Mortgage Loan Portfolio Optimization Using Multi-Stage Stochastic Programming

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Mortgage Loan Portfolio Optimization Using Multi-Stage Stochastic Programming

Rasmussen, Kourosh Marjani; Clausen, Jens

Published in:

Journal of Economic Dynamics and Control

Link to article, DOI:

10.1016/j.jedc.2006.01.004

Publication date:

2007

Link back to DTU Orbit

Citation (APA):

Rasmussen, K. M., & Clausen, J. (2007). Mortgage Loan Portfolio Optimization Using Multi-Stage Stochastic

Programming. Journal of Economic Dynamics and Control, 31(3), 742-766. DOI: 10.1016/j.jedc.2006.01.004

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Multi Stage Stochastic Programming

Kourosh MarjaniRasmussen

∗

and Jens Clausen

†

November 11,2005

Abstract

We consider the dynamics of the Danish mortgage loan system and

proposeseveralmodelstoreectthechoicesofamortgagoraswellas

his attitude towardsrisk. The models are formulated as multi stage

stochasticintegerprograms,whicharediculttosolveformorethan

10 stages. Scenario reduction and LP relaxation are used to obtain

near optimal solutionsfor large problem instances. Our results show

that the standard Danish mortgagor should hold a more diversied

portfolioofmortgageloans,andthatheshouldrebalancetheportfolio

morefrequentlythancurrentpractice.

1 Introduction

1.1 The Danish mortgage market

The Danish mortgage loan system is among the most complex of its kind

intheworld. Purchaseof mostpropertiesinDenmarkis nancedbyissuing

xedrate callable mortgagebondsbasedon an annuity principle.It is also

possibletoraiseloans,whicharenancedthroughissuingnoncallable short

term bullet bonds. Such loans may be renanced at themarket rate on an

ongoingbasis. Theproportion of loansnanced byshortterm bulletbonds

hasbeen increasing since 1996. Furthermore it is allowed to mix loans in a

mortgageloan portfolio, but this choice hasnot yetbecomepopular.

Callable mortgage bonds have a xed coupon throughout the full term of

the loan. The maturities are 10, 15, 20 or 30 years. There are two options

∗

Informatics and Mathematical Modelleing, Technical University of Denmark, Bldg.

305,DK-2800Lyngby,[email protected]

†

Informatics and Mathematical Modelleing, Technical University of Denmark, Bldg.

(3)

i.e. he can redeem the mortgage loan at par at four predetermined dates

eachyearduringthelifetimeofthe loan.Whentheinterestratesarelowthe

call option can be used to obtain a new loan with less interest payment in

exchange for an increase inthe amount of outstanding debt. The borrower

hasalso a delivery option. When the interest rates arehighthis option can

be used to reduce theamount of outstanding debt, in exchange for paying

higherinterestratepayments.There areboth xedandvariabletransaction

costsassociated withexercising any ofthese options.

Noncallable shortterm bullet bonds are used to nance the adjustable

rate loans. The bonds' maturities range from one to eleven years and the

adjustablerate loans are oered as 10,15, 20 or 30year loans. Since 1996

themostpopularadjustablerateloanhasbeentheloannancedbytheone

yearbond.From 2001,however, therehasbeena newtrend, where demand

for loans nanced by bullet bonds with 3 and 5year maturities has risen

substantially.

1.2 The mortgagor's problem

It is known on the investor side of the nancial markets that investment

portfolios should consist of a variety of instruments in order to decrease

nancial riskssuch asmarket, liquidityand currencyriskwhilemaintaining

axedlevelofreturn.Theportfolioisalsorebalancedregularlyto takebest

advantageof themovesinthe market.

Theportfoliodiversicationprincipleandrebalancingis,however,not

com-moninthe borrowerside ofthe mortgage market. Mostmortgagors nance

theirloans inone type ofbondonly.Besides they do not always rebalance

theirloan whengoodopportunitiesfor this have arisen.

There aretwomajor reasonsfor the mortgagors reluctance to better taking

advantageoftheiroptions(thattheyhavefullypaidfor)throughthelifetime

ofthemortgage loan.

1. The complexity of the mortgage market makes it impossible for the

average mortgagorto analyzeallthealternativesandchoosethebest.

2. Themortgagecompaniesdonotprovideenoughquantitativeadviceto

theindividualmortgagor. Theyonlyprovide generalguidelines,which

are normally not enough to illuminate all dierent options and their

consequences.

The complexity of the mortgage loan system makes it a nontrivial task

(4)

([19], [14],[18],[20],[21 ],[22 ]).

We assumeinthefollowing thatthereader isfamiliarwiththedynamics of

a mortgage loan market such as the Danish one, aswell as the basic ideas

behind themathematical modeling concept ofstochasticprogramming.

TheDanishmortgagor'sproblemhasbeenintroducedbyNielsenandPoulsen

(N&P, [13]). They use a two factor term structure model for generating

interestratescenarios.Theyhavedevelopedanapproximativepricingscheme

to price the mortgage instruments in all nodes of the scenario tree and on

top of it have built a multistage stochastic program to nd optimal loan

strategies. The paper, however, does not describe the details necessary to

have afunctional optimization model, and itdoesnot dierentiate between

dierenttypesofrisksinthemortgagemarket.Themaincontributionofthis

article is to make Nielsen & Poulsen's model operational by reformulating

parts oftheir modeland addingnew featuresto it.

We reformulate the Nielsen & Poulsen model in section 2. In section 3 we

modeldierent options available to the Danish mortgagor, and insection 4

wemodelmortgagor'sriskattitudes.Hereweconsiderbothmarket riskand

wealth risk.

Inthe basic modelweincorporate xedtransaction costsusing binaryv

ari-ables.We usea noncombining binomial treeto generate scenarios inan 11

stageproblem. Thisresultsin51175binaryvariables,makingsome versions

of the problem extremely challenging to solve. Dupa

ˇc

ová, GröweKuska, Heitsch and Römisch (Scenred, [7 , 8]) have modeled the scenario

reduc-tionproblem asa setcovering problem andsolvedit using several heuristic

algorithms. We review these algorithms in section 5 and use them in our

implementation to reducethesizeof the problemandherebyreducethe

so-lutiontimes. Anotherapproachtogettingshortersolutiontimesisproposed

insection6,where we solveanLPapproximated versionoftheproblem.In

section7wediscuss andcommenton ournumericalresults andweconclude

thearticle withsuggestions forfurther research insection8.We useGAMS

(General Algebraic Modeling System) to model the problem and CPLEX

9.0astheunderlying MPandMIPsolver.Forscenario reductionwe usethe

GAMS/SCENRED module(scenredmanual,[9 ]).

