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\pubyear{2009}
\volume{0}
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\firstpage{347}
\lastpage{351}
\doi{00019/0000.0000}
\arxiv{math.PR/00000019}

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\def\turk{Wilhelmine von T$\ddot{\mbox{u}}$rk}
\def\sturk{W. von T$\ddot{\mbox{u}}$rk}
\def\cturk{Von T$\ddot{\mbox{u}}$rk, W.}


\begin{document}
\date{empty}
\begin{frontmatter}
\title{A Note on the Investment Proportions of a
Minimum-Variance Equity Portfolio}%\protect\thanksref{T1}}
\runtitle{Minimum-Variance Equity Portfolio}
%\thankstext{T1}{Prepared with partial support of N.S.F. Grant G.P. 2593.}

\begin{aug}
%\thanksref{a,t2}
\author{\fnms{\turk}\thanksref{a1,t1} \ead[label=e1]{wvonturk@sbcglobal.net}}
\address[a1]{43 Del Mar Avenue, Berkeley, CA  94708, email:
\url{wvonturk@sbcglobal.net}}
\thankstext{t1}{The author gratefully acknowledges her intellectual indebtedness to the tradition of the Department of Statistics, University of California at Berkeley, where she completed the dissertation with the guidance and advice of Professor Erich L. Lehmann.  The responsibility for any shortcomings in the present paper rests squarely on the shoulders of the author.}

\runauthor{\sturk}

\affiliation{American Century Investments}

%\end{center}

\contributor{\cturk}{American Century Investments}


\end{aug}

\begin{abstract}
It is shown that the capitalization-weighted portfolio is mathematically required to coincide with the minimum-variance portfolio, provided both portfolios are defined with respect to the same (arbitrary) collection of equities having linearly independent returns.  This result is a logical consequence of the law of iterated expectations and has important implications for equity return covariance structure.
\end{abstract}

\begin{keyword}[class=AMS]
\kwd[Primary ]{91B28}
\kwd[; secondary ]{62H25}
\end{keyword}

\begin{keyword}
\kwd{minimum-variance portfolio}
\kwd{capitalization-weighted portfolio}
\kwd{equity return covariance structure}
\kwd{Law of Iterated Expectations}
\kwd{equity portfolio optimization.}
\end{keyword}

\tableofcontents



\end{frontmatter}

\section {Introduction}
Samuelson (1965) was the first to recognize the relevance of the Law of Iterated Expectations (LIE) in the area of finance.  His celebrated paper shows how this law can be applied in the context of the price of a single risky asset.  Essentially, the LIE reveals that forecasting error is not predictable.  An illuminating discussion of this law can be found, e.g., in Campbell, Lo and MacKinlay (1997).

This paper extends Samuelson's insight by developing an application of the LIE in the context of an arbitrary collection of equity returns.  Specifically, it will be shown that the LIE implies that the capitalization-weighted portfolio is mathematically required to coincide with the minimum-variance portfolio, provided both portfolios are defined with respect to the same (arbitrary) collection of equities having linearly independent returns. This result has important implications for equity return covariance structure, as summarized in technical appendix A.

The paper is organized as follows.  Section 2 introduces the notation and presents a sufficient condition for proving the main result.  The population view of the surprises (forecasting errors) is developed in Section 3.  This population view yields an important building block for the proof of the main result.  Section 4 applies the LIE in the context of a cross-section of equities, and exhibits the implication of this law from which the main result logically flows.  The proof of the main result is summarized in Section 5.  Section 6 concludes.  

%Technical appendix A reviews the definitions of market capitalization and return, and the operation of repricing shares, which is at the heart of the population view developed in Section 3.  
Technical appendix A summarizes the implications of this paper's main result for equity return covariance structure.  

\section {Notation and Preliminary Results} Consider an arbitrary collection of $n$ equities with linearly independent returns. Denote their returns by $\textbf{R} = (R_1 \dots R_n)'$ and denote the covariance matrix of $\bf{R}$ by $\Sigma$. Since the returns are assumed to be linearly independent, $\Sigma$ is nonsingular, so that its inverse $\Sigma^{-1}$ is well defined. Let $\bf{1}$ denote the $n$ x 1 vector all of whose elements are equal to 1.

