A Theory of Credit Scoring and Competitive Pricing of Default Risk Satyajit Chatterjee Dean Corbae Jos´ e V´ ıctor R´ ıos-Rull Philly Fed, University of Wisconsin, University of Minnesota Mpls Fed, CAERP, CEPR, Oslo Labor Workshop, April 3, 2012 Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 1 / 43
Goal • Develop a competitive quantitative-theoretic model of unsecured consumer credit where: borrowers can legally default, 1 the punishment for default is a drop in the credit score or perceived 2 creditworthiness, and the theory is consistent with other key credit scoring facts. 3 • Use the model as a laboratory for evaluating regulations regarding information use by creditors Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 Low score raises interest rate and/or limits access to credit MustoFig 3 Record of default lowers score, removal of record raises it 4 Increasing/decreasing indebtedness lowers/raises score 5 Scores are mean reverting 6 MustoMR Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 4 / 43
Credit Scores and Delinquency Rates model Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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. • Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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) Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 9 / 43
People • Unit measure of people comprising of two types i ∈ { g, b } ; Γ i ′ i = Pr { 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 . Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 . Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 ) • Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 ( ¯ e + x min − y max > 0 ), this will keep the Bayesian updating function well-defined (and avoid supplying off-the-equilibrium path beliefs). Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 14 / 43
Recursive Formulation of HH Problem Cont. The current return function is given by � u i ( e + x − q ( p ( y, ψ (0 ,y ) ( x, s )) · y )) if d = 0 R (0 ,y ) ( e, x, s ; q, p, ψ ) = i u i ( e ) if d = 1 The value function is given by � � R ( d,y ) � ( e, x, s ) + β i θW i ( y, ψ ( d,y ) ( x, s ))) · m ( d,y ) V i ( e, θ, x, s ) = max (2) i m ∈ M i ( d,y ) where � � W i ( x, s ) = Γ j i V j ( e, θ, x, s )Φ j ( de )Λ( θ ) ∀ i ∈ { g, b } E j ∈{ g,b } ,θ • The optimal decision correspondence is denoted M ∗ i ( 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 ) . Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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: � q ( p ) = p/ (1 + r ) if y < 0 π ( y, p ) = 0 ⇔ (3) q (1) = 1 / (1 + r ) if y ≥ 0 More Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 16 / 43
Intermediary’s Problem Cont. The credit scoring function is given by � � p ( y, s ′ ) = s ′ · � Λ( θ ′ ) P (1 , 0) ( θ ′ , y, s ′ ; q, p, ψ ) 1 − g θ ′ (4) � � � Λ( θ ′ ) P (1 , 0) + (1 − s ′ ) · ( θ ′ , y, s ′ ; q, p, ψ ) 1 − , b θ ′ Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN 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 ) = � θ Λ( θ ) P ( d,y ) � � ( θ, x, s ) s g Γ gg θ Λ( θ ) P ( d,y ) θ Λ( θ ) P ( d,y ) � ( θ, x, s ) s + � (5) ( θ, x, s )(1 − s ) g b � θ Λ( θ ) P ( d,y ) � � ( θ, x, s )(1 − s ) b + Γ gb θ Λ( θ ) P ( d,y ) θ Λ( θ ) P ( d,y ) � ( θ, x, s ) s + � ( θ, x, s )(1 − s ) g b Chatterjee, Corbae, R´ ıos-Rull Philly Fed, Wisconsin, MN Credit Scoring Labor Workshop April 3, 2012 18 / 43
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