TheobtainedresultsshowthattheaverageDanishmortgagor wouldbenet

from choosing more than one loan in a mortgage loan portfolio. Likewise

he should readjust the portfolio more often than is the case today. The

developedmodelandsoftwarecanalsobeusedtodevelopnewloanproducts.

Such products will consider the individual customer inputs such as budget

constraints, risk prole,expected lifetimeofthe loan,etc.

Even thoughwe considerthe Danishmortgage loanmarket, theproblem is

(5)

2 The basic model

In this section we develop a riskneutral optimization model which nds a

mortgageloanportfoliowiththeminimum expected total payment.

Weconsideraniteprobabilityspace

(Ω, F, P )

whoseatomsaresequencesof realvaluedvectors (coupon rates andprices ofmortgage backed securities)

t

inthetree.

In the scenario tree,every node

n

∈ N

for

1 ≤ t ≤ T

has a unique parent denoted by

a

(n) ∈ N

t−1

, and every node

n

∈ N

t

for

0 ≤ t ≤ T − 1

has a nonempty set of child nodes

C(n) ⊂ N

t+1

. The probability distribution

P

is modeled by attaching weights

p

n

>

0

to each leaf node

n

∈ N

T

so that

P

n∈N

T

p

n

= 1

.For each nonterminalnode one has,recursively,

p

_n

=

X

m∈C(n)

p

_m

∀n ∈ N

_t

,

t

= T − 1, · · · , 0

andsoeachnodereceivesaprobabilitymassequal tothecombined massof

thepathspassingthrough it.

We assumethatwe havesucha treeat hand withinformation on priceand

coupon rate for all mortgage bonds available at each node as well as the

probability distribution

P

for the treeat hand.

In the basic model we only consider xedrate loans, i.e. loans where the

interest rate does not change during the lifetime of the loan. For the sake

ofdemonstration we consider anexample with4 stages,

t

∈ {0, 1, 2, 3}

,and 15 decisionnodes,

n

∈ {1, · · · , 15}

,with theprobability

p

n

for being at the node

n

.

Wewant the basic modelto ndan optimalportfolioofbondsfromanite

numberofxedratebonds.Considerthe4bondsshowninFigure(1).Each

bond is representedas (Index:TypeCoupon/Price), so(3:FRM3206/98.7)

isa xedrate callable mortgage bond withmaturityin 32 years, acoupon

rateof 6%anda price of98.7.

To generate bonds information we can useterm structure and bond pricing

theories (see [10 , 12 , 4] for an introduction to these topics and further

(6)

n=2

n=4

n=5

t = 1

t = 0

t = 2

n=3

n=1

n=6

n=7

t = 3

n=8

n=9

n=10

n=11

n=12

n=13

n=14

n=15

1:FRM31−05/96.8

1:FRM30−05/101.8

1:FRM30−05/92.35

1:FRM29−05/88.8

2:FRM32−06/95.3

1:FRM29−05/95.4

3:FRM32−06/98.7

1:FRM29−05/95.4

3:FRM32−06/98.7

1:FRM29−05/105.4

4:FRM32−04/98.3

1:FRM28−05/108.4

4:FRM31−04/101.4

4:FRM31−04/94.2

1:FRM28−05/96.9

3:FRM31−06/100.7

1:FRM28−05/96.9

3:FRM31−06/98.4

1:FRM28−05/93.7

3:FRM31−06/100.7

1:FRM28−05/96.9

3:FRM31−06/98.4

1:FRM28−05/93.7

1:FRM28−05/84.4

2:FRM31−06/92.5

2:FRM31−06/98.4

Figure 1: A binomialscenario tree,representingourexpectationof futurebondprices

andcouponrates.Allbondsarecallable xedratebonds.

possible bondpricesinthefuture. A combination oftheoreticalpricing and

expertinformation can also be usedtogenerate suchscenariotrees. Nielsen

andPoulsen(N&P,[13 ])proposeanapproximativeapproachforpricingxed

rate bonds with embedded call and delivery in a scenario tree. In this

pa-perweusethe BlackDerman& Toy ([5 ])interestrate treeto represent the

underlyinginterestrateuncertaintyandestimatethepricesofthemortgage

backedbondsinallthenodesofthetreeusingthecommercialpricingmodule

RIO4.0developedbyScanrate FinancialSystemsA/S ([17 ]).

Given a scenario tree with

T

stages and its corresponding coupon rate and priceinformation onasetofbonds

i

∈ I

wecannowdene thebasicmodel. Parameters:

= min{1, k

in

}

for callablebondsand

Callk

in

= k

in

fornoncallable bonds.

γ

:Tax reductionratefrom interestrate payment.

β

:Tax reductionratefrom administrationfees.

(7)

η

:Transactionfee rate for saleand purchaseof bonds.

m

:Fixedcostsassociated withrebalancing.

Nextwedene the variables usedinour model:

otherwise.

