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Separate alpha from compensated exposures

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Start with the idea

A factor model separates returns associated with common exposures from residual variation. The model you choose defines the meaning of alpha; it cannot reveal skill independently of its specification.

Symbols, units & horizon

Rᵢ,ₜ: asset return
R_f,ₜ: risk-free return
αᵢ: intercept or unexplained average excess return
βᵢ: vector of factor sensitivities
fₜ: factor-return vector
εᵢ,ₜ: residual
R_m: market return
B: matrix of asset factor loadings
Σ_f: factor covariance matrix
Σ_ε: residual covariance
Σ: model asset covariance
T: transpose
Cov,Var: covariance and variance

When and why to use this

Use factor exposure to explain a strategy’s returns and to avoid combining several disguised versions of the same market bet.

R_{i, t} - R_{f, t} = α_{i} + β_{i}^{𝖳} f_{t} + ε_{i, t}

Algebra and arithmetic

Separate fitted factor return from the residual

For one observation, fitted excess return is $\hat{α} + {\hat{β}}^{T} f$ . Rearrange the model to compute $ε = R_{i} - R_{f} - α - β^{T} f$ .
Averaging with zero-mean residuals gives $α = E [R_{i} - R_{f}] - β^{T} E [f]$ . This is conditional on the factors included.

Work it by hand

Excess return 1.2%, beta .8, market excess return 1%, alpha .1% gives residual=.3%.

Returns on the left and factor returns on the right must share frequency, currency, and return conventions. Alpha is the intercept conditional on the chosen model; adding a relevant factor can change it substantially. It is not an intrinsic property of a strategy.

β_{i} = \frac{Cov (R_{i}, R_{m})}{Var (R_{m})}, Σ = B Σ_{f} B^{𝖳} + Σ_{ε}

Algebra and arithmetic

Derive beta and factor covariance

In a one-factor least-squares model, the normal equations give $β = \frac{Cov (R_{i}, R_{m})}{Var (R_{m})}$ .
For centred returns $R = B f + ε$ , expand $E [R R^{T}]$ . If factors and residuals are uncorrelated, the cross terms vanish, leaving $B Σ_{f} B^{T} + Σ_{ε}$ .

Work it by hand

Covariance .018 and market variance .0225 imply beta=.8. One asset with beta .8, factor variance .04 and residual variance .01 has total variance .0356.

The covariance decomposition assumes residuals are uncorrelated with factors. B contains exposures, the factor covariance matrix describes joint factor risk, and residual covariance captures what remains. A diagonal residual model is convenient but can miss common sector or crowded-trade risks.

Use economically motivated factors: market, sector, size, value, momentum, duration, credit, currency, or volatility, depending on the book.
Check rolling exposure stability and use dependence-aware standard errors for estimated alpha.
Compare holdings-based exposures with return regressions; a slow regression can miss a rapid portfolio change.

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

import numpy as np  # dependency: numpy

def factor_residual(excess_return, alpha, betas, factor_returns):
    return excess_return-alpha-float(np.asarray(betas)@np.asarray(factor_returns))

def market_beta(asset_returns, market_returns):
    cov = np.cov(asset_returns, market_returns, ddof=1)
    if cov[1,1] == 0:
        raise ValueError("Nonzero market variance required")
    return cov[0,1]/cov[1,1]

def factor_covariance(loadings, factor_cov, residual_cov):
    b = np.asarray(loadings)
    return b@np.asarray(factor_cov)@b.T+np.asarray(residual_cov)

print(factor_residual(.02,.001,[1.2],[.01]))

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