## What is alpha?

Last week on Slate Money I had a disagreement, or at least a lively discussion, with Felix Salmon and Josh Barro on the definition of alpha.

They said it was anything that a portfolio returned above and beyond the market return, given the amount of risk the portfolio was carrying. That’s not different from how wikipedia defines alpha, and I’ve seen it said in more or less this way in a lot of places. Thus the confusion.

However, while working as a quant at a hedge fund, I was taught that alpha was the return of a portfolio that was *uncorrelated* to the market.

It’s a confusing thing to discuss, partly because the concept of “risk” is somewhat self-referential – more on that soon – and partly because we sometimes embed what’s called the capital asset pricing model (CAPM) into our assumptions when we talk about how portfolio returns work.

Let’s start with the following regression, which refers to stock-based portfolios, and which defines alpha:

Now, the term term refers to the risk-free rate, or in other words how much interest you get on US treasuries, which we can approximate by 0 because it’s easier to ignore them and because it’s actually pretty close to 0 anyway. That cleans up our formula:

In this regression, we are fitting the coefficients and to many instances of time windows where we’ve measured our portfolio’s return and the market’s return Think of market as the S&P500 index, and think of the time windows as days.

So first, defining alpha with the above regression does what I claimed it would do: it “picks off” that part of the portfolio returns that are correlated to the market and put it in the beta coefficient, and the rest is left to alpha. If beta is 1, alpha is 0, and if the error terms are all zero, you are following the market exactly.

On the other hand, the above formulation also seems to support Felix’s suggestion that alpha is the return that is not accounted for by risk. The thing is, it’s true, at least according to the CAPM theory of investing, which says you can’t do better than the market, that you’re rewarded by market your risk in a direct way, and that everyone knows this and refuses to take on other, unrewarded risks. In particular, alpha in the above equation *should* be zero, but anything “extra” that you earn beyond the expected market returns would be represented by alpha in the above regression.

So, are we actually agreeing?

Well, no. The two approaches to defining alpha are very different. In particular, my definition has no reference to CAPM. Say for a moment we don’t believe in CAPM. We can still run the regression above. All we’re doing, when we run that regression, is measuring the extent to which our portfolio’s returns are “explained” by its overlap with the market.

In particular, we do not expect the true risk of our portfolio to be apparent in the above equation. Which brings us to how risk is defined, and it’s weird, because it cannot be directly measured. Instead, we typically infer risk from the volatility – computed as standard deviation – of past returns.

This isn’t a terrible idea, because if something moves around wildly on a daily basis, it would appear to be pretty risky. But it’s also not the greatest idea, as we learned in 2008, because lots of credit instruments like credit default swaps move very little on a daily basis but then suddenly lose tremendous value overnight. So past performance is not always indicative of future performance.

But it’s what we’ve got, so let’s hold on to it for the discussion. The key observation is the following:

The above regression formula only displays the market-correlated risk, and the remaining risk is unmeasured. A given portfolio might have incredibly wild swings in value, but as long as they are uncorrelated to the market, they will be invisible to the above equation, showing up only in the error terms.

Said another way, alpha is not truly risk-adjusted. It’s only market-risk-adjusted.

We might have an investment portfolio with a large alpha and a small beta, and someone who only follows CAPM theory would tell me we’re amazing investors. In fact hedge funds try to minimize their relationship to market returns – that’s the “hedge” in hedge funds – and so they’d want *exactly* that, a large alpha, a tiny beta, and quite a bit of risk. [One caveat: some people stipulate that a lot of that uncorrelated return is fabricated through sleazy accounting.]

It’s not like I am alone here – for a long time people have been aware that there’s lots of risk that’s not represented by market risk – for example, other instrument classes and such. So instead of using a simplistic regression like the one above, people generalize everything in sight and use the Sharpe ratio, which is the ratio of returns (often relative to some benchmark or index) to risks, where risks are measured by more complicated volatility-like computations.

However, that more general concept is also imperfect, mostly because it’s complicated and highly gameable. Portfolio managers are constantly underestimating the risk they take on, partly because – or entirely because – they can then claim to have a high Sharpe ratio.

How much does this matter? People have a colloquial use for the word alpha that’s different from my understanding, which isn’t surprising. The problem lies in the possibility that people are bragging when they shouldn’t, especially when they’re hiding risk, and especially especially if your money is on the line.

Thank you for this; I have a clearer idea now of what is going on than after the discussion on the podcast. 😉

I think that what you have here is an agreement on what alpha “should” be measuring, in a sense, but a disagreement on how to define it. And of course, different definitions may disagree on important points (while still managing to have a large intersection of what they “should” be measuring).

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And to add to the confusion, the two definitions collide under CAPM assumptions.

My take-away is to ignore all claims of alpha.

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Shouldn’t one ignore all claims about a measurement when the exact definition of the measurement is not forthcoming? My funds have extremely high gamma! Better than anyone else! What is gamma? Well, never you mind…

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I mean, theoretically, yes. But the thing is, people would see through that a bit sooner (still not soon enough). The real problems occur when you think you know what it means but you don’t.

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Reblogged this on Matthews' Blog and commented:

Appears an interesting discuss to me.

