**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 will 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
will indicate 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,
will often have 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 may 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.

**Beta vs. Best-fit 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
best-fit index. For Beta
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.

**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.