How diversification cuts portfolio risk

The line is the yearly standard deviation of an equally weighted portfolio as holdings are added, one to fifty. Every holding is 30 percent volatile on its own, an illustrative figure held fixed, and every pair moves together by the correlation you set.

$$\sigma_p = \sigma\sqrt{\frac{1}{n} + \left(1 - \frac{1}{n}\right)\rho}$$

The $1/n$ term is the risk each holding carries alone, and it shrinks as $n$ grows. The $\rho$ term is the risk they all share, and it does not shrink at all. Set the correlation to zero and the floor drops onto the axis, so the line keeps falling with every holding added instead of levelling off. Set it to one and the line is flat, because fifty holdings then behave as one.

The dashed line sits at $\sigma\sqrt{\rho}$, the level the curve approaches and never crosses. Below it is systematic risk, the part every holding is exposed to at the same time. The gap above it is the unsystematic risk still left in the portfolio, the part specific to each holding, which cancels out as holdings are added. The split is exact in variance rather than in standard deviation: total variance is $\sigma^2\rho$ shared plus $\sigma^2(1 - \rho)/n$ specific, and the chart plots the square root of the two together.

Most of the work happens early. At a correlation of 0.3, the first eight holdings remove about 83 percent of the part that can be removed, and twenty holdings remove about 93 percent. The fiftieth holding removes almost nothing, because by then the specific risk is nearly gone and only the shared risk is left.

Equal weights, one volatility for every holding, and one correlation for every pair. The 30 percent volatility is an illustrative teaching figure chosen so the curve reads clearly, not a measurement of any holding. Real portfolios match none of those assumptions exactly, and the shape survives all of them: risk drops quickly at first, then flattens onto a floor set by how much the holdings share.

The reading on the axis is a standard deviation of returns, so it measures spread rather than loss. Adding holdings also does nothing to the expected return, which stays the weighted average of what you hold. Diversification changes the width of the range of outcomes, not its centre.