage holders, can claim the interest paid on their mortgage
loans as a tax deduction.These differences in tax rules have significant implications for the
prepayment behaviour of Australian and U.S. mortgage holders, as will be seen infollowing sections, and particularly in the text under the heading PartialPrepayment and Tax in section 3.1.4: Partial Prepayers.
2.2 Fixed-Rate Mortgage Loan Term Periods
While in the U.S. the interest-rate on a FRM is fixed for the entire loan period(usually 2530 years), in Australia the interest-rate is fixed for sub-periods,(1, 2,3,..., 15 years) of the full loan term period. Loans with a fixed term longer than 5
years are very rare in Australia.
2.3 VRM Rates: Set by Lender (AUS), Indexed (US)
The Australian VRM can be described as a loan to a (home) borrower where theinterest-rate on the loan varies according to the discretion of the lender.Competition amongst lenders ensures that this rate will be the sum of the short-term interest rate (referred to as the cash rate) set by the Reserve Bank of Australia(RBA) plus some margin rate which can vary from lender to lender. In the U.S.VRM interest-rates are adjusted according to indices.
Further discussion of these differences (in how the respective VRM rates of
Australia and the U.S. are reset), and their implications for prepayment, occurs insection 3.1.2: Refinancers.
2.4 Ratio of Fixed-rate to Variable-Rate Mortgages
Campbell and Cocco (2003) showed that over the years 1985 to 2001 in the U.S.the proportion of fixed-rate loans varied approximately between 30% and 90%.Several sources (including: NAB 2002, and, Research Note 2003) document thatthe proportion of loans which are fixed-rate in Australia varies approximately
between ten and twenty percent. Thus the ratio of fixed-rate loans to variable-rateloans in the U.S. is higher than the corresponding ratio in Australia, and usuallymuch higher; for example, most of the time over 80% of U.S. mortgage loans arefixed-rate, whereas in Australia at least 80% of mortgage loans are variable-rate
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(all the time).3
FRMs and VRMs can show significantly different prepaymentbehaviour patterns.
2.5 Partial Prepayments(Curtailments)
In comparison to U.S. mortgage prepayment, Australian prepayments most uniquecharacteristic is the high level of partial prepayment. In the U.S. (where the bulk of
prepayment modelling research has been carried out), partial prepayment is soinsignificant a part of prepayment that inclusion in prepayment models is regardedas unnecessary;4 (as stated in the introduction) the prepayment data used in theempirical tests presented later in the paper has revealed that Australian partial
prepayment is on average approximately one third of full prepayments, makinginclusion of partial prepayment in Australian prepayment models essential.Reasons for the extent of the difference of levels of partial prepayments forAustralia and the U.S., are investigated, appropriate variables are designed, and anAustralian partial prepayment model is formulated in section 3.
2.6 Recourse/Nonrecourse Mortgage Loans
In many states of the U.S., such as California, if the mortgage holder chooses todefault on the loan, then the lender has recourse only to the property that themortgage holder is purchasing with the loan. In Australia, the lender recoveringdebt in cases of default, has recourse to the mortgage holders other assets such asshares, other properties, and so on. This difference in mortgage contracts probablyaccounts for why default on mortgages is comparatively much rarer in Australiathan the U.S.
2.7 Subprime/Low Doc Loans
These are the respective U.S. and Australian categories of mortgages for mortgageholders with substandard credit ratings; (these loans are higher-risk loans, whichusually means the lender charges higher rates). There is not complete agreementabout the true percentage of Australian mortgages that are low-document loans;estimates vary between less than 0.5 percent to 15 percent. The number of low-document loans has been rapidly declining in the wake of the subprime crisis. Asone lender states: the proportion of low-document loans-home loans which
3. Recently the percentage of fixed-rate loans has increased to just over 20%. Fixed-rate loans made up
23:7% of new mortgages last month, according to mortgage broker Australian Finance Group. This wasup from a low of 16:5% in July. (Uren, The Australian, November 6, 2007). (There is no evidence to
suggest that the statement: Loans with a fixed term longer than 5 years are very rare in Australia ; is nolonger accurate.)
4. The research of Hayre, Chaudhary and Young (2000), reveals that partial prepayments of U.S. mortgageholders do in fact become significant in the later years of long-lasting loansthey state:
Information from mortgage services indicates that the prepayment rates from curtailmentsare typically very low. They average about 0:5% CPR [conditional prepayment rate], earlyin the life of an ageing pool . . . but can ramp up sharply to as much as 15%-20% CPR atthe end of a mortgage term.
Because U.S. MBSs typically have lives much less than, say fifteen years, such late partialprepayments of mortgages, (occurring within a typical U.S. MBS pool), would not begin until after the
MBS has been called. Hence such partial prepayments in the U.S. do not invalidate the practice ofignoring partial prepayment in U.S. MBS prepayment models.
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require little or no proof of a borrowers ability to make their repayments-writtenby the broker had plummeted from 15 percent to an insignificant level somewherebelow 2 percent. (Klan, The Australian, September 20-21, 2008). This compares toalmost 15 percent of U.S. loans being categorised as subprime. Data for the
percentage of low-document loans within each Australian MBS pool was notincluded as part of the Reuters prepayment data set used for the model testing inthis paper
2.8 Caps/Floors
There are no Caps or Floors for Australian VRMs as there are for U.S. VRMs.
3. Creating an Australian Variable-Rate Loan Prepayment Model
The two leading U.S. VRM prepayment models considered for adaptation to the
Australian variable-rate mortgage market, are McConnell and Singh (1991), andSanyal (1994). Sanyals prepayment model develops from the modelling theory asestablished by McConnell and Singh. Rather than a review of McConnell andSingh and Sanyal, the following focuses on explanation of how their models areadapted and expanded for Australian variable-rate mortgage prepaymentmodelling.
3.1 Adapting the VRM Literature
McConnell and Singhs VRM model (and hence also Sanyals) is based on the ideathat, for the purposes of estimating a prepayment function, individuals who prepaytheir variable-rate mortgages can be classified as falling into one of threecategories: 1) relocators; 2) refinancers; and, 3) switchers. These three categories of
prepayers are described, along with the variables which represent them in theAustralian version (of McConnell and Singh/Sanyal), in the following threerespective subsubsections.
3.1.1 Relocators (or Home-Sellers) These mortgage holders are prepaying due to job relocation or changes to family circumstances. In prepayment modellingterminology this type of prepayment is referred to as exogeneous prepayment, thatis, prepayment unrelated to changes in interest-rates.
Relocator variables:
Based on the assumption that relocators are more likely to move during the summermonths, a seasonal indicator variable is defined as:
=SEASONAL
2 bb +
1, if the month is a summer month,
A positive correlation between SEASONAL and CPR is expected.Based on the assumption that the quadratic equation: CPR(exogeneous)=
10 expresses the relationship between the average age of the pool, ( ), and
exogeneous prepayment, the transformed variable,
, is defined by:
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= AGEPOOL =
MAX{WAS}
WAS
* 100% if WAS MAX{WAS},
100% if WAS > MAX{WAS},
where WAS (an acronym for weighted average seasoning) is the average age ofthe mortgages in the pool, weighted by the value of the mortgages, and whereMAX{WAS} is the maximum WAS over all pools.
