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The mortgage choice problem

Rasmussen, Kourosh Marjani; Clausen, Jens

Publication date:

2008

Document Version

Publisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):

Kourosh Marjani Rasmussen

KongensLyngby2008

Informaticsand MathematicalModelling

Building321,DK-2800 Kongens Lyngby,Denmark

Phone +4545253351, Fax +4545882673

www.imm.dtu.dk

In the light of the recent years' steep rise in the universe of products

oered by the Danish mortgage banks an advisory model for individual

homebuyers is introduced in this thesis. Taking the existing mortgage

products, homebuyers risk preferences, tax rules and transaction costs

into consideration, the model helps mortgage advisors nd the optimal

choiceofmortgageloanforanindividualhomebuyer.Themodelprovides

thehomebuyer withbasisfor a decision which isbyfar more tailored to

theindividual's needsascompared to current practice.

The number of mortgage products available in the Danish market has

steeply increased during recent years. From a handful of products just

lovgivning) in July 2007, this number is expected to increase even

fur-therinthe future. Itis thereforeanevermorechallenging taskto advise

individual homebuyers on their choice of mortgage strategies. Mortgage

advisors should therefore have access to tools and analysis which in an

easily accessible way convey pros and cons of the decision of potential

homebuyers.

Todaymortgagebanksprovidehomebuyerswithinformationonrstyear

paymentsonly.Withtheintroductionofthenewcoveredbondlegislation

thebanksshouldinsteadprovidethe annualcostsinpercents.The

prob-lem withboth of these keygures is thatthey saynothing about future

risk and assuch they aregrossly misleading. Svend Jakobsen (2007)

ar-guesthatpoliticianshave notbeen suciently ambitiouson homebuyers

behalf. He suggests a consequence analysis overa set of scenarios where

bothincreasinganddecreasinginterestratesareconsidered.Inthisthesis

wegoasubstantialstepfurthertowardsndingthebestpossibledecision

under futureuncertainty for agiven homebuyer.

Thethesisdescribesamodelwhichsolvesthe homebuyersoptimal

mort-gagechoiceproblembasedonanumberofoptimalitycriteria.Themodel

involves modeling interest rate uncertainty, mortgage pricing,

Medudgangspunktidesenereårskraftigestigningirealkredittensprodukt

palette iDanmark introduceres idenne afhandling en rådgivningsmodel,

der på baggrund af bl.a. de eksisterende realkreditprodukter, låntagers

præferencer, beskatningogtransaktionsomkostningerskalhjælpe

rådgiv-eren til at optimere låntagers valg af realkreditlån. Modellen giver

lån-tageren etbeslutningsgrundlag, som i langt højeregrad end hidtil tager

højde for den enkelte låntagers behov.

Realkreditinstitutternes produktpalette er de seneste år vokset kraftigt.

Forbare10årsidenhavdelåntagernekunenhåndfuldforskellige

produk-ter atvælge imellem.I mellemtidenerantalletaf låneprodukter

re-nyeSDOlovgivning, dertrådteikraftjuli 2007,vilformentlig betydeen

yderligereudvidelseafproduktpaletten.Ienrådgivningssituationkandet

derfor både nu,og måske specielt fremover, være svært at nde dethelt

rigtigeprodukttilkunden.Idetlyserdetvigtigt,atrådgivernehar

kend-skab og adgang til værktøjer og analyser, der på en nem og overskuelig

mådekan anskueliggøre fordeleog ulemperved låntagerensvalg.

Førsteårsydelseerdetenestenøgletal, somdeesterealkreditinstitutter

oplyser i rådgivningssammenhænge i dag. I forbindelse med SDO

lov-givningen er der indført skærpede krav om lånerådgivning i form af en

revisionafbekendtgørelsenomgodskikfornansielle virksomheder.Det

pålægger realkreditinstitutter at oplysede årlige omkostningeri procent

(ÅOP). Problemet med begge disse nøgletal er, at der ikke bliver taget

højde for fremtidig risiko. Svend Jakobsen (2007) argumenterer for, at

lovgiverne ikke har været tilstrækkeligt ambitiøse på låntagernes vegne.

I artiklen foreslår Svend Jakobsen, at der skal tages udgangspunkt i en

konsekvensberegning. Vi går her et stort skridt videre i retning af at

stille detbedst mulige beslutningsgrundlag, under fremtidig usikkerhed,

for låntageren.

Denne afhandlingbeskriveren model, derudfra en række kriterierløser

inddragerallerelevante realkreditprodukter ogdissesmarkedspriser,

lån-tagers præferencer for risiko og gevinster, begrænsning af tab ved

om-lægninger samt omkostninger ved optagelse og omlægninger og

This thesis was prepared at IMM, DTU inpartial fulllment of the

re-quirements foracquiring the Ph.D. degree inengineering.

Thethesisdealswithdierentaspectsofmathematicalmodelingfor

nd-ingtheoptimalchoiceofmortgageforanindividualhomebuyer.Themain

focusis on developing and testing a modeling framework to capture the

reallife complexityof themortgagechoiceproblem, but also specialized

interest rate modeling,appropriate choice of riskmeasure and the

inter-pretation ofcertain mortgageproductsasGien goods areconsidered.

The thesisconsistsof a summaryreportand a collectionof ve research

papers written during the period 20042007. The rstthree of these

Lyngby,November 2007

thesis

[A] Kourosh Marjani Rasmussen and Jens Clausen (2007), Mortgage

LoanPortfolioOptimizationUsingMultiStageStochastic

Program-ming. Journal of economic dynamicsand control, 31,pp742766.

[B] Rolf Poulsen and Kourosh Marjani Rasmussen (2007), Financial

Gien Goods: Examples and Counterexamples. European Journal

of Operational Research, inpress.

[C] Kourosh Marjani Rasmussen and Stavros A. Zenios (2007), Well

ARMedandFiRM:Diversicationofmortgageloansfor

homeown-ers. The Journal of Risk,10, pp6784.

Op-http://www2.imm.dtu.dk/kmr/

[E] Kourosh MarjaniRasmussen andRolfPoulsen (2007), Yield curve

eventtreeconstructionformultistagestochasticprogramming

I thank mysupervisorProfessor JensClausen who coauthored therst

paperinthis thesis andhelped shape thedirection for further work.His

approval and support for my close cooperation with the industry has

resultedinaworkwhichhasalreadybeen readbymanyandthathas

be-comethetheoreticalfoundationforanadvisorysystemwhichhasrecently

been putin useinNykreditrealkredit A/S.

Themaininnovationinthisthesisisthecombinationoftheoreticalresults

from mathematical nance with the mathematical programming

frame-work within optimization in nance all put into a reallife application

area.Myothertwoco-authoursProfessorRolfPoulsenfromUniversityof

tre, University of Pennsylvania, Philadelphia have motivated this work

andcontributedgreatly tothe qualityoftheresults.RolfPoulsen's work

inmathematicalnanceiswidelypublishedandacknowledgedwithinthe

mathematical nance community. Stavros A. Zenios is an international

front gureinthe eldof optimizationinnance. Ithank themboth for

their collaboration, forexcellent advice andmany fruitfuldiscussionswe

havehadduringtheworkonthepapers.AnextrathanksgoestoStavros

for hishospitalityduring myseveralvisitsinCyprus in2005 and 2006.I

believe close collaboration in between themathematical nanceand the

optimization in nance communities will result in solving several

inter-esting realistic problems yet unsolved by each of the groups alone. The

work presented hereisthe result ofsuch a collaboration.

