Introducing convexity into optimal compensation contracts

We study when it is appropriate to add a convex component such as stock options to an optimal, managerial compensation contract. We show that convexity is introduced when managers have moderate levels of relative risk aversion and decreasing absolute risk aversion. In addition, we study how convexity is affected as the distribution of outcomes becomes more skewed toward low outcomes. Here we show that while convexity increases when skewness is increased without regard to the effect on mean stock price, the opposite effect results when increases in skewness leave the mean stock price unchanged.