## 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

## Alpha is a point on a straight line, plus more.

December 22, 2016

{update on 12/23/2016: the significance of technical and operational alpha}

Alpha (⍺) is the cherished -but overrated- measurement of superior investment. Here are several interpretations:

• A measurement of how well an investment outperforms its market index (ref 1).
• The observed characteristic of a mutual fund that predicts higher fund performance (ref 2).
• A portfolio’s return that’s independent of market returns (ref 3).
• The excess (or deficit) return compared to the market’s return. Used this way, ⍺ is called Jensen’s Alpha.

Alpha represents a unique risk of outperforming the market’s returns. It is classically calculated as the “Y intercept” of a straight line attributed to the CAPM model (see appendix). In the last century, famous investors outperformed the market either by choosing exceptional investments or by investing in exceptional market sectors. The investment could be a single security (e.g., a stock) or a portfolio of capital assets (e.g., a mutual fund) (footnote 1, refs 1, 2). Now in this century, those alledged ‘alpha’ strategies are increasingly difficult to achieve. There’s an emerging sentiment among investors to avoid wasting time and money on attempting to outperform the market, the so called “loser’s game”. The current “winner’s game” is to seek ‘beta’ (refs 1, 2, 4, 5).

‘Beta’ is the portfolio’s return generated by market returns. Therefore, beta represents the risk of earning the market’s returns. The beta statistic, β, is currently calculated and reported by financial research firms as a coefficient for the incline of a straight line attributed to the CAPM model (see appendix).

### Straight line of imaginary returns

#### (refs 5-8)

A straight line of imaginary returns presumably offers the best possible comparison of investment returns to a market index (footnote 2). ‘Returns’ and ‘performance’ are interchangeable terms that indicate the direction and movement of prices over time. An investment’s rate of return is calculated as the percentage change in price at regular intervals of time [likewise, the market’s rate of return is a percentage change in value of the market’s index at regular intervals of time]. Any rate of return is easily converted to a risk premium by subtracting the guaranteed interest rate for a Treasury bill (“T bill”). The risk premium is an investor’s potential reward for purchasing a security other than the T bill.

The straight line is drawn on a graph that shows actual measurements of investment returns plotted against market returns. The returns may either be measured as the rate of return or the risk premium depending on the goal of analysis. In the following chart, black dots represent a series of investment returns plotted against corresponding market returns.

The blue line of imaginary returns is the best possible comparison of investment returns to market returns. The position of the line on the graph is governed by its incline (β) and intersection (⍺+ε) with the vertical axis.

### ⍺, the intersection

#### (refs 1-3, 5-8)

Alpha resides at the intersection of the theoretical line with the vertical axis for investment returns (chart). Since the vertical axis crosses the horizontal axis at 0% market returns, ⍺ is the theoretical investment return at 0% market returns. A positive value for ⍺ implies that the investment tends to outperform its market index. Likewise, ⍺ = 0 implies no inherent advantage of the investment and a negative value for ⍺ implies that the investment tends to move less than the market index.

There’s a degree of error (ε) involved in drawing the line of imaginary returns, which means that its intersection is defined by the term ⍺+ε. The ε declines when a series of returns lie close to the line. The chart shows plots for 2 different series of returns; one series of black dots and another series of white circles. Both series have an equally small ε as illustrated by the close alignment of data to each straight line. Alpha of the blue line is 0% return and ⍺ of the orange line is 5% return, both occuring when the market return is 0. The series of open-circle returns outperformed the series of black-dot returns by 5%.

### Significance

#### (refs 1, 2, 4, 5)

Alpha measures how well an investment outperforms the market. Yesterday’s ‘technical’ ⍺, shown in the preceding chart, applied to measuring superior stock-picking skills.  Today, the technical ⍺ of stocks is not reported by the most popular financial websites.

Today’s ‘operational’ alpha is really a beta loading factor of multi-factor models (see appendix).  Operational alpha is more relevant to measuring the performance of actively managed mutual funds and investment portfolios. The investment goal of an actively managed mutual fund is to outperform its market index. Active management may be the “loser’s game” of paying excessive fees in contrast to passive management, which may be the “winner’s game” of paying minimal fees.

### Footnotes

1. Capital assets are securities and other forms of property that potentially earn a long term capital gain(loss) for the owner.

2. The straight line has other names that precede my use of the term ‘imaginary returns’. The straight line is also called a regression line or security characteristic line (ref 6).

### References

1. Larry E. Swedroe and Andrew L. Berkin. Is outperforming the market alpha or beta? AAII Journal, July 2015. pages 11-15.

2. Yakov Amihud and Rusian Goyenko. How to the measure the skills of your fund manager. AAII Journal, April 2015. pages 27-31.

3. Daniel McNulty. Bettering your portfolio with alpha and beta. Investopedia. http://www.investopedia.com/articles/07/alphabeta.asp#ixzz4SYJ0rN9q

4. John C. Bogle. The little book of common sense investing. John Wiley & Sons Inc., Hoboken, 2007.

6. Professor Lasse H. Pederson. The capital asset pricing model (CAPM). New York University Stern School of Finance. undated. http://www.stern.nyu.edu/~lpederse/courses/c150002/11CAPM.pdf

7. MoneyChimp. Regression, Alpha, R-Squared. 2016. http://www.moneychimp.com/articles/risk/regression.htm

8. Invest Excel. Calculate Jensen’s Alpha with Excel. undated. http://investexcel.net/jensens-alpha-excel/

### APPENDIX: models for pricing assets and managing portfolios

#### (refs 1-3, 5-8)

The original one-factor model was called the Capital Assets Pricing Model (CAPM). The single factor is market returns (M).  The investment returns (I) are predicted by a best-fit line with incline (βm) and intersection with the vertical axis (⍺ + ε) (equation 1).

I = ⍺ + ε+ βmM,     equation 1, CAPM

Subsequent series of three-factor and four-factor models were sequential upgrades of CAPM. Equation 2 is an example of a four-factor model for the risk premium of an investment fund (F) comprised of separate portfolios for the broad market (M), asset size (S), asset value (V), and asset momentum (U).

F = ⍺ + ε + βmM + βsS + βvV + βuU,     equation 2, four-factor model

⍺ is the excess risk premium attributable to skillful management of the Fund.
ε is the model’s error
βm, βs, βv, and βu are portfolio loading factors assigned by the Fund’s manager

The four-factor model offers a spectrum of possibilities.

• During 1927-2014, the average annual returns of indices for the the four-factor model were 8.4% for the broad stock market, 3.4% for stock size, 5% for stock value, and 9.5% for stock momentum.  The sum of average annual returns, 26.3%, represented the alpha-threshold for superior fund returns (ref 1).
• Passive management could be predicted by setting βm to 1.00, measuring the market index return, and setting the remaining loading factors to 0.  A market index fund would  be expected to generate a risk premium that matches the market index risk premium with an ⍺ of 0 and slight ε for tracking error.
• Active management involves designing loading factors and portfolio assets to outperform the fund’s predicted returns.