Themulti stage stochasticinteger modelcan now be formulated asfollows:

min

T

X

t=0

X

n∈N

t

p

_n

· d

_t

· B

_tn

(1)

X

i∈I

k

_i1

· S

_i01

≥ IA

(2)

X

i01

= S

i01

∀i ∈ I

(3)

X

_itn

= X

_i,t−1,a(n)

− A

_itn

− P

_itn

+ S

_itn

∀i ∈ I, n ∈ N

_t

, t

= 1, · · · , T

(4)

X

i∈I

(k

_in

· S

itn

) =

X

i∈I

(Callk

_in

· P

itn

)

∀n ∈ N

t

, t

= 1, · · · , T

(5)

A

itn

= X

i,t−1,a(n)

h

_r

i,a(n)

1 − (1 + r

i,a(n)

)

−T +t−1

− r

i,a(n)

i

∀i ∈ I, n ∈ N

t

, t

= 1, · · · , T

(6)

B

01 =

X

i∈I

η

· S

i01

+ m · L

i01

(7)

B

tn

=

X

i∈I

A

itn

+ r

i,a(n)

· (1 − γ)X

i,t−1,a(n)

+ b · (1 − β)X

i,t−1,a(n)

+

η

· (S

_itn

+ P

_itn

) + m · L

_itn

∀n ∈ N

t

, t

= 1, · · · , T

(8)

BigM

· L

_itn

− S

_itn

≥ 0

∀i ∈ I, n ∈ N

_t

, t

= 0, · · · , T

(9)

X

_itn

, S

_itn

, P

_itn

≥ 0 , L

_itn

∈ {0, 1}

∀i ∈ I, n ∈ N

_t

, t

= 0, · · · , T

(10)

Theobjective isto minimize theweighted averagepayment throughout the

mortgage period. The payment for all the nodes exceptthe root is dened

inequation(8) asthesum ofprincipal payments, taxreduced interest

pay-ments, taxed reduced administration fees (Danish peculiarity), transaction

fees for sale and purchase of bonds and nally xed costs for establishing

new mortgage loans. The principal payment is dened in equation (6) as

anannuity payment. The payment intheroot (equation 7) is basedon the

(8)

straint (2)makessurethat wesell enough bondsto raisean initial amount,

IA

,needed bythe mortgagor. In equation (3) we initialize the outstanding debt. Equation (4) is the balance equation, where the outstanding debt at

anychildnodefor anybond equalstheoutstandingdebtat theparent node

minus principal payment and possible prepayment (purchased bonds), plus

possiblesoldbondsto establishanewloan.Equation(5)isacashow

equa-tion which guarantees that the money used to prepay comes from thesale

ofnew bonds.

Finallyconstraint(9)addsthexedcoststothenodepayment,ifweperform

any readjustment of the mortgage portfolio. The

BigM

constant might be set to a value slightly greaterthan the initial amount raised. If a too large

valueisused, numerical problemsmayarise.

3 Modeling mortgagor's options

Themodeldescribedinsection2hasthreeimplicitassumptionswhichlimit

its applicability:

1. Weassumethataloanportfolioisheldbythe mortgagoruntiltheend

ofhorizon.

2. We assumethatall bondsarexedrateand callable,i.e. theycan be

prepaidat anytimeat a priceno higher than 100.

3. Themortgagor isassumedto be riskneutral.

We will relaxthe rst two assumptions in this section and the third inthe

following section.

= H

is a posi-tive amount which needsto beprepaid. We dene this prepayment amount

(

P P

Hn

) as:

P P

_Hn

=

X

i

(9)

follows:

min

X

H

t=0

X

n∈N

t

p

_n

· d

_t

· B

_tn

+

X

n∈N

H

p

_n

· d

_H

· P P

_Hn

.

(12)

The objective is now to minimize the weighted payments at all nodes plus

the weighted prepayments at time

H

.

The problemwith the second assumption is more subtle. Consider the

sce-nario tree at Figure (2), where two adjustablerate loans have been added

to our set of loans at time 0. Loan 5 (ARM1) is an adjustablerate loan

withannualrenancing and loan6(ARM2) isan adjustablerateloanwith

renancing every secondyear

n=2

n=4

n=5

t = 1

t = 0

t = 2

n=3

n=1

n=6

n=7

t = 3

n=8

n=9

n=10

n=11

n=12

n=13

n=14

n=15

5:ARM1−04/99.1

6:ARM2−04/98.2

1:FRM31−05/96.8

1:FRM30−05/92.35

5:ARM1−06/101.2

6:ARM1−04/95.8

1:FRM30−05/101.8

5:ARM1−02/99.1

6:ARM1−04/104.7

6:ARM2−03/101.2

5:ARM1−01/99.2

4:FRM32−04/98.3

1:FRM29−05/105.4

6:ARM2−05/100.1

5:ARM1−04/99.2

3:FRM32−06/98.7

1:FRM29−05/95.4

6:ARM2−05/100.1

5:ARM1−04/99.2

3:FRM32−06/98.7

1:FRM29−05/95.4

6:ARM2−08/100.6

5:ARM1−08/102

2:FRM32−06/95.3

1:FRM29−05/88.8

1:FRM28−05/84.4

2:FRM31−06/92.5

6:ARM1−08/97.8

5:ARM1−10/101.5

1:FRM28−05/93.7

2:FRM31−06/98.4

1:FRM28−05/93.7

3:FRM31−06/98.4

1:FRM28−05/96.9

3:FRM31−06/100.7

1:FRM28−05/93.7

3:FRM31−06/98.4

1:FRM28−05/96.9

3:FRM31−06/100.7

1:FRM28−05/96.9

4:FRM31−04/94.2

1:FRM28−05/108.4

4:FRM31−04/101.4

5:ARM1−01/102.4

6:ARM1−03/108.4

6:ARM1−03/98.6

5:ARM1−03/99.9

6:ARM1−05/105.4

5:ARM1−03/99.9

6:ARM1−05/102.9

5:ARM1−07/99.6

6:ARM1−05/105.4

5:ARM1−03/99.9

6:ARM1−05/102.9

5:ARM1−07/99.6

6:ARM1−08/103.2

5:ARM1−07/99.6

Figure2:Abinomialscenario treewherebothxedrateandadjustablerate loansare

considered.

For adjustablerate loans (ARMmloans) the underlying

m

year bond is completely renanced every

m

years by selling another

m

year bond. But unlike normal renancing this kind of renancing does not incur any

ex-tra xed or variable transaction costs since an ARMmloan is issued as a

single loan rather than a series of bulletbonds following each other. We

modelan ARMmloanby using the sameloan index for an adjustablerate

loan throughout the mortgage period.For example index 5 is used for the

loanwithannualrenancing,eventhough theactualbondsbehindtheloan

change every year.Sincewe usethesame index,themodeldoesnot register

any actual sale or purchase of bonds when renancing occurs. We should,

(10)

dierent frompar. To take this into account we introduce the set

I

0 _{⊆ I}

of

noncallable adjustablerateloans.For theseloans weusethefollowing

bal-anceequationinstead ofequation (4).

k

_in

· X

_itn

= X

_i,t−1,a(n)

− A

_itn

− P

_itn

+ S

_itn

∀i ∈ I

0 , n

∈ N

_t

, t

= 1, · · · , T.

