R-Squared

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What is 'R-Squared'

R-squared is a statistical measure that represents the percentage of a fund or security's movements that can be explained by movements in a benchmark index. For example, an R-squared for a fixed-income security versus the Barclays Aggregate Index identifies the security's proportion of variance that is predictable from the variance of the Barclays Aggregate Index. The same can be applied to an equity security versus the Standard and Poor's 500 or any other relevant index.

BREAKING DOWN 'R-Squared'

R-squared values range from 0 to 1 and are commonly stated as percentages from 0 to 100%. An R-squared of 100% means all movements of a security are completely explained by movements in the index. A high R-squared, between 85% and 100%, indicates the fund's performance patterns have been in line with the index. A fund with a low R-squared, at 70% or less, indicates the security does not act much like the index. A higher R-squared value indicates a more useful beta figure. For example, if a fund has an R-squared value of close to 100% but has a beta below 1, it is most likely offering higher risk-adjusted returns.

R-Squared Calculation Example

The calculation of R-squared requires several steps. First, assume the following set of (x, y) data points: (3, 40), (10, 35), (11, 30), (15, 32), (22, 19), (22, 26), (23, 24), (28, 22), (28, 18) and (35, 6).

To calculate the R-squared, an analyst needs to have a "line of best fit" equation. This equation, based on the unique date, is an equation that predicts a Y value based on a given X value. In this example, assume the line of best fit is: y = 0.94x + 43.7

With that, an analyst could compute predicted Y values. As an example, the predicted Y value for the first data point is:

y = 0.94(3) + 43.7 = 40.88

The entire set of predicted Y values is: 40.88, 34.3, 33.36, 29.6, 23.02, 23.02, 22.08, 17.38, 17.38 and 10.8. Next, the analyst takes each data point's predicted Y value, subtracts the actual Y value and squares the result. For example, using the first data point:

Error squared = (40.88 - 40) ^ 2 = 0.77

The entire list of error's squared is: 0.77, 0.49, 11.29, 5.76, 16.16, 8.88, 3.69, 21.34, 0.38 and 23.04. The sum of these errors is 91.81. Next, the analyst takes the predicted Y value and subtracts the average actual value, which is 25.2. Using the first data point, this is:

(40.88 - 25.2) ^ 2 = 14.8 ^ 2 = 219.04. The analyst sums up all these differences, which in this example, equals 855.6.

Lastly, to find the R-squared, the analyst takes the first sum of errors, divides it by the second sum of errors and subtracts this result from 1. In this example it is:

R-squared = 1 - (91.81 / 855.6) = 1 - 0.11 = 0.8

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