Modern Portfolio Theory

The engine of the framework is the variance equation. Portfolio expected return is a simple weighted average of the component expected returns, but portfolio variance is not: it equals the sum of each asset's squared weight times its variance, plus the sum of every pairwise term formed by two weights multiplied by the covariance between that pair. Since covariance equals the correlation coefficient times the two standard deviations, correlation is the lever that determines how much risk survives combination.

For two assets the result is intuitive. With correlation equal to +1 the portfolio standard deviation collapses to the weighted average of the two standard deviations, and no risk reduction occurs. As correlation falls, the cross term shrinks and portfolio standard deviation drops below that weighted average. At a correlation of -1 a specific weighting drives variance to zero. Every diversification benefit in the theory comes from correlations strictly below one.

Plotting all feasible combinations of risky assets in expected return and standard deviation space produces an opportunity set bounded on the left by the minimum-variance frontier: for each level of expected return, the combination with the lowest possible variance. The single leftmost point is the global minimum-variance portfolio. Everything on the frontier above that point forms the efficient frontier, because portfolios below it are dominated, offering the same risk with lower return.

A risk-averse investor's preferences are represented by indifference curves that slope upward, since additional risk must be compensated by additional expected return. The steeper the curves, the greater the risk aversion. The optimal risky portfolio is where the investor's highest attainable indifference curve is tangent to the efficient frontier. Two investors facing identical inputs therefore hold different portfolios only because their risk aversion differs.

Introducing a risk-free asset transforms the problem. Combining risk-free lending or borrowing with any risky portfolio produces a straight line in risk-return space, and the line with the steepest slope is the one tangent to the efficient frontier. That slope is the Sharpe ratio, so the tangency portfolio is the risky portfolio with the highest reward-to-variability ratio available. The resulting capital allocation line dominates the curved frontier at every risk level.

This yields the two-fund separation result: the choice of risky portfolio is separated from the choice of how much risk to take. All investors hold the same tangency portfolio and differ only in the fraction placed in the risk-free asset. When investors share identical expectations, that tangency portfolio becomes the market portfolio, the capital allocation line becomes the capital market line, and the framework leads directly into the CAPM and beta.

Strategic asset allocation is the clearest application: allocating across equities, fixed income, real estate and alternatives depends on their covariances rather than on standalone volatility, which is why low-correlation exposures are valued even when individually risky. The framework also explains why unsystematic risk is uncompensated once diversification is available, and why performance is judged on risk-adjusted rather than absolute terms.

The weaknesses are equally testable. Optimised weights are highly sensitive to estimated means, variances and correlations, and inputs are drawn from historical data that may not persist. Correlations tend to rise in crises, precisely when diversification is needed. Variance treats upside and downside symmetrically, while return distributions exhibit skewness and fat tails. Candidates should be able to compute two-asset portfolio risk, identify efficient portfolios, and explain how correlation governs the outcome.