MATHEMATICAL MODELS FOR MICROLENDING

Francine Diener

Description

MATHEMATICAL MODELS FOR MICROLENDING Francine Diener ? Marc Diener ? Osman Khodr ? Philip Protter † Proceedings of the 16th Mathematical Conference of Bangladesh Mathematical Society 17-19 December, 2009, Dhaka, Bangladesh Abstract Microlending has not yet been placed on a firm mathematical foundation, in contrast to the highly developed theory of Mathematical Finance. Here we propose a first step, modeling a slightly simplified procedure than the one actually used, as a Markov chain. Using this model we compute the expected benefit each borrower gains from his or her activity. We compute the distribution of the beneficiaries among the population involved, and discuss the resultant equilibrium, as well as the issue of strategic defaults. Our proposed model builds on the pioneering work of 2006, by G.A. Tedeschi. 1 Introduction The mathematical formulation of microcredit is in its infancy, in stark contrast to the highly developed theory of Asset Pricing, and even of Credit Risk. We take a first step here, by proposing a Markov chain model to formalize the dynamic model of microcredit lending. In doing so, we recover some of the basic results and formulas of G.A. Tedeschi [3] , who obtained them through economic reasoning alone. Our model leads naturally to an optimization problem which is for the lender to choose the optimal time of exclusion with regards to a given borrower.

• has nothing


• bangladesh mathematical


• rational risk-neutral


• simplified model


• markov chain


• probability ?


• microcredit loan


• can dynamic


• distribution pi?


• been shown

Sujets

• Cornell University
• Diener
• Markov chain

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

Extrait

Proceedingsofthe16thMathematicalConferenceofBangladeshMathematicalSociety
17-19December,2009,Dhaka,Bangladesh

MATHEMATICALMODELSFORMICROLENDING

FrancineDiener
∗
MarcDiener
∗
OsmanKhodr
∗
[e-mail]@[e-mail]
PhilipProtter
†
Abstract
Microlendinghasnotyetbeenplacedonafirmmathematicalfoundation,in
contrasttothehighlydevelopedtheoryofMathematicalFinance.Herewepropose
afirststep,modelingaslightlysimplifiedprocedurethantheoneactuallyused,as
aMarkovchain.Usingthismodelwecomputetheexpectedbenefiteachborrower
gainsfromhisorheractivity.Wecomputethedistributionofthebeneficiaries
amongthepopulationinvolved,anddiscusstheresultantequilibrium,aswellasthe
issueofstrategicdefaults.Ourproposedmodelbuildsonthepioneeringworkof
2006,byG.A.Tedeschi.

1Introduction
Themathematicalformulationofmicrocreditisinitsinfancy,instarkcontrasttothehighlydeveloped
theoryofAssetPricing,andevenofCreditRisk.Wetakeafirststephere,byproposingaMarkovchain
modeltoformalizethedynamicmodelofmicrocreditlending.Indoingso,werecoversomeofthebasic
resultsandformulasofG.A.Tedeschi[3],whoobtainedthemthrougheconomicreasoningalone.Our
modelleadsnaturallytoanoptimizationproblemwhichisforthelendertochoosetheoptimaltimeof
exclusionwithregardstoagivenborrower.Thisarisestoavoidabusebytheborrower,whichstems
ultimatelyfromacompletelackofcreditratingsandthepossibilityofpostedcollateral.(Thislackof
creditratingsisintegraltomicrocredit,whichtriestoextendbeneficiallendingpracticesinafinancially
primitivesetting.)Microcredithasbeenshowntobesustainableinpractice.Indeeditscreatorand
mainpromoter,MuhammadYunus&GrameenBankofBangladeshreceivedtheNobelPeacePrize.
MicrofinancehasreceivedsignificantattentionintheEconomicsliterature.Seeforexample[2]and
thereferencestherein.Thereforeitisabitsurprisingthatweknowofnopreviousattemptstomodel
microcreditwithmodernmathematicaltools.

2DescriptionoftheModel
Oursimplifiedmodelisasfollows:apotentialborrowcanborrowoneunitoveraperiodoftime
t
.At
time
t
theborrowerisobligatedtorepaytheoneunit,plusinterest.Thuswehavethescheme1
→
1+
r
.
Theamount
r
willdependontheinterestratecharged,andthetimeduration
t
.Forthecurrencywe
couldtaketheBangladeshiTaka,butforourmodeltheactualcurrencyused(ornume´raireinfinancial
parlance)isirrelevant.Theborrowerisexpectedtoinvestthe1(Taka
1
)inabusinessproposition,
resultinginanamount
w
attime
t
.If
w>
1+
r
thentheborrowercanrepaytheloan,andwecallthis
a
success
.Aborrowerisassumedtobesuccessfulwithafixedprobability
α
.Wecallthesetwostates
A
(forapplicant)and
B
(forBeneficiary).Notallapplicantsreceiveloans,thusonedoesnotmovefrom
state
A
tostate
B
withcertainty.
∗
Universite´deNiceSophia-Antipolis,LaboratoiredeMath´ematiquesJ.Dieudonne´,ParcValrose,06108Nicecedex2
†
SchoolofOperationsResearchandInformationEngineering,CornellUniversity,219RhodesHall,Ithaca,NY14853.
SupportedinpartbyNSFGrantDMS-0906995.
1
or,morerealistically,one
thousand
Taka,asintheoriginalloandescribedbyM.Yunusin[4]

