Why Capital Market Line Points Map to Distinct Portfolios

Summary

The document considers whether each point on the capital market line (CML) identifies a unique portfolio. It describes a mean-variance argument: for a target expected return, minimize portfolio variance subject to a return constraint. The Lagrange conditions give holdings proportional to the inverse covariance matrix applied to the expected-return vector, with the target return fixing the proportionality constant. Under those conditions, the portfolio for each target return is determined uniquely.

A second explanation focuses on the mix of the risk-free asset and the risky portfolio: different positions along the CML imply different proportions in those components. The discussion assumes a zero risk-free rate in its setup and gives only a sketch of the optimization argument. It does not spell out the assumptions needed for uniqueness, such as an invertible covariance matrix, or address cases with degenerate assets or constraints. The result therefore applies to the stated mean-variance framework rather than every possible portfolio setting.

Key ideas

• A point on the capital market line represents a combination of a risk-free asset and a risky portfolio.
• For a specified expected return, portfolio variance can be minimized subject to a return constraint.
• The first-order conditions make the optimal holdings proportional to the covariance matrix inverse applied to expected returns.
• Different target returns determine different proportions between risky and risk-free holdings.
• The uniqueness argument relies on conditions such as an invertible covariance matrix.

Tags

# How to prove that every point on the capital market line corresponds to a unique portfolio


# How to prove that every point on the capital market line corresponds to a unique portfolio












> Prove that every point on the capital market line corresponds to a unique portfolio.

Attempted proof

I know that each point on the capital market line represents a linear combination of the risk free rate and some portfolio. But I am not really sure how to show that we have uniqueness. Any suggestions or references are greatly appreciated.

Note that we can assume the risk-free rate is zero, which is what my professor presented. Perhaps we would need a different proof if the risk-free rate was not zero but I am not sure, as usual I am lost.

## Answer by Wolfy (score 0, accepted)

https://quant.stackexchange.com/a/31500

I believe to prove this rigorously we need to first note that we want to $$\min{h^T V h } \ \ \text{subject to} \ \ h^T f = f_P$$ Using Lagrange multiplier we have the problem

\begin{cases} 2Vh &= \lambda f\\ h^T f &= f_P \end{cases} Thus, $$h = \frac{\lambda}{2}V^{-1}f = c\times h_Q$$ where $c = \frac{\lambda}{2}$ so, every point on the capital market line corresponds to a unique portfolio.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/31202

You have the right idea:

Two different points A and B on the CML have different proportions in the risk free asset, the point to the left has more in the risk free asset than the point on the right. This means that the composition of the two portfolios is distinct , they are not the same, but rather each is unique.

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