Specialthanks to the Nykredit team aroundthe project Optimus  the

titleforthebusinesscaseandthesoftwaredevelopedincooperationwith

Nykreditwithoutwhomthisworkwouldnothavebeennearlyasapplied

and realistic as it has become. We have discussed the modeling aspects

suchasappropriatechoicesofinterestratemodels,pricingalgorithmsand

optimizationcriteriaonadailybasis.Likewisethetestsetupandthe

anal-ysis of results have been debated intensively within the group. Kenneth

Styrbæk,SørenLolle,ThomasKyhl, SteenH.Bertelsen,TheisIngerslev,

all been involved indierent parts ofthis project.In particular Kenneth

Styrbæk and Søren Lolle have developed a graphical user interface and

specialized code tointeract witha mortgagepricing module(ScanRate's

RIO). Thishaseased the testingprocess immensely.

Also thanks to the people from ScanRate A/S in particular Professor

Svend Jakobsenand Johnni Andersen for providing adviceon theuseof

their mortgage pricing system (RIO) and its integration into our model

framework.

The current work has resulted in some spin o master thesis projects

which I have partially supervised alongside my work on this thesis. I

would like to thank my students for showing interest in this work and I

hope theywill take the research work up wherethis thesis leavesit o.

Finally ahuge thanksgoesto mywife Anne MetteRasmussen.Without

her love and support I would have never nalized this work. I dedicate

thisthesistomywifeandtomytwolovingchildrenAnnaClaraandCarl

Summary i

Resumé v

Preface ix

I Summary report 1

1 Introduction 3

1.1 Background and motivation . . . 4

1.2 Problemstatement . . . 6

2 The Danish mortgage bond market 9

2.1 TheDanish mortgage nancelegislation . . . 10

2.2 Mortgage products . . . 13

2.3 Thedeliveryoption . . . 15

2.4 Renancing andprepayment . . . 16

3 Our approach versus the traditional mortgage advice 19

3.1 Traditional mortgage advice . . . 20

problem 25

4.1 Interestratemodeling . . . 27

4.2 Interestratescenario generation. . . 31

4.3 Mortgage bond pricing . . . 37

4.4 Stochasticprogramming . . . 39

5 Summaryof the papers 47

5.1 Interestratemodeling . . . 48

5.2 Scenario generation . . . 49

5.3 Optimization framework . . . 51

5.4 FinancialGien goods . . . 57

6 Research contributions 61

6.3 A termstructure scenario generationmodel . . . 63

7 New results on model robustness 65

7.1 Comparisonof two scenariogeneration approaches . . . . 67

8 Final remarks 73

8.1 Conclusions andEmpirical ndings . . . 73

8.2 Futurework . . . 75

Financial glossary 76

II Papers 87

A Mortgage Loan PortfolioOptimization UsingMultiStage

Stochastic Programming 89

A.1 Introduction . . . 90

A.4 Modelingrisk . . . 106

A.5 Scenario reduction . . . 113

A.6 LP relaxation . . . 116

A.7 Numericalresults . . . 118

A.8 Conclusions . . . 129

B Financial Gien Goods:

Examples and Counterexamples 137

B.1 Introduction . . . 138

B.2 The MarkowitzModel . . . 140

B.3 The MertonModel . . . 142

B.4 A Mortgage ChoiceModel . . . 144

B.5 Conclusion. . . 148

C.1 Introduction . . . 153

C.2 Are there diversication benets from portfolios of

mort-gageloans? . . . 156

C.3 Someexplanations onDanishmortgages . . . 159

C.4 A diversication model . . . 168

C.5 Taking a longterm perspective . . . 175

C.6 Two interesting observations . . . 179

C.7 Conclusions . . . 183

D Optimal Mortgage Loan Diversication 187

D.1 Introduction . . . 189

D.2 Single mortgagestrategies . . . 195

D.3 Themortgage choice model . . . 202

D.6 Conclusion. . . 224

E Yieldcurveeventtreeconstructionformultistage

stochas-tic programmingmodels 227

E.1 Introduction . . . 229

E.2 Factor analysisof yieldcurves . . . 233

E.3 A vector autoregressive modelofinterest rates. . . 237

E.4 Scenario generation andevent tree construction . . . 241

E.5 An approximative solutionapproach . . . 249

E.6 VasicekversusVAR1 for event tree construction . . . 251

E.7 Conclusions . . . 254

Introduction

Thisthesisconsistsofasummaryreport, chapters 18,andacollection

of ve research papers in the appendices. The purpose of the summary

report canbe summarizedasfollows:

1. Chapter 1motivatesthe problemand givesan overallproblem

de-scription.

2. Chapter2 describesthe Danishmortgage bond market.

4. Chapter4 introduces themethods usedthroughout this thesis.

5. Chapter5summarizes thepapersandclearly statestheir

interrela-tion.

6. Chapter6pointsoutthenovelcontributionsachievedinthisthesis.

7. Chapter7documentsadditional testsand resultsonmodel

robust-ness which have notbeen fullyaddressedinthepapers.

8. Chapter8 draws overall conclusion andshows directionsfor future

work.

1.1 Background and motivation

Homebuyersinmostcountriestakeupmortgagesfortheirhousenancing

needs. In Denmark they may loan up to 80% of the value of the house.

Thisthesisdealswithwhichloanorwhichcombinationofloansisoptimal

for anindividual homebuyer.

Until 1996 callable xed rate mortgages (FRMs) were the only type of

mortgagesavailableintheDanishmarket.Somortgageadvisorswereonly

maininnovationshaveincludedintroductionofadjustableratemortgages

(ARMs) in 1996, then interestonly (IO) versions of both FRMs and

ARMs were introduced in 2003. Finally in 2005 the capped rate

mort-gages (CRMs) entered the market. 1

The number of mortgage products

was added up to no less than 60 according to Skovgaard (2005). With

theintroductionofthe newcoveredbondlegislation(SDO lovgivning)in

July 2007,thisnumberisexpectedto increaseeven furtherinthefuture.

It is therefore an ever more challenging taskto advise individual

home-buyers on their choice of mortgage strategies. Mortgage advisors should

therefore have access to tools and analysis which inan easily accessible

wayconvey prosand consofthehomebuyers decision.

ThetotalamountofoutstandingmortgageloansinDenmarkin2006was

250 billionEURO, correspondingto120%oftheGDP.Thegreatvolume

of theoutstanding debtmeans thatappropriatechoices ofmortgages are

not onlyofinterest forthe individualhousehold butthey alsohavegreat

macroeconomicalimportance.Riskychoicesofmortgages,combinedwith

a house price fall and increased unemployment would result inmass

de-1

OneoftheDanishmortgagebanks(Totalkredit)launchedtherstCRMsin

Den-mark(BoligXlån)alreadyin2000.TheCRMsdidnotgainmuchpopularity,however,

until anothermortgagebank(RealkreditDanmark)introduced theirrstgeneration

faults on the individual homeowner side which in turn may result in a

further devaluation of the housing market and may at the worst case

bringmajor nancial institutions to bankruptcy,which againmayresult

ineconomical depression.Therecentsubprimeloanscrisisisanexample

of howirresponsibleand speculative choice of mortgages for even a

par-tial segment of the US market has threatened nancial and economical

stability inseveral partsof the world.

