How many assets does diversification need?

Why does adding assets cut risk?

Portfolio variance is a sum over every pair of assets. Take an equally weighted portfolio of N assets, each with weight 1/N. Each of the N own-variance terms is multiplied by 1/N², so together they shrink like 1/N. The N² − N covariance terms keep their weight. As the number of assets grows, covariances take over.

Suppose every asset has variance σ², and the average correlation between pairs is ρ̄.

$$\sigma_P^2 = \frac{\sigma^2}{N} + \left(1 - \frac{1}{N}\right)\bar{\rho}\,\sigma^2$$

The same idea works without assuming equal variances. The average covariance replaces the correlation term, and σ̄² is the average variance.

$$\sigma_P^2 = \frac{\bar{\sigma}^2}{N} + \frac{N - 1}{N}\,\overline{\sigma_{i,j}}$$

Where does it stop?

At the average covariance. For large N, σ²/N vanishes and (1 − 1/N) approaches 1.

$$\sigma_P^2 \approx \bar{\rho}\,\sigma^2 \quad\text{so}\quad \sigma_P \approx \sigma\sqrt{\bar{\rho}}$$

For the stocks above, √(0.25 × 0.09) = √0.0225 = 15%. No number of these stocks takes risk below 15%.

How much can be removed? Individual risk is σ² = 0.09 and the floor is ρ̄σ² = 0.0225, so 0.0675 can go. With N assets the share removed is exactly 1 − 1/N, for any σ and ρ̄ in this setup: 90% at N = 10, 95% at 20, 96% at 25, 96.7% at 30 and 99% at 100. That is why a modest number of assets does most of the work.

The diversification rule of thumb says that 20 to 30 assets capture most of the benefit, especially when correlations are low. The curriculum also says 30 or more in places, so treat it as a guide, not a number to quote as exact.

Correlation is the dial. With ρ̄ = 0 the floor is zero, and N = 25 gives 30% ÷ 5 = 6.00%. With ρ̄ = 1 nothing diversifies and σ_P stays at 30%. In a market crisis, correlations rise. At ρ̄ = 0.70 the same 25 stocks give 25.31% instead of 15.87%, just when protection is wanted.

Many questions ask what dominates the risk of a large portfolio: the average covariance, not the individual variances.

Go deeper How many assets for a target risk?

Rearrange the equal-weight formula for N: N = (1 − ρ̄)σ² ÷ (σ_P² − ρ̄σ²). With σ = 30% and ρ̄ = 0.25, a target of 16% needs N = 0.0675 ÷ (0.0256 − 0.0225) = 21.8, so 22 assets (round up). The target must be above the floor of 15%, or no number of assets reaches it. The solver above finds N for you.

Which risk is left?

Diversification removes unsystematic risk : the risk specific to one asset, also called non-systematic or idiosyncratic risk. What remains is systematic risk , the market-wide risk every asset shares, and it cannot be diversified away. The capital asset pricing model lesson shows why only systematic risk earns a higher expected return. Learning Module 2 introduces macro (systematic) and micro (unsystematic) risk.

Total risk of an asset

• Unsystematic risk Memorize asset-specific; also non-systematic or idiosyncratic; shrinks like 1/N
• Systematic risk Memorize market-wide; the ρ̄σ² floor; cannot be diversified away

Equally weighted portfolio risk

• Number of assets N Use it own-variance part falls like 1/N
• Average correlation ρ̄ Memorize sets the floor ρ̄σ²; lower is better
• Asset volatility σ Know the idea scales everything

The same idea shows up with two assets, which the previous lesson covered. At a correlation of +1, the line between the two assets on a risk and return plot is straight, with no benefit. As the correlation falls, the curve bends to the left. At −1 it is two straight lines meeting at zero risk.