Deep makes more money than Wide

Apologies for not blogging this sooner, knowledgeable reader; I have had this bee in my bonnet for some time. It is important, and goes to the very heart of the nature of rationality and investing. Have a look at this:

 Where IR is “Information Ratio”, which I would understand to be longhand for how much money you make, and IC is “Information Coefficient” (whenever I hear the word coefficient something inside me dies), or how good you are at selecting things which go up or down in the right way, and N is how many independent opportunities you have to exercise your IC on. 

So, how much money you make, or your schmalpha, is how clever you are at picking investments times the root of however many different things you can invest in. Clearly, I can make more money by being more clever (more cleverer?), or if I am just a bit clever, I can make more money by spreading that bit of cleverness over lots and lots of different independent opportunities. Looks pretty easy, doesn’t it? 

This, according to a Humble Student of the Markets, (found, like so many other interesting things, during a daily troll through Abnormal Returns) at is viewed as a “Fundamental Law of Active Management.” I would call it a “Recipe for Serious Underperformance”, or alternatively “Arse-Bitingly Wrong”. To Humble’s credit, he also notes that “a blind application of this work may lead to suboptimal results.” 

There are many things which make me cross about this. Where to start? 

Firstly, it fundamentally misunderstands human rationality. We are creatures whose software and hardware has evolved to deal with direct threats and exploit opportunities using a degree of abstract reasoning. Being able to use only finite processing power in a not-obviously-finite world, we need to make abstractions and generalisations, ie to simplify things. This works great even given complex problems which are static, and which can be broken down and dealt with collaboratively, e.g. how to build an atomic bomb or invent computers. However where complex problems are dynamic, ever changing, where rules that work in one instance stop working the next and problems cannot be broken down into finite pieces and dealt with step by step, well, there we get completely screwed. Our simplifications and abstractions cease to accurately describe the situation we are in, and we begin to draw castles in the air.  

As Spinoza would say, we are unable to get an “adequate idea” of things. We have 2 kinds of knowledge, he says. First we form ideas from: 

. . . singular things which have been represented to us through the senses in a way which is mutilated, confused and without order for the intellect . . . and from signs, for example, from the fact that, having heard or read certain words, we recollect things, and form certain ideas of them . . .; these two ways of regarding things I . . . call knowledge of the first kind, opinion or imagination

Knowledge of the second kind is intuitive knowledge, which can occur only when we have adequate ideas of something: we can have an adequate idea of the essence of the number two, of its two-ness, and the four-ness of four follows logically from combining two twos. But there’s not much else to the number two than its two-ness; understanding that doesn’t leave room for a lot of error. 

Knowledge of the first kind is the bad kind of knowledge, the type which tells us “I can make squillions borrowing multiples of my capital at LIBOR and buying ‘AAA packages’ of subprime CDOs yielding me 100bp more; everyone else is, why not me?” 

Do I have an adequate idea of Cisco? If I try real hard, break the business lines into bits and really think about how it all hangs together, maybe I can have a less inadequate one. Can I have a similarly less inadequate idea of the dynamics of the networking space? Well, harder obviously, but if I know CSCO inside out and add my knowledge of Juniper and a couple of others I could get somewhere, with a bit of abstraction and imagination. Actually, that would help my understanding of CSCO, too. I could make a few well-judged bets on some future trends there, with reasonable confidence; I would have a high “Information Coefficient”. What if I add the demand that I learn all there is to know about luxury goods? Exotic credit derivatives? Well, then I have to crowd out a bit of the extra work I was going to do in CSCO. Not only do I know less about credit derivatives than I would, I will also know less that I would about CSCO and networking. 

So do you see where I’m going? If my investment skill is dependent on some degree of knowledge or understanding or having some adequate idea of what I am investing in (a not unreasonable proposition) how can infinitely increasing N, the number of different independent opportunities I need to analyse possibly help? It can’t. It would add too much complexity. The abstractions would become inaccurate, even harmful and misleading, and I would make errors, persist in them, and lose money. IC moves inversely to N, not positively as in the formula above. 

Oh silly Baruch, a defender of the proposition (most likely a quant) would say. You’ve got it completely wrong. N merely means the number of times “at bat”. It is how many times you get to exercise your IC, your skill; it’s not how many sectors you look at or how many stocks you think you know. In this case, yes, if you can push N to infinity you will make more money ceteris paribus. The positive relationship between IC and N in the formula still holds, according to this view of the meaning of N. 

This leads to my second objection: markets adjust, resources are limited, and time is money. Information edges get arbitraged away in direct proportion to their accessibility; no-one can deny that. Why else would hedge funds pay millions to get proprietary information (and almost invariably get fleeced) from 3rd party industry watchers and consultants. What we all want is deep, meaningful data arrived at independently before the hoi polloi get their hands on it. Unless we have unlimited research resources we simply can’t do wide and deep together; and if we did have unlimited resources, the co-ordination between, and sorting of, the resources by a directing intelligence would start to take up as much time as the research itself. Wide knowledge is shallow knowledge, stuff which is unlikely to move the stock as much. 

Not only does it get arbitraged away quicker, then, but there is something more dangerous that happens: it lends itself to a potentially explosive combination of leveraged models, and crowded trades. This goes to the heart of what I was banging on about in my posts on the Demise of the Quants last year, for surely there is no purer exponent of this pernicious formula than a high turnover stat arb model. Here is the lie: if I am just a bit clever, I will ultimately NOT make more money by spreading that bit of cleverness over lots and lots of different independent opportunities. What is more likely to happen is that I will lower my “Information Coefficient”, first to levels only just above zero, and then ultimately, below zero. In the real world, again, IC moves inversely to N. 