Let $\textbf{M} = (M_1 \dots M_n)'$ denote the market capitalizations of the equities.  Let $w_i = M_i/{\bf{M'1}}$ and let $\textbf{w} = (w_1 \dots w_n)'$. The vector $\bf{w}$
corresponds to the \textit{capitalization weights}.  Note that all of the elements of $\bf{w}$ are positive. Let $\textbf{p} = (p_1 \dots p_n)'$ denote an $n$ x 1 vector of constants such that $\textbf{p}'\textbf{1}=1$. Then the elements of $\bf{p}$ correspond to the investment proportions of a portfolio fully invested in the $n$ equities, and
$\bf{p'R}$ denotes its return. If $\bf{p} = \bf{w}$, the investment proportions are those of the capitalization-weighted portfolio. The expression $\bf{w'R}$ denotes the return of the capitalization-weighted portfolio. 

Among all possible fully invested portfolios that can be formed from this collection of equities, there is one portfolio whose return has minimum variance.  This portfolio is the minimum-variance portfolio.  It is well known that, if $\Sigma$
is nonsingular, the investment proportions of the minimum-variance portfolio are given by $\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$ (see, e.g., Roll (1977), Campbell, Lo and MacKinlay (1997) or Grinold and Kahn (2000)).

The desired result is that
\begin{equation*}
\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}.
\end{equation*}
For the purpose of finding a proof of this result, it is convenient to define
\begin{equation*}
\beta_i = \textrm{Cov}(R_i, \bf{w'R})/{\textrm{Var}(\bf{w'R})}.
\end{equation*}
Let $\boldsymbol{\beta} = (\beta_1 \dots \beta_n)'$. Then
\begin{equation*}
\boldsymbol{\beta} = \Sigma\textbf{w}/{\textbf{w}'\Sigma\textbf{w}}. 
\end{equation*}
Note that $\textbf{w}'\textbf{1}=1$, so that
\begin{equation*}
\textbf{w}'\Sigma\textbf{w} = 1/{\boldsymbol{\beta}'\Sigma^{-1}\textbf{1}} 
\end{equation*}
and
\begin{equation*}
\textbf{w}=\Sigma^{-1}\boldsymbol{\beta}/{\boldsymbol{\beta}'\Sigma^{-1}\textbf{1}}.
\end{equation*}
With this notation it is possible to establish the following preliminary result.

\begin{lemma} \textit{The following assertions are equivalent}:\\
\indent (i) $\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$\\
\indent (ii) $\boldsymbol{\beta}=\bf{1}$
\end{lemma}

\noindent\textit{Proof. }
$\textbf{w}=\Sigma^{-1}\boldsymbol{\beta}/{\boldsymbol{\beta}' \Sigma^{-1}\textbf{1}}$  implies that $\Sigma\textbf{w}=\boldsymbol{\beta}/{\boldsymbol{\beta}'\Sigma^{-1}\textbf{1}}$,  and $\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$  implies that 
$\Sigma\textbf{w}=\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$. Therefore, $\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$ if and only if $\boldsymbol{\beta}=c\bf{1}$ for $c = \boldsymbol{\beta}'\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$. Since $\textbf{w}'\boldsymbol{\beta}=\textbf{w}'\textbf{1}=1$, this constant $c$ is equal to 1.  

\medskip
Next, consider an investment period which begins at time $t = 0$ and ends at time $t = T$. Note that at time $t = 0$, $\bf{w}$ is a vector of known constants while $\bf{R}$ is a vector of random variables. Let E[$\bf{R}$] denote the $n$ x 1 vector whose elements correspond to E[$R_i$], where E[$R_i$] is the expected value of $R_i$ at time $t = 0$ ($i=1,\dots,n$). The expression E[$R_i$] corresponds to the best forecast at time $t = 0$ of the realized value of $R_i$ observed at time $t = T$.

