What Is Mean-Variance Optimisation?
In quantitative finance, Mean-Variance Optimization (MVO) is a portfolio construction framework that balances risk and return. Developed by Harry Markowitz in 1952, it underpins Modern Portfolio Theory (MPT) and emphasizes diversification to achieve efficient portfolios.
Key Concepts
Mean (Expected Return): The average return an asset is anticipated to generate.
Variance (Risk): A measure of how much an asset’s returns fluctuate over time.
Covariance: Indicates how two assets’ returns move relative to each other (e.g., rising or falling together).
Efficient Frontier: A curve representing portfolios that offer the highest return for a specific risk level or the lowest risk for a target return.
Portfolio Return and Risk
The expected return of a portfolio is the weighted average of its individual assets’ returns.
Portfolio risk depends not only on individual asset risks but also on how assets interact (covariance). Diversification reduces risk when assets are not perfectly correlated.
Optimization Goal
Maximize returns for a given risk tolerance or minimize risk for a target return.
Investors specify constraints, such as budget limits (weights summing to 100%) or restrictions on short-selling.
Efficient Frontier
Portfolios on the efficient frontier are optimal—no other portfolio offers a better risk-return tradeoff.
Lower correlation between assets improves diversification, pushing the frontier upward (higher returns for the same risk).
Assumptions
Investors are rational and prefer lower risk for the same return.
Returns follow a normal distribution (symmetrical, no extreme outliers).
No transaction costs, taxes, or liquidity constraints.
Single-period investment horizon (no rebalancing over time).
Implementation Steps
Estimate Inputs: Forecast expected returns, risks (variances), and pairwise covariances for all assets.
Define Constraints: Ensure portfolio weights add to 100% and comply with restrictions (e.g., no short-selling).
Optimize: Use computational tools to find weights that align with the investor’s risk-adjusted return goals.
Limitations
Input Sensitivity: Small errors in return or covariance estimates can drastically alter the optimal portfolio.
Simplified Risk Measure: Variance ignores tail risks (e.g., market crashes) and liquidity risks.
Static Approach: Assumes market conditions and investor goals remain constant over time.
Normality Assumption: Real-world returns often exhibit skewness and fat tails.
Extensions and Alternatives
Hierarchical Risk Parity: Allocates risk equally across assets instead of capital.
Black-Litterman Model: Combines market equilibrium with investor views.
Tangency Portfolio: Maximizes return per unit of risk (Sharpe ratio).
Multi-Period Models: Address dynamic rebalancing and changing goals.
See also