Modern Portfolio Theory and the Efficient Frontier

Modern portfolio theory (MPT), introduced by Harry Markowitz in his 1952 paper "Portfolio Selection", changed investing by showing that what matters is not each asset's risk in isolation but how assets combine. By mixing assets that do not move perfectly together, an investor can build portfolios with better return for a given level of risk. Markowitz shared the 1990 Nobel Prize in economics for this work. MPT underpins much of modern portfolio management, even though its practical use requires care.

Core ideas#

1. Investors care about expected return and risk, measured as variance or volatility.
2. Portfolio risk depends on correlations between assets, not just their individual volatilities. See Covariance and Correlation.
3. For each level of risk, some portfolio offers the highest expected return.
4. The set of those best portfolios forms the efficient frontier.

Two asset portfolio maths#

Portfolio return = w1 × R1 + w2 × R2
Portfolio variance = w1² × σ1² + w2² × σ2² + 2 × w1 × w2 × ρ × σ1 × σ2

The efficient frontier#

Plotting expected return against volatility for every possible mix of assets produces a curved boundary. Portfolios on the upper edge of the curve are efficient: no other portfolio offers more return for the same risk. Portfolios below the edge are inefficient.

From MPT to CAPM#

Building on MPT, William Sharpe, John Lintner and others developed the Capital Asset Pricing Model in the 1960s. If all investors hold the tangency portfolio, it must be the whole market, and an asset's expected return depends only on its beta to the market. See Alpha and Beta.

Criticisms and limitations#

Richard Michaud famously described mean variance optimisers as "error maximisers", because they put the most weight on assets whose expected returns are overestimated.

Practical responses#

• Constraints on weights to prevent extreme positions. See Portfolio Optimization.
• Shrinkage of estimates toward simpler, more stable values.
• Black Litterman blending of market implied returns with views. See Black-Litterman Model.
• Risk based methods that avoid return forecasts, such as minimum variance and risk parity. See Risk Budgeting and Risk Parity.
• Resampling and robust optimisation.

Portfolio maths uses vectors and matrices for weights and covariances; Linear Algebra for Traders explains the essentials.

Frequently asked questions#

What is modern portfolio theory?#

A framework from Harry Markowitz showing how to combine assets to maximise expected return for a given level of risk, using their correlations.

What is the efficient frontier?#

The set of portfolios that offer the highest expected return for each level of risk.

Why is MPT hard to use in practice?#

It depends on uncertain estimates of expected returns, volatilities and correlations, and small errors can produce extreme, unstable portfolios.

Next, learn a way to make optimisation more stable in Black-Litterman Model.