The Fundamental Law
Before analyzing data, we must understand the mathematical bedrock of active management. This equation defines the upper limit of your potential performance.
The Equation of Skill
The Fundamental Law of Active Management, formulated by Grinold and Kahn, breaks down performance into two independent components: Skill and Opportunity.
Information Ratio (IR)
Your risk-adjusted return depends on how accurate you are (IC) and how many independent bets you place (Breadth).
- IR: Information Ratio (Active Return / Active Risk)
- IC: Information Coefficient (Correlation of signal to return)
- BR: Breadth (Number of independent bets per year)
The Extended Law (Reality Check)
In practice, constraints (liquidity, risk limits, long-only) prevent you from fully expressing your signal. We introduce a "Transfer Coefficient" to measure this leakage.
TC (Transfer Coefficient): Correlation between your ideal weights and actual weights.
- TC = 1.0: Pure Long/Short, no friction.
- TC = 0.3: Long-only, sector neutral, low turnover. (Most mutual funds).
Why Theory Fails
- •The Independence Illusion: 500 stocks ≠ 500 bets if they all crash together. BR is usually lower than you think.
- •Non-Linearity: Pearson IC assumes linear relationships. It fails to capture "smile" curves (e.g., extreme growth and extreme value both outperforming).
Implementation Reality
High IC does not guarantee profit. Frictions, costs, and constraints eat Alpha for breakfast.
Implementation Shortfall
The difference between your "Paper Return" and "Realized Return". It has three main components:
Square Root Law of Impact
Market impact is not linear. It scales with the square root of trade size relative to volume.
Impact Cost Model
Doubling your trade size doesn't double cost—it increases it by ~1.41x (√2). This non-linearity is why high-turnover strategies have a hard AUM cap.
- σ: Daily Volatility
- Q: Trade Size
- V: Daily Volume
Portfolio Construction
How do we turn a Z-score into a dollar weight? The simplest robust method is Inverse Volatility Scaling.
The Backtest Lie
Why do Sharpe 2.0 strategies fail in production? Overfitting, P-Hacking, and Point-in-Time errors create a 'Paper Wealth' illusion.
P-Hacking Simulator
We will generate 100 completely random strategies (noise). Statistically, one of them will look amazing just by luck. This is "Selection Bias".
Look-Ahead Bias
Using data in your backtest that wasn't available at the time. This is the #1 reason for backtest failure.
Always use "Point-in-Time" databases.
Overfitting / P-Hacking
If you test 100 random signal variations, 5 will look profitable by pure chance (p=0.05). If you keep only those 5, you are lying to yourself.
Solution: Deflated Sharpe Ratio (DSR)
DSR penalizes the Sharpe Ratio based on the number of trials (N).