Theliberalizationofthemortgagemarketsshould thereforebe

accompa-nied by sucient individual advice for homebuyers in order to suit the

individual'sneedsandpreferenceswhileatthesametimereducingdefault

risk.Theadvicegiventodayisbyfarnotsucientanditiscertainlynot

tailored to theneedsof theindividuals.

1.2 Problem statement

The central question to be answered inthis thesis can be formulated as

follows:

The problem statement above needs more clarication. What is the

op-timality criteria for a given homebuyer? What isan appropriate horizon

for optimization?Howarefutureinterestrateandmortgage price

uncer-tainties captured?

Weneedtoanswerthesequestionsbeforeanyattemptsforjustifyingwhy

weconsideramortgagestrategy optimal.Webelievethatthesequestions

do not have acompletely objective answer. There isno standard

frame-work for modeling interest rate and mortgage price uncertainty. Most

homebuyers have noclearideaoffor howlongtheyaregoingtokeep the

property and the notion of optimality for a mortgage cashow given its

price isunderstooddierently bydierent groupsof homebuyers.

Never-theless mortgage bank advisors should provide homebuyers with advice

on their mortgagechoice.

In thisthesis we dene what we understand by appropriateassumptions

on these essentiallysubjective questions. Whenthe assumptionsare set,

we will move on to introducing a modeling framework in which several

optimality criteria, several horizons, as well several models for interest

rateandmortgagepriceuncertaintymayeasilybeimplementedandtheir

The Danish mortgage bond

market

The Danish mortgage bond market is Europe's second largest covered

bond marketafter the German market. Real property nancing in

Den-mark ismainly basedon mortgage loans raisedthroughmortgage banks

whose lending is funded exclusively through the issuance of mortgage

bondscovered bonds.

complex natureand therisksinvolved inthese productsshould convince

the reader that the research done in this thesis on advising homebuyers

on aproperchoice ofmortgage loaniswell justied.

2.1 The Danish mortgage nance legislation

Themain principles behindtheDanishmortgage nance systemare:

•

All loansaregranted againstmortgages onreal property.

•

Thebalanceprinciplewhichimpliesthatalllendingisfundedthrough theissuanceofbondsandthattherepaymentsontheloansandthe

paymentstothebondholdersmustalwaysbebalanced.Thisbalance

between funding and lending eliminates the interest rate, liquidity

andcurrency risks relatingto themortgage bankbalance sheets.

•

Mortgage bankshave no inuence onlending rates which are com-pletely marketdependent.

The balance principle, the backbone of Danish mortgage nance, has

basicallynot beenchanged since1850. Iteliminatesthe mortgagebank's

borrowerdefault,however,thevalueofthepropertytypicallycoversmost

of the charges. Even though borrowers default from time to time, the

mortgage bondholders have not faced a single case of insolvency on the

mortgagebanksideduringthe200yearlonghistoryofDanishmortgage

bonds.

TheDanishmarketischaracterizedbyahighdegreeofconcentrationat

present,fourmajorissuersaccountfor95%ofthebonddebtoutstanding.

The liquidity of the Danishmortgage bonds is further supported by the

factthatallmortgagebanksissuebondswithalmostidentical

character-istics resulting in a unitylike market. In practice, bonds from dierent

issuers arethereforetraded on equalterms.

TheliquidityoftheDanishmortgagebonds,thebalanceprincipleandthe

longhistoryofthe Danishmortgagebanks(withnoinsolvencycases)has

resultedinanextremelyecientmarketitwouldnotbeanexaggeration

toconsideritworld'smostecientmarket.Thismeansthattheinvestors

enjoya highdegree ofsecurity ontheir investments onDanish mortgage

bonds and that the borrowers experience extremely attractive rates on

their home nancing. Danish homebuyers, due to the balance principle,

eectively issue mortgage bonds via the mortgage banks.The mortgage

riskontheborrowersideaswellastheadministrationcosts.Thismargin

istheloweston any mortgagemarketinthe world.

It isnoteworthythaton the rstof July 2007 anewamendment, known

astheDanishcoveredbondlegislation,wasaddedtotheDanishmortgage

nancelegislation.Amongotherthingsthenewlegislation allows

separa-tion oflending and fundingwithin certain limits.The new lawopensup

fordesigningnewmortgageproductswhicharenotsimplypassthroughs.

Thismeansthatmortgagebanksshouldmakeadecisionastowhetheror

not they arewilling to assumesome degree ofthe interest rate, liquidity

and currency risks when issuing bonds to investors and lending money

to homebuyers. So far none of the Danishmortgage banks have utilized

this feature ofthenewlaw.But shouldthey considerto make useof the

newpossibilities,theworkdoneinthisthesisisofeven moreimportance

not only for advising homebuyers on their mortgage choice but also for

optimal productdesign andriskmanagement.

Themortgageproductsintroduced inthischapterandanalyzed

2.2 Mortgage products

Fixed rate mortgage loans (FRMs) which arefunded inlongterm xed

ratecallableannuitybondshavetraditionallydominatedtheDanish

mort-gage bond market. However, the introduction of xed rate noncallable

bulletbondsandrelatedadjustableratemortgages(ARMs)inthesecond

halfofthe1990'sand,mostrecentlyin2004,thesuccessfulintroductionof

capped longterm oating rate Cibor 1

linked bondsand relatedoating

rate mortgage loans withinterest rate caps (CRMs)have diversied the

Danish mortgage bond market, providing investors as well as borrowers

withfar more investment opportunities. Inthe following we give a short

outline ofthemain featuresofthese typesof mortgages.

Fixed rate mortgages

Fixed rate mortgages (FRMs)are funded by xed rate callable annuity

bonds with a strike price at par. That means that the borrower should

never pay more than the face value of the outstanding debt in case of

prepaymentofthemortgage.FRMscomebothwithandwithoutinterest

only options. Interestonly periods have a maximum period of 10 years.

Maturitiesavailablefor FRMs are10,20 or 30years.

Fixed ratecallable bond series have an openingperiod oftypically three

years.Thatmeansthatwhenabondserieiscreatedbythemortgagebank

ithasamaturitywhichis3yearslongerthan thematuritiesavailablefor

FRMs.Theserieremainsopenforlending toborrowersupto threeyears

unless thebond price goes above par due to interest rate decreases or if

theprice falls way below par due to interest rate increases.This process

ensures a high volume of the outstanding debt in the individual bond

series andtherebyreduce liquidityrisk.

Adjustable rate mortgages

An adjustable rate mortgage (ARM) is funded by issuing one or more

underlying bulletbonds. A bullet bond is a noncallable coupon paying

bond with a single repayment of principal on the maturity date. The

Danish bullet bonds have maturities of 1 to 11 years, and the Danish

borrower may choose between ARMswith coupon xing periods of 1 to

10 years (ARM1to ARM10).

Since bullet bondsare per construction interestonly the Danish ARMs

can be oered with an interestonly option without incurring anyextra

bullet bonds. The annuity ARM does not incur any extra costs to the

borrowereither.

