Credit price optimisation using survival analysis
The competitive nature of the financial industry requires the effective use of prescriptive models to assist with strategic decision-making. One of the challenges in managing consumer credit portfolios is determining the optimal prices (i.e. interest rates) that maximise both the loan take-up probability of a potential borrower and the expected net present interest income (NPII) to the lender, while still adhering to certain risk distribution constraints on the portfolio. According to Phillips (2013) the miss-allocation and miss-pricing of consumer credit may have a severe impact on the global economy as seen in the 2008 global financial crisis. This impact can mainly be attributed to the high cost associated with an unexpectedly high number of defaults. Traditionally, risk-based pricing has been used to determine the price of consumer credit. For this type of credit, the price included a risk premium which is dependent on the risk category of the bor-rower (or customer). However, this approach does not account for the demand of the customers i.e. the willingness of the customers to pay for a product or service. Hence, in recent years, pricing method-ologies have moved away from risk-based pricing towards price optimisation (Phillips, 2013). In price optimisation, the willingness of a customer to pay for a product (or demand) is mathematically repre-sented by a price response function, where the demand is expressed as a function of price (Terblanche and De la Rey, 2014). In this study, a price response model that not only relates loan take-up probabilities to price but also to loan-to-value (LTV), is presented. This allows one to relate the demand of a borrower to both a change in price and LTV. Furthermore, by including the LTV in the credit price optimisation problem, constraints can be imposed to limit the proportion of loans with a high LTV, since these loans are considered more risky as apposed to loans with lower LTVs (see Phillips, 2013 and Caufield, 2012). Two different approaches, namely a non-linear and a piece-wise linear approximation approach, were used to simultaneously determine the optimal price and LTV, when the objective is to maximise the expected value of the NPII. Although both approaches yield similar results, the latter provides proven optimal solutions and, in addition to this, it allows for the inclusion of binary decision variables, which facilitate logical decision-making capability on a portfolio level. The results indicate that, when constraints are imposed on the risk distribution to limit the take-up proportion of high risk customers, a higher average price is offered to customers deemed more risky in conjunction with a lower average LTV. Conversely, the low and medium risk customers are offered a lower average price together with a higher average LTV, subsequently increasing the take-up proportion of these risk gradings. In addition to this, when constraints are imposed on the loans with a high LTV, the average LTV of the high risk customers were substantially lower when compared to the low and medium risk customers. To make provision that a borrower may default during a loan, a parametric mixture cure model (which is a generalisation of the well-known Cox Proportional Hazard model) was used to estimate the probability that a borrower is still repaying the loan (not defaulting on the loan). The use of the mixture cure model permits one to take into account the relatively large proportion of customers that are not susceptible to default, when solving the optimisation problem. This newly developed optimi-sation model was applied to two simulated data sets. The results demonstrate that a clear interaction between price, LTV, take-up and survival probabilities exist. On average the price offered to the low risk customers were lower than the price offered to the high risk customers. On the other hand, the LTV proposed to the low risk customers was, on average, higher than that of the high risk customers. As a result of including logical decision making variables, customers with lower survival probabili-ties (or higher default probabilities), were excluded from the portfolio when the price offered to these customers were too high. The literature study on survival analysis, and more specifically on the parametric families of distri-butions present in survival analysis, led to the development of a new goodness-of-fit test for exponen-tiality. The newly proposed test performed favourably in terms of powers relative to several existing tests. The test was also applied to a simulated data set to determine whether the parametric assumptions underlying the Cox Proportional Hazard model were violated.