## R-squared, the linearity of investment returns.

December 24, 2016

[updated 12/25/2016: R2 is a useful measure of indexing]

The R-squared (R2) statistic describes a pattern of plotted data with respect to a straight line. R-squared is called the coefficient of determination (ref 1,2).

The black dots in figure 1 represent investment returns that are poorly related to market returns. There is a random distribution of investment returns with respect to market returns. The blue line is an inadequate representation of the relationship simply because there is no relationship. The R2 score for this distribution is 0.03. Conversely, the black dots in figure 2 show the ‘herding’ of data around a straight line.

Figure 2’s investment returns are highly related to market returns with an R2 of 0.997.

### Significance

The R2 score represents the degree of alignment of data to a best-fit line as determined by regression analysis. The lowest possible score of 0 indicates a random pattern of data with absolutely no alignment. The highest possible score of 1 represents complete alignment.

The product of R2 X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R2 measures the relavance of the best-fit line to a set of data. Relavance increases as the R2 score varies from 0 to 1.

The lowest score of 0 defies any financial analyst to draw a meaningful line for investment returns as they relate to market returns. In figure 1, the incline (β) and Y-intercept (⍺) of the blue line are unreliable measurements of investment performance.

The highest R2 score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R2 above 0.75 identifies a straight line for making predictions of returns. Lower scores represent increasingly random events. In figure 2, the incline (β) and Y-intercept (⍺) are reliable measurements of investment performance.

R-squared is an excellent measure of index fund performance.  Websites for index mutual funds and ETFs publish R2 as a measure of alignment between fund returns and the market index.   Funds that have an R2 score of nearly 1.00 track the index very closely.

### References

1.  Lain Pardoe, Laura Simon, and Derek Young. STAT 501, Regression Methods. 1.5- The coefficient of determination, r-squared. Pennsylvania State University, Eberly College of Science, Online courses. https://onlinecourses.science.psu.edu/stat501
2.  R-squared. 2016, Investopedia http://www.investopedia.com/terms/r/r-squared.asp?lgl=no-infinite

## ETF Valuation

September 24, 2011

Figure 2 (below) was obtained from the interactive chart provided by Bloomberg.com.  The chart shows price trends for 3 different valuations of SPY (SPY is the stock market’s ticker symbol for the SPDR S&P 500 ETF Trust).

• SPYNV:IND (red) represents the S&P 500 Depository Receipts Index Net Asset Value, which measures the NAV per share of SPY
• SPTR:IND (green) epresents the S&P United States 500 Total Return 1988, which measures the total return of 500 stocks in the S&P 500 Index in reference to 1988.
• SPY:US (orange) represents the share price of SPY.

Fig. 2 SPY’s Index, NAV, and Price movements

In the legend above the chart, the numbers are the closing prices on 9/30/2011.  In the chart, the price movements are expressed as ‘percentage changes’ on the vertical axis in relation to ‘time’ on the horizontal axis.  The parallel behavior of the graphs provide evidence to support (1) the investment goal of the Fund, which is to match the NAV to the performance of the Index, and (2) the effectiveness of arbitrage in matching the share price to the NAV.

Fund documents report the following discrepencies between ETF valuations:

• Discrepencies between NAV and Index performance are reported as “tracking error“.
• Discrepencies between share price and NAV are reported as “premium(discount)“.