In this paper we extend the concept of Value-at-risk (VaR) to bivariate return distributions in order to obtain measures of the market risk of an asset taking into account additional features linked to downside risk exposure. We first present a general definition of risk as the probability of an adverse event over a random distribution and we then introduce a measure of market risk (b-VaR) that admits the traditional b of an asset in portfolio management as a special case when asset returns are normally distributed. Empirical evidences are provided by using Italian stock market data.