Market-Based Portfolio Selection

2025-04-22 0 0 388.79KB 26 页 10玖币

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Victor Olkhov

Independent, Moscow, Russia

victor.olkhov@gmail.com

ORCID: 0000-0003-0944-5113

March 31, 2025

Abstract

We show that Markowitz’s (1952) decomposition of a portfolio variance as a quadratic form

in the variables of the relative amounts invested into the securities, which has been the core of

classical portfolio theory for more than 70 years, is valid only in the approximation when all

trade volumes with all securities of the portfolio are assumed constant. We derive the market-

based portfolio variance and its decomposition by its securities, which accounts for the impact

of random trade volumes and is a polynomial of the 4th degree in the variables of the relative

amounts invested into the securities. To do that, we transform the time series of market trades

with the securities of the portfolio and obtain the time series of trades with the portfolio as a

single market security. The time series of market trades determine the market-based means and

variances of prices and returns of the portfolio in the same form as the means and variances of

any market security. The decomposition of the market-based variance of returns of the portfolio

by its securities follows from the structure of the time series of market trades of the portfolio

as a single security. The market-based decompositions of the portfolio’s variances of prices

and returns could help the managers of multi-billion portfolios and the developers of large

market and macroeconomic models like BlackRock’s Aladdin, JP Morgan, and the U.S. Fed

adjust their models and forecasts to the reality of random markets.

Keywords : portfolio theory, random market trades, portfolio variance, covariance

JEL: C0, E4, F3, G1, G12

This research received no support, specific grant, or financial assistance from funding agencies in the public,

commercial, or nonprofit sectors. We welcome valuable offers of grants, support, and positions.

1. Introduction

More than seventy years ago, Markowitz (1952) described the principles of portfolio selection.

Since then, many researchers have contributed to the further development of the portfolio

theory (Markowitz, 1991; Rubinstein, 2002; Cochrane, 2014; Elton et al., 2014). However,

since 1952, Markowitz’s decomposition of the portfolio variance by the covariances of the

securities that compose the portfolio, the core result of modern portfolio theory, remains

unchanged. We reconsider that classical Markowitz’s result. His paper is the only reference

required for the understanding of our contribution to the portfolio theory.

We believe that Markowitz’s results are well known and need no additional clarifications. For

convenience, we almost reproduce Markowitz’s notations and present the mean return R(t,t0)

(1.1) of the portfolio at time t that was composed at time t0 in the past of j=1,2,.. J securities:





  (1.1)

As Rj(t,t0), we denote the mean returns of the security j at time t. Coefficients Xj(t0) denote the

relative amounts invested into security j at time t0. It is assumed that all prices are adjusted to

the current time t. Markowitz (1952) presented the variance Θ(t,t0) (1.2) of returns of the

portfolio as a quadratic form by the relative amounts Xj(t0) invested into security j:





  (1.2)

The functions θjk(t,t0) (1.3) in (1.2) denote the covariance of returns of securities j and k:

  (1.3)

For decades, the relations (1.1-1.2) that were derived by Markowitz in 1952 served successfully

as a basis of the portfolio theory.

Actually, since 1952, the mean and variance of any portfolio that is composed of

tradable market securities is presented through its components as (1.1-1.2). Probably, that

originates implicit beliefs in substantial differences between the descriptions of the market

securities and the portfolio they compose. However, the properties of market securities that

compose the portfolio are determined by the random time series of their market trades. The

random time series of market trades define the means, variances, and covariances of prices and

returns of tradable market securities. All factors that can impact the randomness of prices and

returns of market securities, like agents’ expectations, risks, market shocks, etc., are already

accounted for and reflected by the time series of the performed market trades. The time series

of market trades completely determines the statistical moments of prices and returns of market

securities. However, market trades take time. For simplicity, we assume that trades with all

market securities that compose the portfolio occur simultaneously at the same time ti with a

short time span ε>0 between trades that is constant and is the same for trades with all securities.

To estimate the means, variances, or covariances of prices and returns of securities that are

determined by random time series of market trades, one should choose the time averaging

interval Δ (4) and consider the N terms of time series of market trades during Δ:

  

 

         (1.4)

The contribution of our paper to portfolio theory consists of the description of how the

time series of market trades with all securities that compose the portfolio determines the time

series of trades with the portfolio as a single tradable market security. We show that the time

series of market trades determines the means and variances of the prices and returns of the

portfolio completely in the same form as for each of the market securities. There are no

differences between the expressions of the means and variances of the portfolio and of the

market securities that compose that portfolio. We show that market trade time series equally

describe any portfolio and any market security.

Further, we show that the decomposition of the variance Θ(t,t0) (1.2) (Markowitz, 1952)

of the portfolio by the covariances of its securities describes a rather limited approximation in

which all volumes of market trades with all j=1,2,..J securities of the portfolio are assumed

constant during the averaging interval Δ (1.4). We show that the classical decomposition of the

variance Θ(t,t0) (1.2) of the portfolio, the core of modern portfolio theory, neglects the impact

of random volumes of market trades with the securities. We derive a market-based

decomposition of the variance Θ(t,t0) of returns of the portfolio by its securities that accounts

for the influence of the random volumes of trades with the securities of the portfolio. The

decomposition of variance is a polynomial of the 4th degree in the variables of the relative

amounts invested into securities, and that differs it significantly from the classical quadratic

form (1.2) derived by Markowitz (1952). The distinctions of the market-based decomposition

of the portfolio variance from the classical case (1.2) reveal that the selection of the portfolio

with higher returns under lower variance is a much more complex problem than it is assumed

now.

