Portfolio Variance

Unwitting Markowitz’ Simplification of Portfolio Random Returns ArXiv ID: 2508.08148 “View on arXiv” Authors: Victor Olkhov Abstract 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. ...

Market-Based Portfolio Variance ArXiv ID: 2504.07929 “View on arXiv” Authors: Unknown Abstract The variance measures the portfolio risks the investors are taking. The investor, who holds his portfolio and doesn’t trade his shares, at the current time can use the time series of the market trades that were made during the averaging interval with the securities of his portfolio and assess the current return, variance, and hence the current risks of his 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 trades with the portfolio determine its return and variance 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 the portfolio trades define the decomposition of the portfolio variance by its securities, which is a quadratic form in the variables of relative amounts invested into securities. Its coefficients themselves are quadratic forms in the variables of relative numbers of shares of its securities. If one assumes that the volumes of all consecutive deals with each security are constant, the decomposition of the portfolio variance coincides with Markowitz’s (1952) variance, which ignores the effects of random trade volumes. The use of the variance that accounts for the randomness of trade volumes could help majors like BlackRock, JP Morgan, and the U.S. Fed to adjust their models, like Aladdin and Azimov, to the reality of random markets. ...