Using quantile regression for optimal risk adjustment

This paper analyzes 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 used in most of the contest literature, optimal transfers for a risk adjustment scheme have to be determined by means of a restricted quantile regression, irrespective of whether insurers primarily engage 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 within a discrete choice model for a new class of contest success functions, but only if positive and negative DRS are equally important; if not, optimal transfers have to be calculated from a restricted asymmetric least squares regression. Using data from a German and a Swiss health insurer, we find considerable differences between the three types of regressions. Optimal transfers therefore critically depend on which contest success function represents insurers' incentives for DRS and whether positive and negative DRS are equally important or not. Results from the two data sets indicate that if a regulator does not know which case applies, transfers should rather be calculated by means of a quantile than a least squares regression.