Choosing a Tangency Portfolio and Allocating to the Risk-Free Asset

Summary

The document asks how to handle zero weights from a Markowitz allocation when every asset must be included, and how to determine the risk-free allocation when constructing a portfolio on the capital market line. The response focuses on selecting the risky portfolio that gives the steepest capital market line: maximize excess return over the risk-free rate divided by portfolio volatility. This identifies the tangency portfolio among candidates on the efficient frontier.

For required weight restrictions, the response characterizes the allocation as a constrained quadratic optimization problem that generally needs a numerical solver. It does not provide an explicit formula for the risk-free asset’s weight or a method for imposing positive minimum weights on every risky asset. The practical guidance is therefore limited to choosing the tangency portfolio and recognizing that additional weight constraints require optimization machinery.

Key ideas

• The tangency portfolio maximizes excess return per unit of volatility.
• The slope of the capital market line is determined by the portfolio’s Sharpe ratio relative to the risk-free rate.
• Weight restrictions generally turn the allocation into a constrained optimization problem.
• A numerical solver can be used for constrained quadratic portfolio optimization.

Tags

# MPT Efficient portfolio /Asset allocation


# MPT Efficient portfolio /Asset allocation












When finding the optimal allocation using markovitz, the model will return '0' weights for assets that are "inefficient". What is the standard way for dealing with these weights if all assets have to be included in the portfolio and a simple minimum is not an option? Also when introducing the CML the textbook says one should add a risk free asset to produce the optimal portfolio but i cant seem to find a formula to calculate the weight of the risk free asset. I understand the optimal portfolio should be a point on the CML but how can i find it?

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/42011

You want the CML to be as steep as possible (greater reward for lesser risk), so find the corresponding point, $T$, from all of those on the frontier that satisfies:

$$ \max_{T} \quad \nabla = \frac{R_T - R_f}{\sigma_T} $$

If you impose constraints on your weights then you, typically, have a quadratic optimisation problem with constraints, and will need some form of numerical solver. The docs for cvxopt in python have this sort of problem as an example.

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