Modern Portfolio Theory

In 1952, a 25-year-old PhD student named Harry Markowitz published a 14-page paper in the Journal of Finance titled "Portfolio Selection." It is one of the most cited papers in the history of economics, and it established the mathematical foundation for how rational investors should build…

Mean-variance optimisation is the mathematical procedure that constructs the efficient frontier. The inputs are expected returns for each asset, variances (standard deviations squared) for each asset, and covariances between every pair of assets. The output is the set of portfolios that minimise…

The Capital Asset Pricing Model (CAPM) is the bridge between Markowitz's portfolio theory and a theory of how assets are actually priced in markets. Markowitz told investors how to combine assets optimally; CAPM asks what must be true about prices if all investors do exactly that.

Beta is a measure of an asset's sensitivity to the movements of the market as a whole. Technically, it is the covariance of the asset's returns with the market's returns, divided by the variance of the market's returns. More intuitively, it answers the question: when the market moves by 1%, how…

In the CAPM framework, alpha is the return an investment earns above what its level of market risk (beta) would predict. If CAPM predicts a portfolio with a given beta should return 8% in a year, and the portfolio actually returns 11%, the alpha is +3 percentage points. Negative alpha means the…

Comparing investment returns without accounting for risk is misleading. A fund returning 20% per year sounds impressive — but if it achieves that by taking twice the market's volatility, it may actually be delivering inferior performance on a risk-adjusted basis. A more cautious fund returning 10%…

The Sharpe ratio is simple, intuitive, and widely used — but its simplicity involves assumptions that are often violated in practice. Understanding these limitations is essential for interpreting risk-adjusted return figures correctly and avoiding the traps that have caught professional investors…

Mean-variance optimisation is mathematically rigorous and intellectually compelling. In practice, it has a notorious reputation among professional investors for producing portfolios that are extreme, unstable, and practically unusable. The problem is not with the mathematics — it is with the…

Modern Portfolio Theory is built on the normal distribution — the symmetric bell curve familiar from statistics. In a normal distribution, outcomes close to the mean are common, and outcomes far from the mean become exponentially rarer. Under normality, a return more than three standard deviations…