The variables: AGEPOOL and SQR(AGEPL), (= ), when simultaneously present in the prepayment model, are expected to have positive, and negativecorrelations with CPR, respectively.
2
3.1.2 Refinancers Refinancers, whether FRM or VRM refinancers, prepay totake advantage of lower market interest-rates; however unlike switchers theyremain within their loan type, fixed-rate or variable-rate.
Before describing refinancers in more detail some further background
information is necessary. Unlike Australian VRMs whose rates are reset almostimmediately following an RBA induced rate changefor example,the banksusually reset their interest-rates on mortgages within a week or two of an RBAinterest-rate changethe U.S. VRMs are indexed to a rate such as a Treasury rateor a LIBOR rate and there is usually a lag of up to a year between index resets andthe mortgage rate resets. On first impression of the relationship between RBA and
bank mortgage rates, in comparison to the U.S. VRM, the Australian VRM couldbe expected to behave almost as a pure floating-rate security.
If the index rate for the U.S. VRM drops sharply during the interval betweeninterest-rate reset dates, refinancers will choose to prepay their current VRM totake out another variable-rate loan linked to the new lower index rate. So their
prepayment behav iour is similar to refinancers of FRMs.These refinancers would not be expected to be present in Australian MBS
pools if they consist only of VRMs because, as was stated above, (bank) mortgagerates are being continuously reset to the RBA rate. Why Australian VRMs do infact exhibit refinancing behavior is revealed through the empirical investigations insection 4 and discussed specifically in subsection 4.5.
Also because Australian MBS pools can be expected to contain FRMs as wellas VRMs, the fixed-rate refinancing effect will need to be accounted for in thevariable-rate model.
Refinancing by variable-rate mortgagors may also be affected by a burnout
factor much as refinancers are in the case of fixed-rate loans. Richard and Roll(1989) report that prepayments on fixed-rate loans are a function of the history ofinterest-rates. They argue that fixed-rate mortgagors are differentially sensitive todeclines in the rate on fixed-rate mortgages when making refinancing decisions.The first time that the market coupon rate on fixed-rate mortgages falls below thecoupon rate of an existing mortgage, for example, the most sensitive mortgagors ina pool will refinance. That is, the most rate-sensitive fixed-rate mortgagerefinancers will burnout of the pool. The second time that the pool is subject to adecline in the current market rate to this same level, prepayments will be lowerthan during the first interest-rate cycle. Only if the current rate on fixed-ratemortgages falls below its previous low will the next level of rate-sensitive fixed-
rate refinancers be induced to refinance their loans. By extension a similar effect
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may be present with refinancers in VRM-backed securities, the difference, in theU.S., being that the variable-rate mortgagors are responding to index changes ratherthan actual rates.
Refinancing Variables:
The differential between pool origination coupon rate and the current marketmortgage rate is a commonly used variable to represent the refinancing incentive infixed-rate mortgage pools in the U.S.. Hence FXDdifl is defined as:
FXDdifl = max{3yrFXD coupon rate at origination current market 3yrFXD , 0}
where 3yrFXD is the three-year fixed mortgage rate. A positive correlation between FXDdifl and CPR is expected. Two variables are used to representburnout:
The first burnout variable functions almost identically to Ne as in
McConnell and Singh and Sanyal and will therefore also be denoted as . Theonly differences between the McConnell and Singh / Sanyal version of NewMin and
NewMin for the Australian prepayment model are: (i) the short-term rate,
wMinr
NewMinrr
r r, is proxied by the -year fixed mortgage rate; and, (ii) the condition over theprevious 12 months, becomes (in the months) since origination of the pool.
3
r
=BURNOUT
A positive correlation between and CPR is expected.NewMinThe second burnout variable (denoted by BURNOUT) is designed to capture
the effect of the reduction of the differential rate effect (on CPR), as the number ofmonths, from the last increase in the rate differential, increases (only when the ratedifferential is positive). Hence
unchanged otherwise.
incremented by 1 if (FXDdifl > 0),
0, if (rNewMin = 1),
A negative correlation between BURNOUT and CPR is expected.
3.1.3 Switchers Switchers prepay to change the loan type, (forexample,switching from fixed to variable or vice versa). McConnell and Singhsuggest that switchers base their prepayment decisions upon their expectations
regarding the future. Two determinants of these expectations are:(i) The fixed mortgage rate: McConnell and Singh hypothesize that every time
the fixed-rate loan rate falls to a new minimum (in the time since poolorigination) some of the switcher individuals will prepay (in order to switch toa fixed-rate loan). McConnell and Singh use the long-term rate as a proxy forthe fixed mortgage rate. The variable is defined as:
=NLM
1, if the long-term interest-rate reaches a new minimum
since origination of the pool,
0 otherwise.
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(NLM is an acronym for New Long Minimum.) A positive correlationbetween NLM and CPR is expected.
(ii) (The change in) the slope of the yield curve: The relative change in spreadover each month, between long- and short-term interest-rates, (denoted by
SLYC), is used as a proxy for changes in the slope of the yield curve.Sanyal and McConnell and Singh disagree about how SLYC will affectCPR; therefore the expected sign of the coefficient of SLYC is left asindeterminate.
3.1.4 Partial Prepayers To complete the Australian prepayment model, the threecategories of prepayers of the McConnell and Singh/Sanyal model aresupplemented by a fourth category: partial prepayers.
The essential difference between mortgage prepayments in the U.S. andmortgage prepayments in Australia is partial prepayments. Following is aninvestigation of partial prepayment in Australia including how and why there is this
difference between partial prepayment behaviour of the U.S. and Australia. Thusthe theoretical foundations for modelling partial prepayment, as is required for thevariable-rate model, are established. In order to model partial prepayment, thevarious reasons for partial prepayment are considered; these include:
tax advantages;
partial prepayment by default; and,
aversion to debt and future interest-rate rises.
Partial-Prepayment and Tax
The information regarding differences between Australian and U.S. mortgage
holder tax regimes, as presented in section 2: Australian Mortgage Market-HowDifferent from U.S., under the heading: Tax, is taken as a basis for the followingdiscussion of partial prepayment and tax.
