Mean-Variance Analysis

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Mean-Variance Analysis

Mean-variance analysis is a framework for evaluating investments by comparing expected return (the mean) with risk (measured by variance or standard deviation). It is a core component of Modern Portfolio Theory (MPT) and helps investors choose portfolios that either maximize return for a given level of risk or minimize risk for a given expected return.

Key concepts

  • Expected return (mean): the probability-weighted average of possible returns for an investment.
  • Variance and standard deviation: measures of return dispersion; higher values indicate greater uncertainty.
  • Covariance and correlation: measures of how two assets move together; important because portfolio risk depends on how asset returns co-vary as well as on individual variances.
  • Portfolio weights: the fraction of total capital allocated to each asset.

Why it matters

Mean-variance analysis formalizes trade-offs between risk and reward. By combining assets with different return and risk characteristics—and especially by combining assets with low or negative correlations—investors can often reduce overall portfolio risk without sacrificing expected return.

Formulas

Expected portfolio return:
R_p = Σ (w_i × R_i)
where w_i is the weight of asset i and R_i is its expected return.

Portfolio variance (two-asset case shown; generalizes to covariance matrix for multiple assets):
σ_p^2 = w1^2σ1^2 + w2^2σ2^2 + 2w1w2Cov(1,2)
Cov(1,2) = ρ12 × σ1 × σ2
where σ1, σ2 are standard deviations and ρ12 is the correlation between assets 1 and 2.

Portfolio standard deviation is the square root of portfolio variance: σ_p = sqrt(σ_p^2).

Example

Portfolio with two investments:
* Investment A: $100,000 (25% weight), expected return 5%, standard deviation 7% (σ_A = 0.07)
* Investment B: $300,000 (75% weight), expected return 10%, standard deviation 14% (σ_B = 0.14)
* Correlation between A and B: ρ = 0.65

  1. Expected portfolio return:
    R_p = 0.25×5% + 0.75×10% = 8.75%

  2. Portfolio variance (using decimals for percentages):
    σ_p^2 = (0.25^2 × 0.07^2) + (0.75^2 × 0.14^2) + 2×0.25×0.75×0.07×0.14×0.65
    σ_p^2 ≈ 0.0137

  3. Portfolio standard deviation:
    σ_p = sqrt(0.0137) ≈ 0.1171 = 11.71%

Interpretation: the portfolio’s expected return is 8.75% with an estimated annual volatility of about 11.7%.

Practical considerations and limitations

  • Assumptions: mean-variance analysis assumes returns are adequately summarized by their mean and variance and that investors are risk-averse and rational. It typically assumes normally distributed returns or quadratic utility, which may not hold in practice.
  • Sensitivity: results depend heavily on inputs (expected returns, variances, correlations). Small changes can materially alter optimal allocations.
  • Estimation error: forecasting returns and covariances is difficult; noisy estimates can lead to unstable portfolio recommendations.
  • Extensions: robust optimization, Bayesian approaches, resampling, and the use of alternative risk measures (e.g., CVaR) can address some limitations.

Conclusion

Mean-variance analysis provides a clear, quantitative way to balance expected return against risk and to construct diversified portfolios that reduce risk through asset selection and allocation. Despite practical drawbacks related to input estimation and distributional assumptions, it remains a foundational tool in portfolio construction and asset allocation.