---
description: "Combines multiple stock-selection factors (e.g., value and momentum) by blending factor rankings or allocating capital across factor sub-portfolios, reducing single-factor risk."
tags: [stocks, multifactor, momentum, value]
---

# Multifactor Portfolio

**Section**: 3.6 | **Asset Class**: Stocks | **Type**: Multifactor

## Overview
A multifactor portfolio buys and shorts stocks based on multiple factors simultaneously — such as value and momentum — which are often negatively correlated with each other, providing diversification benefits. Combining factors can add value relative to any single-factor strategy. The holding period depends on which factors are combined.

## Construction / Signal
Two primary approaches to combining F factors:

**Approach 1 — Capital allocation across sub-portfolios**

Each of F factor portfolios is built independently (as in Sections 3.1–3.5). Capital is allocated with weights `w_A` (A = 1,...,F):

```
sum_{A=1}^{F} w_A = 1                                     (275)
```

Investment level for factor A: `I_A = w_A * I`

Simple uniform weights: `w_A = 1/F`

Volatility-weighted: `w_A ∝ 1/sigma_A` or `w_A ∝ 1/sigma_A^2`, where `sigma_A` is the historical volatility of factor portfolio A (uniformly normalized per dollar invested).

Alternatively, optimize weights using an invertible F×F covariance matrix of the F factor portfolio returns.

**Approach 2 — Blended ranking scores**

Define demeaned ranks for factor A across N stocks:

```
s_{Ai} = rank(f_{Ai}) - (1/N) * sum_{j=1}^{N} rank(f_{Aj})   (276)
```

where `f_{Ai}` is the numeric value of factor A for stock i. Average the ranks across factors:

```
s_i = (1/F) * sum_{A=1}^{F} s_{Ai}                        (277)
```

Sort stocks by the combined score `s_i` and construct a long/short portfolio (top decile long, bottom decile short).

## Entry / Exit Rules
- **Entry**: At rebalance date, compute factor scores, blend them (via capital allocation or rank averaging), and enter long/short positions.
- **Exit**: Hold for the relevant factor horizon; rebalance monthly (or per factor schedule).
- **Tie-breaking**: If ambiguity exists at decile boundaries (e.g., tied combined scores), resolve by preferring one factor's ranking.

## Key Parameters
- **Number of factors F**: Typically 2–5 (e.g., value + momentum; or value + momentum + low-vol)
- **Factor weights w_A**: Uniform (1/F) or volatility-suppressed
- **Combining method**: Capital allocation vs. rank averaging
- **Holding period**: Depends on the factors combined

## Variations
- **Two-factor momentum + value**: Sort top/bottom quintiles by momentum, then split by value (or vice versa), creating 4 sub-portfolios
- **Weighted rank averaging**: Non-uniform weights in Eq. (277) using Manhattan or Euclidean distance minimization
- **Portfolio optimization**: Fix weights w_A by optimizing expected returns using an invertible F×F covariance matrix

## Notes
- Value and momentum are empirically negatively correlated, making them natural complements that reduce portfolio volatility.
- Uniform rank averaging (Eq. 277) minimizes the sum of squared Euclidean distances between the combined N-vector s_i and the K individual N-vectors s_{Ai}.
- Holding period depends on the slowest factor; mixing monthly and annual factors requires careful rebalancing scheduling.
- Transaction costs increase with the number of factors if rebalancing frequencies differ.