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# The Propagation of Errors in Alpha: A Cascade of Errors for Portfolio Construction

Dr. Paul White

“For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.”
- Richard P. Feynman on the Space Shuttle Challenger disaster1

Alpha is that elusive and rare quantity often denoted as manager skill.  Detecting it is harder than just fitting returns to an OLS (ordinary least squares) model and extracting the intercept.  The industry has done a good marketing job in programming investors that this is the goal but less of a good job in telling investors how to look for it.

Finding it requires some work.  Part of the issue is that it is a delicate statistic in the sense that small perturbations in the output.  This is sometimes referred to as sensitivity, which has a mathematical definition.  The sensitivity is actually a compounding of sensitivities, that is, if alpha depends on a parameter which depends on yet another parameter, then the error in the very first parameter filters through to the error in the second parameter which filters through to the error in alpha.  This is akin to a domino effect, or a small pebble can cause a landslide!

Our pursuit of the sensitive of alpha leads us to turn a very simple equation, y = a + bx , into a complicated thought and make a mess by using calculus.  Because of that, we will skip to the end for the sake of brevity and state the sensitivity, or the errors, in the respective parameters.  Borrowing liberally from “Data Reduction and Error Analysis for the Physical Sciences” by Philip R. Bevington, we have the following equations with the usual definitions:

The sensitivities are these:

The astute reader will note the obvious result that σ appears in both σα and σb, so the errors propagate.  Further, we can express the sensitivity in a in terms of the sensitivity of b.

This was more math than a novice may want, and the practitioner may find little insight, but it helps to say it outright that a (or alpha) depends on b (or beta) and that depends on σ (or the quality of the line fit).

HFRIWRLD vs S&P500

In some cases, a picture is worth a thousand words.  In the graph above, we take the HFRIWRLD index (Hedge Fund Research World Index) and plot its returns against those of the S&P 500 for 60 months (about 5 years).  Raw data are in blue and fitted data are in orange.  A casual observer will note that the line fits the data reasonably well.  The interpretation of this is the S&P 500 “explains” the returns of the hedge fund index fairly well.  However, we are not interested in the benchmark, we are interested in alpha, or the INTERCEPT of the graph.  Reality would defy that same casual observer to pick out the intercept of the orange line as it crosses the y-axis.  Luckily, the computer output is below.

Without turning this into a lesson on regression, we note the t-static, which tries to identify a rough proxy of signal to noise.  If the noise (error in the fit) is too large, it will wash out the measurement of the intercept (propagation of errors).  Since the t-statistic is below 2 (a common threshold value), we say that even though the measurement is positive, it still is indistinguishable from zero.  The error from the fit was not large enough to throw off our beta (course measurement), but it was large enough to drown out our ability to detect alpha.  Alpha was the original aim of this work.  We need to keep looking....

Building a portfolio around alpha is hard.  If the errors propagate, we won’t have enough confidence that we are building a portfolio around true alpha and not just some noise.  Care needs to be taken in the measurement and in the usage of alpha.  In the best case, one will have used a source for which there is no alpha; in the worst case one will have used a source for which there is negative alpha.  Portfolio construction absolutely demands that the inputs be identified correctly as positive alpha.  Failing to check this initial step will result in unintended consequences.

At Trinary Capital, we recognize the difficulty in identifying alpha and construct our portfolios accordingly because the cascade of errors can be disastrous.

1 Dr. Feynman is credited with having discovered a flaw in the tiny O-rings that lead to a cascade of failures.  The insistence of safety by numerous political and corporate figures gave way to the laws of Nature.

Past performance is not indicative of future results. Remember, there can be no assurance that the future performance of any specific investment, investment strategy, or product (including the investment strategies discussed in this article) will be profitable, equal any corresponding indicated historical performance level(s), be suitable for your portfolio or individual situation, or prove successful. Please remember to always speak with your individual advisor before making any investment decisions.

Past performance is not indicative of future results. Remember, there can be no assurance that the future performance of any specific investment, investment strategy, or product (including the investment strategies discussed in this article) will be profitable, equal any corresponding indicated historical performance level(s), be suitable for your portfolio or individual situation, or prove successful. Remember to always speak with your individual advisor before making any investment decisions.