 Mutual fund separation theorem

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.
Contents
Portfolio separation in meanvariance analysis
Portfolios can be analyzed in a meanvariance 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 meanvariance efficient 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 meanvariance analysis, it can be shown^{[3]} that every varianceminimizing 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 riskfree asset
To see twofund separation in a context in which no riskfree asset is available, using matrix algebra, let σ^{2} be the variance of the portfolio return, let μ be the level of expected return on the portfolio that portfolio return variance is to be minimized contingent upon, let r be the vector of expected returns on the available assets, let X be the vector of amounts to be placed in the available assets, let W be the amount of wealth that is to be allocated in the portfolio, and let 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 σ^{2}
 subject to
 X^{T}r = μ
 and
 X^{T}1 = W
where the superscript ^{T} denotes the transpose of a matrix. The portfolio return variance in the objective function can be written as σ^{2} = X^{T}VX, where V is the positive definite covariance matrix of the individual assets' returns. The Lagrangian for this constrained optimization problem (whose secondorder conditions can be shown to be satisfied) is
 L = X^{T}VX + 2λ(μ − X^{T}r) + 2η(W − X^{T}1),
with Lagrange multipliers λ and η.This can be solved for the optimal vector X of asset quantities by equating to zero the derivatives with respect to X, λ, and η, provisionally solving the firstorder condition for X in terms of λ and η, substituting into the other firstorder conditions, solving for λ and η in terms of the model parameters, and substituting back into the provisional solution for X. The result is
where

 Δ = (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
 X^{opt} = αW + βμ
where α and β are parameter vectors based on the underlying model parameters. Now consider two benchmark efficient portfolios constructed at benchmark expected returns μ_{1} and μ_{2} and thus given by
and
The optimal portfolio at arbitrary μ_{3} can then be written as a weighted average of and as follows:
This equation proves the twofund separation theorem for meanvariance analysis. For a geometric interpretation, see the Markowitz bullet.
One riskfree asset
If a riskfree asset is available, then again a twofund separation theorem applies; but in this case one of the "funds" can be chosen to be a very simple fund containing only the riskfree asset, and the other fund can be chosen to be one which contains zero holdings of the riskfree asset. (With the riskfree asset referred to as "money", this form of the theorem is referred to as the monetary separation theorem.) Thus meanvariance efficient portfolios can be formed simply as a combination of holdings of the riskfree asset and holdings of a particular efficient fund that contains only risky assets. The derivation above does not apply, however, since with a riskfree asset the above covariance matrix of all asset returns, 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 σ^{2}
 subject to
 (W − X^{T}1)r_{f} + X^{T}r = μ,
where r_{f} is the known return on the riskfree asset, X is now the vector of quantities to be held in the risky assets, and 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 (W − X^{T}1) is the quantity held in the riskfree asset, thus incorporating the asset addingup constraint that in the earlier problem required the inclusion of a separate Lagrangian constraint. The objective function can be written as σ^{2} = X^{T}VX, where now 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
Of course this equals a zero vector if μ = Wr_{f}, the riskfree portfolio's return, in which case all wealth is held in the riskfree asset. It can be shown that the portfolio with exactly zero holdings of the riskfree asset occurs at and is given by
It can also be shown (analogously to the demonstration in the above twomutualfund case) that every portfolio's risky asset vector (that is, X^{opt} for every value of μ) 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 riskfree asset.
Portfolio separation without meanvariance analysis
If investors have hyperbolic absolute risk aversion (HARA) (such as is true for the power utility function and the exponential utility function) , separation theorems can be obtained without the use of meanvariance analysis. For example, David Cass and Joseph Stiglitz^{[4]} showed in 1970 that twofund 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. A similar result is given by Schmedders.^{[7]}
References
 ^ Chamberlain, G. 1983."A characterization of the distributions that imply meanvariance utility functions", Journal of Economic Theory 29, 185201.
 ^ Owen, J., and Rabinovitch, R. 1983. "On the class of elliptical distributions and their applications to the theory of portfolio choice", Journal of Finance 38, 745752.
 ^ Merton, Robert. September 1972. "An analytic derivation of the efficient portfolio frontier," Journal of Financial and Quantitative Analysis 7, 18511872.
 ^ Cass, David, and Joseph Stiglitz, "The structure of investor preferences and asset returns, and separability in portfolio allocation", Journal of Economic Theory 2, 1970, 122160.
 ^ Huang, Chifu, and Robert H. Litzenberger, Foundations for Financial Economics, NorthHolland, 1988.
 ^ Çanakoğlu, Ethem, and Süleyman Özekici (March 2010), "Portfolio selection in stochastic markets with HARA utility functions", European Journal of Operational Research 201(2), 520536. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VCT4VXDTWH5&_user=10&_coverDate=03%2F01%2F2010&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1572358725&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c24c04131ff627766be9dc38e04726d2&searchtype=a
 ^ Schmedders, Karl H. (June 15, 2006) "Twofund separation in dynamic general equilibrium," SSRN Working Paper Series. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=908587
Categories: Finance
 Financial economics
 Portfolio theories
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