Diversification — the only free lunch

Diversification is often called 'the only free lunch in finance'. Properly understood, it is a statement about how the variance of a portfolio behaves under combination, and how correlation among constituents determines how far that variance can fall.

Two-asset variance

σ²_p = w₁² σ₁² + w₂² σ₂² + 2 w₁ w₂ σ₁ σ₂ ρ₁₂

For w₁ = w₂ = 0.5 and σ₁ = σ₂ = σ: σ²_p = 0.5 σ² (1 + ρ). With ρ = 1: σ_p = σ — no diversification. With ρ = 0: σ_p = σ/√2 — variance cut in half. With ρ = -1: σ_p = 0 — riskless portfolio.

Equal-weighted portfolio of n i.i.d. assets

Suppose all assets have variance σ² and pairwise correlation ρ. The equal-weighted portfolio has variance:

σ²_p = σ²/n + σ² ρ (n - 1)/n  =  σ²[(1 - ρ)/n + ρ]

As n → ∞, σ²_p → ρ σ². Diversification eliminates the idiosyncratic (n-specific) variance but leaves the systematic ρ σ² floor. Equity-market average pairwise correlation is around 0.3-0.4, so a fully diversified equity portfolio retains about √0.35 ≈ 60% of the single-stock vol.

Why 30 stocks is 'enough'

Plot σ_p as a function of n for typical equity ρ ≈ 0.35. By n = 30 you've captured about 95% of the achievable variance reduction; doubling to 60 stocks shaves another 2%. This is the empirical content behind the textbook claim that '30 stocks is enough for diversification'. Important caveats: (1) you must be reasonably diversified across sectors / countries / sizes; (2) the average ρ rises in crises, eroding the 30-stock conclusion exactly when it matters.

Marginal contribution to risk

MCR_i = ∂σ_p / ∂w_i = (Σw)_i / σ_p
RC_i  = w_i · MCR_i           (risk contribution of asset i)
Σ RC_i = σ_p                  (Euler decomposition)

Risk decomposition tells you which holdings drive portfolio variance. RC_i / σ_p is the share. In an equal-weighted portfolio, the largest σᵢ contributes the most; in a risk-parity portfolio, all RC_i are equal.

Diversification vs hedging

Diversification (passive) lowers expected variance via low or zero average correlation. Hedging (active) seeks negative correlation against a specific risk factor. Holding gold against inflation: hedging. Holding 500 stocks: diversification. The Sharpe-ratio improvement from hedging is bounded by your information advantage; from diversification, by the average pairwise correlation in your universe.

Correlation breakdown — the big asterisk

Correlations between risky assets rise toward 1 during crises (October 1987, autumn 2008, March 2020). A portfolio diversified at normal correlations of 0.4 retains 60% of single-stock vol; in a crisis with ρ → 0.9, it retains 95%. The 'free lunch' has a hard temporal limit. Tail-aware portfolio construction (CVaR, conditional copulas) is the modern response.