Capped rate mortgages

Cappedratemortgages(CRMs)arefundedbyoatingrateannuitybonds

(oaters). The coupons are typically based on sixmonth Cibor plus a

xedspreadandtheyaresubjecttosemiannualcouponxing.CRMsare

oered withor without interestonly options. The interestonly periods

havea maximumperiodof 10yearsandthey areslightly moreexpensive

compared to their annuitycounterparts.

CRMs have maturities of5,10, 20 or30 years, andthe underlying bond

series have openingperiodsof typicallythree years,like theFRMs.

2.3 The delivery option

A distinct and very important feature of the Danish mortgage nance

system is the delivery option also called the buyback option. It means

bank.Thebuybackoptionappliestoallmortgagebondswhethercallable

or noncallable. The buyback option constitutes a signicant dierence

betweentheUSandtheDanishmortgagenancesystem.TheUSsystem

onlyallows mortgageloanprepaymentat par (100).The buybackoption

is an advantage to borrowers in situations with rising interest rates. As

bondpricesfall,themarketvalueofborrowers'debtisreducedalongwith

borrowers'exposuretoincreasingrates.Thisisparticularly usefulincase

ofdecreasing propertypricesormoving toanother propertybeingforced

to renance at the higher interestrate level. For borrowers with30year

xedrate loans,such eect maybesignicant.

2.4 Renancing and prepayment

Renancing refers to the process of changing one or more underlying

bonds behind a mortgage loan with some other bonds. For ARMs

re-nancingusually meansadjustingofthemortgageratetothemarketrate

of the underlying bond. An ARM with yearly adjustments (ARM1) is

renanced oncea year.Inpractice itmeansthattheoutstandingdebtof

thematuring bulletbond ispaid byissuing anew oneyear bullet bond.

bor-Thisincurs some renancing fees.

Renancing FRM'sand CRM's refers to the process of paying back the

outstandingdebtbeforehorizon(prepayment)byissuingnewbonds.The

borrower uses either the call option or the buyback option to prepay a

loan. Prepayment usually occurs as a consequence of the callability of

FRMsatpar.Inthecaseofdecreasinginterestratestheborrowerprepays

themortgagewithahighercouponbyissuinganewmortgagewithalower

coupon.ThenewmortgagemaybeanFRM,anARMoraCRM.Another

reason for prepayment is reduction of outstanding debt. When interest

ratesincreasethepricesofFRMsandCRMsfall,sotheunderlyingbonds

may be bought back at a cheap price. This transaction is funded by

either an ARM, FRM or CRM of a higher price and probably higher

rate. The result is an outstanding debt reduction which approximately

corresponds to thedierence of theoldand the newbonds. Prepayment

also occurs simplydue to selling the property. Prepayment incurs extra

fees as compared to renancing of ARMs to dierent xing periods at

Our approach versus the

traditional mortgage advice

A valid question at this point would be what is the value added by

introducing anewmortgageadvisingsystem? Thischapteranswersthis

3.1 Traditional mortgage advice

Today mortgage banks are only required to provide homebuyers with

information on rst year payments. With the introduction of the new

covered bond legislation the banks should also provide the annual costs

in percent. The problem with both of these key gures is that they say

nothingabout futureriskandassuchtheyaregrosslymisleading. Svend

Jakobsen (2007) argues that politicians have not been suciently

ambi-tious on homebuyers behalf. He suggests a consequence analysis over a

set of scenarios where both increasing and decreasing interest rates are

considered. Indeed some mortgage banks have taken up theidea and as

an extraadvisoryservicetheyprovide payment calculationsunder afew

interest rate scenarios for a given choice of mortgage loan. Even though

this approach provides more information to homebuyers than rst year

payments and annualcostsinpercent, ithasthefollowing aws:

1. Theinterestratescenariosaregeneratedonanadhocbasis.Market

information is not used to capture the overall tendencies in the

dynamics ofthe term structure of interest rates.

italongtheway.Thereforeadecisiononthechoiceofmortgageloan

hereandnowshould considerfuturerebalancing possibilitiesunder

dierent market conditions.

3. The analysis is done for one mortgage loan at a time. Even if one

allows for a combination of loans and perhaps some ad hoc

rebal-ancings along the way, the analysis will not reveal what the best

strategy is according to some criteria for example lowest average

payments,least variability,leastmaximumpayments, etc.

3.2 Our approach

Inthisthesiswegoalargestepfurtherfromtheexistingmethodstowards

nding the best possible decision under future uncertainty for a given

homebuyer.

Figure(3.1) givesan exampleof whatwe understandbyanoptimalloan

strategy.

For simplicity of this illustrative example we have made the following

with yearly adjustments (ARM1) and a xed rate mortgage with

4%coupon payments(FRM 4%).

2. Wewishtocomparetheholdingperiodcostsoveraveyearperiod.

3. We consideronly issueand holdstrategies, i.e. norebalancings are

allowed.

4. Wewishtondthecombinationofloanswhichresultsinthe

small-est average holding period cost for thehighest 10% of the holding

period costscenarios.

Comparing thetwofrequency distributions forARM1 andFRM4%itis

obvious thatthe ARM1 distribution hasa smaller right tail. Now given

that the homebuyer ofour example wishto minimize the average of the

10%righttail,thequestioniswhetheracombinationofthetwoloanswill

result in a smaller right tail than that of ARM1. In the existing

conse-quence analysis systems one maysimulate several combinations of these

two loans and compare the right tails obtained. We have tried this once

witha 5050combination of the twoloans,which clearly doesnot result

in a smaller right tail than that of the ARM1 alone. We could continue

thesecalculationsfor severalothercombinationsuntilsome thresholdfor

nding the optimal combination. Applying our optimization model we

can withinafewsecondsndtheoptimalcombination which isan 8119

combination ofARM1 and FRM4%.

The overall theme of this thesis is to make such an example as realistic

ascomputationalresourcesandtheexistinguncertaintyembeddedinthe

nature ofthis problemallow us.Ourmodelframework allowsfor several

mortgage products, futurerebalancing possibilities underuncertainty as

well asseveraldierentoptimizationcriteria.Inthe nextchapterwetake

astepbackandintrodue themethodsneededtoachieve theobjectivesof

Figure3.1: Comparisonofsingleloanstrategieswithloanportfolios. Top:An

arbi-trary combination of thetwo loansis comparedwith thetwo singleloan strategies.

Down:Theoptimalcombinationofthetwoloansiscomparedwiththetwosingleloan

Fundamental elements and

methods for the mortgage

choice problem

Thisworkcanbecharacterizedasanintegrationofdierentmodelsintoa

systemwhichprovidesmortgagorswithindividualadvice.Theintegration

is illustrated inFigure 4.1.

26 problem

Interest rate scenario generation

Stochastic programming

Mortgage bond pricing

Mortgage loan strategies

Interest rate modeling

Figure4.1: Themodeling paradigmsand theirinteractions inthis thesis.

and7aswellasinthepapersintheappendices.Wewill,however,briey

gothrough thebasic terminologyand theintuitionbehindtheparticular

4.1 Interest rate modeling

The predominant risk aecting the cashow payments of a mortgagor

is therisk associated withchanges in the general level of interest rates.