Now shift attention from the returns $R_i$ to the surprises $S_i$, where $S_i=R_i-E[R_i]$. The surprises correspond to \textit{forecasting errors}. Let
$\textbf{S}=(S_1,\dots,S_n)'$, let $\mathcal{A}$ denote the $\sigma$-field induced by $\bf{S}$, and let $\mathcal{F}$ denote the $\sigma$-field induced by $\bf{w'S}$. Since $\bf{w'S}$ is a function of $\bf{S}$, $\mathcal{F} \subset \mathcal{A}$. The conditional expectation of $S_i$ given $\bf{w'S}$ can then be written as E$^{\mathcal{F}}[S_i]$, which is defined almost surely (a.s.) in the sense that any two versions agree, except possibly on a null set in $\mathcal{A}$. With this notation, it is possible to simplify the task of proving the desired result as follows. 

\begin{lemma} \textit{For the purpose of showing that } $\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$, \textit{it is sufficient to show that } E$^{\mathcal{F}}[S_i] = \textbf{w}'\textbf{S}$ ($a.s.$) ($i=1,\dots,n$). 
\end{lemma}


\noindent\textit{Proof. }In light of Lemma 2.1, it is sufficient to show that E$^{\mathcal{F}}[S_i] = \textbf{w}'\textbf{S}$ ($a.s.$) ($i=1,\dots,n$) implies that $\boldsymbol{\beta}=\textbf{1}$. 

By definition, 
\begin{equation*}
\textrm{Cov}(R_i, \textbf{w}'\textbf{R})= \textrm{E}[S_i\textbf{w}'\textbf{S}]
\end{equation*}
and
\begin{equation*}
\textrm{Var}(\textbf{w}'\textbf{R})= \textrm{E}\left[(\textbf{w}'\textbf{S})^2\right].
\end{equation*}
Therefore it suffices to show that E$^{\mathcal{F}}[S_i] = \bf{w'S}$ ($a.s.$) ($i=1,\dots,n$) implies that
\begin{equation*}
\textrm{E}[S_i\textbf{w}'\textbf{S}]= \textrm{E}\left[(\textbf{w}'\textbf{S})^2\right]\textrm{ } (i=1,\dots,n).
\end{equation*}
This is easily accomplished by recalling the usual properties of conditional expectation operators, which yields 
\begin{equation*}
\textrm{E}[S_i\textbf{w}'\textbf{S}]= \textrm{E}\left[\textbf{w}'\textbf{S}\textrm{E}^{\mathcal{F}}[S_i]\right],
\end{equation*}
so that E$^{\mathcal{F}}[S_i] = \bf{w'S}$ (a.s.) implies $\textrm{E}[S_i\textbf{w}'\textbf{S}]= \textrm{E}\left[(\textbf{w}'\textbf{S})^2\right]$ ($i=1,\dots,n$). 

\medskip
Before it can be shown that the assertion E$^{\mathcal{F}}[S_i] = \bf{w'S}$ (a.s.) ($i=1,\dots,n$) flows logically from the LIE, it is first necessary to develop the population view of the surprises (forecasting errors).  This is done in the next section.

\section{The Population View} Consider an arbitrary collection of $n$ publicly traded companies and an investment period which begins at time $t = 0$ and ends at time $t = T$. The development of the population view of the surprises depends on the operation of repricing the shares of each company at the beginning of the investment period, while adjusting the shares outstanding so as to leave the market capitalization of each company unchanged.  This operation, very familiar to equity investors, guarantees that there is no loss of generality in repricing the shares of each company at the beginning of the investment period to have a value of one dollar. 

As in Section 2, let $M_i$ denote the market capitalizations at the beginning of the investment period ($i = 1,\dots,n$). If the price of one share is one dollar (and if the market capitalizations are rounded to the nearest dollar), then $M_i$ corresponds to the number $N_i$ of shares outstanding for the $i$-th company.  Each share of the $i$-th company has return $R_i$ and associated surprise (forecasting error) $S_i$. Therefore, at the end of the investment period, the observed values $s_i$ of the random variables $S_i$ form a population in which $s_i$ occurs with frequency $N_i$. The capitalization weights $w_i$ then correspond to the relative frequencies with which the $s_i$ are observed.  This population has mean $\bf{w's}$, where $\textbf{s}=(s_1,\dots,s_n)'$ is the observed value of the random vector $\textbf{S}=(S_1,\dots,S_n)'$.

