A Theory of Credit Scoring and Competitive Pricing of Default Risk

A Theory of Credit Scoring and Competitive Pricing of Default Risk Satyajit Chatterjee Dean Corbae José Víctor Ríos-Rull Philly Fed, University of Wisconsin, University of Minnesota Mpls Fed, CAERP, CEPR, Oslo Labor Workshop, April 3, 2012 Credit Scoring Labor Workshop April 3, 2012 1/43

Goal Develop a competitive quantitative-theoretic model of unsecured consumer credit where: 1 borrowers can legally default, 2 the punishment for default is a drop in the credit score or perceived creditworthiness, 3 and the theory is consistent with other key credit scoring facts. Use the model as a laboratory for evaluating regulations regarding information use by creditors Credit Scoring Labor Workshop April 3, 2012 2/43

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 to condition a credit score Credit Scoring Labor Workshop April 3, 2012 3/43

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 Scoring Labor Workshop April 3, 2012 4/43

Credit Scores and Delinquency Rates model Credit Scoring Labor Workshop April 3, 2012 5/43

FICO Scores Lenders assess creditworthiness of borrowers using FICO credit scores (higher score, higher creditworthiness) Over 75% of mortgage lenders and 80% of the largest financial institutions use FICO scores. FICO scores are calculated from data in the individual s credit report in five basic 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. Credit Scoring Labor Workshop April 3, 2012 6/43

Model Infinite horizon, discrete time model with uninsured idiosyncratic iid shocks to endowments and preferences 2 types of people (g and b): Type affects preferences and the distributions from which iid shocks are drawn; follows a persistent Markov process People can save or borrow to smooth consumption; if they borrow they have the option to default; (no pecuniary costs or exogenous restriction on ability to borrow) Neither type nor shocks are directly observable to lenders; lenders can only see an individual s credit market transactions (including default) going back T periods Lenders accept deposits that pay the risk-free rate and extend non-contigent loans at an interest rate that exactly covers the expected loss from default Credit Scoring Labor Workshop April 3, 2012 7/43

Type Score and Credit Score Lenders observe a person s credit market behavior and assess the likelihood that the borrower will be of type g next period this probability is labeled the 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 one key input into the construction of a person s credit score; the other key input is the amount borrowed Credit Scoring Labor Workshop April 3, 2012 8/43

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) Credit Scoring Labor Workshop April 3, 2012 9/43

People Unit measure of people comprising of two types i {g, b}; Γ i i = P r{i t+1 = i i t = i}. A person of type i draws iid endowment e and iid time preference shock θ in from distributions Φ i with support E = [e, e] Λ with finite support Θ contained in [0,1] Type can also affect preferences u i (c) and β i. Credit Scoring Labor Workshop April 3, 2012 10/43

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 the loan. To determine p, lenders assess the probability that a person will be of type g at 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 scoring function Credit Scoring Labor Workshop April 3, 2012 11/43

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 person is currently of type g. Credit Scoring Labor Workshop April 3, 2012 12/43

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) Credit Scoring Labor Workshop April 3, 2012 13/43

Recursive Formulation of the Individual Problem The set of feasible actions is a finite set B(e, x, s; q, p, ψ) such that c = e + x q(p) y 0. We permit randomization over feasible actions: m (d,y) [0, 1] denotes the probability mass on the element (d, y) and m is the associated vector. We assume that all budget feasible actions are chosen with at least some small probability (i.e. people make tiny mistakes as in Selten). Together with an assumption on primitives (ē + x min y max > 0), this will keep the Bayesian updating function well-defined (and avoid supplying off-the-equilibrium path beliefs). Credit Scoring Labor Workshop April 3, 2012 14/43

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 = 0 u i (e) if d = 1 The value function is given by [ V i (e, θ, x, s) = max m M i (d,y) R (d,y) i ] (e, x, s) + β i θw i (y, ψ (d,y) (x, s))) m (d,y) (2) where W i (x, s) = Γ j i V j (e, θ, x, s)φ j (de)λ(θ) i {g, b} j {g,b},θ E The optimal decision correspondence is denoted Mi (e, θ, x, s; q, p, ψ) and a given selection from this correspondence is denoted m i (e, θ, x, s; q, p, ψ). Credit Scoring Labor Workshop April 3, 2012 15/43

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 < 0 q(1) = 1/(1 + r) if y 0 (3) More Credit Scoring Labor Workshop April 3, 2012 16/43

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) Credit Scoring Labor Workshop April 3, 2012 17/43

