Variance is the average of squared deviations of returns from their mean — a fundamental measure of return dispersion; Standard Deviation is simply the square root of Variance.
Variance = (1/n) × Σ(Ri − R̄)², where Ri is each period's return and R̄ is the average return. It quantifies how spread out the return distribution is around its mean. A higher variance means returns are more unpredictable and spread across a wider range. Variance is additive for independent assets — making it useful in portfolio construction (portfolio variance accounts for correlations between assets). Standard deviation is used in practice because it shares the same units as the original return data.
If a fund's returns over 5 years were: 8%, 12%, -3%, 15%, 9% — the mean is 8.2%. Variance calculates how much each year's return deviated from 8.2%, squares each deviation (to make them all positive), and averages them. Standard deviation = √Variance — brings it back to percentage terms for interpretability. Variance itself is rarely quoted to retail investors but is the mathematical foundation for portfolio risk calculations.
Step 1: Calculate mean return R̄ = (Σ Ri) / n.
Step 2: For each period, compute (Ri − R̄)².
Step 3: Average these: Variance = Σ(Ri − R̄)² / n.
Step 4: Standard Deviation = √Variance.
Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂ (includes correlation term).
σ² = (1/n) × Σ(Ri − R̄)²