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Comparing Portfolio Risk with Variance and Standard Deviation

Article Quant Q&A · Author: Thomas Johnson
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Summary

The document frames a risk-measure question using two positively correlated, mean-reverting assets. An investor already holds one unit of asset A and is considering adding one unit of B. It asks whether the existing exposure should make the investor more cautious about buying B, and whether the comparison should be made using standard deviation or variance of the combined position.

The question also connects these measures to a common spread-trading convention: entering when a spread moves a specified number of standard deviations from a reference level. It asks why such thresholds are expressed in standard deviation units rather than variance units. No answer is included, so the document does not resolve which measure is appropriate or derive a decision rule. In practice, the covariance matters to portfolio variance, while standard deviation expresses risk in the same units as returns or price changes; the relevant choice also depends on whether the goal is portfolio risk comparison or signal normalization.

Key ideas

  • The covariance between two assets affects the risk of their combined position.
  • The question compares incremental portfolio risk using variance and standard deviation.
  • Spread entry rules are commonly expressed in standard deviation units.
  • The source raises the issue but does not supply a derivation or definitive answer.

Tags

Risk managementStatisticsMean reversion
Full text
# Units of Risk: Variance vs Standard Deviation


# Units of Risk: Variance vs Standard Deviation












Suppose you are trading two mean-reverting assets, A and B, and that $Covar(A, B) > 0$. You are currently long one unit of A, and are considering buying one unit of B. Compared to the situation where you have no position, should you be...

(A) More eager to buy B, because it adds some diversification.

That is: $\sigma_{A+B} - \sigma_B < \sigma_A$

(B) Less eager to buy B, because you already have correlated exposure to A.

That is: $ \sigma_{A+B}^2 - \sigma_B^2 > \sigma_A^2$

(C) Something else?

If (B), then why do people use $2\sigma$ as a rule of thumb for entering a spread rather than $X \sigma^2$ ?

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.