A few decades ago, as modern financial theory was being developed, academics looked for a way to describe the expected return of an asset as a function of its risk. Eventually they settled on a model that describes an investor's returns as the sum of the risk-free return and the premium they received for taking risk, plus an error term to account for the fact that different investors will get different returns (hint: that error term is important).

So their formula looks like this: returns = risk free rate + (risk premium for 100% exposure to a given asset class) * (actual exposure to that asset class's risks) + whatever deviation, positive or negative, beyond the amount they're "paid" to accept volatility.

The math should be familiar to anyone who has taken a high school-level statistics class; it's just a linear regression model with different variable names. If you plotted a bunch of real returns (y-axis) by their level of market exposure (x-axis) and ran a linear regression, you’d end up with something that looks like “ŷ = b₀ + b₁x”. That generated equation just describes a predicted value with an intercept, a variable, and a known coefficient. Any gap between the value that equation predicts (a point on the line) and the actual return (a real datapoint) is would be denoted with an "ε".

“ER_i = R_f + β(E(R_m)-R_f) is basically the same thing; R_f is the risk-free rate, β is the level of market exposure, and E(R_m)-R_f is the expected return from 100% market exposure, less the risk free rate (i.e. it's the premium you get for taking the risk). When this model leads to imperfect results, like when you're fully-invested in stocks for a decade and have been compounding at something abnormal like 30% or -5%, the gap is denoted with an "α".

Aside from using a brand new Greek letter in the notation, we're doing exactly the same thing. We're looking at a model that describes an average case, and we’re comparing it to deviations. But it's useful to re-label the same concept, because the average market participant is trying to maximize that error term in their own case, which is exactly opposite of what we normally want when we come up with error terms.

There's a constraint to doing this, though: alpha, in the aggregate and by definition, must sum to zero before transaction costs and management fees. If a market returns 10% some year, total returns for all investors in that market must be 10%, minus the cost of trading in and out of specific parts of the market, and minus whatever they paid someone for managing their money. And that leads to two corollaries about alpha, especially sustainable alpha:

1. In a competitive market, the only source for repeatably good decisions on your part is repeatably bad decisions on somebody else's part.

2. "Transaction costs" are not just commissions, but also the cost of executing a trade. Anyone who successfully buys a share of stock is, at least for a moment, the high bidder in the world's most competitive dynamic auction, and anyone who buys or sells a lot in a hurry is going to get an unfavorable price. So one source of persistent alpha is to be on the other side of those rushed transactions.

In the first corollary, "bad decisions" implies that a good investor is passively waiting for someone else to make a mistake, but many of the most hyperkinetic market participants, like pod shops, are waiting for errors of omission—if the consensus in January was that some company would earn $0.50 per share this quarter, and the evidence now points to $0.48, but estimates won't be updated until the last minute, then the pod shops will want to get there first. In other words, the repeatable mistake they're looking for is the mistake of not continuously ingesting all possible information that could affect a company's performance, or, having done so, not pricing accordingly.¹

There are also bad decisions that some people and companies tend to make, which traders can reliably bet on. One convenient result of the growth of online market chatter is that it's much easier to see exactly what retail investors are excited about, and to bet against it. There are, of course, some recent examples of this ending poorly for funds that shorted meme stocks, but the aggregate profits from trading against retail investors exceed the profits of riding on their coattails.

Companies, too, make bad decisions that can be repeatably exploited. For example, they tend to keep buybacks and (especially) dividends going for longer than is optimal. There's also typically some mix of debt and equity that maximizes the market value of a company, and in theory the company's capital-raising tries to solve for this, but in practice they can overshoot, which means a company can engage in value-destroying buybacks and dividends.

The second corollary is also a helpful one: you can produce repeatably good returns if you're generally supplying liquidity to counterparties who a) want it, and b) have bad reasons to want it. This can mean many different things: evaluating lots of companies and then strategically buying when there's a short-term dislocation (someone is selling a large position) or a longer-term one (a bad quarter that other market participants over-extrapolate). Or it can mean pure liquidity provision. Market-making firms don't typically frame their results as "alpha,"² but if you count it as alpha, then aggregate alpha is less negative than it looks, because what's a transaction cost to one firm is transaction-based revenue to another.³

At an even higher level of abstraction, one way to think about alpha is that we should expect it to shrink over time, at least relative to the pure return from taking risk. Once there's a metric, a label for it, and a set of compensation arrangements based on it, there's a lot of money in making it go away. But reversing this works, too: there's more outperformance in fields where it's harder to come up with firm expectations about what normal performance is. So the alpha to look for is in places where it's hard to measure alpha.

Read More in The Diff

The Diff uses the concept of “alpha” from time to time:

• We talked about how as easy and quantifiable sources of alpha disappear, what’s left is hard to feed into a model (or measure against historical returns).

• Along those same lines, being able to slice up differentiated datasets is a great way to generate alpha.

• We did a breakdown once on how alpha and beta apply to buybacks.

• It showed up when we talked about operational alpha and alpha capture at multi-strategy funds ($).

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1. Incidentally, this is probably one reason pod shops, even good ones, tend to be overweight momentum (see this piece on factors for more about momentum). If you cut your losing positions frequently, whatever you don't cut is generally moving in the right direction, and the better those selections are the more this is true. Said differently, these portfolios will be more weighted towards momentum, especially if the winning trades were right.

2. It doesn't make sense in a model where the target market exposure is zero, and where human effort and technology rather than capital are the limiting input for returns.

3. One way to make alpha a useful tool for prop trading firms is to think of it on a per-trade basis. If the spread between the bid and ask for an asset is 10 basis points, the theoretical maximum a market-maker can make is 10 basis points every two trades, i.e. every time they buy shares for $10.00, they can promptly turn around and sell them for $10.01. The actual edge they have on each trade is smaller, and one way to think about that is that if they're making 1 basis point of profit when there's a 10-point spread, their counterparties generate 9 points of gross alpha from being on the right side of the trade, but lose 10 points of alpha from executing the trade.

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