We propose a measure for extreme downside risk (EDR) to investigate whether bearing such a risk is rewarded by higher expected stock returns. Constructing an EDR proxy with the left tail index in the classical generalized extreme value distribution, we document a significantly positive premium on firm-specific EDR in cross-section of stock returns even after controlling for market, size, value, momentum, and liquidity effects. The EDR premium is more prominent among glamour stocks and when high market returns are expected. High-EDR stocks generally have high idiosyncratic risk, large downside beta, lower coskewness and cokurtosis, and high bankruptcy risk. The EDR premium persists after these characteristics are controlled for. EDR is also closely related to firm-specific Value at Risk (VaR) which substantially impacts EDR's effect on expected stock returns. EDR supplements VaR in predicting stock returns by exhibiting additional explanatory power.

In Part I of this dissertation, I propose a measure for the extreme downside risk (EDR) to investigate whether bearing such a risk can be rewarded by higher expected stock returns. By constructing an EDR measure with the left tail index in the classical generalized extreme value distribution, I find a significant positive premium on firm-specific EDR in cross-section of stock returns even after I control for size, value, return reversal, momentum, and liquidity factors. EDR serves as a good indicator of extreme market plunges. High-EDR stocks generally exhibit high idiosyncratic volatility, large value-at-risk, large negative co-skewness, and high bankruptcy risk. I also controlled for these characteristics to find that the EDR premium remains robust. Furthermore, the EDR effect exhibits long-run persistence and is not subsumed by business cycles. In Part II, I apply the concept of extreme downside risk to a policy-related issue: In August 1991, NASDAQ introduced a controversial SI minimum bid price threshold as part of its listing maintenance criteria (the dollar delisting rule or one-dollar rule). This part empirically evaluates this rule through an extreme value approach. Utilizing the Generalized Extreme Value distribution model to capture extreme price plummets, I find NASDAQ stocks frequently trading below S1 in the pre-rule period are extremely vulnerable to catastrophic losses. The implementation of the one-dollar rule effectively curbs the extreme downside price movements, which helps to protect investors' interest, uphold their faith in the exchange, and improve the credibility of the market. Such a pattern is prevalent across all industries and is not affected by market movements. The S1 benchmark serves as an appropriate cutoff point in screening the issues listed on the exchange. The minimum price continued listing standard on NASDAQ is justified and has proved to be successful.

This thesis investigates the relationship between downside risk and stock returns. In an economy with agents that are more concerned about downside losses than upside gains (downside risk averse investors), stocks that covary strongly with systematic risk factors in adverse states are expected to earn higher returns. The premium on downside sensitive stocks reflects a compensation for the risk of high negative returns in unfavorable states. Analyzing different risk factors that are proposed in the literature to systematically affect stock returns, I find strong evidence for a downside risk-return relationship for three factors: the returns on the market portfolio, the liquidity innovation factor and a factor reflecting unanticipated changes in the risk premium. I estimate that the premium for bearing market downside risk is approximately 4-6%, for liquidity downside risk 3-5% and 2-3% for stocks that covary strongly with unanticipated (negative) changes in the risk premium.

"Does extreme downside risk require a risk premium in the pricing of individual assets? Extreme downside risk is a conditional measure for the co-movement of individual stocks with the market, given that the state of the world is extremely bad. This measure, derivedfrom statistical extreme value theory, is non-parametric. Extreme down-side risk is used in double-sorted portfolios, where I control for the five Fama-French and various non-linear asset pricing factors. I find that the average annual excess return between high- and low-exposure stocks is around 3.5%"--Abstract.

'We decompose total variance into its bad and good components and measure the premia associated with their fluctuations using stock and option data from a large cross-section of firms. The total variance risk premium (VRP) represents the premium paid to insure against fluctuations in bad variance (called bad VRP), net of the premium received to compensate for fluctuations in good variance (called good VRP). Bad VRP provides a direct assessment of the degree to which asset downside risk may become extreme, while good VRP proxies for the degree to which asset upside potential may shrink. We find that bad VRP is important economically; in the cross-section, a one-standard-deviation increase is associated with an increase of up to 13% in annualized expected excess returns. Simultaneously going long on stocks with high bad VRP and short on stocks with low bad VRP yields an annualized risk-adjusted expected excess return of 18%. This result remains significant in double-sort strategies and cross-sectional regressions controlling for a host of firm characteristics and exposures to regular and downside risk factors'--Abstract, p. ii.

If investors are more averse to the risk of losses on the downside than of gains on the upside, investors ought to demand greater compensation for holding stocks with greater downside risk. Downside correlations better capture the asymmetric nature of risk than downside betas, since conditional betas exhibit little asymmetry across falling and rising markets. We find that stocks with high downside correlations with the market, which are correlations over periods when excess market returns are below the mean, have high expected returns. Controlling for the market beta, the size effect, and the book-to-market effect, the expected return on a portfolio of stocks with the greatest downside correlations exceeds the expected return on a portfolio of stocks with the least downside correlations by 6.55% per annum. We find that part of the profitability of investing in momentum strategies can be explained as compensation for bearing high exposure to downside risk.