Intermediary s Problem Cont. The (Bayesian) type scoring function is given by s = ψ (d,y) (x, s; q, p, ψ) = [ (d,y) θ Λ(θ)P g (θ, x, s)s Γ gg (d,y) θ Λ(θ)P g (θ, x, s)s + θ [ (d,y) θ Λ(θ)P b (θ, x, s)(1 s) + Γ gb (d,y) θ Λ(θ)P g (θ, x, s)s + θ Λ(θ)P (d,y) b (θ, x, s)(1 s) ] Λ(θ)P (d,y) b (θ, x, s)(1 s) ] (5) Credit Scoring Labor Workshop April 3, 2012 18/43

Equilibrium A recursive competitive equilibrium is: (i) a pricing function q (p); (ii) a credit scoring 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}. Credit Scoring Labor Workshop April 3, 2012 19/43

Existence of Equlibrium Since for every borrowing level y, q is just a linear function of the repayment probability p, we apply Schauder s fixed point theorem to the credit scoring function 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 the action set ({0, 1} L) is finite. Credit Scoring Labor Workshop April 3, 2012 20/43

Model with Information Restrictions There are more restrictions on information used in FICO scores than assumed above; no asset holdings or adverse events past T periods. Denote an individual s finite history by h T = (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, h T ) is the partition block in which (x, h T ) belongs A(y, d) is the partition block in which (y, d) belongs An individual s state space is now (i, e, θ, x, h T ). 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 distribution µ i (e, θ, x, ht ) to construct priors and assessments must condition out unobservable positive asset choices. Credit Scoring Labor Workshop April 3, 2012 21/43

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, x 1, x 2 } Credit Scoring Labor Workshop April 3, 2012 22/43

Mapping Model to Data Cont. Earnings of type i is Beta-distributed e Be(ν i, η i ). Table: Earnings Statistics (PSID 1996-2001) and Parameter Values Statistics Target Model Parameter Estimate Gini index 0.54 0.50 ν b 1.0153 (0.0616) Mean/median 1.40 1.21 η b 24.4051 (2.1358) Autocorrelation 0.67 0.60 ν g 2.6570 (0.1440) 1st quintile share 0.17 0.99 η g 4.0642 (0.2208) 2nd quintile share 6.77 4.52 Γ gb 0.0149 (0.0009) 3rd quintile share 14.73 16.30 Γ bg 0.0104 (0.0007) Percentage of yth quintile is endowments received for agents within yth quintile over total endowments. (Γ bg, Γ gb ) imply 59% of agents are type g. (ν i, η i ) imply type g earns 10 times more on average than type b. Credit Scoring Labor Workshop April 3, 2012 23/43

Mapping Model to Data Cont. Table: Model Statistics (TransUnion and SCF) and Parameter Values Statistics Target Model Parameter Estimates Overall delinquency rate 29.23% 31.28% x -0.0033 Subprime (bottom 27%) del. rate 75.74% 54.56% x 1 0.1078 Debt to earnings ratio 0.002 0.001 x 2 0.5683 Asset to earnings ratio 1.36 1.35 Λ(0) 0.0500 Percentage in debt 6.7 5.4 ϕ 6.4618 While the model matches the overall delinquency rate well, it fails to account for all of the subprime delinquency rate (72%). Credit Scoring Labor Workshop April 3, 2012 24/43

Equilibrium Decision Rules When θ = 0 (i.e. people are temporarily myopic), anyone with debt defaults and anyone without debt borrows. When θ = 1, With debt, type g default for low earnings or save (to x 1 or x 2 ) with higher earnings type b default for a larger set of low earnings or save to x 1 With zero assets, type g continue with zero assets or save (to x 1 or x 2 ) type b borrow when earnings are very low, continue with zero assets for intermediate earnings, or save to x 1 at high earnings With savings, both types continue to save. P vs Psi Credit Scoring Labor Workshop April 3, 2012 25/43

Model distribution of Credit Scores As in the data, the distribution puts more weight on high scores which have lower likelihood of default. data Credit Scoring Labor Workshop April 3, 2012 26/43

Credit Scoring Fact: Low scores raise interest rates Credit Scoring Labor Workshop April 3, 2012 27/43

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 very high FICO score could expect a huge drop in their score.... someone with many negative items already listed on their credit report might only see a modest drop in their score (FICO). On average, credit scores drop by 48% after default (from 0.82 to 0.43). Credit Scoring Labor Workshop April 3, 2012 28/43

Credit Scoring Fact: Removal of default flag raises 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 the distribution in the model versus 5% in Musto s data. Credit Scoring Labor Workshop April 3, 2012 29/43

Credit Scoring Fact: Decreasing indebtedness raises score Red bars correspond to optimal actions, while blue bars correspond to trembles. On average, credit scores rise by 59% after hhs pay off debt (from 0.49 to 0.78). Credit Scoring Labor Workshop April 3, 2012 30/43