The main underlying driving force which is respon sible for the differencebetween U.S. and Australian partial prepayment is that, while interest payments onmortgages are tax deductible in the U.S. (regardless of whether the mortgage holderis an owner/occupier or an investment mortgage holder), mortgage interest is nottax deductible in Australia unless the mortgage holder is an investment mortgageholder. This difference creates the incentive for owner/occupier mortgage holdersin Australia to partially prepay. Excess funds used to prepay mortgages effectivelyearn the mortgage rate as an after-tax interest-rate. In comparison the US mortgageholder effectively has an interest free loan and partial prepayment does not saveinterest payments.5 Australian owner/occupier mortgage holders will only betempted to consider alternative investments (such as shares or government bonds)for their excess funds when the after-tax return on the alternative investmentsexceeds the current mortgage rate. The following (hypothetical) example is
5. The Australian investmentmortgage holder, (as distinct from an owner/occupier), is in the same positionas all US mortgage holders in being able to claim interest paid on their mortgage loans as a taxdeduction, and would therefore be expected not to partially prepay; (that is if they do prepay, they would
be expected only to make full prepayments). However there is, as yet, no direct statistical evidence tosubstantiate such a proposition.
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designed to illustrate how partial prepayments on Australian mortgages effectivelyearn the mortgage rate as an after-tax interest-rate.
Example: Partial-Prepayment And Tax Let a mortgagor have a marginal rate oftax of 50% . To simplify let the mortgage rate equal the risk-free rate of10% . Themortgagor might consider:
Option 1 Market Investment:
(a) Invest in Share Market, (RISKY):
Make assumptions on risk premium, say . Hence:5%
Expected return on Market: 15.0%
Marginal Tax Rate on return of 50%: 7.5% After Tax Return: 7.5%
(b) Invest in Money Market(Goverment Loans), (NOT RISKY):
Return on Money Market: 10.0%Marginal Tax Rate on return of 50%: 5.0%
After Tax Return: 5.0%
Option 2 Partial Prepayment (NOT RISKY):
Return on Mortgage(Opportunity Cost): 10.0%Marginal Tax Rate on Mortgage 0.0%: 0.0%
After Tax Return: 10.0%
So the mortgagors best financial option, (given these assumptions), is partialprepayment.6
From consideration of example 3.1.4.1 the variable ALTINV is defined:
,1,0SVR
Rate)TaxMarginal(1*Rmax=ALTINV
AOA
(1)
where AOAR
is the average continuous compounded return on the All OrdinariesAccumulation Index (over the previous twelve months), and SVR is an acronymfor standard variable rate, the standard variable-rate in Australia, calculated, (bythe RBA), by averaging the variable-rates of Australian commercial lenders.7
A positive correlation between ALTINV and CPR(Partial) is expected.
Default Partial-Prepayment
The following quote from Westpac Bank (2002), explains why and how partialprepayment by default occurs:
6. A comparable example for the U.S. would have a return on mortgage, (opportunity cost), of zero percent
for partial prepayment, so the U.S. mortgagor is better off investing in shares or government bonds.7. Technical Note: Since the Australian marginal rate of taxwas, over the sample data period, 42 cents for
every $1 earned between $58,000 and $70,000, and 47 cents for every $1 earned over $70,000, the
marginal rate of tax used in calculations, in the empirical investigations later in this article, was set,(somewhat arbitrarily), at 45%, (unlike example 3.1.4.1 where the marginal rate of tax is taken as 50%).
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...as interest-rates fall, borrowers have the option to reduce their loanrepayments or maintain them at previous levels. If repayments are leftunchanged as rates fall, the difference between the old and newrepayment level becomes partial prepayment instead of interestrepayment. Of course not all borrowers will maintain repayment levels
when interest-rates fall, however, on the whole we can expect this tohave an impact, especially as the majority of mortgages in Australia arevariable-rate and attract little, if any penalties for early repayments.
To illustrate this point, consider the simple example of a loanwith an annual interest-rate of 10% . Total interest payments for 1 yearwould be $10,000 initially but if interest-rates fall to 9% , then thiswould fall to $9,000 . If the repayment was maintained at $10,000 however, the difference of would become [partial] prepayment.Based on this example a 1% decrease in interest-rates could be expectedto see up to a 1% increase in (partial) prepayment rates.
$100,000
$1,000
(Australian) banks have adopted the practice of encouraging borrowers to maintainrepayment levels in a falling interest-rate environment in the way described by the
preceding quote. The direct debit method of repayment of loans, (where thelender deducts mortgage payments directly from the borrowers account, and whichis the repayment method used for most loans), has facilitated this process (ofencouraging borrowers to maintain repayment levels in a falling interest-rateenvironment).
From consideration of the preceding quote the variable: partial prepaymentby default, (denoted by ParDFLT), is defined as:
(2),{SVR}max=ParDFLTt
currentSVR
where max t denotes the maximum mortgage interest-rate over the period frompool origination8 until the current month t. The max t in (2) is included to accountfor increases in rates after pool origination; that is, the borrower paying by directdebit is assumed to increase repayments in response to rate increases as iscontractually required, and maintain repayments at the new higher level (even ifrates subsequently fall).
A positive correlation between ParDFLT and CPR(Partial) is expected.
Aversion to Future Rate Rises
The volatility variable, (VOL ), expresses the idea that the variable-rate mortgageholders aversion to (mortgage) debt increases as the risk of interest-rate risesincreases, that is, as interest-rate volatility increases. Mortgage holders will beexpected to increase partial prepayments in response to increasing rate volatility.Hence:
y
months.12previousoverSVRofvolatility=VOLy
(3)
8. Pool origination here, refers to that time when the average age of the mortgages in the pool is zero;calculation of this time is further discussed in subsubsection 4.1.1.
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A positive correlation between VOL and CPR(Partial) is expected.y
Changes in the mortgage holders partial-prepayment behaviour in response torate volatility would best be tested over a long period, probably thirty years ormore; the effect on mortgage holders of the high-volatility (of rates) periods (such
as the early to mid nineteen-nineties) could then be compared to low-volatilityperiods. Unfortunately a sample data set spanning such a breadth of time was notavailable.
Similar arguments to those used to validate volatility as a partial prepaymentregressor variable could be used to validate the level of interest-rates as a partial
prepayment regressor variable. There would also be similar data requirements fortesting the latter variable.
Also considered (but not implemented in empirical work) was a one-sidedrisk measure, since the mortgage holder is only worried about rate rises, not falls;an upper partial moment might be more appropriate.
Summary of partial prepayment in Australia :The preceding proposals for variables can be summarized by the followingfunctional relationship:
(4)).VOL,,ParDFLTf(=Australia)inprepaymentpartialCPR(for y
y
),SEASONAL,NLM,SLYC,VOL,BURNOUT
,FXDdifl,ALTINV,ParDFLT,SQR(AGEPL),AGEPOOLf(CPR
NewMiny
r
=
ALTINV
Model (4) is expected to comprise the essential core of the Australian prepaymentmodel; that is, the variables: ParDFLT, ALTINV, and VOL should comprise themain explanation of Australian MBS prepayment.
