Markowitz’s Portfolio Variance Describes Only a Limited Case of Constant Trade Volumes

We show that Markowitz’s (1952) portfolio variance describes only a limited case when all volumes of successive trades at exchange with shares of all securities of the portfolio are assumed constant during the averaging interval. We overcome this limitation and derive market-based portfolio variance that depends upon the means, variances, and covariance of random values and volumes of consecutive trades. To derive that, we convert the time series of trades made at the exchange with shares of portfolio securities and obtain the time series that model the trades with the portfolio like trades with one security. That establishes an equal description of market-based variance of any security and portfolio. The time series that model the trades with the portfolio, like trades with one security, reveal that the market-based equation that describes random portfolio return depends on random returns of its securities and on random volumes of market trades. Markowitz hasn't accounted for the impact of random trade volumes on random portfolio returns. If all trade volumes are assumed constant, the market-based equation on random portfolio returns coincides with the corresponding Markowitz equation. We use market-based portfolio variance that accounts for random trade volumes and describe the dependence of the Sharpe Ratio of the portfolio on random trade volumes. We prove that correlations between random prices and trade volumes always equal zero. To study market-based statistical dependence between random trade volumes and prices, one should empirically calculate correlations between prices and squares of trade volumes.