1

Ifshe
2
issuccesful,sheisentitledtoget(withcertainty)anewloanof1,sosheisagaininthestate
B
.Ifnot,sheentersa
credit-exclusionperiod
oflength
T
atleast.Acountdownof
applicationstates
proceeds,
A
T
,
A
T
1
,
...A
1
,andaslongsheisinstate
A
i
,with
i>
1,shecannotgetanyloan.After
T
1stepssheobtains(forsure)
A
1
=:
A
whenshecanapplyagain,withprobability
γ
tobecome
beneficiary
B
ofaloanatthenextstep.Withprobability(1
γ
)shewillstayapplicantforthenextstep
andcanapplyagainwiththesamechancestobecomebeneficiary.Pleaseobservethattheoutcome(
B
stays
B
orbecomes
A
T
,
A
1
becomes
B
orstays
A
1
)isknownattheendofthetimeperiod,sobecoming
A
T
impliesawaitof
T
timestepstoobtainthepossibilityofanewloan.
TheserulescanbesummarizedinaMarkovchain(
X
t
)
t
∈
N
with
X
t
∈S
:=
{
B,A
1
,A
2
,...,A
T
}
=
{
A
0
,A
1
,A
2
,...,A
T
}
,letting
A
0
:=
B
.ThetransitionmatrixofthisMarkovchainisgivenby
α
00
∙∙∙
01
α
γ
1
γ
0
∙∙∙
00
010
∙∙∙
00
P
=001
∙∙∙
00where
...000
∙∙∙
10
P
(
X
t
+1
=
B
|
X
t
=
B
)=
α
(succesfulbeneficiary)
P
(
X
t
+1
=
A
T
|
X
t
=
B
)=1
α
(unsuccesfulbeneficiary,leadstocreditexclusionforatleast
T
)
P
(
X
t
+1
=
A
i
1
|
X
t
=
A
i
)=1,
i
=2
...,T
(countdownofcreditexclusionperiod)
P
(
X
t
+1
=
B
|
X
t
=
A
1
)=
γ
(applicantgetsaloan)
P
(
X
t
+1
=
A
1
|
X
t
=
A
1
)=1
γ
(applicantfailstogetaloanandstaysanapplicant)
AssoonasaMarkovchainmodelizessomedynamic,itisnaturaltocheckifitadmitsalimit
stationarydistribution.Inourcase,thereisoneanditgivesthelimitdistributionofthetotalpopulation
intothedifferentstates
B
,
A
1
,...,
A
T
.
Proposition1
Foranyinitialstatedistribution
π
0
=(
π
00
,...,π
0
T
)
theMarkovdynamictendstothe
distribution
π
∗
,with
1π
∗
=(
γ,
1
α,γ
(1
α
)
,...,γ
(1
α
))
.
(1)
(1
α
)(1+
γ
(
T
1))+
γ

Proof:
Thematrix
P
isstochasticand1isitslargest(dominant)eigenvalue.Itiseasytocomputean
associated(dominant)lefteigenvector
π
∗
withpositivecoefficientsaddingupto1.As
P
isprimitive,the
Perron-Frobeniustheoremshowsthatlim
n
→
+
∞
π
0
P
n
=
π
∗
.
✷
Noticethatthedistribution
π
∗
=(
π
∗
0
,π
∗
1
,...,π
∗
T
)isastationarydistribution.Thismeansthat,at
theequilibrium,if
N
isthetotal(largeandfixed)numberofpotentialborrowersinvolved(i.e.inone
ofthestates
B
,
A
1
,...,
A
T
),then,inviewofthelawoflargenumbers,
π
∗
0
N
istheactualnumberof
beneficiarieswhereas(1
π
∗
0
)
N
isthenumberofthepeopleinvolvedwaitingforaloan.Itisnowpossible
tobuildupaprescribed
dynamic
increasingnumber
N
(
t
)ofinvolvedpotentialborrowersor,similarly,a
prescribeddynamicnumber
b
(
t
)ofactualbeneficiariesofaloan.Itsufficetoaddnewcommersineach
statesinordertoputthenumberofpeopleineachstatesto
N
(
t
)
π
∗
.Thiscanbeusefulinordertomeet
somepredeterminedsocial-businessplanofanincreasingnumberofbeneficiaries,takingadvantageof
thenecessarywaitingtimetoinvolvethecandidatesinsomepreparatoryactivity.

3Computingtheexpectedtotaldiscountedreturn
Letusnowdescribethe(linear)modelfortheactivityrelatedtoaloan.Whenlent1,theborrowercan
enteraproductionactivitythatwillproduce,ifnothingbadhappens,anincomeof
w
duringtime1,for
whichshewillhavetopay1+
r
,principalplusinterest:thisiswhattakesplacewithprobability
α
.
Butifsheisunlucky,herincomeis0andshehasnothingtoreimburseforthelent1(thisisthespecific
featureofmicrocredit)andisexcludedofanycreditforatimeperiod
T
atleast.Sothenetincome
availableattime
t
fortheactivityrelatedtotheloan1lentattime
t
1isafunction
f
(
X
t
1
,X
t
)with
f
(
B,B
)=
w
(1+
r
),and
f
(
x,y
)=0forallother(
x,y
)than(
x,y
)=(
B,B
).
2
AsmostoftheGrameenBank’sborrowersarewomen,wehavechosenheretouseafemininpronoun
fortheborrowers.

Letusconsider,forany
s
≥
0,the
expectedtotaldiscountedfutureincome
W
s
! ∞XW
s
=
E
δ
ts
f
(
X
t
1
,X
t
)
|F
s
,
1+s=twhere
F
t
=
σ
(
X
0
,...,X
t
)isthefiltrationassociatedwith
X
t
and
δ
∈
(0
,
1)the(fixed)
discountf