a theory of credit scoring and competitive pricing of default riskvr0j/slides/workshpcdv.pdf ·...

A Theory of Credit Scoring and Competitive Pricingof Default Risk

Satyajit Chatterjee Dean Corbae Jose Vıctor Rıos-Rull

Philly Fed, University of Wisconsin, University of MinnesotaMpls Fed, CAERP, CEPR, Oslo

Labor Workshop, April 3, 2012

Chatterjee, Corbae, Rıos-Rull Philly Fed, Wisconsin, MN

Credit Scoring Labor Workshop April 3, 2012 1/43

Goal

• Develop a competitive quantitative-theoretic model of unsecured consumercredit where:

1 borrowers can legally default,

2 the punishment for default is a drop in the credit score or perceivedcreditworthiness,

3 and the theory is consistent with other key credit scoring facts.

• Use the model as a laboratory for evaluating regulations regardinginformation use by creditors



Outline

1 Key properties of credit scores

2 Informal description of the model

3 Mapping the model to data

4 Properties of the model

5 Welfare consequences of restriction on information that can be used tocondition a credit score



Key Properties of Credit Scores

1 A credit score is an index of the probability of repayment on a loan

2 A score is based mostly on payment behavior and amount borrowed

3 Low score raises interest rate and/or limits access to credit MustoFig

4 Record of default lowers score, removal of record raises it

5 Increasing/decreasing indebtedness lowers/raises score

6 Scores are mean reverting MustoMR



Credit Scores and Delinquency Rates

model



FICO Scores

• Lenders assess creditworthiness of borrowers using FICO credit scores (higherscore, higher creditworthiness)

• Over 75% of mortgage lenders and 80% of the largest financial institutionsuse FICO scores.

• FICO scores are calculated from data in the individual’s credit report in fivebasic categories: PieChart

• Payment history (35%) – includes adverse public records• Amounts owed (30%)• Length of credit history (15%)• Credit limits (10%) and types of credit used (10%)

• Ignores:

• Race, color, national origin, sex, and marital status (prohibited by law)• Age, assets, salary, occupation, and employment history.



Model

• Infinite horizon, discrete time model with uninsured idiosyncratic iid shocksto endowments and preferences

• 2 types of people (g and b): Type affects preferences and the distributionsfrom which iid shocks are drawn; follows a persistent Markov process

• People can save or borrow to smooth consumption; if they borrow they havethe option to default; (no pecuniary costs or exogenous restriction on abilityto borrow)

• Neither type nor shocks are directly observable to lenders; lenders can onlysee an individual’s credit market transactions (including default) going backT periods

• Lenders accept deposits that pay the risk-free rate and extend non-contigentloans at an interest rate that exactly covers the expected loss from default



Type Score and Credit Score

• Lenders observe a person’s credit market behavior and assess the likelihoodthat the borrower will be of type g next period – this probability is labeledthe type score

• The credit score is the probability of repayment on a loan

• Since the propensity to default is closely related to type, the type score is onekey input into the construction of a person’s credit score; the other key inputis the amount borrowed



Some Related Work

• Bankruptcy: Athreya (2002, JME ), Chatterjee, et.al. (2007, ECTA),Livshits, et.al. (2007,AER)

• Reputation and Signalling: Cole, et.al. (1995, IER), Chatterjee, et.al. (2008,JET ), Elul and Gottardi (2007), Athreya, Tam, and Young (2010), Sanchez(2008)



People

• Unit measure of people comprising of two types i ∈ {g, b};Γi′ i = Pr{it+1 = i′|it = i}.

• A person of type i draws iid endowment e and iid time preference shock θ infrom distributions

• Φi with support E = [e, e]

• Λ with finite support Θ contained in [0,1]

• Type can also affect preferences ui(c) and βi.



Intermediaries

• Competitive credit industry in one period discount bonds:

• accepts deposits y > 0 at price 1/(1 + r)

• makes loans y < 0 at price q(p) where p is the probability of repayment of theloan.

• To determine p, lenders assess the probability that a person will be of type gat the time the loan is due

• s is the prior probability that a person is of type g

• s′ = ψ(d,y)(x, s) is the posterior probability that a person who takes action(d, y) in state (x, s) is of type g next period

• p(y, s′) is the credit scoring function and s′ = ψ(d,y)(x, s) is the type scoringfunction



Information

• i, e, θ, or c are not observable.

