Diversification Across Many Assets
How the two-asset matrix generalizes to many assets, why company-specific risk shrinks, and why average covariance remains.
- n×n covariance matrix: n variances, n²−n covariances
- Equal-weight formula: σ²_P = (1/n)σ̄² + ((n−1)/n)Cōv
- Individual variance shrinks as n grows
- Average covariance remains as systematic risk
- Diversification has a floor
From two to many assets
The two-asset matrix generalizes cleanly. With n assets you build an n×n covariance matrix: each diagonal cell holds a variance, each off-diagonal cell holds a covariance.
Portfolio variance equals the weight vector transposed, times the covariance matrix, times the weight vector. It sums every weighted covariance cell.
- Column vector of portfolio weights.
- n×n covariance matrix (uppercase sigma).
- Row vector (weights transposed).
| n | Variance cells | Covariance cells | Total |
|---|---|---|---|
| 3 | 3 | 6 | 9 |
| 100 | 100 | 9,900 | 10,000 |
With 100 assets there are 9,900 covariance cells versus only 100 variance cells. But covariances matter not just because there are more of them — each cell is weighted, and the weight on any single cell shrinks as the portfolio spreads out.
Deriving the equal-weight variance
Set every weight to 1/n. The four-stage derivation below splits portfolio variance into two pieces with very different futures.
With equal weights w_i = 1/n, each of the n² cells receives (1/n) × (1/n) = 1/n². Portfolio variance is the sum of all cells, each divided by n².
Diagonal cells (cyan) hold variances; off-diagonal cells (amber) hold covariances.
Set every weight to one over n so each of the n-squared cells receives one over n-squared. The n diagonal cells sum to n times average variance. The n times n minus one off-diagonal cells sum to n times n minus one times average covariance. Substituting and simplifying yields one over n times average variance plus n minus one over n times average covariance.
- Step 1: Every cell gets 1/n².
- Step 2: n diagonal variance cells.
- Step 3: n(n−1) off-diagonal covariance cells.
- Step 4: Combine and simplify.
Shrinks toward zero as n grows — this is the diversifiable part.
Approaches average covariance Cōv — the floor you cannot diversify below.
Decompose the variance at n = 10
Same scenario: σ̄² = 0.01, Cōv = 0.004, with 10 equal-weight assets. Build σ²_P from its two components, then convert to σ_P.
The curve in numbers
Every asset has σ = 10% (so σ̄² = 0.01), average correlation ρ = 0.40 (so Cōv = 0.004). Watch what happens to the equal-weight portfolio as n grows.
| n | σ²_P | σ_P |
|---|---|---|
| 1 | 0.010000 | 10.00% |
| 2 | 0.007000 | 8.37% |
| 5 | 0.005200 | 7.21% |
| 10 | 0.004600 | 6.78% |
| 20 | 0.004300 | 6.56% |
| 50 | 0.004120 | 6.42% |
| 100 | 0.004060 | 6.37% |
| ∞ (limit) | 0.004000 | 6.32% |
At n = 100 the portfolio sits at 6.37%, close to but not identical to the limiting 6.32%. The bulk of the diversification happens in the first handful of assets; each new position after that removes only a sliver of risk.
How many assets does it take?
See the equal-weight formula in motion. Slide the number of assets and switch correlation presets to watch the curve bend toward its floor — or stay flat when correlation is 1.
At ρ = 0.40 the curve falls quickly then flattens toward the 6.32% covariance floor. At n = 10, σ_P = 6.78%.
The two components are two kinds of risk
The two mathematical pieces map directly to two economic kinds of risk (Module 5 covers this in depth).
Company-specific events — a product failure, accounting fraud, a lawsuit. This is the part that washes away as n grows.
Market-wide forces — recession, inflation, financial crisis. This is the average-covariance floor that remains.
Diversification reduces idiosyncratic risk. It does not eliminate systematic risk — that is the price of participating in the market.
Holding 50 semiconductor companies is not the same as being diversified. Neither is holding many banks, or many oil producers. The covariance floor stays high because the assets share the same risk source. Diversification depends on distinct risk sources, not the number of tickers.
- It assumes equal weights — real portfolios rarely hold every asset at 1/n.
- It assumes variances and covariances are stable — they are estimated and can shift.
- It ignores transaction costs, liquidity, and taxes — adding many assets is not free.
- It treats correlation as fixed — in stress, correlations can rise together.
The diversification curve is a model, not a recommendation. It teaches the shape of the benefit and the existence of a floor — the real portfolio must account for everything the model leaves out.
Decompose σ²_P at n = 20
Same parameters: σ̄² = 0.01, Cōv = 0.004, now with 20 equal-weight assets. Find each component, the total variance, and σ_P.
Questions you may still have
Optional reminders and extensions.
Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.
- 01
Equal-weight, n = 10, σ̄² = 0.01, Cōv = 0.004. What is σ²_P?
- 02
As n → ∞, σ²_P approaches:
- 03
n = 20, σ̄² = 0.01, Cōv = 0.004. σ_P ≈ ?
- 04
ρ = 1, equal weights, σ = 10%. σ_P for any n?
- 05
Why doesn't σ_P approach zero when ρ > 0?
- 06
50 semiconductor companies in a portfolio:
- 1The n×n covariance matrix has n variance entries and n²−n covariance entries.
- 2Equal-weight variance: σ²_P = (1/n)σ̄² + ((n−1)/n)Cōv.
- 3As n grows, the individual-variance component shrinks toward zero.
- 4The average-covariance component approaches Cōv — the systematic risk floor.
- 5Diversification reduces idiosyncratic risk but not systematic risk.
- 6Diversification depends on distinct risk sources, not just the number of tickers.
- 7The equal-weight model is a simplification with stated assumptions and limitations.