3.1.5 The Formulation of an Australian Variable-Rate PrepaymentModel Table 1 summarizes the variable-rate prepayment modelling variables andhow they are structured according to (an augmented version of) McConnell andSinghs prepayer categories. From these variables the Australian variable-ratemodel is formed and can be expressed in functional form as:
(5)
4. Empirical Evaluation of the Australian Variable-Rate ModelSection 4 presents an empirical evaluation of the Australian variable-rate
prepayment model.Subsection 4.1 describes the data used for the empirical testing. Firstly
organization of the data into variable-rate and fixed-rate pool categories is shown insubsubsection 4.1.1. How the prepayment data is divided into componentscorresponding to total, full and partial prepayment is shown in subsubsection 4.1.2.Descriptive statistics and correlation matrix for all variables including total, fulland partial prepayment are given in subsubsections 4.1.3 and 4.1.4, respectively.
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Table 1
Prepayer Categories and their (Proposed) Corresponding Variables
Table 1 summarizes the (VRM)prepayment modellingvariables and how they are structured according to(an augmented version of)McConnell and Singhsprepayer categories. From thesevariables theLINREGmodelis formed.
Relocators Refinancers Switchers Partial Prepayers
AGEPOOL FXDdifl SLYC ParDFLT
SQR(AGEPL) BURNOUT NLM ALTINV
SEASONALNewMinr
y VOL
Legend of variables:
AGEPOOL = the relative weighted average age of mortgages in the pool, (relative to the highestweightedaverageage ofmortgages, (WAS), over all pools in the data set);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;ALTINV = alternative investment;FXDdifl = 3 year fixed differential;
= a new minimum in the 3 year fixedmortgagerate;NewMinBURNOUT = burnout;
y
r
VOL = ratevolatility;SLYC = change in the slope of the yield curve;
NLM = new long minimum; and,SEASONAL = seasonality.
Although in section 3 (Creating an Australian Variable-Rate Loan Prepayment
Model) the variables of the models of McConnell and Singh, and Sanyal werediscussed, the forms of their respective models were not considered. McConnelland Singh uses a proportional hazards model form, whereas Sanyal uses a linearregression model form. Preliminary empirical testing showed that the resultsobtained, for both the McConnell and Singh and the Sanyal model, were so closethat presentation of only one them is necessary; the Sanyal model is presented here.
The variable-rate model is tested with total, partial, and full prepayment as thedependent variable, in sections 4.2, 4.3, and 4.4, respectively.
One of the most noteworthy results for these tests is that the ParDFLT1 andALTINV variables are highly significant as predictors for full prepayments,whereas they were both intended as predict ors for partial prepayments only.
Subsection 4.5 investigates explanations/hypotheses appropriate for these results.
4.1 Data
The interest-rate data (over the period 19962003) were obtained from the RBAwebsite. The data obtained included:
Three-year fixed mortgage rates. (The notation 3yrFXD is used informulae).
Standard variable rate. (The notation SVR is used in formulae).
10-year goverment bond, as proxy for the long-term interest rate.
1-month T-bill, as proxy for the short-term interest rate.
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The latter two data items were used to calculate the SLYC variable (as describedin subsection: 3.1.3 Switchers).
All Ordinaries Accumulation index data was also obtained from the RBA forthe calculation of the ALTINV variable; (see equation 1).
The MBS prepayment data (which were obtained from Reuters via SIRCA
9
)consist of twelve pools of prepayment data ranging over the period June 1998 toSeptember .2003
Prepayment data and test results are presented throughout the EmpiricalEvaluation of the Australian Variable-Rate model section, in whole-number
percentage form (rather than decimal-fraction form).
4.1.1 Fixed and Variable Rate Pools Data Categorization Columns 1 and 2 oftable 2 show respectively: the percent age of fixed-rate loans at the beginningof the data period; and (2) the percent age of fixed-rate loans at the end of the data
period, for the twelve pools; (the numbers in the brackets of column 1 are the
respective beginning dates for the Reuters data of each pool, and the data for allpools end at March 2003.) Column 3 shows the percent age fall in fixed-rate loans,(column 1 minus column 2), over the data period. As shown in column 4, the four
pools corresponding to rows 2 5 were selected as the fixed-rate pools; they wereselected as fixed-rate pools on the basis of the percentage of fixed-rate loans atthe beginning of the data period. Clearly these are the only pools with over 50 % ofthe loans being fixed at the beginning of the data period. The remaining pools arethe variable-rate pools, on which the Australian variable-rate model is tested.
(1)
10
4.1.2 (total)CPR, CPR Full , and CPRPartial In practice the monthly conditional prepayment rate of an MBS pool, which is referred to as the single monthly
mortality, (SMM), is calculated as:
=SMM
).SMM(11=CPR)total(
where: total prepayments=opening monthly outstanding principal, less closingmonthly outstanding principal, less scheduled monthly repayments.
Then the annualized conditional prepayment rate, ((total)CPR), is given by:
12
(6)
CPRFull and CPRPartial components of the (total)CPR data are then calculated usingthe respective formulae:
,)balanceOpen
sprepaymentFull(11=CPRFull
12 (7)
total prepayments
beginning of monthoutstanding principal at
9. Securities Industry Research Centre of Asia-Pacific10. Using such a criterion (to categorize the pools into fixed and variable) may seem somewhat arbitrary
since other criteria (such as the fall in the percent age of fixed-rate loans, with the presumption being thatthe fall is due to refinancing) might also have been used as a basis for categorization of the data.
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and:
,)balanceOpen
FullTotal(11=CPRPartial
12 (8)
where open balance is the aggregated outstanding principal over all mortgages inthe pool at the beginning of each month.
Table 2
Change in Percentage of Fixed-Rate Loans in each Pool, andCategorization of Pools into Fixed and Variable
.
FIXED % at: FIXED %
(at Mar/03):
% CHANGE Pool Category
(Fixed or Variable)
1.03
(6/98)
5.86 4.83 Variable
55.41
(7/98)
33.28 22.13 Fixed
56.66
(2/99)
47.84 8.82 Fixed
70.25
(12/98)
40.94 29.31 Fixed
69.15
(6/99)
54.91 14.24 Fixed
25.13
(9/99)
8.97 16.16 Variable
29.7
(4/99)
14.36 15.34 Variable
1.73
(10/98)
2.04 0.31 Variable
14.53
(8/99)
10.83 3.7 Variable
33.14
(7/00)
31.47 1.67 Variable
9.75
(3/00)
11.88 2.13 Variable
12.21
(3/00)
8.35 3.86 Variable
4.1.3 Descriptive Statistics Table 3 provides the descriptive statistics for thevariable-rate data for each of the variable-rate prepayment model variables; thestatistics for the full and partial prepayments are included. Mean (total)CPR,CPRPartial , and CPR , are approximately 27.02, 6.36 and 22.08 percentrespectively.
Full
FullEvidently (from the mean of CPRPartial / CPR ), partial prepayments areusually approximately 33.91% of full prepayments.
