Pricing of skew risk

Todd Mitton and Keith Vorkink suggest that diversified firms have to offer higher expected returns to compensate for the lack of positive skew.  This is tied to the well known phenomenon of individuals buying lotteries with a small chance of making large gains. 

In this paper, we seek to add to our understanding of why firms with diversification
discounts have higher expected returns. We consider an explanation based on the return
distributions of the stocks of diversified firms relative to single-segment firms. Specifically,
we consider whether investors pay a premium for single-segment firms because the return
distributions of single-segment firms have higher upside potential (positive skewness) than do
the return distributions of diversified firms. If investors have a preference for stocks with
positive skewness, then stocks of diversified firms may have to offer higher returns in order to
compensate investors for a lack of upside potential.

The assumption that investors would place a premium on stocks with greater skewness
exposure is grounded in theory. Arditti (1967) and Scott and Horvath (1980) demonstrate
that investors with typical preferences demonstrate a preference for positive skewness in return
distributions. Kraus and Litzenberger (1976) and Harvey and Siddique (2000) build on these
results to develop asset pricing relationships in a representative agent framework, finding that
an asset’s coskewness with the market portfolio should be priced. Other research shows that
even idiosyncratic skewness may be a priced component of stock returns. Barberis and Huang
(2005) show that when investors have preferences based on cumulative prospect theory, stocks with greater idiosyncratic skewness may command a pricing premium. Mitton and Vorkink (2006), in a model incorporating heterogeneous investor preference for skewness, also predict a pricing premium for stocks with idiosyncratic skewness. The optimal expectations model of Brunnermeier and Parker (2005) also produces qualitatively similar asset pricing implications for skewness as Barberis and Huang (2005) and Mitton and Vorkink (2006).

This seems to be about positive skew.  What does this mean for negative skew.  The obvious implication would be that investors would be more cautious about investments that have a negative skew. However, work on the carry trade suggests that they do not fully take notice of this risk.  Is this a case where risk is considered in an asymmetric fashion?  Could it be considered something akin or equivalent to Prospect Theory. 

Milton, T., & K. Vorkink, 2010, 'Why do firms with diversification discounts have higher expected returns?', Journal of Financial and Quantitative Analysis, 45 (6), pp. 1367-1390