(13)

Note that variables

P

itn

and

S

itn

remain 0 as long we keep an adjustable rateloan

i

∈ I

0

inourmortgageportfolio. Theoutstandingdebtinthechild

nodeis howeverrebalancedbymultiplying thebond price.

Whenwe considertheadjustablerateloansweshouldrememberthatthese

loans arenoncallable,sofor prepayment purposes we have:

Callk

_in

= k

_in

∀i ∈ I

0 , n

∈ N

_t

, t

= 0, · · · , T.

Anotherissuetobedealtwithisthatifabondisnotavailableforestablishing

aloanat agivennode,wehaveto setthe correspondingvalueof

k

in

to0to makesurethatthe bondisnotsoldatthatnodeinanoptimalsolution. For

examplebond(6:ARM104/95.8)atnode2 isnot openforsale butonlyfor

prepayment.

4 Modeling risk

Sofar wehaveonly consideredariskneutral mortgagorwhoisinterestedin

theminimumweightedaverageoftotalcosts.Mostmortgagorshoweverhave

anaversiontowardsrisk.There aretwokindsofriskwhichmostmortgagors

areawareof:

1. Marketrisk:Inthemortgage marketthis istheriskofextrainterest

ratepaymentfor amortgagorwhoholdsanadjustablerateloanwhen

interestrateincreases,orthe riskofextraprepayment foramortgagor

withanykindofmortgageloanwhentheinterestratedecreasessothe

bond price increases.

2. Wealth risk: In the mortgage market this is a potential risk which

can be realized if themortgagor needs to prepay themortgagebefore

a planned date or ifhe needs to use the free value of thepropertyto

take another loan. It can be measured asa deviation froman average

outstandingdebt atanygiven timeduringthelifetime oftheloan.

We will in the following model both kinds of risk. To that end we use the

ideas behind minmax optimization and utility theory with use of budget

(11)

Anextremely riskaverse mortgagorwantsto payleastintheworst possible

scenario.Inotherwordsifwedene themaximumpayment as

M P

thenwe have thefollowing minmax criterion:

min

M P,

(14)

M P

≥

T

X

t=0

X

n∈N P

ts

B

_tn

∀s ∈ S,

(15)

where

N P

ts

is a set of nodes dening a unique path from the root of the tree to one of the leaves. Each of these paths dene a scenario

s

∈ S

. For the examplegiven inFigure2 we have:

N P

t,1

= {1, 2, 4, 8}

N P

t,2

= {1, 2, 4, 9}

· · ·

N P

t,8

= {1, 3, 7, 15}

4.2 Utilityfunction

Insteadofminimizingcostswecandeneautilityfunction,whichrepresents

asaving andmaximizeit.NielsenandPoulsen (N&P[13 ])suggestaconcave

utilityfunction withthesame formasinFigure (3).

Utility

Saving

Figure 3: A concave utility function. An increase of an already big saving is not as

interestingasanincreaseofasmallersaving.

The decreasing interest for bigger savings is based on the idea that bigger

(12)

max

X

T

t=0

X

n∈N

t

p

_n

· log(d

_t

· (B

_tn

max

− B

_tn

)),

(16) where

B

max

tn

isthe maximumamount a mortgagor iswilling to pay. Nielsen and Poulsen x

B

max

tn

to a bigvalue so thatthe actual payment will never rise above this level.

Addingthis nonlinear objective function to our stochastic binaryproblem

makes the problem extremely challenging to solve. There are no eective

general purposesolvers for solving large mixedinteger nonlinear programs

(seeBussieckandPruessner, [6 ]).There arethreewaysofcircumventingthe

problem: Either we usea linear utility function inconjunction withbudget

constraints(mip)orrelaxthebinaryvariablesandsolvethenonlinear

prob-lem (nlp) or both (lp). We demonstrate the rst approach in the following

andcomment onthe second andthird approach insection6.

Instead of maximizing the logarithm of the saving at each node we can

simplymaximize thesaving:

B

max

tn

− B

tn

.If

B

max

tn

issolargethatthesaving is always positive, then we are in eect minimizing the weighted average

costs similar to the risk neutral case presented in section 2. However ifwe

allowthesaving to benegative at timesand add a penaltyto theobjective

function whenever we get a negative saving,we can introduce riskaversion

into the model. For this reason we need to have a goodestimate for

B

max

tn

. The risk neutral model can be solved to give us these estimates. Then we

can usethefollowing objective functionand budgetconstraints.

max

X

T

t=0

X

n∈N

t

p

_n

· d

_t

(B

_tn

max

− B

_tn

) − P R

_tn

· BO

_tn

(17)

B

_tn

max

+ BO

_tn

− B

_tn

≥ 0

∀n ∈ N

_t

, t

= 0, · · · , T

(18)

BO

_tn

≤ BO

max

_tn

∀n ∈ N

_t

, t

= 0, · · · , T.

(19) We allow crossing the budget limit in constraint (18) by introducing the

slack variable

BO

tn

4.3 Wealth risk aversion

So far we have only considered the marketrisk or theinterest rate risk. In

(13)

Wealthriskistheriskthattheactualoutstandingdebtbecomesbiggerthan

theexpectedoutstandingdebtatagiventimeduringthelifetimeoftheloan.

Forexamplesellinga30yearbondatapriceof80,wehaveabigwealthrisk

given that asmall fall inthe interest rate cancause a considerableincrease

inthebond price,which meansaconsiderableincreaseintheamount ofthe

outstandingdebt.

We consider the deviation from the average outstanding debt, which we

dene as

DX

tn

:

DX

_tn

= X

_t

−

X

i

X

_itn

,

∀n ∈ N

_t

, t

= 0, · · · , T,

where

X

t

isthe average outstanding debtfora giventime

t

:

X

_t

=

X

i∈I

X

n∈N

t

p

_n

· X

_itn

,

∀t = 0, · · · , T.

