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Choosing an Optimal Portfolio Without a Risk-Free Asset

Article Quant Q&A · Author: Mahmudul Hassan
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Summary

Without a risk-free asset, investors cannot draw the usual capital allocation line from that asset or identify an optimal risky portfolio as its tangency with the efficient frontier. The document points to Black’s zero-beta CAPM as a framework for this setting. It describes investors choosing among portfolios on the curved efficient frontier, with the market portfolio and a zero-beta portfolio—one uncorrelated with the market—forming the relevant mix.

The preferred point depends on the investor’s preferences: it is where an iso-utility curve is tangent to the efficient frontier. This is a conceptual explanation, not a worked calculation or empirical comparison. It does not specify how to estimate the portfolios or utility, and the brief answer directs readers to the broader model for details.

Key ideas

  • Without a risk-free asset, the standard straight capital allocation line is unavailable.
  • Black’s zero-beta CAPM provides a framework that does not assume a riskless asset.
  • Investors select a point on the curved efficient frontier according to their risk and return preferences.
  • The selected point is where an iso-utility curve is tangent to the efficient frontier.
  • The model describes a mix of the market portfolio and a portfolio uncorrelated with it.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Which one is the optimal risky portfolio in the efficiency frontier in the absense of a risk free asset?


# Which one is the optimal risky portfolio in the efficiency frontier in the absense of a risk free asset?












I know that the tangency point between the CAL line (drawn from risk-free asset's return) and the efficiency frontier is the optimal risky portfolio. But what if there is no risk free asset? Can I draw CAL line without risk free asset? How do I find the optimal risky portfolio on the efficient frontier?

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/54581

"Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset" Source : Wikipedia

In this model there is no straight line CAL, but investors can choose a point on the (curved) efficient frontier that provides a suitable risk/return combination, by holding a mix of the Market Portfolio and the Zero Beta Portfolio, a portfolio orthogonal to the Market Portfolio. The chosen point is where the investor's iso-utility curve is tangent to the efficient frontier.

Look up Black CAPM or Zero Beta CAPM for more details.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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