---
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."
tags: [stocks, machine-learning, alpha-combination, quantitative]
---

# Alpha Combos

**Section**: 3.20 | **Asset Class**: Stocks | **Type**: Machine Learning / Quantitative / Alpha Combination

## Overview
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.

## Construction / Signal
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]:

1. Start with time series of realized alpha returns `R_is`, i=1,...,N, s=1,...,M+1.
2. Calculate serially demeaned returns:
   ```
   X_is = R_is - (1/(M+1)) * sum_{s=1}^{M+1} R_is
   ```
3. Calculate sample variances of alpha returns:
   ```
   sigma_i^2 = (1/M) * sum_{s=1}^{M+1} X_is^2
   ```
4. Calculate normalized demeaned returns:
   ```
   Y_is = X_is / sigma_i
   ```
5. Keep only the first M columns: Y_is, s=1,...,M.
6. Cross-sectionally demean Y_is:
   ```
   Lambda_is = Y_is - (1/N) * sum_{j=1}^{N} Y_js
   ```
7. Keep only the first M-1 columns: Lambda_is, s=1,...,M-1.
8. Compute expected alpha returns E_i and normalize:
   ```
   E_i = (1/d) * sum_{s=1}^{d} R_is                     (360)
   E_tilde_i = E_i / sigma_i
   ```
   (d-day moving average; d need not equal T)
9. Calculate residuals `epsilon_tilde_i` of regression (no intercept, unit weights) of `E_tilde_i` over `Lambda_is`.
10. Set alpha portfolio weights:
    ```
    w_i = eta * epsilon_tilde_i / sigma_i
    ```
11. Set normalization coefficient eta such that:
    ```
    sum_{i=1}^{N} |w_i| = 1
    ```

## Entry / Exit Rules
- **Entry**: At each rebalance time, recompute combination weights w_i using the 11-step procedure and establish positions accordingly.
- **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.
- **Holding period**: Determined by individual alpha holding periods (typically daily, from close to close).

## Key Parameters
- **Number of alphas N**: Can be hundreds of thousands or millions
- **Return history M+1**: Number of time periods used for variance estimation
- **Expected return window d**: Number of days for moving average of alpha returns (Eq. 360; d need not equal M)
- **Universe**: Typically ~2,500 most liquid U.S. stocks
- **Alpha returns R_is**: Daily alpha returns from close to close

## Variations
- **Fewer alphas**: The procedure scales from a few dozen to millions of alphas
- **Different expected return estimator**: Instead of d-day moving average, other estimators for E_i can be used
- **Multiple universes**: Extend to different stock universes or asset classes

## Notes
- Individual alphas are "ubiquitous, faint, and ephemeral" — their signal is too weak to trade profitably alone due to transaction costs.
- 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).
- "Alpha" here follows the practitioner definition: any reasonable expected return signal, not necessarily Jensen's alpha.
- 101 explicit examples of such quantitative alphas are given in Kakushadze (2016).
- This is a cross-sectional multi-stock strategy requiring significant data infrastructure.
- Typical holding period: daily (overnight or close-to-close).