Actual portfolios contain fewer stocks than are implied by standard financial analysis that balances the costs of diversification against the benefits in terms of the standard deviation of the returns. Suppose a safety first investor cares about downside risk and recognizes the heavytail feature...

This paper characterizes the optimal first-price auction (FPA) and second-price auction (SPA) for selling rights, contracts, or licenses that involve ensuing payoff uncertainty for the winning bidder. The distribution of the random payoff is common knowledge, except that bidders have private...

In recent years the Value at Risk (VaR) concept for measuringdownside risk has been widelystudied. VaR basically is a summary statistic that quantifies theexposure of an asset or portfolio tomarket risk, or the risk that a position declines in value withadverse market price changes. Threeparties...