Modern/Markowitz portfolio theory

Portfolio theory deals with the value and risk of portfolios rather than individual securities. It is often called modern portfolio theory or Markowitz portfolio theory.

The key result in portfolio theory is that the volatility of a portfolio is less than the weighted average of the volatilities of the securities it contains. The standard deviation of the expected return on a portfolio is:

√(ΣWi2σi2 + ΣΣWiWjCovij)

where the sums are over all the securities in the portfolio,
Wi is the proportion of the portfolio in security i,
σi is the standard deviation of expected returns of security i, and,
Covij is the covariance of expected returns of securities of i and j.

Assuming that the covariance is less than one (invariably true), this will be less than the weighted average of the standard deviation of the expected returns of the securities. This is why diversification reduces risk.

The other important results in modern portfolio theory are those dealing with the construction of efficient portfolios.

Modern portfolio theory is not universally accepted, despite being the standard textbook description of portfolio risk and return. Markowitz himself thought normally distributed variance an inadequate measure of risk. Models have been developed that use asymettric and fat tailed distributions (post-modern portfolio theory). There are also more radical objections, including an alternative behavioural portfolio theory.

Any theory or strategy that suggests it is possible to outperform the market without taking extra risk contradicts Markowitz portfolio theory, as does the evidence for the value effect or the existence of persistent arbitrage opportunities. Note that only the last of these is necessarily a failure of market efficiency — the two are often confused (at least in the context of their failure).