Apositivevalueof

DX

tn

means thatwe have asavinganda negative value meansalossascomparedtotheaverageoutstandingdebt

X

t

.Weintroducea surplusvariable

XS

tn

torepresent theamountofsavingandaslackvariable

XL

_tn

to represent theamount of loss:

X

_t

−

X

i

X

_itn

− XS

_tn

+ XL

_tn

= 0

∀n ∈ N

_t

, t

= 0, · · · , T,

(20)

To make the model both market riskand wealth risk averse we update the

objective functionwithweightedvalues of

XS

tn

and

XL

tn

asfollows:

max

X

n∈N

t

T

X

t=0

p

_n

· d

_t

(B

max

tn

− B

tn

) − P R

tn

· BO

tn

+ P W

n

· XS

tn

− NW

n

· XL

tn

,

(21)

where

P W

n

isaparameterwhichcanbeusedtoencouragesavingsand

N W

n

isaparametertopenalizealossascomparedtotheaverageoutstandingdebt.

Ifweset

P W

n

= NW

n

,itmeansthatthemodelisindierenttowardswealth risk.On the otherhand

P W

n

< N W

n

,means thatthemodelis wealth risk averse,sinceitpenalizesapotentiallossharderthanitencouragesapotential

(14)

0

1

2

3

4

5

6

7

8

9

10

100

200

300

400

500

600

700

800

900 1000

Time

Scenarios

Figure4:Abinomialscenariotreewith11stages.

5 Scenario reduction

Sincethenumberofscenariosgrowsexponentiallyasafunctionoftimesteps

thestochasticbinary modelis nolonger tractable when we have morethan

10timesteps.For an11stagemodelwehavethescenariotreeinFigure(4).

Asof today there areno general purposesolverswhich can solve stochastic

integerproblemsofthis sizeinareasonable amountoftime.Noticehowever

that a great number of nodes in the last 3-4 time steps have such a close

distance that a reduction of nodes for these time steps might not eect

therststage results. We are inother words interestedin ndinga wayto

optimallyreducethenumberofscenarios.Ifwegetthesamerststageresult

for a reduced and a nonreduced problem, it suces to solve the reduced

problem, and then at each step resolve the problem until horizon. In that

casethenalresultofsolvinganyofthetwoproblemswillbethesame.The

reasonforthisisthatweinitiallyonlyimplement therststagesolution.As

the time passes by and we getmore information we have to solve the new

problemand implement the newrst stage solutioneach time.

NicoleGröweKuska, Holger Heitsch,Jitka Dupa

ˇc

ová and Werner Römisch

(see [7 , 8 ]) have dened the scenario reduction problem (SRP) asa special

setcovering problemand have solvedit usingheuristic algorithms.

TheauthorsbehindtheSCENREDarticleshaveincooperationwithGAMS

SoftwareGmbHand GAMSDevelopmentCorporation,developeda

num-ber ofC++ routines, SCENRED, for optimal scenario reduction ina given

(15)

nario tree in Figure 5 is obtained after using the fast backward algorithm

of the GAMS SCENRED module for a 50% relative reduction, where the

relative reductionis measuredas anaverage of node reductions for all time

step. If we for example remove half of the nodes at the last time step, we

geta50%reductionfor thattimesteponly.Thenwemeasurethereduction

percentages for all other time steps in the same way. The average of these

percentages correspondstotherelativereduction(see[7,8]).Inourexample

thenumberofscenarios is reducedfrom 1024 to 12.

0

1

2

3

4

5

6

7

8

9

10

100

200

300

400

500

600

700

800

900 1000

Time

Scenarios

Figure5: Abinomialscenario treewith11stages aftera50% scenarioreductionusing

thefastbackwardalgorithmoftheSCENREDmoduleinGAMS.

We useGAMS/SCENRED and SCENRED modules for scenarioreduction,

and compare the results with those found by solving the LPrelaxed non

reducedproblem.

6 LP relaxation

Wheneverwerenancethemortgageportfolioweneedto payavariableand

a xed transaction cost. The variable cost is

100 · η

percent of the sum of thesold and purchased amount of bondsand thexed cost issimply DKK

m

(see constraint 8 and 9). The binary variables in the problem (1 to 10) aredue to incorporation of xed costs

m

.The numeric value of these xed costsisaboutDKK2500whereas

η

= 0.15%

.Whilethevalueofthevariable transaction costsdecreases asthetimepassesby,thexedcostsremainthe

(16)

percentage to the variable transaction costs, even if we let this percentage

increase asafunction of timeto adjust for thedecreasingoutstanding debt

ofthetotalloanportfolio.Wethereforesuggestaniterativeupdatingscheme

forthevariabletransactioncosts,sothatwecanapproximatethexedcosts

without using binary variables. We do that by iteratively solving the LP

+ 1

stiteration as follows:

B

_tn

=

X

i

A

_itn

+ r

_in

· (1 − γ)X

_itn

+

b

· (1 − β)X

_itn

+ η · (S

_itn

+ P

_itn

) + ψ

_itn

k+1

· S

_itn

∀n ∈ N

t

, t

= 0, · · · , T

(22)

SolvingtheLPproblem ateach iteration

k

we get

S

∗k

itn

astheoptimalvalue ofthesoldbondsat the

k

thiteration. Beforeeach iteration

k >

0

,theratio

ψ

k

_itn

is thenupdatedaccording to the following rule:

ψ

k

_itn

=

(

_m

S

∗k

itn

∀i, n ∈ N

t

, t

= 0, · · · , T

if

S

∗k

itn

>

0,

ψ

_itn

k−1

otherwise. (23)

This brings us to our approximation scheme for an LP relaxation of the

problem:

1. Dropthe xed costsand solve the LP relaxedproblem.

2. Find theratios

ψ

itn

according to(23).

3. Incorporate the ratios in the model so that DKK

m

is added to the objectivefunctionforeachpurchasedbond,giventhesamesolutionas

theone inthe last iterationis obtained.Solve theproblemagain.

4. Stop if the solution in iteration

k

+ 1

has not changed more than

α

percent as compared to the solution in iteration

k

. Otherwise go to step5.

5. Update

ψ

itn

accordingto (23). 6. Repeatfromstep 3.

