Unwitting Markowitz' Simplification of Portfolio Random Returns

In his famous paper, Markowitz (1952) derived the dependence of portfolio random returns on the random returns of its securities. This result allowed Markowitz to obtain his famous expression for portfolio variance. We show that Markowitz's equation for portfolio random returns and the expression for portfolio variance, which results from it, describe a simplified approximation of the real markets when the volumes of all consecutive trades with the securities are assumed to be constant during the averaging interval. To show this, we consider the investor who doesn't trade shares of securities of his portfolio. The investor only observes the trades made in the market with his securities and derives the time series that model the trades with his portfolio as with a single security. These time series describe the portfolio return and variance in exactly the same way as the time series of trades with securities describe their returns and variances. The portfolio time series reveal the dependence of portfolio random returns on the random returns of securities and on the ratio of the random volumes of trades with the securities to the random volumes of trades with the portfolio. If we assume that all volumes of the consecutive trades with securities are constant, obtain Markowitz's equation for the portfolio's random returns. The market-based variance of the portfolio accounts for the effects of random fluctuations of the volumes of the consecutive trades. The use of Markowitz variance may give significantly higher or lower estimates than market-based portfolio variance.