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

Olkhov, Victor (2026): Markowitz’s Portfolio Variance Describes Only a Limited Case of Constant Trade Volumes.

Abstract

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: portfolio variance, portfolio theory, random trade volumes
• Subjects: G - Financial Economics > G0 - General > G00 - General; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G17 - Financial Forecasting and Simulation; G - Financial Economics > G2 - Financial Institutions and Services > G24 - Investment Banking ; Venture Capital ; Brokerage ; Ratings and Ratings Agencies

References

Berkowitz, S., Logue, D. and E. Noser, Jr. (1988). “The Total Cost of Transactions on the NYSE,” The Journal of Finance 43, (1), 97-112.

Boyd, S., Johansson, K., Kahn, R., Schiele, P. and T. Schmelzer. (2024). “Markowitz Portfolio Construction at Seventy,” Journal Portfolio Management, Special Issue Dedicated to Harry Markowitz 50, (8), 117 – 160.

Campbell, J.Y., Grossman, S.J. and J. Wang. (1993). “Trading Volume and Serial Correlation in Stock Return,” Quarterly Journal of Economics, 108 (4), 905-939

DeFusco, A.A., Nathanson, C.G. and E. Zwick. (2017). “Speculative Dynamics of Prices and Volume”, Cambridge, MA, NBER WP 23449, 1-74

Karpoff, J.M. (1987). “The Relation Between Price Changes and Trading Volume: A Survey,” The Journal of Financial and Quantitative Analysis, 22 (1), 109-126

Markowitz, H. (1952). “Portfolio Selection,” Journal of Finance 7(1), 77-91.

Markowitz, H. (1991). “Foundations of portfolio theory”, Les Prix Nobel 1990, 292.

Merton, R. (1969). “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” Review of Economics and Statistics 247-257.

Olkhov, V. (2023a). “Theoretical Economics as Successive Approximations of Statistical Moments,” SSRN WP 4586945, 1-17.

Olkhov, V. (2023b). “Economic Complexity Limits Accuracy of Price Probability Predictions by Gaussian Distributions,” SSRN WP 4550635, 1-23.

Olkhov, V. (2024). “Lower Bounds of Uncertainty of Observations of Macroeconomic Variables and Upper Limits on the Accuracy of Their Forecasts,” SSRN WP 4914092, 1-18.

Olkhov, V. (2025a). “Market-Based Portfolio Variance,” SSRN WPS 5212636, 1-17.

Olkhov, V. (2025b). “Markowitz Variance Can Vastly Undervalue or Overestimate Portfolio Variance and Risks,” SSRN WPS 5370800, 1-20.

Pogue, G.A. (1970). “An Extension of the Markowitz Portfolio Selection Model to Include Variable Transaction’s Costs, Short Sales, Leverage policies and Taxes,” Journal of Finance, 25, 1005-1027.

Rubinstein, M. (2002). “Markowitz's "Portfolio Selection": A Fifty-Year Retrospective, Journal of Finance,” 57, 1041-1045.

Samuelson, P. (1970). “The Fundamental Approximation Theorem of Portfolio Analysis in terms of Means, Variances and Higher Moments,” Review Economic Studies 37, 537-542.

Sharpe, W. F. (1966). "Mutual Fund Performance," Journal of Business 39(1), 119-138.

Shiryaev, A. (1999). “Essentials of Stochastic Finance,” v3, World Sci. Publ., Singapore

Shreve, S. (2004). “Stochastic calculus for finance,” Springer finance series, NY, USA

Tauchen, G.E. and M. Pitts. (1983). “The Price Variability-Volume Relationship On Speculative Markets,” Econometrica, 51, (2), 485-505.