Market-Based Portfolio Variance

Abstract: The investor, who holds his portfolio and doesn't trade his shares, at current time can use the time series of the market trades that were made during the averaging interval with the securities of his portfolio to assess the current variance of the portfolio. We show how the time series of trades with the securities of the portfolio determine the time series of trades with the portfolio as a single market security. The time series of portfolio trades determine the return and variance of the portfolio in the same form as the time series of trades with securities determine their returns and variances. The description of any portfolio and any single market security is equal. The time series of portfolio trades define the decomposition of the portfolio variance by its securities. If the volumes of trades with all securities are assumed constant, the decomposition of the portfolio variance coincides with Markowitz's (1952) expression of variance. However, the real markets expose random volumes of trades. The portfolio variance that accounts for the randomness of trade volumes is a polynomial of the 4th degree in the variables of relative amounts invested into securities and with the coefficients different from covariances of securities returns. We discuss the possible origin of the latent and unintended assumption that Markowitz (1952) made to derive his result. Our description of the portfolio variance that accounts for the randomness of real trade volumes could help the portfolio managers and the majors like BlackRock's Aladdin and Asimov, JP Morgan, and the U.S. Fed to adjust their models and forecasts to the reality of random markets.