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# R-Squared

## 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 = 15.68 ^ 2 = 245.86. The analyst sums up all these differences, which in this example, equals 763.52.

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 **/** 763.52) = 1 – 0.12 = 0.88