Credit Scoring Fact: Increasing indebtedness decreases score Since borrowing generally arises when hit with θ = 0 and θ shocks are iid, assessment following borrowing rises since the population proportion of good types is 0.59. On average, credit scores rise by 3% after hhs go into debt (from 0.76 to 0.78). Credit Scoring Labor Workshop April 3, 2012 31/43

Credit Scoring Fact: Mean Reversion Slope coefficient for the fitted line equals 0.8. Credit Scoring Labor Workshop April 3, 2012 32/43

Welfare Effects of Restricting Information In a world of incomplete markets and private information, restricting information flow may be welfare improving Question: how much would a household of type i in state (x, h T ) be willing to pay to be in a regime where there are no information restrictions? Table: CE by types and shocks θ\i g b 1 0.0420e-3-0.5266e-3 0 0.0650e-3-0.1072e-3 Credit Scoring Labor Workshop April 3, 2012 33/43

Welfare Effects Cont. For each (i, e, θ, x, h T ) we compute compensating consumption variations λ i (e, θ, x, h T ) that satisfy V i (e, θ, x, s T (x, h T )) = (1 + λ i (e, θ, x, h T =2 )) 1 ϕ V (i, e, θ, x, h T =2 ) The aggregate welfare gain/loss is given by i,e,θ,x,h T =2 λ i (e, θ, x, h T =2 )µ(i, e, θ, x, h T =2 ). We find an aggregate welfare loss of -0.0001, since type b must be compensated more than type g gains from removing information restrictions. Table: CE by types and shocks θ\i g b 1 0.0420e-3-0.5266e-3 0 0.0650e-3-0.1072e-3 Credit Scoring Labor Workshop April 3, 2012 34/43

Conclusions We provide a theory where lenders learn from an individual s borrowing and repayment behavior about the agent s unobservable characteristics and encapsulates this in a credit score. After choosing a sparse set of parameters to match some key credit market data moments, we show the theory is broadly consistent with the way credit scores affect unsecured consumer credit market behavior. We show that for that set of parameters, aggregate welfare would be lower if information restrictions were removed. Credit Scoring Labor Workshop April 3, 2012 35/43

Limited Credit Access Following Default: Change in Credit Limit of Open Bank Cards over yth Postdischarge Year Back Credit Scoring Labor Workshop April 3, 2012 36/43

FICO Score Inputs Back Credit Scoring Labor Workshop April 3, 2012 37/43

Examples of partitions: T = 1 Suppose the assets choices are L = {l, 0, l 1 +, l 2 +} with l < 0 < l 1 + < l 2 +. The action space is The state/history tuple is L H T =1 = {(0, 1), (l, 0), (0, 0), (l 1 +, 0), (l 2 +, 0)}. The partition block is H 1 = {(0, 1)} H 2 = {(l, 0)} H 3 = {(0, 0)} H 4 = {(l 1 +, 0), (l 2 +, 0)} A 1 = {(0, 1)} A 2 = {(l, 0)} A 3 = {(0, 0)} A 4 = {(l 1 +, 0), (l 2 +, 0)} Back Credit Scoring Labor Workshop April 3, 2012 38/43

Each score from the bankruptcy filing group is mapped to a FICO percentile, which is the percent of scores in the non-filing contrast group below that score. E.g. [0, 10) means that less than 10% of the contrast group have scores below the filing group in that bin. Back Credit Scoring Labor Workshop April 3, 2012 39/43

Ausubel Data from randomized pre-approved solicitations allowed access to individual s credit bureau info. Adverse selection on observable characteristics (like credit scores): pool of consumers who accept an inferior contract (shorter introductory rates) exhibit inferior characteristics. E.g. credit scores of respondents to solicitations are 523 while nonrespondents are 643. Adverse selection on hidden information: even after controllinig for observables, the pool who accept inferior contracts default more than the pool who accept a better offer. Back Credit Scoring Labor Workshop April 3, 2012 40/43

Intermediary s Problem The profit π(y, p) on a financial contract of type (y, p) is: π(y, p) = { (1 + r) 1 p ( 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 the intermediary. The intermediary solves max π(y, p)da(y, p) a Back Credit Scoring Labor Workshop April 3, 2012 41/43

Mean Reversion: FICO-percentile Change over the yth Postdischarge Year (Musto, 2004, JOB) Back Credit Scoring Labor Workshop April 3, 2012 42/43

Equilibrium Mapping Between Type Scores and Credit Scores Equilibrium decision rules imply default and borrowing are more likely to come 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 Credit Scoring Labor Workshop April 3, 2012 43/43