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Table 3
Summary Statistics of the Variable-Rate Model Variables,(Variable-Rate Data)
Variable No.Obs Mean Std Dev Min Median Max
CPR 336 27.02426 7.486218 6.810000 27.07000 55.93000
CPRPartial 336 6.355 3.886 3.587 6.016 24.013
CPRFull 336 22.087 7.891 5.269 22.128 58.383
CPRPartial /CPRFull
NewMin
336 33.911 26.852 28.783 30.087 205.521
ParDFLT 336 1.281845 1.260241 0.000 1.250000 4.450000
ALTINV 336 0.045704 0.129503 0.000 0.000 0.761522
FXDdifl 336 0.571875 0.961011 0.0000 0.100000 3.650000
NLM 336 0.023810 0.152683 0.000 0.0000 1.000
r 336 0.107143 0.309756 0.000 0.000 1.000
SEASONAL 336 0.235119 0.424706 0.000 0.000 1.000
BURNOUT 336 3.681548 4.383458 0.000 3.000000 25.000
VOLy
336 1.0836 0.666865 0.000 1.019358 2.167686
SLYC 336 1.414494 1.428279 1.660000 1.540000 3.710000
Legend of variables:
AGEPOOL = the relative weighted average age of mortgages in the pool, (relative to the highestweightedaverageage ofmortgages, (WAS), over all pools in the data set);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;
NLM = new long minimum;
BURNOUT = burnout;SEASONAL = seasonality;
ALTINV = alternative investment;FXDdifl = 3 year fixed differential;
yVOL = ratevolatility;SLYC = change in the slope of the yield curve; and,
= a new minimum in the year fixedmortgagerate.NewMinr 3
Partial
4.1.4 Correlation Matrix Table 4 provides the pairwise correlation matrix for thevariables (total)CPR, CPRPartial , CPRFull , and all the independent variables (except
NLM and SEASONAL, which are not included for reasons of space, and since theyhave the least correlation with (total)CPR).
Clearly the correlation between (total)CPR and CPRFull , (0.849946) is veryhigh, while the correlation between (total)CPR and CPR , (0.324592) is low.
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29
3
Table 4
Pairwise Correlation Coefficients of Variable-Rate Model Variables(for Variab
Variable CPR CPR Partial FullCPR AGEPOOL SQR(AGEPL) ParDFLT ALTINV FXDdif
CPRPartial 0.324592
CPRFull
NewMinr
y
0.849946 0.104148
AGEPOOL 0.500139 0.048650 0.504714
SQR(AGEPL) 0.460684 0.070681 0.478471 0.975954
ParDFLT 0.464493 0.110296 0.408756 0.648268 0.630425
ALTINV 0.183919 0.003032 0.196845 0.321952 0.297609 0.153668
FXDdifl 0.290776 0.094995 0.243122 0.392838 0.416709 0.849569 0.056497
0.013309 0.046097 0.018453 0.069217 0.066748 0.126271 0.005631 0.17352
BURNOUT 0.162340 0.046084 0.137593 0.408773 0.411378 0.304806 0.105423 0.35501
VOL
0.194605 0.014544 0.198229 0.340172 0.210038 0.122502 0.309673 0.03862
SLYC 0.141895 0.030887 0.128697 0.107451 0.129847 0.250238 0.153481 0.02577
Legend of variables:
AGEPOOL = the relative weightedaverageage ofmortgages in the pool, (relative to the highest weightedaverageage ofset);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;ALTINV = alternative investment;FXDdifl = 3year fixed differential;
= a new minimum in the 3 year fixedmortgagerate;NewMinrBURNOUT = burnout;y
VOL = ratevolatility; SLYC = change in the slope of the yield curve.
(SEASONALandNLMwere thevariables leastcorrelated with CPR, (for thevariable-ratedata set), and weretherefore, for sp
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There are strong correlations between the time variables (AGEPOOL andSQR(AGEPL)) and the variables, ParDFLT and ALTINV. Another potentialsource of multi-collinearity is the strong correlation between ParDFLT andFXDdifl.11
4.2 (total)CPR as Dependent Variable
The testing of the Australian Variable-Rate model (as given by equation (5)) beginswith (total)CPR as dependent variable. Equation (5) is repeated here for thereaders convenience:
),SEASONAL,NLM,SLYC,VOL,BURNOUT
,FXDdifl,ALTINV,ParDFLT,SQR(AGEPL),AGEPOOLf(CPR
NewMiny
r
=
y 1
r
y 1
This is the unrestricted form of the model. The tests show how a moreparsimonious, but equally effective, form of the model can be found.
Because the Australian Variable-Rate model is essentially a linear regressionmodel, univariate tests of the respective variables reveal which variables have
potential significance in restricted forms of the model, and which variables can beimmediately eliminated from further consideration.
Univariate Tests:
The following variables were found to have significant effect on CPR at 5% level(or better) in the univariate tests:
AGEPOOL, SQR(AGEPL), ParDFLT1, ALTINV, FXDdifl, VOL , SLYC1,
where the 1 suffixes denote the variables are lagged by one month. The variablesSEASONAL, NewMin , BURNOUT and NLM (and variables derived from their lags)were either not significant or had incorrect signs (for the theory); these variableswere therefore eliminated from further tests. The results for the univariate tests areshown in table 5. The variables AGEPOOL, SQR(AGEPL), and ParDFLT1 havethe highest t-statistics.
Revised Unrestricted Model Tests:
These seven variables:
AGEPOOL, SQR(AGEPL), ParDFLT1, ALTINV, FXDdifl, VOL , SLYC1,
then formed a revised unrestricted model, which was tested on the data. Theresults for this test are shown in column 1 of table 6. Applying Liklihood-RatioTests and the Bayesian Information Criterion to this latter model (as shown in
11. That ParDFLT and FXDdifl are highly correlated is not very surprising; for example , if the SVR and3yrFXD rates are equal (observation shows they are invariably very close) and rates are monotonically
decreasing, then ParDFLT and FXDdifl are identical. However that is hypothetical since over the dataperiod 19982003, the interest rates were certainly not monotonically decreasing.
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columns and of table 6) resulted in the four variables: ALTINV, FXDdifl,VOL , and SLYC1, being found redundant, leaving a best restricted Australianvariable-rate model for (total)CPR with variables: AGEPOOL, SQR(AGEPL),ParDFLT1, and represented in equation form by:
2
(total
3
CPR
y 1
) 1*)(**= 321 ParDFLTbAGEPLSQRbAGEPOOLbconst + ++
Partial Full
y
2
(9)
Attempts at further interpretation of these results are postponed until the results ofthe corresponding tests with CPR and CPR as dependent variable have beencompleted.
4.3 CPR Partial as Dependent Variable
Partial prepayment theory has only been developed for the variables ParDFLT,ALTINV, and VOL (in subsubsection 3.1.4); hence the expected signs, of the
coefficients of all other independent variables for CPR , are indeterminate.Partial
Univariate Tests:
The univariate tests with CPRPartial as dependent variable (table not shown here)find that only the variable ParDFLT1 is significant at 5% level and to have correctsign. Though having no theory for partial prepayment, the FXDdifl variable issignificant at 10% level.
