"This proposal is related to the relationship between extreme returns and expected returns for international stock markets. For the first part, I will focus on the minimum daily return on the aggregate market over various historical time windows. The negative of this variable can be interpreted as a measure of Value at Risk and a significantly positive relation between this measure and expected market returns will provide evidence for the importance of downside risk in determining index returns. Fixed-effects panel regressions will be used to test for the proposed linkage controlling for a set of relevant variables. For the second part, I will focus on the maximum daily return for individual stocks over various historical time windows. The under-diversification of investment portfolios and the preference of investors for lottery-like stocks suggest that there should be a significant relationship between historical maximum returns and expected returns at the firm-level. Finding a significantly negative relation will also provide support for behavioral theories related to investor psychology. In this part, univariate and bivariate portfolio analysis as well as firm-level cross-sectional regressions will be employed. The main data source for this project is DataStream. The proposed analyses will be carried out separately for 52 financial markets with an eye on the differences between developed and emerging countries. The results will have potential implications related to international risk management and portfolio allocation, especially in the aftermath of the financial crisis of 2008."