In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually. Second, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested.

Portfolio separation in mean-variance analysis

Portfolios can be analyzed in a mean-variance framework, with every investor holding the portfolio with the lowest possible return variance consistent with that investor's chosen level of expected return (called a minimum-variance portfolio), if the returns on the assets are jointly elliptically distributed, including the special case in which they are jointly normally distributed.^[1][2] Under mean-variance analysis, it can be shown^[3] that every minimum-variance portfolio given a particular expected return (that is, every efficient portfolio) can be formed as a combination of any two efficient portfolios. If the investor's optimal portfolio has an expected return that is between the expected returns on two efficient benchmark portfolios, then that investor's portfolio can be characterized as consisting of positive quantities of the two benchmark portfolios.

No risk-free asset

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

${\displaystyle X^{\mathrm {opt} }={\frac {W}{\Delta }}+{\frac {\mu }{\Delta }}}$

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:

${\displaystyle X_3^{\mathrm{o}\mathrm{p}\mathrm{t}}=\mathrm{\alpha <!--\; \alpha \; -->}W+\mathrm{\beta <!--\; \beta \; -->}{\mathrm{\mu <!--\; \mu \; -->}}_3=\frac{{\mathrm{\mu <!--\; \mu \; -->}}_3-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_2}{{\mathrm{\mu <!--\; \mu \; -->}}_1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_2}{X}_1^{\mathrm{o}\mathrm{p}\mathrm{t}}+\frac{{\mathrm{\mu <!--\; \mu \; -->}}_1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_3}{{\mathrm{\mu <!--\; \mu \; -->}}_1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_2}}$Zdroj:https://en.wikipedia.org?pojem=Mutual_fund_separation_theorem
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