That FXDdifl variable is significant at 10% level might be taken as a weaksign that partial prepayment is not exclusive to variable-rate mortgage holders. A
positive relationship between partial prepayment and FXDdifl is, on first
impression, difficult to comprehend. If market mortgage rates are lower than fixed-rate borrowers rates (when FXDdifl is larger), the anxieties of these borrowersabout having to pay higher rates in the near future (when their fixed-term expiresand their rates are reset to the current market rate) should be less. However
possibly any rate changes are, to fixed-rate borrowers, a reminder of the volatilityof interest rates, and hence a warning that in a few years time they may be payinghigher rates; hence their reaction of partial prepayment.12
Revised Unrestricted Model Tests:
Column 1 with heading (Revised) unrestricted model in table 7 shows the resultsfor a regression with the two variables ParDFLT1 and FXDdifl as independentvariables. The sign of the coefficient of FXDdifl has changed to negative,
presumably due to multi-collinearity with the ParDFLT1 variable;(correlation(ParDFLT1, FXDdifl) =0.849569 from table 4).
Column (Best (restricted) model) of table 7, shows the results for aLiklihood-Ratio Test of the hypothesis that the variable FXDdifl is redundant to themodel of column 1. Clearly the Liklihood-Ratio Test shows that the hypothesis thatFXDdifl is redundant cannot be rejected.
r12. In Daniel (2006), in the tests on the combined data, two other fixed-rate variables, namely the two
burnout variables: NewMin and BURNOUT, were also found significant for partial prepayment, re-inforcing the evidence for the idea that partial prepayment is not exclusive to variable-rate mortgageholders.
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Table 5
Univariate Variable-Rate Model Variable Coefficients,t-Statistics,p-Values and AdjustedR s, (with (total)CPR as Dependent Variable)2
IndependentVariables:
Expected Signof Coefficient
Coefficient andt-statistic
p-value Adjusted2
R
AGEPOOL + 0.2031 0.0000 0.2479
(10.5553)
SQR(AGEPL)
1
NewMin
0.1908 0.0000 0.2099
(9.4859)
ParDFLT1 + 2.8136 0.0000 0.2236
(9.8286)
ALTINV -- 1.4761 0.0002 0.0375
(3.7497)
FXDdifl1 + 2.2425 0.0000 0.0824
(5.4935)
BURNOUT1 -- 0.2920 0.0017 0.0263
(3.1710)
VOLy
+ 2.0622 0.0007 0.0309
(3.4153)
SLYC1 0.8820 0.0020 0.0254
(3.1196)
SEASONAL1 + 2.5200 0.0056 0.0173
(2.7832)
r + 0.3257 0.8080 0.0028
(0.2432)
NLM1 + 5.4694 0.0410 0.0095
(2.0515)
Note: The coefficient values,t-statistics andp-values are for individual variables only, while the adjustedR
2s are for each of the complete regression models (which include a constant).
Because SQR(AGEPL) is not designed to occur as an independent variable without AGEPOOL, theexpected sign of the coefficient of SQR(AGEPL) is indeterminate.
Legend of variables:
AGEPOOL = the relative weighted average age of mortgages in the pool (relative to the highestweightedaverageage ofmortgages, (WAS) over all pools in the data set);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;ALTINV = alternative investment;FXDdifl = 3-year fixed differential;
NLM = new long minimum;= a new minimum in the 3-year fixedmortgagerate;NewMin
SEASONAL = seasonality;r
BURNOUT = burnout;y
VOL = ratevolatility; and,SLYC = change in the slope of the yield curve.
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Table 6
Variable-Rate Model Coefficients andt-Statistics, (with (total)CPRas Dependent Variable)
(Revised)1
Unrestricted-Model
Restricted-Model Best-(Restricted)Model
Column 1 Column 2 Column 3
Dependent Variable:
(total)CPR (total)CPR (total)CPR
No. of Obs 336 336 336
Independent Variables:
Constant 12.08488 12.4622 13.82566
(4.290840) (5.1328) (7.017232)
AGEPOOL 0.426116 0.4698 7.017232
(3.200806) (4.1741) (4.238887)
SQR(AGEPL) 0.282079 0.3306 0.242452
(2.307838) (3.0792) (2.789269)
ParDFLT1 2.143346 1.4337 1.540579
(2.429146) (3.9014) (4.317186)
ALTINV 0.348102 0.0423
(0.739369) (0.0432)
FXDdifl1 1.158727
(1.203356)
VOL
y
1
0.874345 0.99624(1.267886) (1.4236)
SLYC1 0.249342
(0.798438)
Log-Likelihood 1088.236 1090.492 1091.529
Likelihood Ratio Statistic 4.5126 6.586681
p-val 0.104734 0.159
BIC2 6.616095 6.5949 6.566447
adjusted R2
0.3039 0.2988 0.2987
Note:
1
Revised means after variables rejected by univariate regressions have been eliminated.2 BIC denotes the Bayesian Information Criterion.
Legend of variables:
AGEPOOL = the relative weighted average age of mortgages in the pool, (relative to the highestweightedaverageage ofmortgages, (WAS), over all pools in the data set);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;ALTINV = alternative investment;FXDdifl = 3 -year fixed differential;
NLM = new long minimum, SEASONALisseasonality;= a new minimum in the -year fixedmortgagerate;NewMin
BURNOUT = burnout;r 3
y
VOL = ratevolatility; and,SLYC = change in the slope of the yield curve.
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In this best (restricted) model ParDFLT1 is significant at level and of correctsign. Although the significance level of the best model overall is low (adjustedR = 0.011501 ), both this value and the BIC, (5.569520 ), confirm that the bestmodel is an improvement on the unrestricted model with adjusted R =
and BIC = .
5%
2
2 0.008644
5.586721The best restricted Australian variable-rate model for CPRPartial , in equationform, is given as:
1*= 1Partial ParDFLTbconstCPR + (10)
Neither of our other two partial prepayment variables (the ratio of average after-taxshare market returns to mortgage rates, and the volatility of mortgage rates), as
proposed in subsubsection 3.1.4, showed a significant impact on variable-ratemortgage holders partial prepayment.
Table 7
Variable-Rate Model Coefficients andt-Statistics, (with CPR asDependent Variable), (Variable-Rate Data).
Partial
(Revised)1 Un-restricted
Model
Best (restricted)
Model
Column 1 Column 2
Partial Partial
Dependent Variable:
CPR CPR
No. of Obs 336 336
Independent Variables:
Constant 5.860990 5.879847
(18.50913) (19.54441)
ParDFLT1 0.420482 0.370654
(1.368093) (2.213088)
FXDdifl 0.077862
(0.193468)
Log-Likelihood 929.8435 929.8623
Liklihood Ratio Statistic 0.037765
p-value
BIC2 5.586721 5.569520
adjusted R2
0.008644 0.011501
Note: 1 Revised means after variables rejected by univariate regressions have been eliminated.2 BIC denotes the Bayesian Information Criterion.
