The Mutual Fund Theorem and Risky-Asset Portfolio Weights

Summary

The document presents a question about the mutual fund theorem in a market with one risk-free asset and at least one risky asset. It describes the theorem’s intuition: under the stated assumptions, an investor can combine a risk-free holding with a common portfolio of risky assets rather than choose a separate portfolio of risky assets for every preference. It then gives expressions for a risk-free portfolio and a risky portfolio based on expected returns in excess of the risk-free rate and the covariance structure represented by a volatility matrix.

The passage is incomplete and includes an unresolved concern about the vector definition and whether the portfolios are relative portfolios. It does not state the theorem’s assumptions in full, derive the portfolio weights, or provide a numerical example. The formulas therefore need careful interpretation in the context of the original model, including the meaning and dimensions of the covariance matrix and portfolio vectors.

Key ideas

• The mutual fund theorem describes optimal portfolios as combinations of a risk-free asset and a common risky-asset fund under specified assumptions.
• The risky portfolio formula uses expected excess returns and a matrix derived from asset volatility.
• The document raises a distinction between constant portfolios and relative portfolios.
• The vector notation and its dimensions are not resolved in the passage.
• The theorem’s conclusions depend on assumptions that the document does not fully provide.

Tags

# Mutual fund theorem


# Mutual fund theorem












Theorem (Mutual fund theorem in the case that there is one risk-less asset and at least one risky asset). Suppose that all preced- ing assumptions in this subsection are valid. Consider the constant portfolios $w^r$ and $w^{\mu}$ defined by $\sum_{I=1}^{N} w^{\mu ,I}(0,w^{\mu ,1}, \dots, w^{\mu ,N})^T$ is the relative portfolio that corresponds to $w^{\mu}$ . what is the nature of the constant portfolios $w^r$ and $w^{\mu}$? Are they relative portfolios? \

I start with is problem by thing of the remark: The theorem means that an economic agent in our setting is indifferent between investing in an optimal portfolio con- sisting of the assets in our market model and investing in a portfolio consisting of just two mutual funds, where one of them is the risk-less asset and the other one consists of only risky assets. Also, we have that $$w^r:=(1,0,\dots , 0),$$ $$[0,T]\times \Omega \to R^{1+N}$$ $$w^{\mu}:= \begin{bmatrix}1&0\\ 0 & (\sigma \sigma ^T)^{-1} \end{bmatrix} (0,\mu ^1 -r, \dots, \mu ^N -r),$$ $$[0,T]\times \Omega \to R^{1+N}$$

I know there is problem with the vector $w^{\mu}$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.