• The default decision d ∈ {0, 1} and asset choice y ∈ L are observable.

• Lenders use information (d, y) over time to infer the probability that a personis currently of type g.



Timing

• Enter period with (x, s)

• Type, earnings, and preference shock (i, e, θ) are realized

• Borrowers choose whether to default

• If don’t default, choose next period asset y

• Exit with updated type score s′ = ψ(d,y)(x, s)



Recursive Formulation of the IndividualProblem

• The set of feasible actions is a finite set B(e, x, s; q, p, ψ) such thatc = e+ x− q(p) · y ≥ 0.

• We permit randomization over feasible actions: m(d,y) ∈ [0, 1] denotes theprobability mass on the element (d, y) and m is the associated vector.

• We assume that all budget feasible actions are chosen with at least somesmall probability (i.e. people make tiny mistakes as in Selten).

• Together with an assumption on primitives (e+ xmin − ymax > 0), this willkeep the Bayesian updating function well-defined (and avoid supplyingoff-the-equilibrium path beliefs).



Recursive Formulation of HH Problem Cont.

The current return function is given by

R(0,y)i (e, x, s; q, p, ψ) =

{ui(e+ x− q(p(y, ψ(0,y)(x, s)) · y)) if d = 0ui(e) if d = 1

The value function is given by

Vi(e, θ, x, s) = maxm∈Mi

∑(d,y)

[R

(d,y)i (e, x, s) + βiθWi(y, ψ

(d,y)(x, s)))]·m(d,y) (2)

where

Wi(x, s) =∑

j∈{g,b},θ

Γj i

∫E

Vj(e, θ, x, s)Φj(de)Λ(θ) ∀i ∈ {g, b}

• The optimal decision correspondence is denoted M∗i (e, θ, x, s; q, p, ψ) and agiven selection from this correspondence is denoted m∗i (e, θ, x, s; q, p, ψ).



Intermediary’s Problem

The zero profit condition on a financial contract of type (y, p) implies:

π(y, p) = 0⇔{q(p) = p/(1 + r) if y < 0q(1) = 1/(1 + r) if y ≥ 0

(3)

More



Intermediary’s Problem Cont.

The credit scoring function is given by

p(y, s′) =s′ ·

[1−

∑θ′

Λ(θ′)P (1,0)g (θ′, y, s′; q, p, ψ)

]

+ (1− s′) ·

[1−

∑θ′

Λ(θ′)P(1,0)b (θ′, y, s′; q, p, ψ)

],

(4)



Intermediary’s Problem Cont.

The (Bayesian) type scoring function is given by

s′ = ψ(d,y)(x, s; q, p, ψ) =

Γgg

[ ∑θ Λ(θ)P

(d,y)g (θ, x, s)s∑

θ Λ(θ)P(d,y)g (θ, x, s)s+

∑θ Λ(θ)P

(d,y)b (θ, x, s)(1− s)

]

+ Γgb

[ ∑θ Λ(θ)P

(d,y)b (θ, x, s)(1− s)∑

θ Λ(θ)P(d,y)g (θ, x, s)s+

∑θ Λ(θ)P

(d,y)b (θ, x, s)(1− s)

] (5)



Equilibrium

A recursive competitive equilibrium is: (i) a pricing function q∗(p); (ii) a creditscoring function p∗(y, s′); (iii) a type scoring function ψ∗ (d,y)(x, s); and (iv)decision rules m∗i (e, θ, x, s; q

∗, p∗, ψ∗) such that

1 m∗i (e, θ, x, s; q∗, p∗, ψ∗) is a selection from M∗i (e, θ, x, s; q∗, p∗, ψ∗) which

solves the agent’s DP problem in (2),

2 q∗(p) yield zero profits π(y, p; q∗(p)) = 0 in (3) ∀ (y, p) ,

3 The credit scoring function p∗(y, s′) is consistent with repayment fractions in(4) for m∗i (e, θ, x, s; q

∗, p∗, ψ∗), i ∈ {g, b},4 The type scoring function ψ∗ (d,y)(x, s) satisfies a version of Bayes rule (5)

for m∗i (e, θ, x, s; q∗, p∗, ψ∗), i ∈ {g, b}.



Existence of Equlibrium

• Since for every borrowing level y, q∗ is just a linear function of the repaymentprobability p∗, we apply Schauder’s fixed point theorem to the credit scoringfunction p∗ and the type scoring function ψ∗.

