17.2 Efficient Portfolio

A variance efficient portfolio is one that keeps the risk (17.2) minimal under the constraint that the weights sum to $1$, i.e., $c^{\top}\undertilde{1}_{p}=1$. For a variance efficient portfolio, we therefore try to find the value of $c$ that minimizes the Lagrangian


\begin{displaymath}
{\cal L}=\frac{1}{2}\;c^{\top}\Sigma c-\lambda (c^{\top}\undertilde{1}_{p}-1). %% &\cr
\end{displaymath} (17.3)

A mean-variance efficient portfolio is defined as one that has minimal variance among all portfolios with the same mean. More formally, we have to find a vector of weights $c$ such that the variance of the portfolio is minimal subject to two constraints:

  1. a certain, pre-specified mean return $\overline{\mu}$ has to be achieved,
  2. the weights have to sum to one.
Mathematically speaking, we are dealing with an optimization problem under two constraints.

The Lagrangian function for this problem is given by

\begin{displaymath}{\cal L} =c^{\top}\Sigma c+\lambda_1(\overline{\mu}-c^{\top}\mu )+
\lambda_2(1-c^{\top}1_p).\end{displaymath}

With tools presented in Section 2.4 we can calculate the first order condition for a minimum:
\begin{displaymath}
\frac{\partial {\cal L}}{\partial c}=2\Sigma c-\lambda_1\mu
-\lambda_21_p=0.
\end{displaymath} (17.4)

EXAMPLE 17.1   Figure 17.1 shows the returns from January 1978 to December 1987 of six stocks traded on the New York stock exchange (Berndt; 1990).

Figure 17.1: Returns of six firms from January 1978 to December 1987. 49601 MVAreturns.xpl
\includegraphics[width=1\defpicwidth]{returns.ps}

For each stock we have chosen the same scale on the vertical axis (which gives the return of the stock). Note how the return of some stocks, such as Pan American Airways and Delta Airlines, are much more volatile than the returns of other stocks, such as IBM or Consolidated Edison (Electric utilities).

As a very simple example consider two differently weighted portfolios containing only two assets, IBM and PanAm.

Figure 17.2: Portfolio of IBM and PanAm assets, equal and efficient weights. 49608 MVAportfol.xpl
\includegraphics[width=1\defpicwidth]{portfip.ps}

Figure 17.2 displays the monthly returns of the two portfolios. The portfolio in the upper panel consists of approximately 10% PanAm assets and 90% IBM assets. The portfolio in the lower panel contains an equal proportion of each of the assets. The text windows on the right of Figure 17.2 show the exact weights which were used. We can clearly see that the returns of the portfolio with a higher share of the IBM assets (which have a low variance) are much less volatile.

For an exact analysis of the optimization problem (17.4) we distinguish between two cases: the existence and nonexistence of a riskless asset. A riskless asset is an asset such as a zero bond, i.e., a financial instrument with a fixed nonrandom return (Franke et al.; 2001).


Nonexistence of a riskless asset

Assume that the covariance matrix $\Sigma$ is invertible (which implies positive definiteness). This is equivalent to the nonexistence of a portfolio $c$ with variance $c^{\top}\Sigma c = 0$. If all assets are uncorrelated, $\Sigma$ is invertible if all of the asset returns have positive variances. A riskless asset (uncorrelated with all other assets) would have zero variance since it has fixed, nonrandom returns. In this case $\Sigma$ would not be positive definite.

The optimal weights can be derived from the first order condition (17.4) as

\begin{displaymath}
c=\frac{1}{2}\Sigma^{-1}(\lambda_1\mu + \lambda_21_p).
\end{displaymath} (17.5)

Multiplying this by a $(p \times 1)$ vector $1_p$ of ones, we obtain

\begin{displaymath}1=1_p^{\top}c=\frac{1}{2}1_p^{\top}\Sigma^{-1}(\lambda_1\mu + \lambda_21_p^{\top}),\end{displaymath}

which can be solved for $\lambda_2$ to get:

\begin{displaymath}\lambda_2=\frac{2-\lambda_11_p^{\top}\Sigma^{-1}\mu}{1_p^{\top}\Sigma^{-1}1_p}.\end{displaymath}

Plugging this expression into (17.5) yields
\begin{displaymath}
c=\frac{1}{2}\lambda_1 \left( \Sigma^{-1}\mu-
\frac{1_p^{\to...
...1_p \right) +
\frac{\Sigma^{-1}1_p}{1_p^{\top}\Sigma^{-1}1_p}.
\end{displaymath} (17.6)

For the case of a variance efficient portfolio there is no restriction on the mean of the portfolio $(\lambda_1=0)$. The optimal weights are therefore:

\begin{displaymath}
c=\frac{\Sigma^{-1}1_p}{1_p^{\top}\Sigma^{-1}1_p}.
\end{displaymath} (17.7)