It is well known that the population mean is the expected value of a randomly selected element from that population.  From the standpoint of time $t = T$, the population mean is observed with certainty, while from the standpoint of time $t = 0$, the population mean corresponds to the random variable $\bf{w'S}$. This means that     
\begin{equation*}
\textrm{E}_T[\textrm{a randomly selected surprise}]= \textbf{w}'\textbf{s},
\end{equation*}
where ``E$_T$'' is to be read as ``the expected value at time $t = T$, conditional on the observed value $\bf{w's}$ of $\bf{w'S}$.'' Translating this into $\sigma$-field notation yields the following lemma.
\begin{lemma} E$^{\mathcal{F}}$[a randomly selected surprise]= $\textbf{w}'\textbf{S}$ \textit{$(a.s.)$, where $\mathcal{F}$ denotes the $\sigma$-field induced by $\bf{w'S}$.}
\end{lemma}
As will be seen in Section 5, Lemma 3.1 is an important building block for the proof of the main result.

\section{The Law of Iterated Expectations} 
A key insight provided by the LIE is that forecasting error is not predictable.  The present paper contemplates an arbitrary collection of $n$ equities with returns $R_i$ and associated forecasting errors $S_i = R_i -$E[$R_i$] ($i=1,\dots,n$). In this context, the LIE implies that the expected value of any surprise $S_i$ corresponds to the expected value of a \textit{randomly selected surprise}. From the standpoint of time $t = 0$, this yields the set of equations 
\begin{equation*}
\textrm{E}[S_i] = \textrm{E}[\textrm{a randomly selected surprise}]\textrm{ } (i=1,\dots,n).
\end{equation*}
Conditional on $\bf{w'S}$, the LIE similarly implies the following lemma.

\begin{lemma} E$^{\mathcal{F}}[S_i]$=E$^{\mathcal{F}}$[a randomly selected surprise] \textit{$(a.s.)$ $(i=1,\dots,n)$, where $\mathcal{F}$ denotes the $\sigma$-field induced by $\bf{w'S}$.}
\end{lemma}

Lemma 4.1 exhibits the implication of the LIE from which the main result of the present paper logically flows, as summarized in the next section.

\section{Proof of the Main Result} The main result is stated as the proposition below.\\

\noindent\textsc{PROPOSITION: } \textit{For an arbitrary collection of n equities having linearly independent returns}, $\textbf{w}=\Sigma^{-1}\textbf{1}/{\textbf{1}'\Sigma^{-1}\textbf{1}}$.

\medskip
\noindent\textit{Proof. }From Lemma 4.1,
\begin{equation*}
\textrm{E}^{\mathcal{F}}[S_i]=\textrm{E}^{\mathcal{F}}\textrm{[a randomly selected surprise] (a.s.) } (i=1,\dots,n).
\end{equation*}
From Lemma 3.1,
\begin{equation*}
\textrm{E}^{\mathcal{F}}\textrm{[a randomly selected surprise]} =\textbf{w}'\textbf{S} \textrm{ (a.s.)}. 
\end{equation*}
Combining Lemmas 4.1 and 3.1 yields
\begin{equation*}
\textrm{E}^{\mathcal{F}}[S_i]= \textbf{w}'\textbf{S}\textrm{ (a.s.) } (i=1,\dots,n). 
\end{equation*}
In light of Lemma 2.2, this completes the proof.  

\section{Conclusion} It has been shown that the capitalization-weighted portfolio is mathematically required to coincide with the minimum-variance portfolio, provided both portfolios are defined with respect to the same (arbitrary) collection of equities having linearly independent returns.  This result is a logical consequence of the LIE, and has important implications for equity return covariance structure, as summarized in technical appendix A. 

%\section*{TECHNICAL APPENDIX A} Equity investors are very familiar with the concepts of market capitalization and return.  The definitions of these terms are provided here for readers who may not have encountered them before.  This Technical Appendix also discusses the operation of repricing shares, likewise very familiar to equity investors.  It will be seen that, for the purpose of proving the main result of the present paper, there is no loss of generality in repricing the shares of each company to have a value of one dollar at the beginning of the investment period.  This operation is at the heart of the population view developed in Section 3. 

%Consider any publicly traded company and an investment period which begins at time 
%$t = 0$ and ends at time $t = T$.  Let $N$ denote the number of shares outstanding at the beginning of the investment period and let $P$ denote the price of one share at the beginning of the investment period.  Then the \textit{market capitalization} $M$ is defined as
%\begin{equation*}
%M = NP, 
%\end{equation*}
%and corresponds to the company's total value at time $t = 0$.  The shares of a publicly traded company can also be referred to as $stocks$ or $equities$.