One may consider portfolio selection on the basis of (1.2) as a first approximation that

neglects the impact of random trade volumes. However, the investors and traders who manage

multi-billion portfolios must account for the impact of random volumes of market trades with

securities on the portfolio variance and should consider market-based decomposition. The

developers of large macroeconomic and market models like BlackRock's Aladdin, JP Morgan,

and the U.S. Fed could use our results to adjust their models and forecasts to the reality of

random markets.

In Section 1 we describe how the time series of the values and volumes of market trades

with securities of the portfolio determine the time series of values and volumes of trades with

the portfolio as a single market security. These time series determine the means and variances

of prices and returns of the portfolio completely in the same forms as for any tradable market

security. In Section 2 we derive the means and variances of prices and returns of the portfolio

and their decompositions by the securities of the portfolio. The decomposition of market-based

variance of the portfolio returns is a polynomial of the 4th degree by the relative amounts

invested into securities. It takes the conventional quadratic form (1.2) that was derived by

Markowitz (1952) only in the approximation for which all volumes of market trades with all

securities of the portfolio are assumed constant during the averaging interval Δ (1.4). Finally,

we consider a hypothesis that may explain the origin of the implicit assumption that leads

Markowitz to derive his results. The conclusion is in Section 3. Most calculations are in

Appendices A – D. In App. A, we derive the expressions of the market-based means and

variances of prices and returns of a tradable market security. In App. B, we derive the

covariances between prices and returns of two securities. In App. C, we derive the

decompositions of the means and variances of prices and returns of the portfolio by its

securities. In App. D, we explain the economic sense of the distinctions between the market-

based and the frequency-based assessments of the statistical moments of prices and returns. All

prices are assumed adjusted to the current time t.

2. Time series of trades with the portfolio as a single market security

Let us assume that in the past at time t0 the investor has composed his portfolio of

j=1,2,..J market securities. We denote the portfolio at time t0 by the number of shares Uj(t0),

the values Cj(t0) of these shares, and the prices pj(t0) per share of each security j that obey trivial

equations:

    (2.1)

The prices pj(t) and the values Cj(t) of security j can change in time t, but the number

of shares Uj(t0) of each security in the portfolio remains constant. We denote the total value

QΣ(t0) and the total volume WΣ(t0) or total number of shares of the portfolio at time t0:





 



 (2.2)

The prices pj(t0) of different securities j=1,2,…J in the portfolio can vary a lot from

each other. We introduce the price s(t0) (2.3) per share of the portfolio similarly (2.1):

 



 

 (2.3)

We determine the portfolio at time t0 by its total value QΣ(t0), volume WΣ(t0), price s(t0),

and by the set of corresponding values Cj(t0), volumes Uj(t0), and prices pj(t0) of the securities

that compose the portfolio. Relations (2.3) give the decomposition of the portfolio price s(t0)

by prices pj(t0) (2.1) of its securities. The coefficients xj(t0) define the relative numbers of the

shares Uj(t0) of security j in the total number of shares WΣ(t0) of the portfolio. We repeat that

the numbers of shares Uj(t0) of each security j, j=1,2,..J, and the total number of shares WΣ(t)

of the portfolio remain constant in time t.

We assess the means and variances of the prices and returns of the portfolio at the

current time t, taking into account the results of market trades with securities that compose the

portfolio during the averaging interval Δ (1.4). To assess the mean and variance, one should

select the interval Δ that provides sufficient market trade data for such an assessment. We

consider the time series of the market trades with securities j=1,2,..J, made during Δ (1.4). It is

obvious that to estimate the mean and variances of prices or returns of Uj(t0) shares of a

particular security j of the portfolio, one should consider the averaging interval Δ during which

the total volume of trades with the security j would be much more than its number of shares

Uj(t0) at time t0. To assess the means and variances of prices and returns of the portfolio, one

should choose the averaging interval Δ during which the total volumes of trades with each

security j=1,..J are much more than the number Uj(t0) of shares of each security j in the

portfolio. Hence, the total volume of trades with all securities during Δ (1.4) also would be

much more than the total number WΣ(t0) (2.2) of shares of the portfolio. Indeed, otherwise any

attempts to sell the stake Uj(t0) of shares or all shares WΣ(t0) of the portfolio as a whole would

so strongly disturb the market that the results of the sales would be too different from the initial

assessments.

To highlight the differences between time series (2.2; 2.3) that describe the portfolio

and the time series that describe market trades with the securities that compose the portfolio,

we denote the trade values Cj(ti), volumes Uj(ti), and prices pj(ti) of securities j=1,..J, which

follow the equations (2.4) at time ti similar to (2.1):

       (2.4)

We assume that for each security j, the total volume UΣj(t;1) (1.5) of trades during Δ (1.4) is

much more than the number of shares Uj(t0) of the security j in the portfolio at time t0:





      (2.5)

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1Market-BasedPortfolioSelectionVictorOlkhovIndependent,Moscow,Russiavictor.olkhov@gmail.comORCID:0000-0003-0944-5113March31,2025AbstractWeshowthatMarkowitz’s(1952)decompositionofaportfoliovarianceasaquadraticforminthevariablesoftherelativeamountsinvestedintothesecurities,whichhasbeenthecoreofclassic...

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