This formula is identical to the solution of (17.3). Indeed, differentiation with respect to $c$ gives

\begin{eqnarray*}
\Sigma c&=& \lambda\undertilde{1}_{p}\\ [3mm]
c&=& \lambda\Sigma^{-1}\undertilde{1}_{p}.
\end{eqnarray*}



If we plug this into (17.3), we obtain

\begin{eqnarray*}
{\cal L}
&=& \frac{1}{2}\lambda^2\undertilde{1}_{p}
\Sigma^{...
...1}{2}\lambda^2\undertilde{1}_{p}\Sigma^{-1}
\undertilde{1}_{p}.
\end{eqnarray*}



This quantity is a function of $\lambda $ and is minimal for

\begin{displaymath}\lambda =(\undertilde{1}_{p}\Sigma^{-1}\undertilde{1}_{p})^{-1}\end{displaymath}

since

\begin{displaymath}\frac{\partial ^2 {\cal L} }{\partial c^{\top}\partial c }=\Sigma >0.\end{displaymath}

THEOREM 17.1   The variance efficient portfolio weights for returns $X\sim(\mu,\Sigma)$ are
\begin{displaymath}
c_{opt} = \frac{\Sigma^{-1}\undertilde{1}_{p}}{\undertilde{1}_{p}^{\top}
\Sigma^{-1}\undertilde{1}_{p}}.
\end{displaymath} (17.8)


Existence of a riskless asset

If an asset exists with variance equal to zero, then the covariance matrix $\Sigma$ is not invertible. The notation can be adjusted for this case as follows: denote the return of the riskless asset by $r$ (under the absence of arbitrage this is the interest rate), and partition the vector and the covariance matrix of returns such that the last component is the riskless asset. Thus, the last equation of the system (17.4) becomes

\begin{displaymath}2 \Cov(r,X)-\lambda_1r-\lambda_2=0,\end{displaymath}

and, because the covariance of the riskless asset with any portfolio is zero, we have
\begin{displaymath}
\lambda_2=-r\lambda_1.
\end{displaymath} (17.9)

Let us for a moment modify the notation in such a way that in each vector and matrix the components corresponding to the riskless asset are excluded. For example, $c$ is the weight vector of the risky assets (i.e., assets with positive variance), and $c_0$ denotes the proportion invested in the riskless asset. Obviously, $c_0=1-1_p^{\top}c$, and $\Sigma$ the covariance matrix of the risky assets, is assumed to be invertible. Solving (17.4) using (17.9) gives
\begin{displaymath}
c=\frac{\lambda_1}{2}\Sigma^{-1}(\mu -r1_p).
\end{displaymath} (17.10)

This equation may be solved for $\lambda_1$ by plugging it into the condition $\mu^{\top} c = \overline\mu$. This is the mean-variance efficient weight vector of the risky assets if a riskless asset exists. The final solution is:

\begin{displaymath}
c=\frac{\overline\mu \Sigma^{-1}(\mu -r1_p)}{\mu^{\top} \Sigma^{-1}(\mu -r1_p)}.
\end{displaymath} (17.11)

The variance optimal weighting of the assets in the portfolio depends on the structure of the covariance matrix as the following corollaries show.

COROLLARY 17.1   A portfolio of uncorrelated assets whose returns have equal variances ( $\Sigma =\sigma ^2 {\data{I}}_p$) needs to be weighted equally:

\begin{displaymath}c_{opt} = \frac{1}{p} 1_p.\end{displaymath}

PROOF:
Here we obtain $\undertilde{1}_{p}^{\top}\Sigma^{-1}\undertilde{1}_{p}
= \sigma ^{-2} \undertilde{1}_{p}^{\top}\undertilde{1}_{p} = \sigma
^{-2}p$ and therefore $c=\frac{\sigma ^{-2}\undertilde{1}_{p}}{\sigma ^{-2} p}
=\frac{1}{p}\undertilde{1}_{p}.$ ${\Box}$

