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Zero-Beta Stocks and Portfolio Volatility Versus Risk-Free Assets

Article Quant Q&A · Author: poofidoudou
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Summary

The discussion examines whether replacing a zero-beta stock with a risk-free asset preserves a portfolio’s expected return while changing its volatility. It distinguishes a well-diversified portfolio, where the stock’s remaining contribution is unsystematic risk, from a finite portfolio where covariance with other holdings can still matter.

The variance formula explains the key mechanism: a zero-beta stock may be negatively correlated with the rest of the portfolio, lowering total variance. Replacing it with a risk-free asset removes that covariance contribution, so portfolio variance can rise even if expected return is unchanged. The argument is qualitative and depends on the assumed correlations and degree of diversification; it does not establish that every zero-beta stock is less volatile than a risk-free asset.

Key ideas

  • A zero-beta stock can have negative covariance with other portfolio holdings and reduce total variance.
  • In a perfectly diversified portfolio, unsystematic risk becomes negligible.
  • A risk-free asset is typically treated as uncorrelated with risky holdings in this comparison.
  • Expected return depends on asset weights and expected returns, while variance also depends on covariance.

Tags

EquitiesPortfolio constructionRisk managementStatistics
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# Answer by Alex C (score 1)


# Why we can't lower a volatility of a portfolio (without changing expected return) by substituting a zero beta stock with a risk free asset?












part of the answer is that a zero-beta stock must be negatively correlated with other stocks in the portfolio. So having a zero beta stock can decrease the volatility. Does that mean that the volatility a zero beta stock is lower than a risk free asset?

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/42207

If we assume that the starting portfolio is well diversified, then removing a zero beta stock from it will not reduce risk, since the only risk it carries is unsystematic risk but the only risk that matters for the portfolio is market risk. Of course this is only true in the limit of perfect diversification. For a more realistic case (good but not perfect diversification) the risk reduction would be negligible.

## Answer by numerairX (score 0)

https://quant.stackexchange.com/a/42206

Not necessarily. Recall the form of covariance matrix when we calculate variance of whole portfolio (to make it simple say we have two assets $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 w_1w_2 cov(1,2)$)

if your zero-beta stock is negatively correlated with the rest of the portfolio then of course variance is decreased since $\rho$ is negative.

However if risk free asset is substituted in, it is usually assumed to have 0 correlation with the rest of the asset, hence the overall return doesn't change (because expectation is calculated by the weight*individual stock return) but variance will increase.

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