In the late 1990s, a sophisticated European pension allocated $200 million to a well-respected global macro hedge fund. The fund reported 18% annualized returns over five years with 8% volatility (a Sharpe ratio of 2.0 that attracted capital). The pension's risk team ran a standard ordinary least squares (OLS) regression of fund returns against ten factors and found an alpha of 9% with an R-squared of 0.55. On paper, the fund looked legitimate.
PCA and statistical factors. There are three major categories of asset factors: macro, dynamic, and statistical. Macro and dynamic factors start with a known economic series (inflation, productivity, value, momentum, size) and test it empirically against return data. Statistical factors reverse the logic: the return data themselves identify potential factors. Principal component analysis (PCA) is a linear statistical method that identifies the set of orthogonal (uncorrelated) factors from a data set that maximize the percentage of explained variation.
PCA generates two primary outputs: factor loadings and eigenvalues. The factor loadings of each principal component are a vector of scores, one per asset, showing each asset's responsiveness to that...
Common mistakes
- Naming PCA components as economic factors before reading the loadings. PCA factors are pure variance-maximizing combinations, not theory-driven factors. The fix is to interpret ex-post, as with the level, slope, and curvature of Treasury returns. Trap value: assuming a component "is" a credit factor before examining which assets load on it.
- Running a "kitchen sink" regression with multicollinear factors. Correlated independents, such as a US and a non-US equity index, make slope estimates highly inaccurate and inflate standard errors, so coefficients look insignificant despite high R-squared. Correct: form a return spread or use stepwise regression and avoid overfitting.
- Confusing the stated tax rate with the effective tax rate. When depreciation is disallowed or slower than economic decline, the effective rate exceeds the stated rate and the after-tax IRR falls below pre-tax times one minus t. Trap value: assuming the after-tax IRR is always 60% of the pre-tax IRR.
Bottom line
- PCA outputs factor loadings and eigenvalues; each eigenvalue divided by the sum of eigenvalues gives that component's percent of variance explained.
- PCA differs from factor analysis in model assumptions, loading stability as factor count changes, and single-security identification; factor analysis requires factors driving at least two securities.
- Multicollinearity produces inaccurate slope coefficients and inflated standard errors; detect via correlations among independents and correct with return spreads.
- The second-order partial autocorrelation equals , isolating the marginal lag-2 effect that the Pearson coefficient overstates.
Exam shortcut
Before running a regression, commit to a factor set and check for multicollinearity by examining correlations among the independents; when two return factors are correlated, replace one with a spread and use stepwise selection to avoid overfitting. For PCA, read eigenvalues to choose the factor count and interpret loadings only after the fact (level, slope, curvature for Treasuries).
The full lesson (about 5,398 words, 36 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- model types
- fi models intro
- bdt model
- credit risk economics
- structural model overview
- merton model
- kmv model
- reduced form models
- empirical credit models
- one period binomial
- multi period binomial
- tree prices formation
- convertible valuation
- callable bonds tree
- multifactor asset pricing
- fama french
- empirical mf challenges
- factor investing
- adaptive markets
- efficiently inefficient
- trend following
- divergence
- fundamental directional
- behavioral finance
- directional factors
- digital asset valuation
- pca statistical factors
- multifactor regression
- partial autocorrelations
- dynamic risk exposure
- changing correlation
- multifactor return approaches
- performance persistence
- rv overview
- statistical pairs equities
- pairs commodity spreads
- pairs rates fx
- rv market neutral risks
- depreciation tax shields
- tax deferral gains
- after tax comparisons
- transaction based indices
- appraisal based indices
Browse all free CAIA Level II lessons or jump into free CAIA Level II practice questions.