Wednesday, February 20, 2008

Examining your assumptions: The Fundamental Law of Active Management

This is one of a series of posts on the importance of understanding the assumptions behind a quantitative model. As I understand it, the Richard Grinold paper on the Fundamental Law of Active Management is now part of the CFA reading:

What Grinold means by the above formula is that a manager’s value-added (Information Ratio) is a function of his selection skill (Information Coefficient) and the number of opportunities (N) he has.

No doubt thousands of CFA candidates have read this, memorized the formula and nodded sagely. They may have even tried to apply it in their working lives. Let's look at some of the underlying assumptions behind this model and understand how a blind application of this work may lead to suboptimal results.

What do you mean by IC? Most quants think they know how to measure IC, at least mathematically. However, the Information Coefficient for any selection process will vary according to time horizon. Is your IC the same for 1 day as for 1 month or 1 year? If you assume a flat IC for any time horizon and not incorporate trading cost assumptions this model will generate portfolio turnover that is uncontrollably high. Grinold in his later works elaborated on this turnover issue (see Grinold and Stuckelman, 1993; also Grinold and Kahn, 1995).

What do you mean by N? N is the number of independent opportunities available. If you are running a 100 stock portfolio does that mean that the number of independent opportunities, or ideas, is 100? What if you were picking stocks based on some fundamental criteria (e.g. low P/E) or macro theme (e.g. rising inflationary expectations). Is N equal to 1, 100 or somewhere in between?

Putting it into English

While I am a math geek as much as the next quant, I like to put the ideas into English when I apply them to the real world. The idea behind the Fundamental Law of Active Management is to size the bets according to the edge you have.

Grinold's work is actually a thematic variation on Kelly’s Criterion. John Kelly was a Bell Labs engineer in the 1950s who posed the following problem. Supposing a gambler overheard underworld types fixing a horse race on the telephone, but there was noise on the line. What should the gambler bet given this knowledge and the level of noise (= probability of correct information) on the line? This discussion could then be generalized to a treatise on information content, signal-to-noise ratio, etc.


nodoodahs said...

Well, at least they're not talking about "alpha" and "beta" as if they were mythical beasts from medieval times, instead of just regression equation output.


You're right, though, they need a unit time as the denominator.

Using "1.00" for N implies a complete cycle of inventory or the amount of time that the trader has for their average holding time on a dollar-weighted basis, i.e. the time it takes to do 100% turnover. The problem with this assumption is that you lose comparability, so you can't compare day traders, swing traders, CANSLIM gurus, and value investors – because of the different meanings of N for each of them.

I suggest standardizing the time period to one year and comparing everybody on an annualized basis.

Here's a further suggestion. Expectancy for a trading system (or the actual expectancy of a trader) is the percentage of winning trades times the average percentage gain, minus the percentage of losing trades times the average percentage loss. It equates to an expected value per dollar value attached to a trade and could be used in place of IC. Then use position turnover per unit time (year, in this case) for your N in the "number of opportunities" part of the equation. Voila! Instant comparability!

Grinold's issue is a straw man, because it is taken out of context of the trading system (or trader) being evaluated.

Any evaluation worth looking at would use a time frame long enough to be assured of some statistical accuracy for the IC, in the case of a day trader, maybe some number of trades (but you would still use inventory turnover as the basis for N to allow comparability), and in the case of a value investor, some number of years of results. It's quite frankly idiotic to use a daily IC, unless you're talking about automated computerized trading doing 50-100 trades a day, and even then! you'd STILL want to use an annual N so you could compare it to other styles (like value investing).

Grinold's point about transaction costs is valid but immaterial to the calculation of IC or IR, as those should *always* be calculated net of transactions costs.

Cam Hui, CFA said...

Most investment processes have several components:

1) The selection (or alpha gneration) process
2) The portfolio construction process
3) Trading
4) Diagnostics

Your comment:

Grinold's point about transaction costs is valid but immaterial to the calculation of IC or IR, as those should *always* be calculated net of transactions costs.

confuses the selection process (IC) with the output from selection + portfolio construction + (possibly trading). The selection tells you WHAT to buy and sell. The portfolio construction process tells you HOW MUCH to buy and sell. You can get very different results depending on how you dial up or down the risk control in portfolio construction.

Grinold's "Fundamental Law of Active Management", IMHO, is really trying to measure the goodness of the selection process. The rest is up to you. There is no single answer, it depends on your risk parameters and how you trade.

nodoodahs said...

It's impossible to separate selection from sizing and other aspects of construction, unless one is an academic who isn't interested in a practical evaluation. We're evaluating systems here (or a trader, who is using a system whether he is conscious of it or not), and the system (or trader) isn't making selections in a vacuum, therefore the sizing and construction come with them.

IC, IR, expectancy, etc., are all dependent on the total package, and the total package should always be assessed net of transaction costs (unless such costs are so low as to be well within some margin of error in the rest of the measurement, and they may be discarded without material impact to the evaluation).

If the expectation of a system is to sell holdings for a profit, it is TRADING. Buy and hold for 20 years until junior goes off to college is still trading, buying and selling in order to net a profit. Most of what people call "investing" is really trading, and they're just trying to feel better about it by calling it something that sounds more distinguished. Maybe dividend buyers and some bond buyers are "investors" but precious few other equity or bond buyers are, they are trading. Perhaps on long time frames, but trading nonetheless.

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