A time-dependent proportional hazards survival model for credit risk analysis

Abstract

In the consumer credit industry, assessment of default risk is critically important for the financial health of both the lender and the borrower. Methods for predicting risk for an applicant using credit bureau and application data, typically based on logistic regression or survival analysis, are universally employed by credit card companies. Because of the manner in which the predictive models are fit using large historical sets of existing customer data that extend over many years, default trends, anomalies, and other temporal phenomena that result from dynamic economic conditions are not brought to light. We introduce a modification of the proportional hazards survival model that includes a time-dependency mechanism for capturing temporal phenomena, and we develop a maximum likelihood algorithm for fitting the model. Using a very large, real data set, we demonstrate that incorporating the time dependency can provide more accurate risk scoring, as well as important insight into dynamic market effects that can inform and enhance related decision making.

Appendices

Appendix A

Some data treatment issues

The data are described in Section 5. Ten predictor variables were chosen to include in the model, but due to confidentiality concerns, we do not describe them in this paper. We recommend standardizing all variables before implementing the MLE algorithm, for numerical reasons. Scaling of the variables will affect the direction in which a gradient-based algorithm searches for solutions to the parameter estimates. Regarding missing data, which are inevitable in large data sets, we used the following strategy. Predictor variable values that were missing for a customer were substituted with a linear regression prediction of the missing value, based on a covariance matrix calculated from all data that were not missing.

Appendix B

Derivation of the TDPH cdf

As mentioned in Section 4, the TDPH cdf of T can be written as F(t; x, τ; h ₀, ψ, γ)=1 − exp(−∫^t ₀ h ₀(u)ψ(x) γ(τ + u)du). We break the integral up into components that correspond to the pieces over which γ is constant. Let w(τ, t)=⌈(τ+t)/3⌉−⌈τ/3⌉+1 denote the number of different γ pieces between the time of approval (τ) and the time in question (τ+t). The ‘3’ in the denominator arises from the fact that t is in units of months, and the γ function is modelled as constant over each quarter. For a customer joining in month τ, the intervals (in terms of the relative month t) over which γ is constant can be written as

Here ⌈·⌉ denotes the ceiling function returning the smallest integer not less than the argument. In the preceding expressions for the intervals, it is understood that the lower boundary of each is truncated at 0 and the upper boundary at t. The constant value of γ(·) over the interval I _{τ,  k} is γ _{⌈τ / 3⌉+k−1}. Note that s _τ , ₀=0 and s _τ ,  _w(τ ,  _t)=t by definition. The cdf F(t; x, τ; h ₀ , ψ, γ) of T becomes

where H ₀(t)= ∫^t ₀ h ₀(u)du = −log(1 − F ₀(t)) is the cumulative hazard function of the baseline distribution.

Appendix C

The log-likelihood and its gradient for lognormal h ₀ ( t ) and exponential ψ (x)

In this section, we derive the log-likelihood function and its gradient (for use in a gradient-based algorithm for maximizing the likelihood) for the special case of a lognormal h ₀(t) and an exponential ψ(x). For other baseline distributions or ψ(x) functions, a similar approach can be used. For our special case, the baseline cdf and pdf are F ₀(t)=Φ((log(t)−μ)/σ) and f ₀(t)=φ((log(t)−μ)/σ)/(tσ) with mean parameter μ and standard deviation parameter σ, where Φ and φ denote the standard normal cdf and pdf. In order to avoid identifiability problems due to the confounding between and γ, we define the exponential function as ψ(x)=e^{β ′ x}, instead of For notational simplicity, we denote F(t; x _i , τ _i ; μ, σ, β, γ) by F _i (t). From Section 4, the log-likelihood function for the entire data set is

where expressions for the F _i (t) are given in Appendix B.

To find the partial derivatives of l with respect to μ, σ, β, and γ, notice that l depends on μ, σ, and γ via the terms (i=1, 2, …, N), and l depends on β via the terms ψ(x _i ) (i=1, 2, …, N). The relevant partial derivatives of J _i (t) and ψ(x _i ) are obtained from

,

From the expressions in Appendix B, the partial derivatives of F _i (t) are obtained from

From Equation (C.1), we also have

for j=0, 1, 2, …, M, and

for q=1, 2, …, Q.

Combining all of these gives the partial derivatives that constitute the gradient of l with respect to the parameters. The gradient can be used in an optimization algorithm for calculating the MLEs of the parameters. Matlab code is available upon request from the authors for the special case considered in this appendix.