Tim Olson
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122 lines
4.8 KiB
Markdown
122 lines
4.8 KiB
Markdown
---
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description: "Institutional alpha combination: how to merge multiple weak signals into a single high-conviction output using the 11-step procedure and the Fundamental Law of Active Management."
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tags: [signals, alpha, portfolio, kelly, statistics]
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---
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# Signal Combination — Alpha Stacking
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## Core Law
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Do not search for one perfect signal. Combine many weak, independent signals.
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**Fundamental Law of Active Management:**
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```
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IR = IC × √N
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```
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- `IR` = Information Ratio of the combined system (risk-adjusted edge)
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- `IC` = average Information Coefficient per signal (correlation of prediction to outcome)
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- `N` = number of *genuinely independent* signals
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Real institutional signals have IC = 0.05–0.15. A single signal at IC=0.10 is outperformed by 50 signals at IC=0.05 (IR = 0.05 × √50 = 0.354, over 3× better).
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**Critical:** N counts *effective independent signals*, not raw signal count. Fifty correlated signals may yield only 10–15 effective ones. The 11-step procedure below forces honest accounting.
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---
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## Five Signal Categories
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| Category | What it measures | Why it persists |
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|---|---|---|
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| **Momentum / Price** | Direction/rate of price movement over lookback `d` | Underreaction causes short-term trend persistence |
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| **Mean Reversion** | Deviation from cross-sectional fair value | Related instruments maintain consistent relative pricing |
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| **Volatility** | Implied vs. realized volatility gap | Vol risk premium: sellers demand compensation |
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| **Factor** | Value, momentum, carry, quality, low-vol premiums | Persistent behavioral/structural inefficiencies |
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| **Microstructure** | Order book imbalance, bid-ask spread, VPIN | Informed order flow leads price movement |
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> **Dexorder scope**: Only crypto OHLCV data is available. Factor signals (value, carry, quality) require TradFi data not available here. Momentum, mean reversion, volatility, and microstructure signals are all applicable.
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---
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## 11-Step Combination Engine
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Given N signals with historical returns R(i,s) over M periods:
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**Step 1.** Collect realized return series R(i,s) for each signal i, each period s.
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**Step 2.** Remove drift — serially demean:
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```
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X(i,s) = R(i,s) − mean(R(i,·))
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```
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**Step 3.** Compute variance per signal:
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```
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σ(i)² = (1/M) × Σ X(i,s)²
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```
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**Step 4.** Normalize to common scale:
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```
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Y(i,s) = X(i,s) / σ(i)
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```
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Makes signals with different magnitudes directly comparable.
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**Step 5.** Drop the most recent observation from Y — use only out-of-sample history.
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**Step 6.** Cross-sectionally demean at each time period:
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```
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Λ(i,s) = Y(i,s) − avg_j(Y(j,s))
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```
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Removes any market-wide effect driving all signals simultaneously at that moment.
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**Step 7.** Drop the final period from Λ to eliminate residual look-ahead.
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**Step 8.** Estimate expected forward return per signal using d-day moving average, normalize:
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```
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E(i) = (1/d) × Σ R(i,s) over recent d periods
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E_normalized(i) = E(i) / σ(i)
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```
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**Step 9. (Critical)** Regress E_normalized over Λ(i,s) without intercept, unit weights. Residuals `ε(i)` are each signal's *independent* forward-looking contribution — the component not explained by any other signal in the stack.
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**Step 10.** Set weights:
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```
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w(i) = η × ε(i) / σ(i)
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```
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High independent edge + low noise → high weight. No subjective judgment.
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**Step 11.** Normalize: scale η so `Σ|w(i)| = 1`. No unintended leverage.
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**Combined output = Σ w(i) × signal_i_current_value**
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---
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## Empirical Kelly Sizing
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```
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f_empirical = f_kelly × (1 − CV_edge)
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f_kelly = (p × b − q) / b
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```
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- `CV_edge` = coefficient of variation of edge estimates across 10,000 Monte Carlo path simulations of historical returns
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- Higher uncertainty → smaller fraction. The formula automatically scales confidence to what is warranted.
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---
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## Key Failure Mode: Correlation Blindness
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Believing you have 3 independent reasons for a trade when you have 1 reason expressed 3 times, sized as if for 3. This is the mechanism behind most systematic blowups where the trader was directionally correct but over-sized.
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The cross-sectional demeaning (Step 6) and regression residualization (Step 9) structurally prevent this by exposing shared variance before weights are assigned.
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---
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## Dexorder Application Note
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When combining multiple indicators into a single entry/exit signal:
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1. Each indicator (momentum, RSI divergence, volume profile, spread, etc.) is a signal producing a score or directional estimate.
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2. Run the 11-step engine over backtested signal histories to derive weights.
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3. Combined score = weighted sum of current signal outputs.
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4. Size the resulting position using empirical Kelly with CV_edge from simulation.
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If computing probability estimates (e.g. probability of upward breakout), substitute probability estimates for return estimates at each step — the math is identical.
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