COROLLARY 17.2   A portfolio of correlated assets whose returns have equal variances, i.e.,

\begin{displaymath}\Sigma =\sigma ^2\left (\begin{array}{cccc}
1&\rho& \cdots &...
...ots & 1\end{array}
\right ),\qquad -\;\frac{1 }{p-1 }<\rho <1\end{displaymath}

needs to be weighted equally:

\begin{displaymath}c_{opt} = \frac{1}{p} 1_p.\end{displaymath}

PROOF:
$\Sigma$ can be rewritten as $\Sigma =\sigma ^2\left\{(1-\rho ) {\data{I}}_p+\rho
\undertilde{1}_{p} \undertilde{1}_{p}^{\top}\right\}.$ The inverse is

\begin{displaymath}\Sigma ^{-1}=\frac{\data{I}_p }{\sigma ^2(1-\rho ) }-
\frac{...
...ndertilde{1}_{p}^{\top} }
{\sigma ^2(1-\rho )\{1+(p-1)\rho \}}\end{displaymath}

since for a $(p \times p)$ matrix $\data{A}$ of the form $\data{A} = (a-b)\data{I}_p+b\undertilde{1}_{p}\undertilde{1}_{p}^{\top}$ the inverse is generally given by

\begin{displaymath}\data{A}^{-1}=\frac{\data{I}_p}{\displaystyle{(a-b) }}
-\fra...
...de{1}_{p}^{\top} }
{{\displaystyle (a-b)\{a+(p-1)b\} } }\cdotp\end{displaymath}

Hence

\begin{eqnarray*}
\Sigma ^{-1} \undertilde{1}_{p}
&=&\frac{\undertilde{1}_{p} }{...
...c{\undertilde{1}_{p}}{{\displaystyle \sigma ^2\{1+(p-1)\rho\}}}
\end{eqnarray*}



which yields

\begin{displaymath}\undertilde{1}_{p}^{\top}\Sigma ^{-1}\undertilde{1}_{p}^{\top...
...rac{\displaystyle p}{{\displaystyle \sigma ^2\{1+(p-1)\rho\} }}\end{displaymath}

and thus $c={\frac{1}{p}}\undertilde{1}_{p}.$ ${\Box}$

Let us now consider assets with different variances. We will see that in this case the weights are adjusted to the risk.

COROLLARY 17.3   A portfolio of uncorrelated assets with returns of different variances, i.e., $\Sigma=\mathop{\hbox{diag}}(\sigma_{1}^2,\ldots,\sigma_{p}^2)$, has the following optimal weights

\begin{displaymath}c_{j,opt}=\frac{\sigma ^{-2}_j} {\sum\limits ^p_{l=1}
\sigma ^{-2}_l} , \quad j=1,\ldots,p.\end{displaymath}

PROOF:
From $\Sigma^{-1} =\mathop{\hbox{diag}}(\sigma ^{-2}_1,\ldots ,\sigma ^{-2}_p)$ we have $\undertilde{1}_{p}^{\top}\Sigma^{-1}\undertilde{1}_{p}^{\top}=\sum ^p_{l=1}
\sigma ^{-2}_l$ and therefore the optimal weights are $c_j={\sigma ^{-2}_j}/{\sum\limits ^p_{l=1}\sigma ^{-2}_l}$. ${\Box}$

This result can be generalized for covariance matrices with block structures.

COROLLARY 17.4   A portfolio of assets with returns $X\sim(\mu,\Sigma)$, where the covariance matrix has the form:

\begin{displaymath}\Sigma =\left (\begin{array}{cccc}
\Sigma _1& 0 & \ldots & ...
... & \vdots\\
0 & \ldots & 0 & \Sigma_{r}\end{array}
\right )\end{displaymath}

has optimal weights $c=\left(c_{1},\ldots,c_{r}\right)^{\top}$ given by

\begin{displaymath}c_{j,opt}=\frac{\Sigma ^{-1}_j 1_{}} {1^{T} \Sigma^{-1}_j 1_{}},
\quad j=1,\ldots,r.\end{displaymath}

Summary
$\ast$
An efficient portfolio is one that keeps the risk minimal under the constraint that a given mean return is achieved and that the weights sum to $1$, i.e., that minimizes ${\cal L} =c^{\top}\Sigma c+\lambda_1(\overline{\mu}-c^{\top}\mu )+
\lambda_2(1-c^{\top}1_p).$
$\ast$
If a riskless asset does not exist, the variance efficient portfolio weights are given by

\begin{displaymath}c = \frac{\Sigma^{-1}\undertilde{1}_{p}}{\undertilde{1}_{p}^{\top}
\Sigma^{-1}\undertilde{1}_{p}}.\end{displaymath}

$\ast$
If a riskless asset exists, the mean-variance efficient portfolio weights are given by

\begin{displaymath}
c=\frac{\overline\mu \Sigma^{-1}(\mu -r1_p)}{\mu^{\top} \Sigma^{-1}(\mu -r1_p)}.
\end{displaymath}

$\ast$
The efficient weighting depends on the structure of the covariance matrix $\Sigma$. Equal variances of the assets in the portfolio lead to equal weights, different variances lead to weightings proportional to these variances:

\begin{displaymath}c_{j,opt}=\frac{\sigma ^{-2}_j} {\sum\limits ^p_{l=1}
\sigma ^{-2}_l} , \quad j=1,\ldots,p.\end{displaymath}