Legend of variables:
ParDFLT = partial prepayment by default; and,FXDdifl = 3 -year fixed differential.
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4.4 CPR Full as Dependent Variable
Univariate Tests:
Because the partial prepayment variables have no theory for full prepayment, theexpected sign of their coefficients is prima facie indeterminate when CPRFull is thedependent variable. For this reason ParDFLT1, ALTINV and VOL variables havean indeterminate expected sign in the univariate regression tests (table not shownhere). Taking into account the preceding note, the univariate tests, with CPRFull asdependent variable, select the same seven, significant at 5% level and correct signvariables, as the univariate tests with (total)CPR as dependent variable; namely :
y
AGEPOOL, SQR(AGEPL), ParDFLT1, ALTINV, FXDdifl, VOL , SLYC1.y 1
y 1
y 1
2
Full
1**= 21Full ParDFLTbAGEPOOLbconstCPR
That all three proposed partial prepayment variables, ParDFLT1, ALTINV andVOL , are significant for full prepayments is noteworthy.
The significance of SLYC1 presumably shows evidence of switching fromvariable-rate to fixed-rate. The positive sign for the coefficient of SLYC1 agreeswith the empirical findings of McConnell and Singh rather than Sanyal.
Revised Unrestricted Model Tests:
The seven variables:
AGEPOOL, SQR(AGEPL), ParDFLT1, ALTINV, FXDdifl, VOL , SLYC1,
as selected by the univariate tests, become the variables for the revised unrestricted
model, as shown in column 1 of table 8. Applying Liklihood-Ratio Tests (and theBayesian Information Criterion), again leads to the best restricted model (as shownin column 4 of table 8), with variables AGEPOOL and ParDFLT1.
On the basis of the adjusted R s, there is very little difference between theexplanatory power of the respective models in table 8. However the BayesianInformation Criterion does show a reduction in value (that is, an improvement inexplanatory power) as the models progress from unrestricted to best restrictedmodel.
The best restricted Australian variable-rate model for CPR , in equationform, is given as:
++ (11)
4.5 ParDFLT1 and ALTINV Variables Reviewed as Full Prepayment Variables
The univariate tests (with CPRFull as dependent variable, in subsection 4.4) showedthat ALTINV variable is highly significant, (p-value = ), and negativelyassociated with full prepayment.
0.0003
13 Both the univariate and the multivariate tests ofsubsection 4.4 showed that ParDFLT1 variable is also highly significant as a
13. In Daniel (2006) the empirical testing data, as well as being organised into variable-ratedata, was alsoorganised into fixed-rate and combined data sets. The highly significant and negative association
between full prepayment and ALTINV was evident in these data sets as well.
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Table 8
Variable-Rate Model Coefficients andt-Statistics, (with CPR Full asDependent Variable), (Variable-Rate Data)
(Revised)1
Un-restricted Model
RestrictedModel
RestrictedModel
Best (Restricted)Model
Column 1 Column 2 Column 3 Column 4
Dependent Variable:
CPRFull CPRFull CPRFull CPRFull
No. of Obs 336 336 336 336
Independent Variables:
Constant 9.635591 9.860966 9.925397 12.49372
(3.512665) (4.067029) (4.670235) (11.52214)
AGEPOOL 0.320087 0.335559 0.300318 0.174448
(2.413582) (2.759425) 3.215028 (6.616995)
SQR(AGEPL) 0.159164 0.167117 0.131680
(1.301417) (1.454507) (1.404415)
ParDFLT1 1.113415 0.922059 0.945721 0.940813
(1.259093) (2.337623) (2.456919) 2.440707
VOLy 1 0.336575 0.485149
(0.440806) (0.637013)
ALTINV 3.646505 2.948770
(1.166233) (0.956808)
FXDdifl 0.412077(0.433504)
SLYC1 0.439982(1.325612)
Log-Likelihood 1114.473 1116.391 1116.974 1117.969
L.R.S.2 3.837924 5.003108 6.993346
p-value 0.146759 0.286979 0.221136
BIC3 6.772268 6.749064 6.717907 6.706517
adjusted R2
0.267612 0.263688 0.265581 0.263437
Note: 1 Revised means after variables rejected by univariate regressions have been eliminated.2 L.R.S. denotes Liklihood Ratio Statistic.3 BIC denotes the Bayesian Information Criterion.
Legend of variables:
AGEPOOL = the relative weighted average age of mortgages in the pool, (relative to the highestweightedaverageage ofmortgages, (WAS), over all pools in the data set);
SQR(AGEPL) = AGEPOOLsquared;ParDFLT = partial prepayment by default;ALTINV = alternative investment;
yFXDdifl = 3-year fixed differential, VOL is ratevolatility;
NLM = new long minimum;= a new minimum in the 3-year fixedmortgagerate;NewMin
SEASONAL = seasonality, BURNOUTisburnout; and,r
SLYC = change in the slope of the yield curve.
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predictor for full prepayments, Both these variables were intended as predictors for partial prepayments only. Subsection 4.5 investigates explanations/hypothesesappropriate for these results.
4.5.1 ParDFLT1 as a Full Prepayment Variable The result ParDFLT1 havingstrong significance in relation to explaining the variation in CPRFull is inconsistentwith our (default partial prepayment) theory (as presented in section 3.1.4: PartialPrepayers, under the heading: Default Partial Prepayment), which proposedParDFLT as a partial prepayment variable only. As is evident from equation (2)(which defines ParDFLT), the ParDFLT variable is calculated from the variable-rate, not the fixed-rate. Full prepayments by variable-rate borrowers (which theempirical test results imply are occuring to a significant extent) can be explained asevidence of: variable-rate refinancing; or, switching from variable-rate tofixed-rate loans. Each of these possibilities is now considered.
)(i )(ii
)(i
)(ii
Variable-Rate Refinancing: One of the assumptions implicit in our theory prior to empirical testing has been that there is not enough difference betweenvariable-rate lender rates to make a variable-rate borrower fully prepay theirexisting loan in order to take out another lower-rate variable-rate loan. That is, wehave made the assumption that there is no refinancing from variable-rate loan tovariable-rate loan. There was some evidence in the literature of variable-raterefinancing which we chose to regard as unimportant. For example, Westpac bank(2002), states:
A bigger impact of falling interest-rates [than partial prepayments], isthe associated potential increase in full prepayments via refinancing.Changing interest-rate environments tend to create differences between
competing lenders rates and so tend to act as a trigger for borrowers toreassess their debt position. As loan products have become readilyavailable with lower entry and exit costs, borrowers have becomeincreasingly sophisticated and refinancing has become more and morecommonplace. Borrowers quickly take the opportunity to swap toimproved products or cheaper rates when differentials arise leading to anincrease in full prepayments.