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Nice explanation. So in summary, how about “average return not explained by general market movements” ? Key examples for illustration:

1. .5 Market + .5 Cash : alpha = 0, moves half as much as market always

2. 1 debt + 2 market : alpha = 0 (or negative for debt interest), moves twice as much as market always

3. Instrument A: as volatile as market, but completely uncorrelated: alpha = market return, beta = 0, epsilon = same distribution as R_M

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Thinking back to my B-school days, I was never comfortable with past volatility as a measure of risk, for the reasons you cite. At the same time I can’t deny it’s a pretty useful proxy and I don’t have a better idea how to measure risk. I just think it’s important to remember that volatility is an estimation of risk, not a measure of it.

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One definition of alpha is the active return of a security or asset. It is the amount by which the security or asset outperformed (or underperformed) the benchmark.

Or in the words of Grinold and Kahn: alpha is the expected residual return.

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The difference is that you are taking beta as exogenous. The CAPM is a general equilibrium theory that has an explanation for beta in terms of risk aversion. If you ignore the fact that beta represents “risk” then, yes, alpha is just the uncorrelated part of the return.

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How much should one worry about the linearity assumptions here? If I have some fully-but-nonlinearly-correlated strategy (selling out-of-the-money puts, say), won’t a lot of that show up as alpha? I wonder if this explains anything about hedge funds’ attitudes toward tail risk.

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I’ll say one thing about selling out-of-the-money puts, you think you are selling rich volatility, which you are, 9 times out of 10, but when the s*&t hits the fan, and you have your 18 sigma event, you better hope you are trading OPM (Other People’s Money).

But I guess, each new entrant has to learn that the put buyers (long vol) bleed a lot for a long time until they finally capture all that “alpha” in just one good (from their perspective) event.

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Felix and Josh are referring to a single period measure known as “Net Selectivity” that was defined by Gene Fama in 1972: “Components of Investment Performance”, JOF 27, 3 (Jun), pp. 551-567.

Cathy is referring to a multi-period measure known as “Jensen’s Alpha” that was defined by Mike Jensen in 1968: “The Performance of Mutual Funds in the Period 1945-64”, JOF 23, 2 (May), pp. 389-416.

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Are you related to Kenneth French, as in Fama-French Three Factor Model?

It’s interesting that in Fama’s “Foundations of Finance” (1976), he refers to alpha and beta as merely intercept and slope. Furthermore, in a footnote (p 77) he writes: “In the financial literature beta-i is called the systematic risk of security i and sigma-squared(~e-sub-it) is called the unsystematic risk … We do not use the terms systematic risk and unsystematic risk in this book.”

In the same book, there are no references to alpha and beta in the index.

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Yeah, the Fama-French Three Factor is a predecessor to to the four-factor version I referenced below… IIRC, French maintains a high quality time series for the overall stock market’s dollar-weighted market-cap and value-growth orientation over time.

See, for instance: http://faculty.london.edu/aedmans/Rowe.pdf

To ensure that any outperformance of the BCs does not result from risk, I control for the four Carhart (1997) factors using

R_i_t = α + β_MKT MKT_t + β_HML HML_T + β_SMB SMB_t + β_MOM MOM_t + ε_i_t

where R_i_t is the return on Portfolio I in month t in excess of a benchmark, described below. α is an intercept that captures the abnormal risk-adjusted return. MKT_t, HML_t, SMB_t, and MOM_t are the returns on the market, value, size, and momentum factors, taken from Ken French’s Web site.

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also, this is good: https://www.aqr.com/cliffs-perspective/our-model-goes-to-six-and-saves-value-from-redundancy-along-the-way

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I wish I was related to Ken but unfortunately for me, no.

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The regression process looks linear. Perhaps I missed something? Is it just discretized? In any case, I think your takeaway is spot on; “finance speak” in and outside the industry has created some “off-label” terminology ranging from confusing to debatable to just plain wrong. I usually know what parameters alpha and beta are describing when I see them in the maths. But when analysts, marketers, reporters etc throw around “alpha” and “beta” to describe, for example, higher order sensitivities that are stochastically derived with gamma or rho… then it’s, frustratingly, “all Greek” to me.

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I think your definition, as well, fits better with more sophisticated versions of alpha, like the “four-factor alpha”, where you are looking for the returns that cannot be explained by reference to multiple market factors. The standard four-factor model includes coefficients for volatility (which is the classic Beta), but then also adds market-cap, value-growth orientation (often determined based on the market-to-book ratio), and momentum (often done as a binary or ternary, sometimes done simply as a T-1 shift of the return series).

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Great discussion. Cathy’s definition captures the essence of technical alpha which has ex-ante, E[alpha] = 0, and ex post is the regression intercept. It’s true that it does not depend on CAPM; as somebody else has noted, in the CAPM context, it’s a Jensen’s alpha. But CAPM can be viewed as the special case of a multi-factor model. So, alpha is really a function of which common factors are defined. In CAPM, there is only the single common factor of the market premium, R(m) – riskfree rate, and beta is the exposure to the common factor (aka, systematic risk). But APT produces several factors; alpha is uncorrelated return, or the regression intercept, or put another way, the excess return not explained by any of the common factors. One quantified, the issue is: how much is skill versus luck?

But I think the point that “the remaining risk is unmeasured” is true and key. Nowhere is risk measured implicitly, here! The regression produces partial beta coefficients; a high ex ante beta means the excess return is correlated to the common factor (e.g., excess market premium). It’s only really a risk measure because we assume the compensation is for assuming risk. So, the beta risk is inferred, and further to Cathy’s point, the alpha is just the unexplained excess return. That’s why it is the numerator is a risk-adjusted metric like the information ration: alpha/residual risk, or less likely, alpha/tracking error.

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