# optimal down payment estimation in credit scoring

Knowing I can estimate the risk of default, via logistic regression, of a consumer on a small loan...

what would be the best way to estimate the optimal down-payment amount to ask for in order to reduce that risk ?

• How do you propose that the downpayment is related to risk? How exactly do you define "risk" in this context, anyway? By it do you mean the chance of default or perhaps do you mean the expected value of the loan, or maybe even the variance of the value?
– whuber
Jan 5, 2015 at 16:26
• There are so many details missing, it's impossible to answer the question. If you talk about down-payment, it means that the loan is collateralized. What is the collateral? What is the recovery rate on default? Do you assume that downpayment amount does not impact borrower's behaviour? etc. Jan 5, 2015 at 18:45
• My question arises because I was rather supposing that my estimation on the behaviour of the consumer (PD at t=t0) would change if he accepts to perform a downpayment (PD at t=t1). So, I thought a naive solution was to take downpayments as a feature in my model for PD and then consider the coefficient that multiply the downpayment value, as the coefficient, e.g. dx that define the relationship. So, if want a PD<0.1, then I should have a downpayment of X such that dx*X decrease PD to 0.1. Is is a correct way of doing ? Nothing better ? I don't suppose collateral neither at the moment. Jan 5, 2015 at 19:43
• maybe I am missing a point here...if I am working with loans that are not collaterized, is there another word in english than "down payment" for a first payment necessary for the contract to be issued ? I thought prepayment is not correct as it relates to a payment to close the contract before its expected end ? Jan 6, 2015 at 18:09
• @Serge I understood that as such, but it might be possible that down payment in US is related to some sort of specific schedule for payments. Jan 7, 2015 at 8:47

Down payment might not directly reduce risk but it might reduce risk of loss when customer is in default.

In international risk management framework for the banks, usually called Basel setting, things are governed by the following eq.

Expected loss= Prob of default (PD) * Loss given default (LGD) * Exposure at default (EAD)

where PD is your risk estimate for individual. It is a probability that one defaults.

It might be that LGD is affected by the down payment.

If PD is affected, then it might be a result via some sort of selection mechanism.

I have seen some results where LGD is a function of PD and collateral (C) and LGD is affected by collateral in following way:

LGD=1-RR(C,PD)

where RR is recovery rate.

NOTE:

-Added an explanation concerning Basel framework

• @whuber no. Basel setting is international regulatory framework for risk management in the banking sector. Link: bis.org/publ/bcbsca.htm Jan 5, 2015 at 18:12
• @Serge then you have only PD and EAD components. Can these depend directly on down payment? Jan 5, 2015 at 18:36
• @Analyst Re: "some sort of selection mechanism": it appears that you wonder whether Oscar Wilde's aphorism "A bank is a place that will lend you money if you can prove that you don't need it" can be quantified and incorporated into a model. Jan 5, 2015 at 20:07
• @AlecosPapadopoulos yes, that was what I was thinking. +1 for the funny quote.. :) Jan 7, 2015 at 8:44
• @Aksakal This equation gives you only expected loss, it is not itself yet useful for pricing of risk. And EL is not enought, you have to have some sort of extra risk buffer. If you have non collateralized consumer loans, then EAD will be function of time having highest value at the start of period. There are other things at work here also. Can we believe LGD and PD will not change as time progress? Jan 7, 2015 at 8:49