# Downside risk measure: Theory and applications in portfolio choice and risk management

#### Abstract

Most measures of risk used by financial analysts are based on the standard deviation. But these measures typically assume symmetric normal distributions and ignore the fact that most investors are more concerned with the downside than the upside risk. ^ Vinod (2001) proposed a new measure of risk that attempts to correct these fundamental flaws in the standard measures. He defines a downside standard deviation (DSD) that does not rely on symmetric or normally distributed asset returns. It also better reflects the preferences of risk-averse investors. Focusing only on below normal deviations he defines a Downside Beta, a Down Sharpe ratio, and a Down Treynor index, which incorporates DSD for portfolio choice. This dissertation explores how using down side standard deviation, Down Sharpe ratios, and down Treynor indices would affect portfolio choice. ^ By looking at hypothetical portfolios of asset returns I explore how optimal allocations of portfolios would shift using only downside risk measures. How much will investors sacrifice if downside risk is minimized? ^ I address these questions using monthly returns on 50 stocks for 26 years to empirically assess the properties of the distribution of stock returns. Since returns to these stocks are not symmetric, the portfolio selection using the DSD are quite different from that of implied by the standard deviation based measures. Portfolios based on downside Sharpe and Treynor measures are intended for risk-averse individuals. It is expected that such individuals pay a price in terms of reduced returns; in other words, risk-averse investors are willing to accept lower returns than risk neutral investors are in order to minimize downside risk. ^ However, whether they actually pay such a price is an empirical question. I investigate this by comparing the final wealth of the portfolios based on DSD theory with the current risk theory. Both in-sample and out of sample tests show that, in general, the price paid does not appear to be too large, and Down Sharpe ratios appears to dominate Sharpe ratios substantially in the risk returns space in out of sample tests. ^

#### Subject Area

Economics, Finance

#### Recommended Citation

Agnes-Xiaohong Xie,
"Downside risk measure: Theory and applications in portfolio choice and risk management"
(January 1, 2002).
*ETD Collection for Fordham University.*
Paper AAI3045141.

http://fordham.bepress.com/dissertations/AAI3045141