When the interest rates increase the cashow payments of short term

nancing increase as well. On the other hand the value of outstanding

debt for long term xed rate nancing decreases 1

. Interest rate models

are mathematical descriptions of interest rate dynamics. They describe

possiblemovements ofthe entire term structureof interestrates.

4.1.1 Term structure of interest rates

The term structure of interest rates, or the yield curve, is the set of

interest rates fordierentinvestment periods or maturities.Yield curves

can displaya wide variety ofshape asseeninFigure 4.2.Mostly,ayield

curve slopes upwards, with longer term rates being higher. Such curves

arecallednormal.But severalexamplesof historical inverse yieldcurves

havebeenobservedtoo.Onesuchexample isshowninFigure4.2for the

Danishyield curve on the 30/08/2000.

1

28 problem

Figure4.2: Danishyieldcurvesfrom 4dierent historical timepoints.

Principal component analysis of interest rates in several xed income

markets have shown that changes in level, slope and curvature of the

yield curves can explain almost all variation. Looking at Figure 4.2 one

can seethat parallel shiftsofthe yieldcurvesarenot theonly wayyield

curvesmoveintheDanishmarketeither.Formoredetailsonthissubject

seepaperE inthe appendices.

4.1.2 Examples of interest rate models

Figure 4.3:Historical dataon Danishyieldcurvesfor theperiod 1995 to

2006.

•

Extended Vasicek(timevarying mean,Hull &White (1993)),

dr

_t

= α(θ(t) − r

_t

)dt + σdz

_t

;

•

Extended CIR (timevarying mean, Cox, Ingersoll & Ross (1985) and Jamshidian (1995)),

30 problem

•

CKLS (Chan,Karolyi, Longsta&Sanders (1992)),

dr

_t

= α(µ − r

_t

)dt + σr

γ

_t

dz

_t

.

All models have a mean reversion level  the timevarying

θ

(t)

in the

extendedVasicekandextendedCIRmodelsandtheconstant

µ

inCKLS.

The parameter

α

decides the height of the interest rate jumps at each

step. The models also have variance

σ

and a stochastic Wiener process

z

_t

. The extended CIR model has a factor

r

1

2

t

in its volatility which can ensure that rates do not become negative. The volatility function in the

CKLS modelis slightly moreexible.

A large number of scientic papers have been written on interest rate

models.Themodelsoer numerousvariationsofthesimplemodels

men-tioned above and they add each some specialfeatures to them. As some

ofthemostimportant enhancementsto thesemodels onecouldmention:

1. Adding the number of state variables (nfactor models) to better

capture the dynamics of the whole yield curve of the underlying

market.

only positive rateswithhighvolatilityintimesof verylowinterest

rates.

We will not go any further on exploring these special features. Two

ex-cellent books oninterestrate modeling areBrigo&Mercurio(2006)and

James& Webber(2000).

4.2 Interest rate scenario generation

The mortgage choice problem does not have closedformed solutions in

continuous time and state. This is due to the fact that we have several

instrumentswithcomplexcashowsinadynamicsettingandthatmarket

frictions suchasvariableand xed transaction costsand tax regulations

play an important role on the optimal portfolio choice. The uncertainty

spaceneedstobediscretizedbothintime andstate.Werefer tothe

pro-cess of generating discrete yield curve scenarios asinterest rate scenario

generation.

In the following we introduce some scenario generation methods for use

in stochastic programmingapplications. These methods aregeneral and

32 problem

edge, no comparative studieson the suitability of these methods for use

in stochastic programming have been published up to the time of this

writing.

4.2.1 Bootstrapping

Bootstrapping isthe simplest approach for generating scenarios. It does

not involve using any underlying interest ratemodel. Instead it uses the

available historical data directly as future scenarios. For example yield

curves observed the last 120 months may be used to indicate possible

yield curve scenarios in a month, a year or in ve years. The strength

of this approach, besides being simple, is that it preserves theobserved

historical correlation. However, thereareserious shortcomings:

1. It can only be used for oneperiod models, since there isno

mech-anismto capture the conditional momentsinbetween theperiods.

2. The information about the current level of the stochasticvariable

isignored.

3. Thevolatilityofthe historicaldataisonly correctlycaptured ifwe

curves may only be used for generating scenarios over the next

month.

4. The method neversuggests ascenario not observed historically.

5. Itdoesnotnecessarilygenerateconsistentyieldcurvescenarioswith

for example noexistence ofarbitrage.

4.2.2 Sampling

Themostcommonmethodforgeneratingscenariosinnanceissampling

from an underlying stochastic process such as an interest rate model.

Sampling does not suer from the shortcomings of the bootstrapping

method,since theunderlying stochastic process maybe quiteadvanced.

What is more,sampling is almost as easy as bootstrapping inthat it is

essentiallyaquestionofgeneratingrandomnumbersfromthedistribution

of anunderlying random variable.

Themainproblemwithsamplingisthecurseofdimensionality.Itis

com-monthatover1000 scenariosaregenerated tomatchthestatistical

prop-erties of a continuous onefactor stochastic process. The number grows

34 problem

hasa similareect on the numberofscenarios.

Dueto thiscurseofdimensionalityanumberofvariancereduction

meth-ods have been developed. Variance reduction is a procedure used to

in-crease the precision of the estimates that can be obtained for a given

numberof iterations. Every randomly generated variable fromthe

simu-lationisassociatedwitha variance whichlimitstheprecision ofthe

sim-ulationresults. Variance reductionmethods arethenusedto reduce this

variance. The main methods are: Common random numbers, antithetic

variates, controlvariates, importance samplingand stratied sampling.

4.2.3 Moment matching

Høyland &Wallace(2001) suggesta simple moment matchingapproach

togeneratescenariosforstochasticprograms.Unlikeinsampling,moment

matchingusesoptimization to generate scenarioswhich match some

sta-tistical properties of an underlying stochastic process. Such properties

mayinclude mean,covariance,skewness,kurtosis, percentiles, higher co

momentsand soon.

Given asetof statisticalproperties

s

l

,their estimatedvalues

V AL

sl

and

problem isformulated asthe following optimization problem:

min

sL

X

sl

=s

1

w

_sl

(f

_sl

(x

_n

, p

_n

) − V AL

_sl

)

2

wrt.

X

n

p

_n

= 1

p

_n

≥ 0

for all

n

∈ 1, · · · , N.

Here,

x

n

isthevalueofthestochasticvariablefoundbytheoptimization

modelateveryscenario

n

,thefunction

f

sl

takesallsuchvalueswiththeir

probability

p

n

and returns the valuefor the statistical propertyin

ques-tion. Notethat inthis formulationboth

x

n

and the scenario probability

p

_n

are dened as variables. In many cases the probabilities

p

n

arexed beforehandto reduce thenonlinearityof theproblem.

A moment matching approach ensures statistical accuracy by denition

asitmatchesthe statisticalmoments.Inthatrespectthemethodismuch

more ecient than sampling  fewer scenarios are needed to match the

moments. However, the approach is too general for many applications.

36 problem

inpaperE.