Ourexperimentalresultsshowthatfor

α

' 2%

wendnearoptimalsolutions whichhavesimilarcharacteristicstothe solutionsfromtheoriginalproblem

(17)

We consider an 11 stage problem, starting with 3 callable bonds and 1 1

year bullet bond at the rst stage. We then introduce 7 new bonds every

3 years.An initial portfolio of loans has to be chosen at year 0 and it may

be rebalanced once a year the next 10 years. We assume that the loan is a

30yearloanand thatit isprepaid fullyat year11.

The24callablebondsusedinourtestproblemareseeninTable1.Thetable

only presents the average coupon rates and prices for these bonds at their

datesofissue.Notethatonlytherstthreebondshavealreadybeenissued,

so the start prices for these three bonds are market prices on 20/02/2004,

which isthe datefor the rststage inthestochastic program. Thenext 21

bonds arenot issuedyet, and we ndtheir estimatedprices at their future

datesof issue. Since thereare several statesrepresenting the uncertainty in

thefuturewehaveseveraloftheseestimatedstartprices.InTable1,however,

we only give an average of these prices. Besides these 24 callable bondswe

use a 1year noncallable bullet bond, bond 25. The eective interest rate

on this bond isabout 2%to start with. Using a BDT tree (see [5, 3]) with

theinputtermstructure given inTable2andannualsteps theeective rate

can increase to 21%or decreaseto slightly under1% at the10th year. The

termstructure ofTable2is from20/02/2004andisprovided bytheDanish

mortgagebankNykredit RealkreditA/S.The BDTtree hasalsobeen used

forestimatingthepricesandratesofthe24callablebondsduringthelifetime

ofthemortgage loanusingthe bond pricingsystemRio4.0(see [17 ]).

A practical problem arises when writing the GAMS tables containing the

stochastic data. The optimization problem is a path dependent problem,

whereastheBDTtreeispathindependent.GAMSisnotwellsuitedforsuch

programmingtasksasmapping the data from acombining binomial tree(a

lattice) to a noncombining binomial tree.A general purpose programming

language is better suited for this task. We have used VBA to generate the

input data to the GAMS model, and we have run the GAMS models on a

SunSolaris 9 machine witha 1200 Mhz CPU, 16 GB ofRAM and 4 GBof

MPS.

Thepurposeof our testscan be summarizedasthefollowing:

1. Comparing theresults of the4 versions of our modelwith simple sell

and holdstrategies.

2. Observing the eectsof usingthe GAMS/SCENRED module.

3. Tryingour LP approximationon theproblem.

For each of these objectives we consider all four versions of themodel and

(18)

1 6% 103.06 3/10-02 3/10-35 2 5% 98.5 3/10-02 3/10-35 3 4% 89.4 3/10-02 3/10-35 4 9% 107.33 3/10-05 3/10-38 5 8% 103.16 3/10-05 3/10-38 6 7% 103.09 3/10-05 3/10-38 7 6% 100.51 3/10-05 3/10-38 8 5% 94.01 3/10-05 3/10-38 9 4% 84.55 3/10-05 3/10-38 10 3% 74.46 3/10-05 3/10-38 11 9% 105.4 3/10-08 3/10-41 12 8% 101.98 3/10-08 3/10-41 13 7% 100.3 3/10-08 3/10-41 14 6% 96.19 3/10-08 3/10-41 15 5% 89.5 3/10-08 3/10-41 16 4% 80.74 3/10-08 3/10-41 17 3% 71.32 3/10-08 3/10-41 18 9% 104.41 3/10-11 3/10-44 19 8% 100.9 3/10-11 3/10-44 20 7% 98.51 3/10-11 3/10-44 21 6% 94.07 3/10-11 3/10-44 22 5% 87.49 3/10-11 3/10-44 23 4% 79.25 3/10-11 3/10-44 24 3% 70.26 3/10-11 3/10-44

Table1:Thecallablebondsusedasinputtotheproblem.

7.1 The original stochastic MIP problem

Figure6showsthesolutionsfoundfortherstthreestagesoftheproblemfor

all four instancesof our model, namely theriskneutral model, the minmax

model, the model with interest rate risk aversion with budget constraints

andnallythemodelwithinterestrateandwealthriskaversionwithbudget

constraints. Notice,however,thatnofeasiblesolutioncouldbefoundforthe

model with interest rate and wealth risk aversion with budget constraints

within atime limit of 10hours.

A full prescription of the solution with all 11 stages will not contribute to

a better understanding of the dynamics of the solution, which is why we

presentthesolutiontotherstthreestagesoftheproblemonly.Itisthough

enoughto giveus anindicationofthebehaviourofeachsolution. Intherisk

neutral casewestart bytakinga1yearadjustablerateloan.Iftheinterest

(19)

(Year) (%) (%) (Year) (%) (%) 1 2.23% 16 4.87% 17.25% 2 2.35% 32.20% 17 4.93% 17.00% 3 2.73% 32.10% 18 4.99% 16.85% 4 3.08% 29.50% 19 5.05% 16.75% 5 3.41% 27.00% 20 5.11% 16.70% 6 3.68% 25.00% 21 5.16% 16.65% 7 3.92% 23.00% 22 5.21% 16.60% 8 4.12% 22.00% 23 5.25% 16.56% 9 4.30% 20.90% 24 5.29% 16.52% 10 4.44% 20.10% 25 5.34% 16.48% 11 4.56% 19.40% 26 5.37% 16.45% 12 4.62% 18.80% 27 5.40% 16.42% 13 4.68% 18.30% 28 5.43% 16.39% 14 4.74% 17.90% 29 5.46% 16.36% 15 4.80% 17.55% 30 5.49% 16.34%

Table2:TheinputtermstructuretotheBDTmodel.

xedrateloan.Evenifitmeansanincreaseintheamountoftheoutstanding

debt, it proves to be a protable strategy since if the rates increase again

in the next stage we can reduce the amount of outstanding debt greatly

by renancing the loan to another xedrate loan witha higher price. The

minmaxstrategy chooses a xedrateloanwithaprice close to parto start

with.Thisloan isnot renanced until the9th stage oftheproblem.

