The Fundamental Law
of Active Management

by Ronald N. Kahn

In the previous insight we learned that the information ratio is the key to active management. Given that, how can we achieve high information ratios? Let's begin by looking at a relationship Richard Grinold has called "The Fundamental Law of Active Management." This law expresses the information ratio in terms of two other statistics, the information coefficient and the "breadth":

(1)

Now, expressing one quantity in terms of two others isn't always progress, but let's leave that thought for now.

The information coefficient is a measure of manager skill. In particular, it's the correlation of forecast and realized residual returns. It measures the manager's edge in choosing assets. The breadth measures the number of independent bets the manager takes per year. It measures diversification. We define breadth as bets per year because the information ratio is an annualized quantity.

According to the fundamental law, in order to achieve a high information ratio a manager must demonstrate an edge for every asset chosen and must diversify that edge over many separate assets.

In fact, the fundamental law of active management is an old result from gaming theory. Imagine a roulette wheel where players can bet on red or black. The casino has a small edge because two numbers, 0 and 00, are green. Through every spin of the roulette wheel, the casino maintains that same small edge. Now imagine that during the course of the year, players bet a total of $10 million on this roulette wheel. Here are two possible scenarios: In the first, that $10 million consists of five million spins of the wheel with $2 bet for each spin. In the second scenario, the players all agree to pool resources and bet all $10 million on one spin of the wheel. The casino's expected return is the same in both scenarios. However, it would clearly far prefer the first scenario from a reward-to-risk tradeoff.

Examples

Next consider two investment examples. First, imagine a stock picker with an information coefficient of 0.035, a small but reasonably impressive level of skill for the active management business. This manager follows 200 stocks per quarter, effectively taking 800 bets per year. The fundamental law implies an information ratio of 0.99—indicative of a top decile manager.

Let's compare this example with the performance of a market timer. We'll assume that the market timer has a higher level of skill for every bet, with an information coefficient of 0.05. But this manager times the market by looking at broad macroeconomic trends and devises a new forecast once per quarter—taking four independent bets per year. In this case, the fundamental law implies an information ratio of just 0.10, slightly above the median for active managers. So a higher level of skill per bet does not necessarily translate into a higher information ratio. And given that it's easier and cheaper to increase breadth than skill, stock-picking strategies may have a higher chance for success than market-timing strategies.

Implications

The Fundamental Law of Active Management has several implications:

• Given some skill, bet as often as possible.
  
  
• Combine models, because breadth applies across models as well as assets. For example, an international equity manager can bet on countries, currencies, and individual stocks.
  
  
• Don't market-time. Market-timing strategies are unlikely to generate high information ratios. While such strategies can generate very large returns in a particular year, they're heavily dependent on luck. On a risk-adjusted basis, the value added will be small. This will not surprise most institutional managers, who avoid market timing for just this reason.
  
  
• Tactical asset allocation has a high skill hurdle. This strategy lies somewhere between market timing and stock picking: it provides some opportunity for breadth, but not nearly the level available to stock pickers. Therefore, to generate an equivalent information ratio, the tactical asset allocator must demonstrate a higher level of skill.
  

In summary, information ratios, the key to active management, depend on both skill and breadth.