4.2.4 Optimal discretization

Pug (2001) and Hochreiter & Pug (2002) introduce a number of

op-timization models for scenario tree generation using what they refer to

asoptimal discretization. Optimaldiscretization is essentiallydierent

fromalltheotherdiscretizationmethods,inthatthe focusisnot on

cap-turing the characteristics of an underlying stochastic process as closely

as possible. Instead the method generates scenario trees such that the

discretizationerror intheobjective functionof the underlyingstochastic

programming model is minimized. The discretization error of the

objec-tive function can, however, only be determined within some lower and

upperbounds,whicharenotnecessarilytight,meaning thatoptimal

dis-cretization does not with guarantee overperform other methods such as

4.3 Mortgage bond pricing

Thissectionreviewsthepricingmodelsappliedtoxedratecallable

mort-gagebonds(thebondsbehindFRMs)aswellasCiborlinkedoatingrate

callable mortgage bonds (the bonds behind CRMs). Conceptually, the

pricing ofnon callablebullet mortgagebonds(thebondsbehindARMs)

isstraightforward.Thepaymentsofabulletbondarediscountedwithfor

example the swap curve plus a constant yield curve spread (which

gen-erally increases withthematurityof thebond).The pricingofxedrate

callable mortgagebondsandCiborlinked oatingratecallablemortgage

bondsis, however, morecomplex due to theembedded options.

4.3.1 Pricing of xed rate callable bonds

In principle, a xed rate callable bond constitutes a portfolio of a non

callable bond and a short position in a Bermudan call option on that

bond (with a strike price of 100) reecting the embedded prepayment

option. However, for pricingpurposes, the prepayment option cannot be

treatedasastandardBermudancalloptionsinceborrowersdonotpursue

rationalexercisestrategies.Thereisnoprepaymentriskwhenamortgage

38 problem

thereforethere isa substantial prepayment risk.

Empiricalprepaymentmodelsbasedonhistoricaldataareneededtoprice

xedrate callable mortgagebonds.Suchmodelspredictthe prepayment

rate for a given payment date asa function ofthe yieldcurve and other

factorsaecting the level ofprepayments suchasthesize of theloans.

Themostimportant factoraectingtheprepaymentrateisthegainfrom

renancingtoalowerrate.Thegainisdenedasthepercentagereduction

inthemortgage payments onthe newloan, takingtaxation and

prepay-mentcostsintoaccount.Whenprepayingaloan,borrowersfacebothxed

andvariablecosts.Thegaincalculationisbasedonthetotalpayment for

the next year or the present value of all remaining payments using the

aftertax renancing rate onthe newloanasthe discount rate. On

aver-age,borrowers prepay largeloans more actively than smallerloans.This

facthasto betaken into account bytheprepayment model aswell.

4.3.2 Pricing of capped oaters

Capped oaters carryaoatingrate, arecallable andhavean embedded

payment prole will be of the annuity type where amortization may be

deferred for the rst 10years.Acharacteristic of Danishcapped oaters

is that the annuity rate tracks the sixmonth Cibor. This means that

the repayment prole of the bonds is stochastic as the annuity rate is

xed on the basisof thedevelopment insixmonth Cibor. Asthebonds

haveembeddedoptions,astochasticyieldcurvemodelisrequiredfor the

pricing. This model must be calibrated to basis options (such as caps

and swaptions) matching the implied options embedded in the capped

oaters.

With such a model at hand the pricing of capped oaters is done in a

straightforwardmanner,i.e.withoutaneedforaprepaymentmodel.The

embedded call option incapped oaters is insignicant andwill

theoret-ically or practically never goabove thestrike price of105.

4.4 Stochastic programming

Stochasticprogrammingis aframeworkfor modeling optimization

prob-lems thatinvolve uncertainty. Whereas deterministic optimization

40 problem

known only within certain bounds, one approach to tackling such

prob-lems is called robust optimization. Here the goal is to nd a solution

which is feasible for all such dataand optimal in some sense. Stochastic

programming models aresimilar instyle but take advantage of thefact

thatprobabilitydistributions governingthe dataareknownorcanbe

es-timated.Thegoalhereistondapolicythatisfeasibleforall(oralmost

all) the possible data instances and maximizes the expectation of some

function ofthe decisions andtherandom variables.Moregenerally, such

models areformulated, solved analytically or numerically, and analyzed

inorderto provideusefulinformation toadecisionmaker. 2

Twoclassical

bookson stochasticprogrammingareBirge&Louveaux (1997)andKall

&Wallace (1994).

4.4.1 Twostage stochastic programs

Themostwidelyapplied andstudiedstochasticprogrammingmodelsare

twostage linearprograms. Herethedecisionmakertakessome action in

therst stage, after which a random event occurs aecting theoutcome

of the rst-stage decision. A recourse decision can then be made in the

second stage thatcompensatesfor anybad eects thatmight have been

2

Program-experiencedasaresultoftherststagedecision.Theoptimalpolicyfrom

such a model is a single rststage policy and a collection of recourse

decisions (a decision rule) dening which secondstage action should be

taken inresponseto each randomoutcome.

Let

(Ω, P )

be aprobability space,

ω

∈ Ω

be therealizationof the

uncer-tain dataparameters and

p

(ω)

thecorresponding probability. Let

A, b, c

be deterministic parameters and

x

the rst stage deterministic decision

variable. We dene atwo-stage stochasticprogram as:

min Z =cx + E

_ω

Q

(x, ω)

wrt.

Ax

= b

x

≥ 0

42 problem

Q

(x, ω) = min fy(ω)

wrt.

D

(ω)y(ω) = d(ω) + B(ω)x

y

(ω) ≥ 0

Theparameters

D

(ω)

,

d

(ω)

and

B

(ω)

aswellastherecoursevariable

y

(ω)

arestochasticanddenedover

Ω

.Thetwostage stochasticprogrammay

nowberewritten as:

min Z =cx + E

_ω

[fy(ω)]

wrt.

Ax

= b

− B(ω)x + D(ω)y(ω) = d(ω)

4.4.2 Thedeterministicequivalentofthetwostage

stochas-tic program with recourse:

Once the uncertainty space is represented as a set of discrete scenarios

thenthestochasticprogramscanbeformulatedasdeterministicones.For

the twostage stochastic program the deterministic equivalent is

formu-lated asfollows:

min Z =cx + p

1

f y

1

+ p

2

f y

2

+ · · · + p

k

f y

k

wrt.

Ax

= b

− B

1

x

+ D

1

y

1

= d

1

− B

2

x

+

D

2

y

2

= d

2

. . . . . . . . .

− B

k

_x

₊

_D

k

_y

k

_{= d}

k

x, y

1

, y

2

,

· · · , y

k

≥ 0;

0 ≤ p

ω

_{≤ 1}

and

X

ω

p

ω

= 1.0

44 problem

secondstage,wetakesomefunction

f

oftherecoursevariables

y

1

_,

_{· · · , y}

k

rst andthenaverage thereturnvalues.

4.4.3 Multistage stochastic programs

Twostage stochastic programs can be extended to several stages in the

following manner:

min

x

1

=

n

c

₁

x

₁

+ E

_ω

₂

h

min

x

2

c

2

x

2

+

E

_ω

₃

_|ω

₂

h

min

x

3

c

3

x

3

+ · · · + E

ωT

|ωT −1|···|ξ

2

min

xT

c

T

x

T

iio

wrt.

