A short discussion of this important metric.

**Related Articles**

- StumpGrinder: Performance where we track the performance of the actually traded portfolio.

**Background: 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 that will go up in value. On the downside, they try to exit positions early before they lose money. But some experts argue that investors would do better to invest in the market broadly than by 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 efficient, you can’t improve performance by timing stock purchases or sales.
- The market is best represented by major indices like the Dow Jones Industrial Average or the S&P 500.
- It is risky to hold a portfolio that deviates from the “the market portfolio.”

This argument is compelling, especially when you consider the performance of the market up until 2001. The concept

was supported theoretically by the Capital Assets Pricing Model (CAPM) advanced 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. These funds and ETFs provide everyday investors easy access to simple and powerful investment instruments. 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,* meaning that the portfolio is not actively managed. There are hundreds of articles on the net that advocate passive investing over market timing. Here’s one.

In contrast, the philosophy of *active management* suggests that allocations between assets 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 in comparison to passive index investing, how might we measure this value?

*Assessing 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 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.

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. For the purposes of this article, we’ll treat SPLV as a managed portfolio and compare it to 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**

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 each day than would be expected based on it’s beta. If this rate continues all year, for 252 trading days, we would expect SPLV to outperform 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.

**Disclosure:**

Tucker Balch manages a fund with a long position SPLV.

*research, strategy, technology*

mike stepenaskie

April 27, 2013

I dont see the need for a fee, the cost for non-completers is 0. I took this the first offer, and completed none of the homeworks. I took a couple of python tutorial , took it a second time and completed all the homeworks, with all correct scores. Had there been a charge I would not have taken it the first time, and been interested enough to work on python.

Tucker Balch

April 27, 2013

I think your comment is on the wrong article.

DFleer

May 7, 2013

One caution when using regression to estimate parameters – the “true” value is unknown and the regression constants are just a best-guess estimate. Using confidence intervals from statistics helps estimate a range that the “true” value lies in with a stated degree of confidence (typically 95%). In the case of SPLV vs SPY daily returns, for the data covering 1Q13 I found that the 95% confidence interval for the intercept includes zero – meaning that an alpha for this data set of zero is just as reasonable as the regression coefficient of 0.076% (I didn’t use the exact date range that you had so I have slightly different regression coefficients). For actively managed portfolios where the intercept coefficient confidence interval does not contain zero, you could then state that you’re fairly confident (at a 95% level, say) that the portfolio has generated alpha.

It’s probably a minor point, but the detail is somewhat important if using the technique to evaluate a specific individual portfolio.

Tucker Balch

May 7, 2013

Yes, good point. As we gain more data the confidence interval should close.

Nick Iversen

June 7, 2013

I’m not sure that alpha is a measure of skill.

You can regress your fund returns against the market returns and get a statistically significant positive alpha. So you are beating the market.

But in the same situation you can regress the market returns against your fund returns and still get a statistically significant positive alpha. So the market is beating you.

How do you explain that?

(Mathematically, the regression of y vs x is not the same as x vs y and it is possible for both regressions to have positive alpha).

Tucker Balch

June 9, 2013

If y = beta * x + alpha, then x = y/beta – alpha/beta. So, if we have positive alpha in one case we necessarily have negative alpha in the other case. Unless beta is negative, in which case it would be a weird benchmark to chose.

Nick Iversen

June 18, 2013

Try this example (in MATLAB but easy to use R or whatever):

n = 100000; % a big number so that our alphas are statistically significant

x = 1 + randn(n, 1); % market returns are normally distributed with a mean of 1

y = 0.5 + x + randn(n, 1); % manager returns have a mean of the market return plus 0.5 due to skill

Now regress y on x (alpha = 0.5) and x on y (alpha = 0.25)

Statistically significant positive alpha in both cases. So the manager beat the market and the market beat the manager.

Nick Iversen

August 9, 2013

I guess you aren’t going to try my example.

It’s worth it, though. You will be surprised.

Tucker Balch

August 9, 2013

Hi Nick, Will do. Just been busy.

Mike

September 2, 2013

I believe your model is incorrect. You have 1+. This will throw off your numbers for beta and alpha.

Even if we leave that alone, I don’t believe you did the regression right. It is a mathematical impossibility to have a positive point estimate of alpha in both cases (unless the two regressions had different sets of data). In fact, the .25 result for x on y alpha in a scenario where your beta is 1 further shows that your regression method was flawed.

Nick Iversen

September 3, 2013

Mike, your reply illustrates my point nicely.

You think it is impossible – try it. Report back when you have. As I say, you will be surprised.

The reason for the 1+ is that the market has to have a positive mean return. This doesn’t throw off the alpha and beta since they are applied to whatever return the market produces. The 0.25 is because alphas are usually smaller than the mean market return.

I like forward to your next posting after you have tried it.

Mike

September 5, 2013

Nick I don’t think you understand my point about .25 A beta of 1 would not allow different magnitudes of alpha regardless of whether you do y over x or x over y. They would both be .5 (positive in one case and negative in the other). IE the inverse of a line with slope (beta) of 1 would have y intercept (alpha) of equal magnitude but opposite sign. So it would be -.5 There is no need to run that in matlab. It’s basic algebra. If you run your script in matlab and get .25, there is an error in your script.

Nick Iversen

September 5, 2013

Mike, please, please try it yourself.

My point is that you get a different result from what you expect. But if you don’t it for yourself you won’t experience this.

My regressions aren’t wrong. I get a positive alpha in both cases AND YOU WILL TOO.

“There is no need to run that in matlab” – sorry, Mike, but you do need to run it. Trust me.