The risk neutral and the minmax model represent the two extreme

mort-gagors as far as the risk attitude is concerned. The third model reects a

mortgagor witha riskattitude between therst two mortgagors. The

solu-tion to this model guarantees that the mortgagor will not pay more than

what his budget allows at any given node. Table 3 indicates the dierence

inthecharacteristics ofthe solutions forthe threedierent models.

Loanstrategy Totalcosts Std.dev. max min time

1-Riskneutral 1.281.857 92.289 1.502.042 1.004.583 276s

2-Minmax 1.353.713 19.729 1.374.183 1.117.084 10h

3-Int.rateriskaverse 1.288.405 66.019 1.431.857 1.005.412 10h

4-Int./Wealthriskaverse Nosolutionfoundwithin10h.

5-Loan25(ARM1) 1.310.495 115.085 1.821.388 1.120.053 <10s

6-Loan2(Fixedrate5%) 1.353.438 72.582 1.410.190 993.056 <10s

Table 3:Comparisonofthefourstrategiesfortheoriginalproblem.

The risk neutral model gives the lowest average total cost. The standard

(20)

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1128

p25=975

s1=916

p3=1107

Time (Year)

Scenarios

Risk neutral model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s2=1018

Time (Year)

Scenarios

Mimimax model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s2=1018

s3=1040

p2=1003

p2=987

s1=880

s25=952

p3=968

Time (Year)

Scenarios

Interest risk averse model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

NO SOLUTION WITHIN 10 HOURS

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure 6:Presentationof thesolutionsfor therst3stagesoftheproblem.Variable

s

isfor saleand

p

for purchaseandthe unitsaregivenin1000 DKK,so

s3 = 1128

means thatthemortgagorshouldsellapproximately1.128.000DKKatthegivennode.Theshort

ratesfromtheBDTtreeareindicatedusingtheletter

r

.

hasamuchsmallerstandard deviation. Thishigherlevelof securityagainst

variationhasthough anaveragecostof about 72000DKK. Thethirdmodel

hasreducedtheriskconsiderablywithouthavingincreasedthetotalaverage

costwithmore than about 7000 DKK.

We see also that these results outperform the simple sell and hold

strate-gies (strategies 5 and 6). A traditional market riskneutral mortgagor who

chooses an ARM1 loan and keep it until horizon (year 11) is better o

fol-lowing eitherstrategy 1or 3andatraditionalmarket riskaverse mortgagor

whochoosesaxedrateloanandkeepsituntilhorizonisbetterofollowing

eitherstrategy 2 or3.

Regarding thebudget constraints inmodel 3 and4 we use theconstantsin

Table 4. Note that we are reporting these budget constraints on an

aggre-gatelevel. Furthermore wedene theconstants

P P

max

Hn

and

P P O

max

Hn

asthe targetprepaymentamount andmaximumdeviationallowedfromthistarget

(21)

Constant Denition Value BMAX

P

_H−1

t=0

P

n∈N

t

p

n

· B

tn

max

570842 PPMAX

P

n∈N

H

p

n

· P P

Hn

max

711015 BOMAX

P

_H−1

t=0

P

n∈N

t

p

n

· BO

max

tn

50000 PPOMAX

P

n∈N

H

p

n

· P P O

max

Hn

100000

Table 4:Budgetlimitsusedinmodel3and4fortheoriginaldata.

Theseconstantsare chosen after considering theaverage payments and the

standarddeviations fromthese inthe riskneutral model.

The major problem with these solutions is the computing time taken to

ndnearoptimal solutions byCPLEX 9.0.Except for therst strategy,we

cannotndsolutionswithin1%ofalowerboundafter10hoursofCPUtime.

For the fourth strategy no feasiblesolution is found at all.Strategies 5and

6take afew seconds to calculate,howeverwe do not need theoptimization

modelfor thesecalculations.

7.2 The reduced stochastic MIP problem

Afterreducingthenumberofscenarios from1024to 12we getthesolutions

given inFigure7 and Table6.

Regarding thebudget constraints inmodel 3and 4 we usethe constants in

Table5.

Constant Denition Value

BMAX

P

_H−1

t=0

P

n∈N

t

p

n

· B

tn

max

565915 PPMAX

P

n∈N

H

p

n

· P P

Hn

max

601983 BOMAX

P

_H−1

t=0

P

n∈N

t

p

n

· BO

max

tn

50000 PPOMAX

P

n∈N

H

p

n

· P P O

max

Hn

35000

Table 5:Budgetlimitsusedinmodel3and4forthereduceddata.

Modeltype Totalcosts Std.dev. max min time

1-Riskneutral 1.169.173 49.765 1.274.079 1.064.525 12s

2-Minmax 1.187.938 0.00 1.187.938 1.187.938 52.2s

3-Int.rateriskaverse 1.171.926 24.270 1.229.897 1.136.655 300s

4-Int./Wealthriskaverse 1.172.479 25.610 1.229.742 1.128.412 300s

5-Loan25(ARM1) 1.301.237 120.958 1.560.244 1.129.983 <1s

6-Loan2(Fixedrate5%) 1.356.228 59.356 1.410.190 1.249.483 <1s

(22)

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s1=1009

p25=953

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Risk neutral model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s3=144

s25=868

s2=264

p25=245

s3=877

p25=846

s6=340

p2=259,p3=138

s3=924

p25=259

s25=907

p3=999

s25=395

p3=402

Time (Year)

Scenarios

Mimimax model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s1=380

p25=358

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Interest risk averse model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s1=378

p25=356

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure 7: Presentation of the solutions for the rst 3 stages of the reduced problem.

Unitsaregivenin1000 DKK.

We can see inTable 6 that the behaviour of the solutions for the dierent

models is similar to that of the original problem. Notice also that we get a

feasiblesolutionhereforthefourthmodelwithinterestrateandwealthrisk

aversion.

