R-Squared vs. Standard Index

R-squared ranges from 0 to 100 and reflects the percentage of a fund's movements that are explained by movements in its benchmark index. An R-squared of 100 means that all movements of a fund are completely explained by movements in the index. Thus, index funds that invest only in S&P 500 stocks have an R-squared very close to 100. Conversely, a low R-squared indicates that very few of the fund's movements are explained by movements in its benchmark index. An R-squared measure of 35, for example, means that only 35% of the fund's movements can be explained by movements in its benchmark index. Therefore, R-squared can be used to ascertain the significance of a particular beta or alpha. Generally, a higher R-squared indicates a more useful beta figure. If the R-squared is lower, then the beta is less relevant to the fund's performance.

Beta vs. Standard Index

Beta, a component of Modern Portfolio Theory statistics, is a measure of a fund's sensitivity to market movements. It measures the relationship between a fund's excess return over T-bills and the excess return of the benchmark index. Equity funds are compared with the S&P 500 index; bond funds are compared with the Lehman Brothers Aggregate Bond index. Morningstar calculates beta using the same regression equation as the one used for alpha, which regresses excess return for the fund against excess return for the index. This approach differs slightly from other methodologies that rely on a regression of raw returns.

By definition, the beta of the benchmark (in this case, an index) is 1.00. Accordingly, a fund with a 1.10 beta has performed 10% better than its benchmark index—after deducting the T-bill rate—than the index in up markets and 10% worse in down markets, assuming all other factors remain constant. Conversely, a beta of 0.85 indicates that the fund has performed 15% worse than the index in up markets and 15% better in down markets. A low beta does not imply that the fund has a low level of volatility, though; rather, a low beta means only that the funds market-related risk is low. A specialty fund that invests primarily in gold, for example, often has a low beta (and a low R-squared), relative to the S&P 500 index, as its performance is tied more closely to the price of gold and gold-mining stocks than to the overall stock market. Thus, though the specialty fund might fluctuate wildly because of rapid changes in gold prices, its beta relative to the S&P might remain low.

Alpha vs. Standard Index

Alpha measures the difference between a fund's actual returns and its expected performance, given its level of risk (as measured by beta). A positive alpha figure indicates the fund has performed better than its beta would predict. In contrast, a negative alpha indicates a fund has underperformed, given the expectations established by the fund's beta. Some investors see alpha as a measurement of the value added or subtracted by a fund's manager. There are limitations to alpha's ability to accurately depict a manager's added or subtracted value. In some cases, a negative alpha can result from the expenses that are present in the fund figures but are not present in the figures of the comparison index. Alpha is dependent on the accuracy of beta: If the investor accepts beta as a conclusive definition of risk, a positive alpha would be a conclusive indicator of good fund performance. Of course, the value of beta is dependent on another statistic, known as R-squared. (Alpha, beta, and R-squared statistics are all provided on Morningstar.com.)

For Alpha vs. the Standard Index, Morningstar performs its calculations using the S&P 500 as the benchmark index for equity funds and the Lehman Brothers Aggregate as the benchmark index for bond funds. Morningstar deducts the current return of the 90-day T-bill from the total return of both the fund and the benchmark index. The difference is called the fund's excess return. The exact mathematical definition of alpha that Morningstar uses is listed below.

Alpha = Excess Return - ((Beta x (Benchmark - Treasury))

Benchmark = Total Return of Benchmark Index

Treasury = Return on Three-month Treasury Bill

R-Squared vs. Best-fit Index

R-squared ranges from 0 to 100 and reflects the percentage of a fund's movements that are explained by movements in its benchmark index.

An R-squared of 100 means that all movements of a fund are completely explained by movements in the index. In this instance, the benchmark index is the best-fit index. To obtain the best-fit index, Morningstar regresses the fund's monthly excess returns against monthly excess returns of several well-known market indexes. Best fit signifies the index that provides the highest R-squared.

Alpha vs. Best Fit Index

Alpha measures the difference between a fund's actual returns and its expected performance, given its level of risk (as measured by beta).

A positive alpha figure indicates the fund has performed better than its beta would predict. In contrast, a negative alpha indicates a fund has underperformed, given the expectations established by the fund's beta. Some investors see alpha as a measurement of the value added or subtracted by a fund's manager. There are limitations to alpha's ability to accurately depict a manager's added or subtracted value. In some cases, a negative alpha can result from the expenses that are present in the fund figures but are not present in the figures of the comparison index. Alpha is dependent on the accuracy of beta: If the investor accepts beta as a conclusive definition of risk, a positive alpha would be a conclusive indicator of good fund performance. Of course, the value of beta is dependent on another statistic, known as R-squared. (Alpha, beta, and R-squared statistics are all provided on Morningstar.com.)

For Alpha vs. the Best-Fit Index, Morningstar first determines the fund's best-fit index. Morningstar regresses the fund's monthly excess returns against monthly excess returns of several well-known market indexes. Best fit signifies the index that provided the highest R-squared when the fund was regressed against it. After determining the best-fit index, Morningstar deducts the current return of the 90-day T-bill from the total return of both the fund and the best-fit index. The difference is called the fund's excess return. The exact mathematical definition of alpha that Morningstar uses is listed below.

Alpha = Excess Return - ((Beta x (Benchmark - Treasury))

Benchmark = Total Return of Benchmark Index

Treasury = Return on Three-month Treasury Bill