Statistically Speaking

Transcription

1 Statistically Speaking August 2001

2 Alpha a Alpha is a measure of a investment instrument s risk-adjusted return. It can be used to directly measure the value added or subtracted by a fund s manager. It is calculated by measuring the difference between a fund s actual returns and its expected performance given its level of market risk as measured by beta. An alpha of 1.0 means the instrument produced a return 1% higher than its beta would predict. An alpha of 1.0 means the fund produced a return 1% lower than it s beta would predict. The accuracy of an alpha rating depends on 2 factors; 1.) the assumption that market risk as measured by beta, is the only risk measure necessary and 2.) the strength of the instrument s correlation to a chosen benchmark such as the S&P 500. Correlation is measured by R-Squared*. The alpha rating can be useful in analyzing a manager s specific contribution or valueadded to an investment instrument s performance. It is a measure of the historical movement of the instrument s performance not explained by movements of the market. It can be used to explain what a manager brings to the table.

3 Beta b Beta measures the level of systematic risk associated with a specific security or portfolio. It is the coefficient measuring an investment s relative volatility and helps define the tendency of an investment s returns in response to swings in the market measured by an index like the S&P 500. The beta statistic is used to measure the marketrelated risk of a stock or portfolio. It helps to determine the sensitivity of an investment s return relative to changes in the market return. A stock/portfolio with a beta equal to one is thought to move in step with the market. An investment with a lower beta can be expected to rise or fall more slowly than the market where a higher beta would indicate an investment that rises or falls at a quicker pace than the market. The greater an investment s beta, the higher the systematic risk associated with the investment. The beta statistic helps in defining risk. It is important however, to note the benchmark used as a comparison. As a stock/portfolio s beta will change relative to the market indices it is measured against, the user should decide which index is most appropriate for the given investment.

4 Sharpe Ratio Formula: S = R p R f p The Sharpe Ratio is a risk-adjusted performance measure that divides a portfolio s average excess return over a sample period of time by the standard deviation of returns over that same period of time. It measures the reward to total volatility relationship. The Sharpe Ratio is used to measure reward per unit of risk or the risk versus return relationship. Most clients prefer to invest with the manager most able to consistently obtain the highest Sharpe Ratio regardless of a client s aversion to risk because the higher the value, the better the portfolio s historical risk-adjusted performance relative to the risk taken. This measure quantifies a portfolio s return in excess of a guaranteed investment (i.e. T-Bill), relative to its risk. However, the measure should be compared to that of another investment or benchmark to clearly define the alternative investment s benefit.

5 Treynor Ratio Formula: R p R f b The Treynor Ratio is a risk-adjusted performance measure that divides a portfolio s average excess return over a sample period of time by the beta over that same period of time. Risk is measured as the beta of the series or instrument relative to the market (CAPM line). The formula assumes CAPM (Capital Asset Pricing Model-an industry model that represents the relationship between expected risk and expected return) as valid. This ratio considers only systematic risk, not total risk (as seen with the Sharpe Ratio). This measure helps to ascertain the return to volatility relationship. It is very similar to the Sharpe ratio. However, the Treynor ratio focuses on beta and systematic risk where the Sharpe ratio focuses on overall risk through use of the standard deviation.

6 R-Squared R-Squared CORRELATION 2 R-Squared represents the percentage of an investment instrument's performance that can be explained by the behavior of a particular benchmark. The higher the percentage, the greater the correlation of the investment s performance pattern to that of the benchmark. The R-Square measure will always fall within a value range of 0 to 1.0 R-Squared represents the portion of variability in the manager s excess returns correlated with the variability in the market s excess returns. The higher the R-Squared value, the more variability of the manager s excess returns is explained by the market s excess returns. R-Squared can also help in measuring how well a portfolio is being diversified. In a CAPM Regression for instance, R-Squared will indicate the proportion of the manager s risk that is systematic and unavoidable due to market exposure. If R-Squared = 1, the portfolio is fully diversified and unsystematic risk in the portfolio will equal zero. The higher the R-Squared value, the more diversified and similar to the index the portfolio should be.

7 Standard Deviation r The standard deviation of a data series (portfolio returns over a specific time period) is a measure of the extent to which observations in the series differ from the distribution mean of the series. The standard deviation of a series of asset returns is the measure of the volatility, or risk, of the asset. A statistical measure of the range of a product s performance used to measure the volatility of an investment instrument over a specified period of time. A higher standard deviation indicates a product with more risk or volatility. A product s portfolio is expected to differ positively or negatively from the mean by no more than the standard deviation amount of 68% of its cycle. *One r away from the mean in either direction on the x axis accounts for 68% of the group or series (red area), two r away from the mean account for 95% (red and green areas) and three r account for 99% (red, green and blue areas).