R-Squared

• Measure of how close the relationship is between a portfolio and its benchmark.

• Can be thought of as a percentage from 1 to 100.

• It is not a measure of the performance of a portfolio. A great portfolio can have a very low R-squared. It is simply a measure of the correlation of the portfolio�s returns to the benchmark's returns. If you want a portfolio that moves a lot like the benchmark, you'd want a portfolio with a high R-squared. If you want a portfolio that doesn't move at all like the benchmark, you'd want a low R-squared.

• The value of R-squared is it tells us how meaningful Beta and Alpha are. A low R-squared would mean that the Beta and Alpha numbers are not very accurate or meaningful.

• General Range for R-Squared:

70-100% = good correlation between the portfolio's returns and the benchmark's returns  

40-70% = average correlation between the portfolio�s returns and the benchmark's returns

1-40% = low correlation between the portfolio�s returns and the benchmark's returns

An R-squared of 100 indicates that all movements of a portfolio can be explained by movements in the benchmark. 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 portfolio's movements can be explained by movements in its benchmark index. An R-squared measure of 35, for example, means that only 35% of the portfolio�s movements can be explained by movements in the benchmark index.

CVxy = covariance between portfolio and benchmark

Vx = variance of benchmark

Vy = variance of portfolio

R2 = CVxy2 /(Vx *Vy)

For this example, the plots are near the best-fit line so the fund would have a high R2 in relation to this index:

 

This fund's points are not very close to the line, and don't describe a line well. Therefore, it would have a low R2:

 

  Modern Portfolio Statistics Research Paper