🔀 Diversification

Diversification is the most fundamental risk management strategy: by combining assets that don't move in perfect lockstep, an investor can reduce portfolio volatility without necessarily reducing expected return.

📐 The Mathematics

📊 Two-Asset Portfolio Variance

For a portfolio of two assets with weights \(w_1\) and \(w_2 = 1 - w_1\):

\[ \sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{12} \]

where:

\(\sigma_1, \sigma_2\) are the individual asset volatilities
\(\rho_{12}\) is the correlation coefficient (\(-1 \leq \rho \leq +1\))

The magic of diversification lies in the cross-term: when \(\rho_{12} < 1\), the portfolio variance is less than the weighted average of individual variances.

🔑 Correlation Effects

Correlation \(\rho\)	Effect	Example
\(+1\)	No diversification benefit — assets move identically	Two S&P 500 ETFs
\(0\)	Significant variance reduction	Stocks vs Gold
\(-1\)	Perfect hedge — variance can reach zero	Long stock + put option

📈 N-Asset Generalization

For \(N\) assets:

\[ \sigma_p^2 = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \sigma_i \sigma_j \rho_{ij} \]

As \(N\) increases, the contribution of individual variances shrinks (proportional to \(1/N\)), but the contribution of covariances remains. This leads to the concept of systematic risk.

🎯 Systematic vs Idiosyncratic Risk

📊 Idiosyncratic (Diversifiable) Risk

Risk specific to a single company or asset. Examples:

CEO departure
Product recall
Patent expiration

This risk can be diversified away by holding many assets. With ~30 uncorrelated stocks, idiosyncratic risk approaches zero.

🌍 Systematic (Non-Diversifiable) Risk

Risk affecting the entire market. Examples:

Interest rate changes
Recessions
Pandemics
Geopolitical events

This risk cannot be eliminated through diversification. It is the risk that investors are compensated for bearing — the foundation of the Capital Asset Pricing Model (CAPM).

\[ \sigma_{portfolio}^2 = \underbrace{\sigma_{systematic}^2}_{\text{cannot remove}} + \underbrace{\sigma_{idiosyncratic}^2}_{\xrightarrow{N \to \infty} 0} \]