A

₁₁

x

₁

= b

₁

A

₂₁

x

₁

+ A

₂₂

x

₂

= b

₂

A

₃₁

x

₁

+ A

₃₂

x

₂

+ A

₃₃

x

₃

= b

₃

. . . . . . . . .

A

₃₁

x

₁

+ A

₃₂

x

₂

+ A

₃₃

x

₃

+ · · · + A

_{T T}

x

_T

= b

_T

,

where

x

1

is a deterministic rst stage decision variable and

x

2



x

T

are

stochastic recourse variables for periods

2



T

.

ω

t

is the realization of

unfolds at a given time

t

conditioned on the states of the uncertainty

realized at time

t

− 1

.

4.4.4 Two formulations of a stochastic program

Consider thescenario trees inFigure4.4:

n=1

n=2

n=3

n=4

n=5

n=6

n=7

Year 1

Year 2

Year 3

s1

s2

s3

s4

Year 1

Year 2

Year 1

Year 2

Year 3

Year 3

s1

s1

s1

s2

s2

_s2

s3

s3

s3

s4

s4

s4

Figure4.4:Ascenariotreemayberepresentedeitherbyanumberofnodes(top)or

byanumberofscenariosandtimepoints(down).

46 problem

sented inharmony with the waythe uncertainty isunfolded. Inthe

sce-nario formulation (also called splitvariable formulation) the number of

uncertainty variables at each time point is multiplied by the number of

scenarios.Sointhis formulationweneed explicitlytomakesurethatnot

morethanonedecisionismadeatanygivennode.Thisisdonebyadding

a number of constraints known as "nonanticipativity" constraints. For

theexample shownin Figure4.4 (down) and given a stochasticvariable

x

_t,s

dened overall times

t

andscenarios

s

weneed to add thefollowing constraints:

x

_t

₁

_,s

₁

= x

_t

₁

_,s

₂

= x

_t

₁

_,s

₃

= x

_t

₁

_,s

₄

x

_t

₂

_,s

₁

= x

_t

₂

_,s

₂

x

_t

₂

_,s

₃

= x

_t

₂

_,s

₄

Summary of the papers

Theresearcheortsinthisworkarewithinthedomainofoptimizationin

nanceandappliedmathematicalnance.Thefocushasbeenonrealistic

problemsolving.Thatinvolveddevelopingandtestingseveral

mathemat-ical modelsinthe onehandand nancialanalysisandinterpretationand

discussion of the ndings in the other. Along the way the research has

also resulted in a theoretical proof. In this chapter we review the main

features of our work as presented in papers A through E and point out

5.1 Interest rate modeling

Interest rate dynamics is a very well researched area. Thousands of

re-search papers and several books are written with the main focus on

in-terest rate modeling. These models are, however, developed mostly in

orderto providethe underlying dynamics forpricingof interest rate

sen-sitiveinstrumentshereandnowratherthan ensuringthatfutureinterest

ratedynamicsarecaptured inarealisticmanner.Theirsuccesscriteriais

resultinginrealisticpresent valuesfor interestratesensitiveinstruments.

In our setting we not only need prices of mortgages here and now but

we also need approximative prices under dierent market conditions for

rebalancing purposes in some future scenarios. Our contribution within

interest ratemodeling ispresented inpaperE. Our model isto thebest

of our knowledge the only one which uses the three factors level, slope

andcurvaturedirectlyandtherebyproducesareallifelikevariationover

termstructure predictions.Themodelisanspecializationofavector

au-toregressive model withlag 1 (VAR1). It is easy to calibrate to historic

time series with some time step, say weekly observations. The length of

theprediction steps doesnot need to be equal to thestep lengthfor the

of same length asin thecalibrationdata. This means,besides

computa-tional eciency, that the scenario trees generated based on this model

arereproducible whichis an important qualityfor testing.

5.2 Scenario generation

The scenariogeneration isa twostep process:

1. An event treeof the term structuresof interest rates isbuilt.

2. Mortgagesarepriced ineverynode ofthescenario tree.

5.2.1 Interest rates

Withan interest ratemodelat hand we need to generate a scenariotree

of interest rates. Our scenario generation approach is explained fully in

paperE. We dene a numberof qualityrequirementsfor ascenario tree

of term structures. Our scenario generation approach is an extension of

themoment matchingapproach of Høyland &Wallace(2001).

tionmodelsfromthepapersrepresentedinthisthesis.InpaperAweuse

theonefactormodelofBlack,Derman&Toy(1990)andinpapersCand

D weusea specialized versionof the VasicekmodeldevelopedbyJensen

& Poulsen (2002). We have, however, compared the results of the

opti-mizationmodelsbasedonthenewscenariogenerationapproachwiththe

results from theabove mentioned papers. These results arepresented in

chapter7ofthissummaryreportunderadiscussiononmodelrobustness.

5.2.2 Mortgage bond prices

Once a scenario tree of interest rates is built the universe of available

mortgage bondsneed to be priced inthe nodesofthe tree. Whilethis is

anstraight forward calculationforbulletbondswhicharethefunding

in-strumentsbehindARMs,itbecomesanextremely challenging taskwhen

itcomestopricingcallablexedratemortgagebondswhicharelongterm

annuitieswithembeddedBermudancalloptionsaswellasbuyback

deliv-eryoptions. Pricingsuchbondsasksfor aproperprepayment (burnout)

model which predictsthe exerciseof theembedded options under

dier-entinterestratescenarios.Besidesthemodelsusedforpricingsuchbonds

normallyaddasocalledoptionadjustedspread(OAS)to thetheoretical

dependent optionpricing.

WedonotdevelopnewpricingalgorithmsforthebondsbehindFRMsand

CRMs, since we believe this would take us far from thecentral question

inthisproject.Insteadweapplyexisting"stateoftheart"pricingmodels

to everypath ofthe scenario tree.InpaperA weuseNykredit's internal

mortgagebondpricingmodel(Nyklib),whereasinpapersCandDweuse

approximativepricingapproachessimilartothosesuggestedinNielsen&

Poulsen (2004). Finally we have tried ScanRate's RIO pricing system

(see http://www.scanrate.dk) on our VAR1 interest rate trees and the

optimization resultsbasedonthesescenariosarereportedinchapter7of

this summaryreport.

5.3 Optimization framework

Withascenario treeof mortgagebond ratesand pricesat handwe want

to nd optimal mortgagestrategies for homebuyers withdierent

objec-tives. We develop an optimization framework which is completely

sepa-rated from the scenariogeneration process. Agiven scenario treeis only

max-uncertaintymodel.

Butwhydoweneedoptimization?Afterallonemightarguethatifweall

agree on a complete representation of the uncertainty which reproduces

market pricesof mortgages, then all mortgages areequally attractive in

average.Theansweris,thatevenundertheseunrealisticassumptionsthe

homebuyers personal risk preferences ask for an optimization model in

orderto ndthe bestmortgagechoice.Insection3.2wesawan example

ofahomebuyerwhowasinterestedinndingamortgageportfoliowhich

yieldsthe smallestaverage ofthehighest10%of theholdingperiodcosts

over 5years.Answeringsuch questionsissimplynot possible withoutan

optimization model. But even ifwe donot consider personalrisk

prefer-ences,itisbyfaraquestionableassumption thatallmortgagesshouldbe

equallyattractive inaverage. We give thefollowing reasons:

•

Themortgagemarketisincomplete,i.e.therearemorestatesofthe world than mortgages.