ParDFLT is a measure of the differential rate between rates of a (variable-rate)borrower repaying by direct debit and the current market variable-rate. In reviewthe possibility that some variable-rate borrowers find refinancing to another more
competitive variable-rate loan clearly cannot be excluded.
Switching from Variable-Rate to Fixed-Rate Loans: As already mentioned insubsection 4.4, the significance of SLYC as a predictor for full prepaymentsimplies that switching from variable-rate loans to fixed-rate loans is probablyoccurring.
In conclusion (of this review of ParDFLT), we have to remain open aboutwhether these full prepayments of variable-rate loans are just refinancings to othervariable-rate loans, or just switches to fixed-rate loans, or some combination ofthese two modes of full prepayment.
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Whereas previously all variables have represented only one type of prepayerexclusively, now we have a variable which could possibly be representing threetypes of prepayers. ParDFLT possibly simultaneously represents:
(i) partial prepayers;
(ii) refinancers (variable-rate to variable-rate); and,(iii) switchers (variable-rate to fixed-rate).
Our aim, of creating a model which is as parsimonious as possible, is well servedby the multi-faceted nature of the variable ParDFLT. 14
4.5.2 ALTINV as a Full Prepayment Variable We have noted previously (at thebeginning of this section) that our proposed partial prepayment variable ALTINV,as a predictor for full prepayments, is univariately significant and has negative sign(and that this result also occurred in the univariate tests for both fixed-rate and
combined data, as shown in Daniel 2006).In preparation for the following discussion in relation to the ALTINV variablewe note:
(i) that (as previously mentioned in section: 3.1.4 Partial prepayers) theAustralian investor mortgage holder is in the same position as the USmortgage holder in being able to claim interest paid on their mortgage loansas a tax deduction, and would therefore be expected not to partially prepay;(that is, if they do prepay, they would be expected only to make full
prepayments).
(ii) that the variation in ALTINV is mainly determined by the variation in the (All
Ordinaries) share market index. The basis of the definition of the ALTINVvariable (see equation (1)) is the ratio of average after-tax share market returnover the previous year to the current mortgage variable rate. Because themortgage rate is relatively static and the average share-market return morevolatile, the variation in ALTINV is mainly determined by the variation in theshare-market index.
Possibly the ALTINV variable is showing a negative association between theaverage share-market returns and the willingness of professional investor mortgageholders to sell their properties; alternatively stated, the professional investormortgage holder recognizes that a time when the share market is booming is not a
good time to sell because then shares are a competing investment for potentialbuyers. If the preceding explanation is correct, the professional investor mortgageholder will also be watching property returns. The best time to sell an investment
property would be when property returns are high relative to share returns; theworst time to sell would be when share returns are high relative to property returns.
A robustness check on the ALTINV variable was made to check our formersurmise(s).
14. The preceding discussion might be regarded as an argument which implies that the name ParDFLT is
not entirely appropriate for this variable, which is now recognized as not just a partial prepayment (bydefault) variable .
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Using the listed property index GICS22, a variable: AOAR GICS22R (thedifference between average return on shares and average return on property) wasformed and tests, with CPRFull as dependent variable, carried out.
15 Althoughunivariately very significant and correct sign, in multiple tests the variable became
in significant and wrong sign; correlation testing showed these problems were dueto multi-collinearity with the time variables, AGEPOOL and SQR(AGEPL).While the latter results do not corroborate our former surmised
hypothesis/explanation for ALTINV in relation to full prepayments, we need totake into account the fact that ALTINV is a complex variable. ALTINV does not
just reflect share-market index variation; there are several factors in the definition/calculation of ALTINV which might account for the lack of corroboration by therobustness tests.
4.6 Summary of Variable-Rate Model Results
The most noteworthy result of the empirical tests is how well the restrictedprepayment model:
(12)),ParDFLT,SQR(AGEPL),AGEPOOLf(=CPR
,1*= 1Partial ParDFLTbconstCPR
performs relative to the unrestricted model (5).The empirical tests also found that the best variable-rate models for partial
and full prepayments of Australian MBSs are, respectively:
+
and
1**= 21Full ParDFLTbAGEPOOLbconstCPR ++
The prepayment data revealed that in Australia partial prepayment is on averageapproximately one third of full prepayments for variable-rate loans.
The ParDFLT variable (representing the amount by which the mortgageholders highest interest-rate level attained over the prior course of the loan,exceeds the current market mortgage variable rate) was found to have a highlysignificant effect on both partial prepayments and full prepayments for thevariable-rate mortgage holder.
The ALTINV variable (representing the ratio of average after-tax sharemarket returns to mortgage rates) was found to have a high negative associationwith variable-rate mortgage holder full prepayments in the univariate tests.(However ALTINVs significance was not great enough to register as a best modelvariable (in the multivariate tests)).
Other univariately significant and correct sign interest-rate variables, for full prepayments of variable-rate mortgage holders, were FXDdifl, VOL , and
SLYC1.
y 1
AOAR GICS22R15. The notation here for and is consistent with the notation for equation (1), (which definesALTINV).
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That FXDdifl is (weakly) significant for CPRPartial tends to imply that partialprepayment is not confined to variable-rate borrowers, and that partial prepaymentvaries proportionately with the differential between the fixed-rate mortgageholders interest rate and the current market mortgage fixed rate.
The empirical tests of our model(s) have revealed that our independentvariables explain full prepayment much more effectively than partial prepayment.This conclusion is reached through comparing the relatively low adjusted R s,
and for the unrestricted and restricted models respectively, withCPRPartial as dependent variable (as shown in table 7) with the correspondingadjusted R s for the unrestricted and restricted models with CPRFull as dependentvariable, and respectively (as shown in table 8). In Daniel(2006), using the combined data set, more independent variables are foundsignificant as predictors for partial prepayment (namely VOL , NewMin andBURNOUT); however the overall significance of the prepayment model(s) for
partial prepayment (0.057220 and for the unrestricted and restricted
models respectively) are only a small improvement on the variable-rate data testsresults as shown here.
2
0.008644 0.011501
0.267612
2
0.263437
yr
0.046583
5. Conclusion
The objective of this paper has been to investigate Australian mortgage prepaymentby developing and testing prepayment models for loans of Australian MBSs.
While the foundations for the new Australian prepayment model developed inthis paper are in U.S. MBS variable-rate loan prepayment models (notablyMcConnell and Singh 1991; and Sanyal 1994), the model is structured to account
for the differences in the Australian mortgage market (for example, theramifications of the different tax regimes for the mortgage holder as an investor).The new Australian prepayment model proved successful when tested on
(Reuters) Australian MBS data; parsimonious forms of the model were able tosuccessfully explain both total prepayment and the components of total
prepayment: full and partial prepayment.
(Date of receipt of final transcript: February 26, 2008.Accepted by David Gallagher & Garry Twite, Area Editors.)
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