• Key part of proof is establishing that P(d,y)i , which depends on decision rules,

is Lipschitz in s.

• Proof uses the fact that earnings distribution is continuous and that theaction set ({0, 1} × L) is finite.



Model with Information Restrictions

• There are more restrictions on information used in FICO scores than assumedabove; no asset holdings or adverse events past T periods.

• Denote an individual’s finite history byhT = (d−1, x−1, d−2, ..., x−(T−1), d−T ).

• To account for information assumptions as above, we introduce partitions(measurability restrictions):

• H(x, hT ) is the partition block in which (x, hT ) belongs• A(y, d) is the partition block in which (y, d) belongs

• An individual’s state space is now (i, e, θ, x, hT ).

• µ∗i (e, θ, x, hT ) is the equilibrium measure of type i people over the state

space Example

• The only real difference is that partitions require the population distributionµ∗i (e, θ, x, h

T ) to construct priors and assessments must “condition out”unobservable positive asset choices.



Mapping Model to Data

• Model period is 5 years and memory is 2 years

• Discount factor, β = 0.99.

• The utility function is u(c) = c1−ϕ

1−ϕ .

• Time preference shock Θ = {0, 1}.

• Probability of choosing a sub-opitmal action ε = 0.0001.

• L = {x, 0, x1, x2}



Mapping Model to Data Cont.

• Earnings of type i is Beta-distributed e ∼ Be(νi, ηi).

Table: Earnings Statistics (PSID 1996-2001) and Parameter Values

Statistics Target Model Parameter EstimateGini index 0.54 0.50 νb 1.0153 (0.0616)Mean/median 1.40 1.21 ηb 24.4051 (2.1358)Autocorrelation 0.67 0.60 νg 2.6570 (0.1440)1st quintile share 0.17 0.99 ηg 4.0642 (0.2208)2nd quintile share 6.77 4.52 Γgb 0.0149 (0.0009)3rd quintile share 14.73 16.30 Γbg 0.0104 (0.0007)

“Percentage of yth quintile” is endowments received for agents within yth quintile over total endowments.

• (Γbg,Γgb) imply 59% of agents are type g.

• (νi, ηi) imply type g earns 10 times more on average than type b.



Mapping Model to Data Cont.

Table: Model Statistics (TransUnion and SCF) and Parameter Values

Statistics Target Model Parameter EstimatesOverall delinquency rate 29.23% 31.28% x -0.0033Subprime (bottom 27%) del. rate 75.74% 54.56% x1 0.1078Debt to earnings ratio 0.002 0.001 x2 0.5683Asset to earnings ratio 1.36 1.35 Λ(0) 0.0500Percentage in debt 6.7 5.4 ϕ 6.4618

• While the model matches the overall delinquency rate well, it fails to accountfor all of the subprime delinquency rate (72%).



Equilibrium Decision Rules

• When θ = 0 (i.e. people are temporarily myopic), anyone with debt defaultsand anyone without debt borrows.

• When θ = 1,• With debt,

• type g default for low earnings or save (to x1 or x2) with higher earnings• type b default for a larger set of low earnings or save to x1

• With zero assets,• type g continue with zero assets or save (to x1 or x2)• type b borrow when earnings are very low, continue with zero assets for

intermediate earnings, or save to x1 at high earnings

• With savings, both types continue to save.

P vs Psi



Model distribution of Credit Scores

• As in the data, the distribution puts more weight on high scores which havelower likelihood of default. data



Credit Scoring Fact: Low scores raiseinterest rates



Credit Scoring Fact: Default lowers score

• Occurs because type b are more likely to default than type g.

• Consistent with the fact that “Someone that had spotless credit and a veryhigh FICO score could expect a huge drop in their score. ... someone withmany negative items already listed on their credit report might only see amodest drop in their score” (FICO).

• On average, credit scores drop by 48% after default (from 0.82 to 0.43).



Credit Scoring Fact: Removal of default flagraises score

• Equilibrium decisions imply that the action (d, y) = (0, 0) by agents with(x, 0, 0, 1) and ({x}, 0, 0, 1) arise as trembles.

• Removal of the default flag jumps hh’s credit score ahead of 1.2% of thedistribution in the model versus 5% in Musto’s data.



Credit Scoring Fact: Decreasing indebtednessraises score

• Red bars correspond to optimal actions, while blue bars correspond totrembles.