Those are basically the biggest problems I have: 1) as N increases, multiplying factors leads to complexity, and 2) shallow knowledge tends to be unhelpful.  

Other objections would be 

3) It does not help us with the most interesting bit, the nature of the “Information Coefficient”. The answer to the question “how often should people who are good at investing actually invest?” is not at all as useful as the answer to “what makes some people good at investing?” I’d rather get to the bottom of that.  

4) It does not take friction into account. This is alluded to in Humble’s posting, and also in the comment thread with a certain Mr Nodoodahs who is also known to lurk around here sometimes. Clearly the higher the N the higher the transaction cost, which would also tend to lower one’s IR. This might be partially why the formula uses a square root over N, showing that by adding more Ns the marginal impact on returns is less. But it is still a positive relationship, and I think that is wrong. 

What should IR=ICx√N look like instead? Well that’s not my job guv’nor, and I am spectacularly ill-qualified to do anything in algebra, having ceased my maffema’cal studies at the age of 14. But some expression to do with time, some inclusion of the benefits (or not) or diversification – which is also highly relevant to the discussion, creating some benefit to early values of N, and some other stuff would make it better, probably. Bento, you may have some suggestions; I know it is not necessarily your field, but what do you think?


8 thoughts on “Deep makes more money than Wide”

  1. Isn’t risk proportional to 1/N, and that is what makes this equation mistaken? The more stocks, the lower the risk., but the lower the potential reward

    There ought to be a ‘schmeta’ which measures your ability to understand complexity–so that the equation looks something like IR=IC*((N)^-schmeta)*(1/N). If you have a large schmeta, you can deal with complex multi-part problems well. If you have a really high schmeta (schmeta>.5). If you have a schmeta of <.5, wider spread always causes you to do poorly.

    If we add enough variables, we can easily fit all possible opportunities perfectly. Maybe schmalpha is schmeta-(market-average-schmeta).

    I am, however, totally uniformed in the monetary arts, and merely here for the philosophy.



  2. Indeed, B, the impact of diversification, which, as you say degrades returns at higher levels of N, would seem to cock it all up. We would need to have a term, say aN, where a is the level of N after which it all went pear-shaped. Otherwise I liked your version of the equation very much indeed, and had no idea what it was on about by the time I got to the end of it.

    Wouldn’t IC be partially dependent on Schmeta as well, however? Probably correlated, if not causal. If I can handle lots of complexity I may also have a large brain and therefore more IC than the average bear, therefore might there not be 2 schmetas in the equation were we to break down IC definitionally?

  3. The hope is that IC is dependent on your ability to focus and determine detail, where schmeta is our ability to understand complexity. IC is our inner accountant (grey and nameless), while schmeta is our inner steve jobs.

    The far more intriguing question is how to measure all of this. To do so, I think we need luck An appropriate model might thus be

    As we all know, luck is always amplified by schmeta. Better yet, I have a 4 component model to fit lines, which makes me feel that even I can manage this.

  4. B, if the point is that our ability to understand complexity is not great, surely out schmetas will be low. Can we square root those schmetas? We may also need to introduce a term H ( or hubris)for our PERCEIVED ability to understand complexity, where H does not equal Schmeta. N rises in proportion to H, as we feel we are jolly clever and can understand more things. But of course, this adds to N which at higher levels reduces our IC! Throw that in somewhere, pet mathmo of the Spinozists, and I think we may be ready to publish.

    Felix, interesting you bring up Soros, who is a prime inspiration of the reduction of complexity point. He would work intensively to reduce his N. His m.o. was always to isolate a strategic issue out of the many available, and then throw as much resource as possible at it, really concentrating on it, to see if it was worth playing one way or another. I don’t know how many themes he would play at one time, but I would be surprised if it was very many.

    My complexity point is also very informed by Popper and Hayek, and as you know, Soros and Popper in a tree.

    But in general, you are right: we don’t hear so much about the big macro funds any more. So long as they have epistemologically sound approaches, I see no reason why they shouldn’t prosper.

  5. Charlie Munger, ““A Lesson on Elementary, Worldly Wisdom As It Relates To Investment Management & Business”. Way back in 1994.

  6. A number of points:
    1. What is ‘IR’ ? I am assuming it is returns above the average market return of a passive fund invested in a portfolio replicating a benchmark index?
    2. The equation ignores the impact of other participants. What happens if the majority of other market participants are trading off another set of (false) premises? (i.e. Can you be too clever / have too much information) ?
    3. What is this equation trying to say about ‘N’?
    I seem to have a choice between:
    £100 * IC of 1 * sqrt 1 = £100 (Given trading opportunity A)
    or [£50 * IC of 1* sqrt 2] *2 = £141 (Given trading opportunity B and C)
    Therefore increasing the number of trades always increases my returns.
    But this assumes my ‘IC’ is equal for A, B and C and remains constant regardless of how many trades I carry out. If the IC differs between trading opportunity, then couldn’t the relationship between increasing ‘N’ and increasing returns break down very easily:
    £100 * IC of 2 * sqrt 1 = £200
    [ £50* IC of 1 * sqrt 2] * 2 = £141
    Substituting, ‘research effort’ for ‘IC’: Does this imply if I spent twice as much effort researching one stock, I could significantly increase my above market returns compared to splitting my effort equally between two stocks?

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