•

Market frictions such as transaction costsand tax aects have an impacton the mortgagors choice.

Given this background, using optimization techniques for the mortgage

choice problem is indeed well justied. Most of the work in the papers

A, C and D is concentrated around developing and testing optimization

models for the mortgagechoice problem.

The work was inspired by a paper of Nielsen & Poulsen (2004). They

design a trinomialscenariotree usingan underlying twofactor modelof

interestratesfor pricingexistingandsynthetic mortgagebonds.F

urther-moretheyintroduceastochasticprogrammingmodelto ndtheoptimal

initial loanstrategy among anumber ofARMsand FRMsand toadvise

the mortgagor on optimal readjustments along theway. Their

optimiza-tion model, however, does not include a risk measure and the eects of

xedmortgageorigination costswereignored.In paperAweextend the

modeltoincludexedmortgageoriginationcostsandbudgetconstraints.

Dierent objective functionsaretried inthis paper:

1. Minimizing average holding periodcosts.

2. Minimizing thehighestholding periodcost scenario. (Minmax)

3. Minimizingtheaverageholdingperiodcostwithbudgetconstraints.

out-Theconclusion isthat aminmax mortgagoror a mortgagorwithbudget

constrains benets from choosing an initial portfolio of an ARM and a

FRM, given that there are only these two types of products to choose

from.The budgetconstraintsprovideindirectmeans forriskcontrol,but

noexplicitriskmeasureisconsideredinthispapereither.Weincorporate

thescenario reductionalgorithm ofHeitsch &Römisch (2003) to reduce

the size of the tree. We observe, however, that the scenario reductions

introduces a high degree of arbitrage opportunities in the scenario tree

andeventhougharbitrage isnot allowedto beexercisedinour problem,

theoptimal solutions foundinthe reducedtrees becomebiased. We also

introduce asimple iterative algorithm forsolving theLPrelaxedversion

of the01stochasticprogram justusing afewiterations.

WeaddanexplicitriskmeasureforthisclassofproblemsinpaperC.Here

we develop a singleperiod stochastic programming model to trade o

thepresent valueofaverageholding periodcostsagainsttheConditional

Valueat Risk(CVaR 1

)value. We introduce the notionof aMean/CVaR

ecient frontierfor amortgagorandshowthatdiversiedmortgageloan

strategies outperform single mortgage loan strategies. Figure D.1

high-lightsour ndings which speak stronglyinfavor of diversication.

1

Rock-Figure 5.1: For a mortgagor with a seven year horizon a mix of

vari-able and xedrate mortgages provide low payments and low risk, here

FinallyinpaperD we developa multistageversionof our earliermodel

andshowthatimproved results canbeobtained byintroducingdynamic

tradingintothemodel.Itwillbeseenthatthebudgetconstrainedmodel

of paperA is subsumed by thebilinear Mean/CVaR minimizing model.

Furthermore, we consider Capped RateMortgages CRMs aspart of our

universe of loans and suggest a simple approach to determine whether

thecap optioncomes at afairprice for agivenmortgagor withacertain

riskappetite.Figure(5.2)comparesamean/CVaRecient frontierfor a

singleperiod modelwiththatof amultistage model.

Figure5.2:Asmoredecisionstagesareaddedtotheproblemthesolution

Moreoptimizationresultsarecomparedbyusingdierentscenario

gener-ationapproaches, severalloansandmanyoptimizationmodelsinchapter

7.Theseresults have yetnot been publishedin anypaper.

5.4 Financial Gien goods

Paper B may at a rst reading seem to be a deviation from the central

theme of thisthesis. That isnot the case. We showinthis paperthat

-nancial Giengoods cannot existinaMarkowitz meanvariance setting.

We argue that it makes good nancial sense to allow their existence in

optimal portfolio models and we show thatsuch goods do exist inmore

realistic models such as those developed in papers A,C and D.In other

words weprovide additionalevidence asto whywe donotconsider

port-foliovariancebutratherbudgetconstraintsormoregenerallyConditional

Valueat Riskasour measure ofrisk.

A Giengood isonefor whichdemand goesdownifitsprice goesdown.

At rst, itis counter intuitive that such goods exist at all.But most

in-troductorytext books ineconomics will tellyou thattheydo;some with

con- by which we mean a negative relation between expected return and

demandinportfoliochoicemodels.Surprisingdependenceonexpected

rates of return is not uncommon innance. In complete models, option

prices do not depend on the stock's growth rate. And quite generally

call option pricesincrease withthe interestrate; immediately you would

think that cashows are discounted harder, but in fact the replicating

strategywhichentailsashortpositioninthebankaccountbecomesmore

expensive,and hencethe call optiondoestoo.

WerstshowthatinthebasicMarkowitzmean/variancemodel,thereare

noGiengoods;ifastock'sexpectedrateofreturngoesup,its weight in

anyecientportfoliogoesup.Thisseemsatext-bookcomparativestatics

result. We have,however, only been able to nd itindirectly stated, for

instanceonecouldviewitasacorollaryorlemmarelatedtotheHarmony

TheoremfromLuenberger (1998,Section 7.8).So wegiveasimpleproof.

We then look at Merton's dynamic investment framework. In its basic

version demand for any asset depends positively on its expected rate of

return,butifasubsistencelevelisincluded,demandfortheriskfreeasset

mayfall withtheinterestrate.

Skeptics wouldsay thatGien goods existinand only in economic text

uses athemultistage stochasticprogrammingframework frompapers A,

C and D and shows that some  completely rational  mortgagors react

to lowercostsoflong-term nancing(reecting asmallermarketpriceof

risk) by usingmore shortterm nancing.

In the next chapter the main features and novelties of this thesis are

Research contributions

Theresearcheortsinthisworkarewithinthedomainofoptimizationin

nanceandappliedmathematicalnance.Thefocushasbeenonrealistic

problemsolving.Thatinvolveddevelopingandtestingseveral

mathemat-ical modelsaswell asnancial analysisandinterpretationand discussion

ofthendings.Alongthe waythe researchhasalsoresultedina

theoret-ical proof on lackof Gien goods ina Markowitzmean variance setting.

We showthen thatsuch goods do exist inmore realistic models such as

contri-6.1 Optimization models

Theoptimizationmodelsdevelopedinthisprojectarenovel.Inparticular:

•

InpaperA wedevelopanumberofmultistage stochasticprograms to represent the homebuyers mortgage choice problem. The

em-phasis of the modeling work is its realism, i.e variable xed and

transaction costs, tax eects, mortgage rebalancings and early

re-paymentsaremodeled.Likewisehomebuyersbudgetconstraintscan

beadded.

•

InpaperC we generalize the budgetconstraintsbyintroducing an explicitmeasureofrisk(CVaR).Themodelisdevelopedasasingle

stagemodelinordertostudytheincrementaleectsofmovingfrom

single loan issue and hold strategies to optimal portfolios of loans

thoughstill inanissueand holdsetting.

•

In paper D we introduce the multistage version of themodel from paperCandshowthatinitialdiversicationandfuturerebalancings

improves the optimal payment/risk frontiers from the single stage

setting.