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Portfolio Value as a Sum of Squared Standard Normal Variables

Article Quant Q&A · Author: kanbhold
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Summary

The document considers a portfolio that invests one unit in each of several stocks whose future prices are modeled as independent squared standard normal variables. Since each squared standard normal variable has a chi-square distribution with one degree of freedom, their sum has a chi-square distribution with degrees of freedom equal to the number of stocks. The portfolio value is therefore characterized by that distribution under the stated assumptions.

The conclusion relies on both the normality and independence assumptions, as well as the specified price model. It does not account for correlations among stocks, differing investment amounts, or other price dynamics. The responses give the distributional identification but provide little discussion of how the result might change when those assumptions do not hold.

Key ideas

  • Squaring a standard normal variable produces a chi-square variable with one degree of freedom.
  • The sum of independent squared standard normal variables has a chi-square distribution.
  • For equal one-unit holdings, the degrees of freedom equal the number of stocks.
  • Dependence across stock prices would invalidate the simple sum-of-independent-variables argument.

Tags

EquitiesPortfolio constructionStatistics
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# Distribution of the value of a portfolio


# Distribution of the value of a portfolio












Suppose there are k different stocks in a stock market. All of their prices are independent from each other. One year from now the price of the i-th stock will be $X_i^2$, where $X_i \sim \mathcal{N}(0,1) $ What is the distribution of the value of a portfolio after a year if I buy a $1 piece from each stock?

Will it be simple the Chi-squared distribution with parameter k?

## Answer by emcor (score 1, accepted)

https://quant.stackexchange.com/a/15012

Its Chi-Square distribution ($k=$ number of portfolio assets): http://en.wikipedia.org/wiki/Chi-squared_distribution#Definition

## Answer by Hans (score 0)

https://quant.stackexchange.com/a/15011

Yes. It is a simple transformation of the product of the Gaussians from the Cartesian to the hyper-spherical coordinate.

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.