Fundamental Law of Active Management – Value Investing vs. Trading

I have been taking the Computational Investing, Part I course at Coursera taught by Dr. Tucker Balch from Georgia Tech (also co-founder and CTO of Lucena Research).

In one of the classes, he talks about the Fundamental Law of Active Management from Grinold and Kahn and uses it to link the performance between a value investor like Warren Buffett and a HFT trader like Jim Simons of Renaissance Technologies. Thought I’ll share this interesting tidbit.

Fundamental Law of Active Management

  • CAPM
    • Return = Return due to market (beta * market excess return) + Residual return (alpha)
  • Information Ratio (IR)
    • IR = Mean(alpha) / Stddev(alpha) = Return due to skill / Residual risk
  • Information Coefficient (IC)
    • IC = Skill of the fund manager. For each forecast he makes, what is the correlation of that forecast to the actual return.
  • Breadth (BR)
    • BR = Number of opportunities that you execute your method on
  • Fundamental Law
    • IR = IC * Sqrt (BR)
    • IR measures the performance.
    • IC measures the skill
    • BR measures the number of opportunities

Illustrative Examples

  • Warren Buffett
    • Reward / risk = 3.0, 120 trades per year
    • IR = IC * Sqrt (trades)
    • 3.0 = Skill * Sqrt (120)
    • Skill = 0.27
  • Renaissance Technologies
    • Skill = 0.27 / 100 = 0.0027 (assume 100 times lower than Buffett’s)
    • To achieve a 3.0 Reward / Risk ratio:
      • IR = IC * Sqrt (trades)
      • 3.0 = 0.0027 * Sqrt (trades)
      • Trades = 1,234,567 (10,000 times more trades than Buffett)


  • Two ways to succeed
    • Improve your skill
      • Warren Buffett has great skill but his method allows few number of trades
    • Improve your breadth
      • If you have less skill but you can scale it (i.e. it is repeatable over lots and lots of bets), you can also succeed. However you need to make a lot more bets to overcome the Sqrt on the breadth.




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