Modern Portfolio Theory and Mean-Variance Optimization › The Mathematics of Diversification

Diversification is often called the "free lunch" of finance because combining different exposures can improve risk-return tradeoffs. Common market or economic factors affect many assets with different exposure magnitudes and signs. Specific risk can diminish when holdings become sufficiently granular and residual dependence is weak; simply increasing the count of holdings is not enough.

Consider an equally-weighted portfolio of $n$ assets with identical variance $\sigma^2$ and pairwise correlation $\rho$. While this is a simplification, it reveals the essential mechanics of diversification. The portfolio variance is:

$$\sigma_p^2 = \sum_{i=1}^n \sum_{j=1}^n \frac{1}{n^2} \sigma_{ij}$$

where:

• $\sigma_p^2$: variance of the portfolio
• $n$: number of assets
• $\sigma_{ij}$: covariance between asset $i$ and asset $j$

For the diagonal terms ($i = j$): $\sigma_{ii} = \sigma^2$

For the off-diagonal terms ($i \neq j$): $\sigma_{ij} = \rho \sigma^2$

There are $n$ diagonal terms and $n(n-1)$ off-diagonal terms. The diagonal terms represent each asset's own variance contribution, while the off-diagonal terms capture the covariance contributions from all pairs of different assets:

$$\begin{aligned} \sigma_p^2 &= \frac{n \sigma^2}{n^2} + \frac{n(n-1) \rho \sigma^2}{n^2} \\ &= \frac{\sigma^2}{n} + \frac{n-1}{n} \rho \sigma^2 && \text{(simplify fractions)} \end{aligned}$$

where:

• $\sigma^2$: variance of each individual asset
• $\rho$: pairwise correlation between all assets
• $n$: number of assets in the portfolio

The displayed terms are diagonal variance and off-diagonal covariance contributions. For a common-factor interpretation, assume $0\leq\rho\leq1$ and write centered returns as $R_i-\mu_i=\sigma\sqrt{\rho}F+\sigma\sqrt{1-\rho}\epsilon_i$, where $F$ and the residuals have unit variance and are mutually uncorrelated. Then $\sigma_p^2=\rho\sigma^2+(1-\rho)\sigma^2/n$: common-factor variance is constant, while specific variance declines with $n$. The finite equicorrelation matrix permits $-1/(n-1)\leq\rho\leq1$ for $n>1$, but a fixed negative correlation is not valid as $n$ tends to infinity.

For a family of homogeneous portfolios with fixed $0\leq\rho\leq1$, as $n\to\infty$:

$$\sigma_p^2 \to \rho \sigma^2$$

where:

• $\sigma_p^2$: limiting portfolio variance
• $\rho$: pairwise correlation between assets
• $\sigma^2$: variance of individual assets

For fixed $0\leq\rho\leq1$, the specific component $(1-\rho)\sigma^2/n$ tends to zero, leaving $\rho\sigma^2$. The limit is positive when $\rho>0$ and $\sigma>0$; it is zero when $\rho=0$. Common-factor risk remains here because every holding retains the same factor exposure. In a market crash, many stocks can fall together, although individual stocks can move differently.

In this equal-weight, equal-variance model, 20 and 30 assets eliminate 95% and 96.7% of diversifiable variance. That normalized fraction is $1-1/n$, independent of correlation when $\rho<1$; correlation still determines the remaining total variance. Real diversification needs depend on concentration, exposures, dependence, and the chosen risk criterion.