• On average, credit scores rise by 59% after hhs pay off debt (from 0.49 to0.78).



Credit Scoring Fact: Increasing indebtednessdecreases score

• Since borrowing generally arises when hit with θ = 0 and θ shocks are iid,assessment following borrowing rises since the population proportion of goodtypes is 0.59.

• On average, credit scores rise by 3% after hhs go into debt (from 0.76 to0.78).



Credit Scoring Fact: Mean Reversion

• Slope coefficient for the fitted line equals 0.8.



Welfare Effects of Restricting Information

• In a world of incomplete markets and private information, restrictinginformation flow may be welfare improving

• Question: how much would a household of type i in state (x, hT ) be willingto pay to be in a regime where there are no information restrictions?

Table: CE by types and shocks

θ\i g b1 0.0420e-3 -0.5266e-30 0.0650e-3 -0.1072e-3



Welfare Effects Cont.

• For each (i, e, θ, x, hT ) we compute compensating consumption variationsλi(e, θ, x, h

T ) that satisfy

Vi(e, θ, x, sT (x, hT )) = (1 + λi(e, θ, x, h

T=2))1−ϕV (i, e, θ, x, hT=2)

• The aggregate welfare gain/loss is given by∑i,e,θ,x,hT=2

λi(e, θ, x, hT=2)µ(i, e, θ, x, hT=2).

• We find an aggregate welfare loss of -0.0001, since type b must becompensated more than type g gains from removing information restrictions.

Table: CE by types and shocks

θ\i g b1 0.0420e-3 -0.5266e-30 0.0650e-3 -0.1072e-3



Conclusions

• We provide a theory where lenders learn from an individual’s borrowing andrepayment behavior about the agent’s unobservable characteristics andencapsulates this in a credit score.

• After choosing a sparse set of parameters to match some key credit marketdata moments, we show the theory is broadly consistent with the way creditscores affect unsecured consumer credit market behavior.

• We show that for that set of parameters, aggregate welfare would be lower ifinformation restrictions were removed.



Limited Credit Access Following Default:Change in Credit Limit of Open Bank Cards

over yth Postdischarge Year

Back



FICO Score Inputs

Back



Examples of partitions: T = 1

• Suppose the assets choices are L = {`−, 0, `1+, `2+} with `− < 0 < `1+ < `2+.• The action space is

L×HT=1 = {(0, 1), (`−, 0), (0, 0), (`1+, 0), (`2+, 0)}.

• The state/history tuple is

H1 = {(0, 1)}H2 = {(`−, 0)}H3 = {(0, 0)}H4 = {(`1+, 0), (`2+, 0)}

• The partition block is

A1 = {(0, 1)}A2 = {(`−, 0)}A3 = {(0, 0)}A4 = {(`1+, 0), (`2+, 0)}

Back



• Each score from the bankruptcy filing group is mapped to a “FICOpercentile”, which is the percent of scores in the non-filing contrast groupbelow that score.

• E.g. [0, 10) means that less than 10% of the contrast group have scoresbelow the filing group in that bin.

Back



Ausubel

• Data from randomized pre-approved solicitations allowed access toindividual’s credit bureau info.

• Adverse selection on observable characteristics (like credit scores): pool ofconsumers who accept an inferior contract (shorter introductory rates)exhibit inferior characteristics.

• E.g. credit scores of respondents to solicitations are 523 whilenonrespondents are 643.

• Adverse selection on hidden information: even after controllinig forobservables, the pool who accept inferior contracts default more than thepool who accept a better offer.

Back



Intermediary’s Problem

The profit π(y, p) on a financial contract of type (y, p) is:

π(y, p) =

{(1 + r)−1p · (−y)− q(y, p) · (−y) if y < 0

q(y, 1) · y − (1 + r)−1 · y if y ≥ 0(3)

Let a(y, p) be the measure of financial contracts of type (y, p) sold by theintermediary. The intermediary solves

maxa

∫π(y, p)da(y, p)

Back



Mean Reversion: FICO-percentile Change overthe yth Postdischarge Year (Musto, 2004, JOB)

Back



Equilibrium Mapping Between Type Scores andCredit Scores

• Equilibrium decision rules imply default and borrowing are more likely tocome from type b agents.

• Agents with low type scores are more like to be type b.

• Hence, agents with low type scores are more likely to have lower credit scores.

• The correlation coefficient weighted by the distribution measure is 0.9948.

Back



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