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.
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.
To see two-fund separation in a context in which no risk-free asset is available, using matrix algebra, let
be the amount of wealth that is to be allocated in the portfolio, and let
Then the problem of minimizing the portfolio return variance subject to a given level of expected portfolio return can be stated as where the superscript
The portfolio return variance in the objective function can be written as
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 with Lagrange multipliers
of asset quantities by equating to zero the derivatives with respect to
in terms of the model parameters, and substituting back into the provisional solution for
as follows: This equation proves the two-fund separation theorem for mean-variance analysis.
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.
, would have one row and one column of zeroes and thus would not be invertible.
is now the vector of quantities to be held in the risky assets, and
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
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.
This optimization problem can be shown to yield the optimal vector of risky asset holdings Of course this equals a zero vector if
It can be shown that the portfolio with exactly zero holdings of the risk-free asset occurs at
and is given by 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,
For a geometric interpretation, see the efficient frontier with no risk-free asset.
If investors have hyperbolic absolute risk aversion (HARA) (including the power utility function, logarithmic function and the exponential utility function), separation theorems can be obtained without the use of mean-variance analysis.
For example, David Cass and Joseph Stiglitz[4] showed in 1970 that two-fund monetary separation applies if all investors have HARA utility with the same exponent as each other.
[5]: ch.4 More recently, in the dynamic portfolio optimization model of Çanakoğlu and Özekici,[6] the investor's level of initial wealth (the distinguishing feature of investors) does not affect the optimal composition of the risky part of the portfolio.