Mutual fund separation theorem

To see two-fund separation in a context in which no risk-free asset is available, using matrix algebra, let ${\displaystyle \sigma ^{2}}$ be the variance of the portfolio return, let ${\displaystyle \mu }$ be the level of expected return on the portfolio that portfolio return variance is to be minimized contingent upon, let ${\displaystyle r}$ be the vector of expected returns on the available assets, let ${\displaystyle X}$ be the vector of amounts to be placed in the available assets, let ${\displaystyle W}$ be the amount of wealth that is to be allocated in the portfolio, and let ${\displaystyle 1}$ be a vector of ones. Then the problem of minimizing the portfolio return variance subject to a given level of expected portfolio return can be stated as

Minimize ${\displaystyle \sigma ^{2}}$

subject to

${\displaystyle X^{T}r=\mu }$

and

${\displaystyle X^{T}1=W}$

where the superscript ${\displaystyle ^{T}}$ denotes the transpose of a matrix. The portfolio return variance in the objective function can be written as ${\displaystyle \sigma ^{2}=X^{T}VX,}$ where ${\displaystyle V}$ is the positive definite covariance matrix of the individual assets' returns. The Lagrangian for this constrained optimization problem (whose second-order conditions can be shown to be satisfied) is

${\displaystyle L=X^{T}VX+2\lambda (\mu -X^{T}r)+2\eta (W-X^{T}1),}$

with Lagrange multipliers ${\displaystyle \lambda }$ and ${\displaystyle \eta }$. This can be solved for the optimal vector ${\displaystyle X}$ of asset quantities by equating to zero the derivatives with respect to ${\displaystyle X}$, ${\displaystyle \lambda }$, and ${\displaystyle \eta }$, provisionally solving the first-order condition for ${\displaystyle X}$ in terms of ${\displaystyle \lambda }$ and ${\displaystyle \eta }$, substituting into the other first-order conditions, solving for ${\displaystyle \lambda }$ and ${\displaystyle \eta }$ in terms of the model parameters, and substituting back into the provisional solution for ${\displaystyle X}$. The result is

${\displaystyle X^{\mathrm {opt} }={\frac {W}{\Delta }}[(r^{T}V^{-1}r)V^{-1}1-(1^{T}V^{-1}r)V^{-1}r]+{\frac {\mu }{\Delta }}[(1^{T}V^{-1}1)V^{-1}r-(r^{T}V^{-1}1)V^{-1}1]}$

where

${\displaystyle \Delta =(r^{T}V^{-1}r)(1^{T}V^{-1}1)-(r^{T}V^{-1}1)^{2}>0.}$

For simplicity this can be written more compactly as

${\displaystyle X^{\mathrm {opt} }=\alpha W+\beta \mu }$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are parameter vectors based on the underlying model parameters. Now consider two benchmark efficient portfolios constructed at benchmark expected returns ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{2}}$ and thus given by

${\displaystyle X_{1}^{\mathrm {opt} }=\alpha W+\beta \mu _{1}}$

and

${\displaystyle X_{2}^{\mathrm {opt} }=\alpha W+\beta \mu _{2}.}$

The optimal portfolio at arbitrary ${\displaystyle \mu _{3}}$ can then be written as a weighted average of ${\displaystyle X_{1}^{\mathrm {opt} }}$ and ${\displaystyle X_{2}^{\mathrm {opt} }}$ as follows:

${\displaystyle X_{3}^{\mathrm {opt} }=\alpha W+\beta \mu _{3}={\frac {\mu _{3}-\mu _{2}}{\mu _{1}-\mu _{2}}}X_{1}^{\mathrm {opt} }+{\frac {\mu _{1}-\mu _{3}}{\mu _{1}-\mu _{2}}}X_{2}^{\mathrm {opt} }.}$

This equation proves the two-fund separation theorem for mean-variance analysis. For a geometric interpretation, see the Markowitz bullet.

If a risk-free asset is available, then again a two-fund separation theorem applies; but in this case one of the "funds" can be chosen to be a very simple fund containing only the risk-free asset, and the other fund can be chosen to be one which contains zero holdings of the risk-free asset. (With the risk-free asset referred to as "money", this form of the theorem is referred to as the monetary separation theorem.) Thus mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and holdings of a particular efficient fund that contains only risky assets. The derivation above does not apply, however, since with a risk-free asset the above covariance matrix of all asset returns, ${\displaystyle V}$, would have one row and one column of zeroes and thus would not be invertible. Instead, the problem can be set up as

Minimize ${\displaystyle \sigma ^{2}}$

subject to

${\displaystyle (W-X^{T}1)r_{f}+X^{T}r=\mu ,}$

where ${\displaystyle r_{f}}$ is the known return on the risk-free asset, ${\displaystyle X}$ is now the vector of quantities to be held in the risky assets, and ${\displaystyle r}$ is the vector of expected returns on the risky assets. The left side of the last equation is the expected return on the portfolio, since ${\displaystyle (W-X^{T}1)}$ is the quantity held in the risk-free asset, thus incorporating the asset adding-up constraint that in the earlier problem required the inclusion of a separate Lagrangian constraint. The objective function can be written as ${\displaystyle \sigma ^{2}=X^{T}VX}$, where now ${\displaystyle V}$ is the covariance matrix of the risky assets only. This optimization problem can be shown to yield the optimal vector of risky asset holdings

${\displaystyle X^{\mathrm {opt} }={\frac {(\mu -Wr_{f})}{(r-1r_{f})^{T}V^{-1}(r-1r_{f})}}V^{-1}(r-1r_{f}).}$

Of course this equals a zero vector if ${\displaystyle \mu =Wr_{f}}$, the risk-free portfolio's return, in which case all wealth is held in the risk-free asset. It can be shown that the portfolio with exactly zero holdings of the risk-free asset occurs at ${\displaystyle \mu ={\tfrac {Wr^{T}V^{-1}(r-1r_{f})}{1^{T}V^{-1}(r-1r_{f})}}}$ and is given by

${\displaystyle X^{*}={\frac {W}{1^{T}V^{-1}(r-1r_{f})}}V^{-1}(r-1r_{f}).}$

It can also be shown (analogously to the demonstration in the above two-mutual-fund case) that every portfolio's risky asset vector (that is, ${\displaystyle X^{\mathrm {opt} }}$ for every value of ${\displaystyle \mu }$) can be formed as a weighted combination of the latter vector and the zero vector. For a geometric interpretation, see the efficient frontier with no risk-free asset.