Thenumericvaluesofthetotalcostsfortherstfourstrategieshavehowever

decreased considerably. Except for therisk neutral modelwe do not obtain

the same rst stage solutions as we saw for the original problem. It seems

thatthereducedproblemgivesamoreoptimisticview ofthefutureas

com-paredto the original problem. By testingthe scenarioreduction algorithms

fordierentlevelsofreductiononourproblemwenoticethatevenmuchless

aggressive scenario reductions do not guarantee that thesame initial

solu-tions as found for the original problem are found. Oneexplanation for this

more optimistic view of the futureis that since scenario reduction destroys

the binomial structure of the original tree, signicant arbitrage

opportuni-ties ariseinpartsof the newtree structure.Another explanation isthatfor

(23)

We thereforeneed a methodwhich 1) optimally reduces thenumber of

sce-narios while the tree remains balanced and 2) modies bond prices in the

reduced tree so that the arbitrage opportunities which are introduced as a

resultofscenario reductionareremoved.

Thequestionhereiswhetherpoints1and2playanequallyimportantrolein

gettingsimilarrststagesolutions fortheoriginalandthereducedproblem.

Comparing strategies 5 and 6 in Tables 3 and 6 indicates that performing

point two might remove most of the dierence between thesolutions inthe

reducedproblemascomparedto theoriginalproblem. Apparently the

aver-agetotal costsforthe ARM1 loanareslightlydecreased inthereducedtree

whereastheaveragetotalcostsforthexedrateloanareslightly increased.

This slight change in opposite directions can only be explained by the

ob-servation that the reduced tree has an overweight of scenarios with lower

interestrates,since notrading is allowed for these two strategies and

there-fore the arbitrage opportunities cannot be used. We are currently working

onbetter ways of reducing scenario trees takinginto accountspoints1 and

2.

7.3 The reduced and LPapproximated problem

When we use our LPapproximation algorithm on this problem we get the

solutionaspresentedinTable 7and Figure8.

The algorithm uses 1018 runs for the dierent problems to nd solutions

which areoveralllessthan 2%dierentfromthesolutions foundinthelast

iteration.

Modeltype Totalcosts Std.dev. max min time

1-Riskneutral 1.169.147 49.775 1.274.078 1.064.524 25s

2-Minmax 1.179.654 11.150 1.185.795 1.154.602 22s

3-Int.rateriskaverse 1.172.364 26.436 1.239.168 1.130.196 28s

4-Int./Wealthriskaverse 1.174.038 29.128 1.249.520 1.131.185 44s

5-Loan25(ARM1) 1.301.237 120.958 1.560.244 1.129.983

6-Loan2(Fixedrate5%) 1.356.228 59.356 1.410.190 1.249.483

Table7:ComparisonofthefourstrategiesforthereducedproblemwithLP

approxima-tion.

It is important to point out that simply dropping thexed costs results in

solutionswhichdeviateconsiderablyfromtheproblemswiththexedcosts,

whereasapproximatingthexedcostsusingouralgorithm givesverysimilar

(24)

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s1=1009

p25=953

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Risk neutral model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s2=174

s25=829

s3=203

p25=175

s3=102

p25=808

p2=171

s25=159

p2=169,p3=22

s3=681

p25=618

s25=905

p3=997

Time (Year)

Scenarios

Mimimax model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s2=441

p25=372

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Interest risk averse model

0

1

2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=1.66%

s25=1000

s3=1011

p25=975

s2=552

p25=465

s3=1049

p25=953

s25=900

p3=992

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure 8: Presentation of the solutions for the rst3 stages of the LP approximated

reducedproblem.Unitsaregivenin1000DKK.

7.4 Comments on results

The results presented in this section are in agreement with the nancial

arguments used in the Danish mortgage market. Even though the original

problemis hard to solve we have shown thatuseful results can be found by

solvingthereducedproblems.Thereducedscenariotreesrepresentedamore

optimisticpredictionofthefuture,buttheresultsfoundarestillquiteuseful.

Inpracticethemortgageportfolio managershould tryseveralscenariotrees

with dierent risk representations as an input to the model. This way the

optimization modelcan be usedasan analyticaltoolfor performingwhat

(25)

Wehave developed afunctionaloptimization modelthatcanbeusedasthe

basis for a quantitative analysis of the mortgagors decision options. This

modelin conjunction with dierent term structures or marketexpert

opin-ions on the development of bond prices can assist market analysts in the

following ways:

Decisionsupport:Insteadofcalculatingtheconsequencesofthesingleloan

portfoliosforsingleinterestratescenarios,theoptimizationmodelallowsfor

performingwhatifanalysisonahigherlevelofabstraction.Theanalystcan

provide the systemwith dierent sets of information such asthe presumed

lifetimeof theloan, budgetconstraints and riskattitudes.The systemthen

nds the optimalloan portfoliofor each setof inputinformation.

Product development: Traditionally, loan products are based on single

bondsor bondswithembedded options. In some mortgagemarkets such as

theDanishone itisallowed to mixbondsinamortgageportfolioandthere

are even some standard products which are based on mixing bonds. The

product

P

33

isforexamplealoanportfoliowhere 33%oftheloanisnanced in3yearnoncallablebondsandtherestinxedratecallablebonds.These

mixedproductsarecurrently notpopularsince therationale behindexactly

this kind of mixis not well argued. The optimization model givesthe

pos-sibility to tailor mixed products that, given a set of requirements, can be

arguedto be optimal for acertainmortgagor.

Thegreatest challenge insolving the presentedmodelsis on decreasing the

computing times. We have experimented with scenario reduction (scenred,

[7,8,9])andwe have suggestedanLPapproximation methodto reducethe

solution times while maintaining solution quality. It is, however, an open

problem to develop tailored exact algorithms such as decomposition

algo-rithms (see [1 , 2]) to solve the mortgagors problem. Another approach for

gettingreal time solutions is to investigate dierent heuristic algorithms or

make useof parallel programming (see [15,16 ]) to solve theproblem.

Integrationofthetwodisciplinesofmathematicalnanceandstochastic

pro-grammingcombinedwithuseofthestateoftheartsoftwarehasa great

po-tential,whichhasnotyetbeenrealizedinallnancialmarketsingeneraland

inmortgagecompanies inparticular. There is a need for more detailedand

operational models andhighperformingeasy to useaccompanying software

topromote useof the mathematical modelswithspecialfocuson stochastic

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We wouldlike to thank Nykredit Realkredit A/S for providing us withthe

termstructure andmortgageprice data.We arealsogratefultothereferees

for many good comments on an earlier version of this paper and to Rolf

Poulsen for reading and commentingthe revised versionof thepaper.

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