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Lesson 5 of 13 · 13 min

Scenario-based portfolio risk and the many-asset limit

Portfolio risk can be forecast directly from economic scenarios, and in a large portfolio it shrinks toward the average covariance, the part diversification cannot remove.

In short

With scenarios: compute the portfolio return in each scenario, then its probability-weighted mean and variance.
No covariance matrix is needed for the scenario method; co-movement is already built into each scenario's portfolio return.
Equal weights, common $\sigma^2$ and average correlation $\bar\rho$ : $\sigma_P^2 = \sigma^2/N + (1 - 1/N)\bar\rho\sigma^2$ .
As N grows, the first term vanishes and $\sigma_P^2 \to \bar\rho\sigma^2$ , the average covariance.
Most of the benefit arrives with roughly 20 to 30 holdings; what remains is systematic risk.

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