%It will be assumed that $N$ does not change during the investment period, and that companies do not merge or break up.  This assumption is customary in the Finance literature (see, $e.g.$, Fernholz (2002)); its purpose is to keep the presentation simple. Let $P^{(T)}$ denote the price of one share at time $t = T$.  Then 
%$M^{(T)}=NP^{(T)}$ is the market capitalization at time $t = T$. The percentage change in the market capitalization during the investment period can then be written as
%\begin{equation*}
%\%\Delta M = \frac{M^{(T)}-M}{M}
%\end{equation*} 
%Dividing numerator and denominator of $\%\Delta M$ by $N$ reveals that
%\begin{equation*}
%\%\Delta M = \frac{P^{(T)}-P}{P} = \%\Delta P,
%\end{equation*} 
%where $\%\Delta P$ denotes the percentage change in the price of one share.  (Here $\%\Delta M$ and $\%\Delta P$ are expressed as x.xx, not xxx $\%$.)

 %$Return$ is the sum of two components, namely $\%\Delta P$ and $dividend$ $yield$. Let $D$ denote the total amount of dividends paid during the investment period; then 
%\begin{equation*}
%\textrm{dividend yield} = \frac{D}{M} = \frac{\textrm{dividends paid per share}}{P}
%\end{equation*} 
%and
%\begin{equation*}
%\textrm{return} = \%\Delta P + \textrm{dividend yield}.
%\end{equation*} 
%Equity investors are very familiar with the operation of repricing shares.  For example, when a stock splits two for one, the price of one share is halved, the dividends payable per share are halved, and the number of shares outstanding is doubled.  This operation leaves $\%\Delta P$, dividend yield and return unchanged.

%The main result of the present paper is articulated in terms of the capitalization weights $\bf{w}$ and the equity return covariance matrix $\Sigma$. A moment's reflection reveals that neither $\bf{w}$ nor $\Sigma$ is changed when the shares of each company have been repriced at the beginning of the investment period, provided the number of shares outstanding has been adjusted so as to leave market capitalization of each company unchanged.  Thus it can be seen that, for the purpose of proving the main result of the present paper, there is no loss of generality in repricing the shares of each company at the beginning of the investment period to have a value of one dollar.  This operation is at the heart of the population view of the surprises developed in Section 3. 

\section*{Technical Appendix A}
The main result of this paper has important implications for equity return covariance structure, as summarized in the following proposition.   \\

\noindent \textsc{proposition:} \textit{For any collection of n equities with linearly independent returns, the covariance matrix $\Sigma$ of $\textbf{R}=(R_1,\dots,R_n)'$ is of the form}
\begin{equation*}
\Sigma = \textbf{11}'k+\textbf{U}^2
\end{equation*} 
\textit{where $\bf{U}^2$ is a diagonal matrix such that the i-th diagonal element is positive and inversely proportional to $M_i$ ($i=1,\dots,n$), and where $k$ corresponds to a constant which can be positive, negative or zero.}

Note that for positive values of $k$, general equity return covariance structure exhibited in the proposition above corresponds to the covariance matrix of a 1-factor model in which
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item the factor loadings of the unique common factor are all equal to one; 
     and
\item the variances of the specific factors are inversely proportional to the 
     market capitalizations of the equities.
\end{enumerate}

\bibliographystyle{plain}
\begin{thebibliography}{0}
\bibitem{Campbell(1997)}
\textsc{Campbell, J. Y., Lo, A. W.} and \textsc{MacKinlay, A. C.} (1997). \textit{The Econometrics of Financial Markets.} Princeton University Press, Princeton, New Jersey.

%\bibitem{Fernholz(2002)}
%\textsc{Fernholz, E. R.} (2002). Stochastic Portfolio Theory. Springer Science + Business Media, Inc.

\bibitem{Grinold(2000)}
\textsc{Grinold, R. C.} and \textsc{Kahn, R. N.} (2000). \textit{Active Portfolio Management.} McGraw-Hill, New York.

\bibitem{Roll(1977)}
\textsc{Roll, R.} (1977). A critique of the asset pricing theory's tests. \textit{Journal of Financial Economics} \textbf{4}, 129--176.

\bibitem{Samuelson(1965)}
\textsc{Samuelson, P.} (1965). Proof that properly anticipated prices fluctuate randomly. \textit{Industrial Management Review} \textbf{6}, 41--49.

\end{thebibliography}






\end{document}