Market-Based Portfolio Selection

We show that Markowitz's (1952) decomposition of a portfolio variance as a quadratic form in the variables of the relative amounts invested into the securities, which has been the core of classical portfolio theory for more than 70 years, is valid only in the approximation when all trade volumes with all securities of the portfolio are assumed constant. We derive the market-based portfolio variance and its decomposition by its securities, which accounts for the impact of random trade volumes and is a polynomial of the 4th degree in the variables of the relative amounts invested into the securities. To do that, we transform the time series of market trades with the securities of the portfolio and obtain the time series of trades with the portfolio as a single market security. The time series of market trades determine the market-based means and variances of prices and returns of the portfolio in the same form as the means and variances of any market security. The decomposition of the market-based variance of returns of the portfolio by its securities follows from the structure of the time series of market trades of the portfolio as a single security. The market-based decompositions of the portfolio's variances of prices and returns could help the managers of multi-billion portfolios and the developers of large market and macroeconomic models like BlackRock's Aladdin, JP Morgan, and the U.S. Fed adjust their models and forecasts to the reality of random markets.