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 ?
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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:
where RR is recovery rate.
-Added an explanation concerning Basel framework