Tim Olson
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75 lines
4.1 KiB
Markdown
75 lines
4.1 KiB
Markdown
---
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description: "Combines hundreds of thousands of weak individual alpha signals into a single tradeable mega-alpha by optimizing combination weights using a structured 11-step procedure based on demeaned returns and regression residuals."
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tags: [stocks, machine-learning, alpha-combination, quantitative]
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---
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# Alpha Combos
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**Section**: 3.20 | **Asset Class**: Stocks | **Type**: Machine Learning / Quantitative / Alpha Combination
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## Overview
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Alpha combo strategies combine a large number of weak quantitative trading signals ("alphas") into a single tradeable "mega-alpha" portfolio. Each individual alpha is too faint to trade profitably on its own after costs, but a sufficiently large combination can generate a viable signal. The combination weights are optimized using a structured procedure that handles serial correlation, cross-sectional demeaning, and risk normalization.
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## Construction / Signal
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Assume N alphas (possibly hundreds of thousands), all trading the same universe of ~2,500 liquid U.S. stocks. Each alpha produces desired stock holdings at a sequence of times t_1, t_2, .... The procedure for fixing combination weights w_i [Kakushadze and Yu, 2017b]:
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1. Start with time series of realized alpha returns `R_is`, i=1,...,N, s=1,...,M+1.
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2. Calculate serially demeaned returns:
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```
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X_is = R_is - (1/(M+1)) * sum_{s=1}^{M+1} R_is
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```
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3. Calculate sample variances of alpha returns:
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```
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sigma_i^2 = (1/M) * sum_{s=1}^{M+1} X_is^2
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```
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4. Calculate normalized demeaned returns:
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```
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Y_is = X_is / sigma_i
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```
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5. Keep only the first M columns: Y_is, s=1,...,M.
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6. Cross-sectionally demean Y_is:
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```
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Lambda_is = Y_is - (1/N) * sum_{j=1}^{N} Y_js
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```
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7. Keep only the first M-1 columns: Lambda_is, s=1,...,M-1.
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8. Compute expected alpha returns E_i and normalize:
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```
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E_i = (1/d) * sum_{s=1}^{d} R_is (360)
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E_tilde_i = E_i / sigma_i
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```
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(d-day moving average; d need not equal T)
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9. Calculate residuals `epsilon_tilde_i` of regression (no intercept, unit weights) of `E_tilde_i` over `Lambda_is`.
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10. Set alpha portfolio weights:
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```
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w_i = eta * epsilon_tilde_i / sigma_i
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```
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11. Set normalization coefficient eta such that:
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```
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sum_{i=1}^{N} |w_i| = 1
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```
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## Entry / Exit Rules
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- **Entry**: At each rebalance time, recompute combination weights w_i using the 11-step procedure and establish positions accordingly.
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- **Exit**: Positions are updated at each rebalance; individual alpha positions change according to the alpha's own signals, and the overall mega-alpha weight adjusts.
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- **Holding period**: Determined by individual alpha holding periods (typically daily, from close to close).
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## Key Parameters
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- **Number of alphas N**: Can be hundreds of thousands or millions
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- **Return history M+1**: Number of time periods used for variance estimation
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- **Expected return window d**: Number of days for moving average of alpha returns (Eq. 360; d need not equal M)
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- **Universe**: Typically ~2,500 most liquid U.S. stocks
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- **Alpha returns R_is**: Daily alpha returns from close to close
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## Variations
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- **Fewer alphas**: The procedure scales from a few dozen to millions of alphas
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- **Different expected return estimator**: Instead of d-day moving average, other estimators for E_i can be used
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- **Multiple universes**: Extend to different stock universes or asset classes
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## Notes
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- Individual alphas are "ubiquitous, faint, and ephemeral" — their signal is too weak to trade profitably alone due to transaction costs.
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- The 11-step procedure handles: serial correlation (steps 1-2), scale normalization (steps 3-4), cross-sectional neutrality (steps 5-7), expected return estimation (step 8), residualization (step 9), and final weight normalization (steps 10-11).
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- "Alpha" here follows the practitioner definition: any reasonable expected return signal, not necessarily Jensen's alpha.
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- 101 explicit examples of such quantitative alphas are given in Kakushadze (2016).
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- This is a cross-sectional multi-stock strategy requiring significant data infrastructure.
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- Typical holding period: daily (overnight or close-to-close).
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