Investment·Case Study 01
Case Study 01 · PCA · Eigenvalues · Covariance Matrices
The Illusion of Diversification
Most investors believe that holding many stocks automatically reduces risk. This case study uses Principal Component Analysis to prove that is wrong: when assets share the same underlying risk factor, the number of names is irrelevant.
The Setup
Portfolio A · The Illusion
Big Tech: same underlying risk factor
Portfolio B · True Diversification
Equities · Commodities · Fixed Income · Energy · Healthcare · Staples
The Method
PCA decomposes the of daily into and . Each eigenvalue represents the variance captured by one independent risk factor, a Principal Component.
The fraction of total by PC1 is the key metric: λ₁ / Σλ. If it is large, one factor drives the portfolio, regardless of how many assets it holds.
Σ = Q Λ Qᵀ
covariance matrix = eigenvectors · eigenvalues · eigenvectorsᵀ
explained[i] = λᵢ / Σλ
fraction of total variance captured by Principal Component i
Key Findings
64.3%
One factor drives almost all risk
34.5%
Risk distributed across 3+ components
30.7×
Highly sensitive to rising discount rates
5.0%
Up from 1.7%, compressing every tech multiple
Correlation Heatmaps
Portfolio A shows near-uniform correlations above 0.5: the visual signature of a single-factor portfolio. Portfolio B distributes across uncorrelated regimes, with several pairs near zero or negative, including TLT/XOM at −0.116.
· by Principal Component
Portfolio A: PC1 captures 64.33% of all . The steep drop after the first bar is the mathematical fingerprint of the Illusion. Portfolio B: a gradual slope confirms risk spread across multiple independent economic forces.
Fundamental Tear Sheet
Side-by-side valuation and balance-sheet comparison. ETFs (GLD, TLT) display N/A: they hold no earnings. Portfolio A carries an average of 30.7×, resting on future earnings acutely sensitive to the .
Macro Overlay · Portfolio A Return vs.
Portfolio A surged during near-zero rates in 2020–2021. When the Federal Reserve began hiking in March 2022, climbed from 1.7% to a 16-year peak of 4.93%, and Portfolio A fell in lockstep. Not because the businesses failed, but because the applied to their future earnings had doubled.
Conclusion
Six names is not six risks.
Diversification is not about the number of assets you hold. It is about the number of independent risk factors your portfolio is exposed to.
AAPL, MSFT, GOOGL, AMZN, NVDA, and META are six companies, but they share one driver: the technology cycle and the multiple-expansion regime that sustains it. When the Federal Reserve raised rates from 1.7% to 5.0% between 2022 and 2023, all six fell in unison, not because the businesses failed, but because the applied to their future earnings had doubled.
Portfolio B, by holding assets from genuinely different economic regimes, spreads exposure across at least three independent principal components. PCA confirms what intuition suggests.
Count your risk factors, not your tickers.