Deriving Basket Variance Under Perfect Correlation

Summary

The document works through the variance of a weighted basket when all asset returns have correlation one. For two assets, it writes the usual portfolio variance expression, including the covariance term, and recognizes that the expression becomes the square of the weighted sum of individual volatilities. It then generalizes that result to a basket with multiple assets.

The author’s difficulty is understanding a further simplification presented in a dispersion trading primer. They point out that the formula they derive does not appear numerically equivalent to the primer’s final formula and ask for an explanation. The text contains no answer, numerical example, or resolution of the discrepancy, so it teaches the initial algebraic relationship but not what the later simplification means. Its result also depends on the perfect-correlation assumption and the stated portfolio weights and volatilities; it does not describe a general basket with imperfect correlations.

Key ideas

• With two assets and correlation one, the covariance term makes basket variance a perfect square.
• The corresponding multi-asset expression is the square of the weighted sum of asset volatilities.
• The author identifies a mismatch with a further formula in a dispersion trading primer.
• The document does not provide an explanation of that final simplification.
• The derivation applies under the perfect-correlation assumption.

Tags

# Basket variance with correlation 1


# Basket variance with correlation 1












I'm reading the famous nuclear phynance primer on dispersion trading and finding difficulty in understanding the author's simplification for the variance of a basket with correlation 1. See below:

I can't see where the second formula comes from. Let's assume we have a 2-asset portfolio and $\rho = 1$. We would have:

$$\sigma_{Basket}^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2$$

which, recognizing this as the expansion of $(w_1\sigma_1 + w_2\sigma_2)^2$ and generalizing for $n$ assets (assuming again that $\rho = 1$), we can say:

$$\sigma_{Basket}^2 = (\sum_{i=1}^n w_i\sigma_i)^2$$

From this point on though, I have no idea how the author could simplify down to the final formula. They are not numerically equivalent. Maybe I am misreading, or perhaps it's just a math hurdle, but any explanation of what the author is attempting to say here would be wonderful. Thank you!

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