Using Quantile and Asymmetric Least Squares Regression for Optimal Risk Adjustment.

In this paper, we analyze optimal risk adjustment for direct risk selection (DRS). Integrating insurers' activities for risk selection into a discrete choice model of individuals' health insurance choice shows that DRS has the structure of a contest. For the contest success function (csf) used in most of the contest literature (the Tullock-csf), optimal transfers for a risk adjustment scheme have to be determined by means of a restricted quantile regression, irrespective of whether insurers are primarily engaged in positive DRS (attracting low risks) or negative DRS (repelling high risks). This is at odds with the common practice of determining transfers by means of a least squares regression. However, this common practice can be rationalized for a new csf, but only if positive and negative DRSs are equally important; if they are not, optimal transfers have to be calculated by means of a restricted asymmetric least squares regression. Using data from German and Swiss health insurers, we find considerable differences between the three types of regressions. Optimal transfers therefore critically depend on which csf represents insurers' incentives for DRS and, if it is not the Tullock-csf, whether insurers are primarily engaged in positive or negative DRS.