Alpha: A Measure of Fund Management Skill

  • By Paul Wilcox
  • January, 07
Blog Alpha: A Measure of Fund Management Skill

Passive Investing Versus Active Portfolio Management


Many investors like to take an active role in the management of their portfolios. On the upside, they look for “hot” stock picks. On the downside, they try to exit positions early before they lose money.

Some experts argue that investors would do better to invest in the market broadly rather than trying to selectively buy and sell individual stocks. The ultimate expression of this view is index investing which emerged as a preferred method in the 1980s and 1990s. The central ideas of index investing include:

  • The market is best represented by major indices like the Dow Jones Industrial Average or the S&P 500.
  • The market is efficient, you can’t improve performance by timing stock purchases or sales.
  • It is risky to hold a portfolio that deviates from the “the market portfolio.”

Capital Assets Pricing Model

This argument is compelling, especially when you consider the performance of the market over the last century. The concept was described theoretically as the Capital Assets Pricing Model (CAPM) by Nobel Prize winner William Sharpe (also creator of the Sharpe Ratio). 

The CAPM’s success inspired the creation of mutual funds and ETFs designed to track major indices and to allow retail investors easy access to a simple and powerful investment instrument. SPY and DIA are among the leading examples, representing the S&P 500 and the Dow Jones Industrial Average respectively.

Investing in index funds is a form of passive investing. In contrast, the philosophy of active management suggests that allocations between investments ought to to be actively managed. This approach has gained favor recently, especially since the significant market drawdowns of 2001 and 2008. Grinold and Kahn’s book Active Portfolio Management presents the case for active management well.

If active management does add value, how might we measure its value with regard to the market?


The Relationship Between a Managed Portfolio and “The Market Portfolio”

The CAPM suggests that if the allocations to stocks in your portfolio deviate from those in “the market portfolio” then you should expect different returns than the market. That part seems obvious, but the CAPM provides an even more specific prediction. Namely, the difference you can expect depends on the beta of your portfolio.

Beta is a measure of how much your portfolio’s value changes when the market changes. A beta of 1.0 implies that your portfolio moves in direct proportion with the market. On the other hand if your portfolio’s beta is less than 1.0, your portfolio should change more slowly than the market, if it is greater than 1.0, your portfolio moves in larger swings than the market.


SPLV (blue) versus the S&P 500 (red) 2013 year to date (as of 19, April 2013).
SPLV (blue) versus the S&P 500 (red) 2013 year to date (as of 19, April 2013).


If you believe the CAPM, the only way you can outperform the market in an upward trend is to hold a portfolio with a beta larger than 1.0. In a downward trend you can only outperform the market with a lower beta portfolio. In the downward market you lose money with the market, but at a reduced rate.

As an example of these principles, let’s take a look at SPLV. We’ll treat SPLV as a managed portfolio measured against SPY as a benchmark. SPLV is an ETF that represents the 100 lowest volatility members of the S&P 500. We’ll use it as a stand in for an actively managed portfolio. I chose SPLV for this example because it holds a specific subset of the S&P 500, and therefore it represents a portfolio choice to include some stocks and exclude others in the portfolio.

Yes, SPLV is an index fund, not a managed portfolio. But I argue that we can consider that it represents an “active” investment choice in low volatility stocks. CAPM will tell us that we’re taking a risk by doing this.

The chart at right depicts the year to date performance of SPLV versus the S&P 500 (as of 19 April 2013). As you can see, they tend to move in the same direction most days, but over time SPLV has accumulated higher returns.

Let’s dive deeper into the quantitative relationship between SPLV and the benchmark. We’ll start with beta.


Measuring Beta



Alpha: A Measure of Fund Management Skill
A scatterplot illustrating daily returns of SPLV versus the benchmark returns of SPY (the S&P 500). The red line is a list squares linear fit to the data.

 As we mentioned above, beta is a numerical value that represents how much we would expect a stock (or portfolio) to move when the market moves. For example, for a stock with a beta of 1.0, if the market goes up 1%, we expect the stock to go up 1%. If beta were 2.0, we’d expect the stock to go up 2%.

The chart at right shows a scatter plot of the daily returns for the S&P 500 (x axis) versus returns for SPLV on the same days (y axis). We’ve fit a line to the data (the red line) using linear regression. The equation of that line is shown in the upper left corner of the chart.

If our portfolio (in this case SPLV) were perfectly in step with the market, we’d expect all the little blue dots to be arranged in a straight line with a slope of 1.0. Note two things: 1) The blue dots are generally aligned, but they aren’t in a straight line, so sometimes the market zigs and SPLV zags; 2) The slope is 0.75 (not 1.0).

Beta is simply the slope of the line. In this case with beta = 0.75, we expect the portfolio to move up only 0.75% when the market goes up 1.0%. That should be expected because we already know that SPLV represents the low volatility members of the S&P 500.

Recall that the CAPM would predict in this case (an upward market) that SPLV should have lower returns than the S&P 500 because it’s beta is less than 1.0. But of course we know that SPLV has higher cumulative returns this year (10% for SPLV versus 4% for S&P 500). To be fair to CAPM, it provides a random component that can explain returns better than the market. So it could be that SPLV is performing better than the market because of this random component.


Measuring Alpha

The outperformance of a fund with respect to its benchmark may be due to chance (as the CAPM asserts), or it may be due to skill (as active management asserts). In either case, alpha is the way to measure that outperformance. In the hedge fund world, investors assume a portfolio manager has skill, or lacks it, and alpha is the primary measure of that skill. How to measure alpha?

Take another look at the scatterplot again. Pay close attention to the red line. The CAPM predicts that the red line should go through the origin. Note that it does not cross through the origin, but instead it crosses the Y axis a little bit above 0.0.

That difference, the Y-intercept, is alpha. In this case, alpha = 0.10%. That means that on average, SPLV returns a tenth of a percent more than the market each day. If this rate continues all year, for 252 trading days, we would expect SPLV to outperform the market by 29%.

I don’t think we’ll see that though. Overall, I think SPLV is a better choice than the market long term, but I don’t expect to see 29% excess returns. Note that this article isn’t intended to be an SPLV promotion, we’ve just focused on a particular period of time for this example. In 2012 for instance, SPLV lagged the market by 4%. Still, it was significantly less volatile and a sported a higher Sharpe Ratio that year.


The Risk Free Rate

Some quant experts will note that I did not discuss the risk free rate in these discussions. The formal definition of alpha includes this detail. I skipped it in this case because the risk free rate at present is approximately zero.


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