MATHEMATICAL MODELS FOR MICROLENDING

Proceedingsofthe16thMathematicalConferenceofBangladeshMathematicalSociety
17-19December,2009,Dhaka,Bangladesh

MATHEMATICALMODELSFORMICROLENDING

FrancineDiener
∗
MarcDiener
∗
OsmanKhodr
∗
diener@unice.frdiener@unice.frosman.khodr@unice.fr
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