Why a Weighted Stock Portfolio Is Not Generally Lognormal

Summary

The document asks what distribution and price process arise when a fixed-weight portfolio combines stocks modeled as lognormal processes. It writes the portfolio increment as the weighted sum of each stock’s drift and diffusion terms, then points out that sums of lognormal variables are not generally lognormal. This raises a practical modeling question: why portfolio values are often treated as lognormal, and how accurate that approximation is.

The text does not provide a derivation, empirical evidence, or an answer assessing the approximation. It therefore serves mainly as a clearly framed question about portfolio aggregation, rather than a method for evaluating it. Any answer would need to account for the stocks’ correlations, weights, and changing composition, and distinguish the distribution at a particular time from the dynamics of the portfolio process. The document leaves these issues open.

Key ideas

• A fixed-weight portfolio’s change is the weighted sum of its component stock changes.
• A sum of lognormally distributed variables is not generally lognormal.
• Applications may approximate the portfolio as lognormal, but the document does not explain the rationale or accuracy.
• Correlation, weights, and portfolio dynamics are relevant to assessing the approximation.

Tags

# Process for a portfolio of stocks where each share follows a log-normal process


# Process for a portfolio of stocks where each share follows a log-normal process












Given a portfolio of shares $I = \sum{w_iS_i}$ for some fixed weights $w_i$ where each stok $S_i$ has a log-normal distribution, what is the process / distribution followed by the portfolio? That is, what is the distribution of the process

$${dI} = \sum{w_idS_i} = \sum{w_i(\mu_iS_idt+\sigma_iS_idW_i)} ?$$

Given that the log-normal distribution is not closed under the summation, it would imply that the process followed by the portfolio is NOT log-normal. However, it seems that it is, nevertheless, still modeled as log-normal in applications. Why is this so? And how bad/good this approximation is to the "true" process given above?

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